Finite Group Databasebeta

← #30827367

Subgroups — #30827367

Order 17152.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569224C211yesmaximal, characteristic, complemented
2C4×#14524071C211yesmaximal, characteristic
3M16×D268C211yesmaximal, characteristic, complemented
4#22559113C211yesmaximal, characteristic, complemented
5#22559101C211yesmaximal, characteristic, complemented
6#22559075C211yesmaximal, characteristic, complemented
7#22559068C211yesmaximal, characteristic, complemented
8#22559047C211yesmaximal, characteristic, complemented
9#22559040C211yesmaximal, characteristic, complemented
10#22559034C211yesmaximal, characteristic, complemented
11C8×D536C211yesmaximal, characteristic, complemented
12#22558976C211yesmaximal, characteristic, complemented
13#22558653C211yesmaximal, characteristic, complemented
14#22558649C211yesmaximal, characteristic, complemented
15C67×#1455C211yesmaximal, characteristic, complemented
16C2×#14524071C411yescharacteristic
17C2×M16×D67—21no—
18C4×#11761628C2211yescharacteristic
19C2×C4×D268C2211yescharacteristic
20#16863526C2211yescharacteristic, complemented
21#16863526C2211yescharacteristic, complemented
22#16863509C411yescharacteristic
23#16863509C411yescharacteristic
24#16863498C411yescharacteristic
25#16863498C411yescharacteristic
26C4×#11761617C2211yescharacteristic
27C4×#11761617C2211yescharacteristic
28C4×D536C2211yescharacteristic
29C4×D536C2211yescharacteristic
30#16863475C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863469—21no—
33#16863469—21no—
34#16863465—21no—
35#16863465—21no—
36C8×D268C2211yescharacteristic
37C8×D268C2211yescharacteristic
38C8×D268C2211yescharacteristic, complemented
39C8×D268C2211yescharacteristic, complemented
40#16863453C2211yescharacteristic
41#16863451—21no—
42#16863438C411yescharacteristic
43#16863420C2211yescharacteristic
44#16863402C411yescharacteristic
45#16863398C411yescharacteristic
46#16863359—21no—
47#97×D67—21no—
48#16863149C2211yescharacteristic, complemented
49#16863149C2211yescharacteristic, complemented
50#16863147C2211yescharacteristic, complemented
51#16863147C2211yescharacteristic, complemented
52#16863128C2211yescharacteristic, complemented
53#16863128C2211yescharacteristic, complemented
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863125C2211yescharacteristic
57#16863125C2211yescharacteristic
58#16863125C2211yescharacteristic, complemented
59#16863125C2211yescharacteristic, complemented
60C4×M16×C67C2211yescharacteristic
61C4×M16×C67C2211yescharacteristic
62C4×M16×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic, complemented
64#322×C67C2211yescharacteristic, complemented
65#322×C67C2211yescharacteristic, complemented
66C82×C67C2211yescharacteristic, complemented
67C2×#11761628C2×C411yescharacteristic
68C22×D268C2×C411yescharacteristic
69C22×C4×D67—21no—
70#14524071—24no—
71C2×#11761617C2×C411yescharacteristic
72C2×#11761617C2×C411yescharacteristic
73C2×D536C2×C411yescharacteristic
74C2×D536C2×C411yescharacteristic
75C2×#11761612—21no—
76C2×#11761611—21no—
77C2×#11761611—21no—
78M16×D67—44no—
79C2×C8×D67—21no—
80C2×C8×D67—21no—
81C2×C8×D67—21no—
82C2×C8×D67—21no—
83C4×Dic134C2×C411yescharacteristic
84C4×Dic134C2311yescharacteristic
85C4×D268—21no—
86C4×D268—21no—
87C4×D268C2×C411yescharacteristic
88C4×D268C2×C411yescharacteristic
89C4×D268C2×C411yescharacteristic
90C4×D268C2311yescharacteristic
91C4×D268C2311yescharacteristic
92C4×D268C2311yescharacteristic
93C4×C22⋊D67—21no—
94C4×C22⋊D67—21no—
95#14524002C2×C411yescharacteristic
96#14524000—21no—
97C2×C4⋊Dic67C2×C411yescharacteristic
98C2×(C2×D67)⋊C4—21no—
99C42×D67—21no—
100#14523977C2×C411yescharacteristic
101#14523977C2×C411yescharacteristic
102#14523975C2×C411yescharacteristic
103#14523975C2×C411yescharacteristic
104C4⋊(C67⋊C8)—21no—
105C4⋊(C67⋊C8)C2311yescharacteristic
106C4⋊(C67⋊C8)C2311yescharacteristic
107C4⋊(C67⋊C8)C2311yescharacteristic
108C4⋊(C67⋊C8)C2311yescharacteristic
109Dic67⋊C8—21no—
110Dic67⋊C8—21no—
111#14523968C2×C411yescharacteristic
112#14523968C2×C411yescharacteristic
113#14523965C2×C411yescharacteristic
114#14523965C2×C411yescharacteristic
115#14523965C2×C411yescharacteristic
116#14523965C2×C411yescharacteristic
117#14523965C2×C411yescharacteristic
118#14523965C2×C411yescharacteristic
119C22⋊(C67⋊C8)—21no—
120(C2×D67)⋊C8—21no—
121(C2×D67)⋊C8—21no—
122(C2×D67)⋊C8—21no—
123(C2×D67)⋊C8—21no—
124(C2×D67)⋊C8—21no—
125(C2×D67)⋊C8—21no—
126#14523954—21no—
127#14523953—21no—
128#14523953—21no—
129C2×M16×C67—21no—
130C2×M16×C67C2×C411yescharacteristic
131C2×M16×C67C2×C411yescharacteristic
132C2×M16×C67D411yescharacteristic
133C2×M16×C67D411yescharacteristic
134C2×C42×C67C2311yescharacteristic
135#97×C67—21no—
136#97×C67C2×C411yescharacteristic
137#97×C67C2×C411yescharacteristic
138#97×C67D411yescharacteristic
139#97×C67D411yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C4×C8×C67C2311yescharacteristic
145C4×C8×C67C2311yescharacteristic
146C23×D67—21no—
147#11761628—22no—
148#11761628—41no—
149C2×Dic134C22×C411yescharacteristic
150C2×D268—22no—
151C2×D268—21no—
152C2×D268—41no—
153C2×D268C22×C411yescharacteristic
154C2×D268C22×C411yescharacteristic
155C2×D268C22×C411yescharacteristic
156C2×C22⋊D67—21no—
157C22×Dic67—21no—
158C2×C4×D67—21no—
159C2×C4×D67—21no—
160C2×C4×D67—21no—
161C2×C4×D67—21no—
162C2×C4×D67—21no—
163C2×C4×D67—21no—
164C2×C4×D67—21no—
165C2×C4×D67—42no—
166C2×C4×D67—42no—
167#11761617—22no—
168#11761617—22no—
169D536—22no—
170D536M1614yescomplemented
171#11761612—42no—
172#11761611—42no—
173#11761611—42no—
174C2×C67⋊C8—21no—
175C2×C67⋊C8—21no—
176C2×C67⋊C8—21no—
177C2×C67⋊C8—21no—
178C8×D67—42no—
179C8×D67—42no—
180C8×D67—42no—
181C8×D67—42no—
182C4⋊Dic67—21no—
183C4⋊Dic67C22×C411yescharacteristic
184C4⋊Dic67C22×C411yescharacteristic
185C4⋊Dic67C22×C411yescharacteristic
186C4⋊Dic67C22×C411yescharacteristic
187Dic67⋊C4—21no—
188Dic67⋊C4—21no—
189C22⋊Dic67—21no—
190(C2×D67)⋊C4—21no—
191(C2×D67)⋊C4—21no—
192(C2×D67)⋊C4—21no—
193(C2×D67)⋊C4—21no—
194(C2×D67)⋊C4—21no—
195(C2×D67)⋊C4—21no—
196C4×Dic67—21no—
197C4×Dic67—21no—
198C4×Dic67—21no—
199C22×C4×C67C22×C411yescharacteristic
200C22×C4×C67C2×D411yescharacteristic
201C22×C4×C67#4111yescharacteristic
202M16×C67—22no—
203M16×C67—22no—
204M16×C67—41no—
205M16×C67—41no—
206M16×C67—42no—
207C2×C8×C67—21no—
208C2×C8×C67—21no—
209C2×C8×C67C22×C411yescharacteristic
210C2×C8×C67C22×C411yescharacteristic
211C2×C8×C67C22×C411yescharacteristic
212C2×C8×C67C22×C411yescharacteristic
213C2×C8×C67C2×D411yescharacteristic
214C2×C8×C67C2×D411yescharacteristic
215C2×C8×C67C2×D411yescharacteristic
216C2×C8×C67C2×D411yescharacteristic
217C42×C67C22×C411yescharacteristic
218C42×C67C2×D411yescharacteristic
219C42×C67#4111yescharacteristic
220C42×C67C2411yescharacteristic
221C23×C67C4×D411yescharacteristic
222C22×D67—21no—
223C22×D67—21no—
224C22×D67—21no—
225C22×D67—42no—
226C22×D67—42no—
227C22×D67—81no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—41no—
233C2×C4×C67C4×D411yescharacteristic
234C2×C4×C67C4×D411yescharacteristic
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C23×C411yescharacteristic
237C2×C4×C67C22×D411yescharacteristic
238C2×C4×C67C2×#4111yescharacteristic
239C22⋊D67—42no—
240C2×Dic67—21no—
241C2×Dic67—21no—
242C2×Dic67—21no—
243C2×Dic67—21no—
244C2×Dic67—21no—
245C2×Dic67—42no—
246D268—21no—
247D268—21no—
248D268—21no—
249D268—21no—
250D268—42no—
251D268C2×M1612yes—
252D268C2×M1612yescomplemented
253C4×D67—42no—
254C4×D67—42no—
255C4×D67—42no—
256C4×D67—41no—
257C4×D67—41no—
258C4×D67—41no—
259C4×D67—41no—
260C4×D67—41no—
261C4×D67—41no—
262C4×D67—81no—
263Dic134—21no—
264Dic134—21no—
265C8×C67—22no—
266C8×C67—21no—
267C8×C67—21no—
268C8×C67—21no—
269C8×C67—21no—
270C8×C67—21no—
271C8×C67—21no—
272C8×C67—41no—
273C8×C67C2×M1612yescomplemented
274C8×C67AGL(1,ℤ/8)12yescomplemented
275C67⋊C8—41no—
276C67⋊C8—41no—
277C67⋊C8—41no—
278C67⋊C8—41no—
279C22×C67—21no—
280C22×C67—21no—
281C22×C67—21no—
282C22×C67C2×C4×D411yescharacteristic
283C2×D67—22no—
284C2×D67—22no—
285C2×D67—41no—
286C2×D67—41no—
287C2×D67—41no—
288C2×D67—41no—
289C2×D67—41no—
290C2×D67—41no—
291C2×D67—41no—
292C2×D67—84no—
293C4×C67—21no—
294C4×C67—21no—
295C4×C67—21no—
296C4×C67—21no—
297C4×C67C22×M1611yescharacteristic
298C4×C67#56811yescharacteristic
299C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C67#57511yescharacteristic
301Dic67—41no—
302Dic67—41no—
303Dic67—41no—
304Dic67—41no—
305Dic67—41no—
306#18046—671nomaximal, Sylow
307C2×C67—41no—
308C2×C67D4×M1611yescharacteristic
309C2×C67C4×AGL(1,ℤ/8)11yescharacteristic
310C2×C67#295611yescharacteristic
311D67—42no—
312D67—42no—
313D67—81no—
314#2970—671no—
315C4×AGL(1,ℤ/8)—671no—
316D4×M16—671no—
317#1592—671no—
318#1590—671no—
319#1586—671no—
320#1585—671no—
321#1582—671no—
322#1581—671no—
323#1580—671no—
324C8×D8—671no—
325#1569—671no—
326#1490—671no—
327#1489—671no—
328#1455D6711yescharacteristic, complemented
329C67#1804611yescharacteristic, complemented, Sylow
330C2×AGL(1,ℤ/8)—671no—
331C22×M16—1341no—
332C4×#41—671no—
333C2×C4×D4—671no—
334#446—671no—
335#446—671no—
336#442—671no—
337#442—671no—
338#440—671no—
339#440—671no—
340C4×SD16—671no—
341C4×SD16—671no—
342C4×D8—671no—
343C4×D8—671no—
344#436—671no—
345#436—671no—
346#435—1341no—
347#435—1341no—
348#435—1341no—
349#435—1341no—
350C8×D4—671no—
351C8×D4—671no—
352C8×D4—671no—
353C8×D4—671no—
354#432—671no—
355#432—1341no—
356#428—671no—
357#423—671no—
358#418—671no—
359#417—671no—
360#407—1341no—
361C4×M16C2×D6711yescharacteristic
362C4×M16C2×D6711yescharacteristic
363C4×M16C2×D6711yescharacteristic
364C2×#97—1341no—
365#335—671no—
366#335—671no—
367#334—671no—
368#334—671no—
369Q8⋊C8—671no—
370Q8⋊C8—671no—
371D4⋊C8—671no—
372D4⋊C8—671no—
373D4⋊C8—671no—
374D4⋊C8—671no—
375D4⋊C8—671no—
376D4⋊C8—671no—
377#322C2×D6711yescharacteristic, complemented
378#322C2×D6711yescharacteristic, complemented
379#322C2×D6711yescharacteristic, complemented
380C82C2×D6711yescharacteristic, complemented
381C2×#41—671no—
382C22×D4—671no—
383C23×C4—1341no—
384AGL(1,ℤ/8)—1344no—
385C2×SD16—671no—
386C2×SD16—671no—
387C2×D8—671no—
388C2×D8—671no—
389C2×M16—21no—
390C2×M16—2684no—
391C2×M16—1341no—
392C2×M16—1341no—
393C2×M16—1341no—
394C2×M16C4×D6711yescharacteristic
395C2×M16C4×D6711yescharacteristic
396C2×M16D26811yescharacteristic
397C2×M16D26811yescharacteristic
398C22×C8—1341no—
399C22×C8—1341no—
400C22×C8—1341no—
401C22×C8—1341no—
402C4×Q8—671no—
403C4×Q8—671no—
404C4×D4—671no—
405C4×D4—671no—
406C4×D4—671no—
407C4×D4—671no—
408C4×D4—671no—
409C4×D4—671no—
410C4×D4—1341no—
411C4×D4—1341no—
412C4×D4—1341no—
413C4×D4—1341no—
414#117—671no—
415#117—1341no—
416C2×C4⋊C4—671no—
417C2×C22⋊C4—1341no—
418C2×C42—1341no—
419C2×C42C22×D6711yescharacteristic
420#107—671no—
421#107—671no—
422#106—671no—
423#106—671no—
424C4⋊C8—671no—
425C4⋊C8—671no—
426C4⋊C8—671no—
427C4⋊C8—671no—
428C4⋊C8—1341no—
429C4⋊C8—1341no—
430C4⋊C8—1341no—
431#103—671no—
432#103—671no—
433#102—671no—
434#102—671no—
435#102—671no—
436#102—671no—
437#102—671no—
438#102—671no—
439C22⋊C8—1341no—
440C22⋊C8—1341no—
441C22⋊C8—1341no—
442C22⋊C8—1341no—
443C22⋊C8—1341no—
444C22⋊C8—1341no—
445C22⋊C8—1341no—
446#97—21no—
447#97—1341no—
448#97—1341no—
449#97—1341no—
450#97C4×D6711yescharacteristic
451#97C4×D6711yescharacteristic
452#97D26811yescharacteristic
453#97D26811yescharacteristic
454C4×C8C22×D6711yescharacteristic
455C4×C8C22×D6711yescharacteristic
456C4×C8C22×D6711yescharacteristic
457C4×C8C22×D6711yescharacteristic
458C4×C8C22×D6711yescharacteristic
459C4×C8C22×D6711yescharacteristic
460C24—1341no—
461#41—2681no—
462#41—1342no—
463C2×Q8—671no—
464C2×D4—2681no—
465C2×D4—671no—
466C2×D4—671no—
467C2×D4—671no—
468C2×D4—1342no—
469C2×D4—1341no—
470C2×D4—1341no—
471C22×C4—2682no—
472C22×C4—2682no—
473C22×C4—1341no—
474C22×C4—1341no—
475C22×C4—1341no—
476C22×C4—1341no—
477C22×C4—1341no—
478C22×C4—1341no—
479C22×C4—1341no—
480C22×C4—1341no—
481C22×C4C2×C4×D6711yescharacteristic
482C22×C4C2×D26811yescharacteristic
483C22×C4#1176162811yescharacteristic
484SD16—1342no—
485SD16—1342no—
486D8—674no—
487D8—1342no—
488M16—22no—
489M16—22no—
490M16—41no—
491M16—41no—
492M16—42no—
493M16—2682no—
494M16—2682no—
495M16—2682no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—2682no—
499C2×C8—2682no—
500C2×C8—2682no—
501C2×C8—2682no—
502C2×C8—1341no—
503C2×C8—1341no—
504C2×C8—1341no—
505C2×C8—1341no—
506C2×C8C2×C4×D6711yescharacteristic
507C2×C8C2×C4×D6711yescharacteristic
508C2×C8C2×C4×D6711yescharacteristic
509C2×C8C2×C4×D6711yescharacteristic
510C2×C8C2×D26811yescharacteristic
511C2×C8C2×D26811yescharacteristic
512C2×C8C2×D26811yescharacteristic
513C2×C8C2×D26811yescharacteristic
514C4⋊C4—671no—
515C4⋊C4—671no—
516C4⋊C4—671no—
517C4⋊C4—671no—
518C4⋊C4—1341no—
519C4⋊C4—1341no—
520C4⋊C4—1341no—
521C22⋊C4—1341no—
522C22⋊C4—1341no—
523C22⋊C4—1341no—
524C22⋊C4—1341no—
525C22⋊C4—1341no—
526C22⋊C4—1341no—
527C22⋊C4—1341no—
528C42—1341no—
529C42—1341no—
530C42—1341no—
531C42C2×C4×D6711yescharacteristic
532C42C2×D26811yescharacteristic
533C42#1176162811yescharacteristic
534C42C23×D6711yescharacteristic
535C23—2682no—
536C23—2682no—
537C23—5361no—
538C23—1341no—
539C23—1341no—
540C23—1341no—
541C23C4×D26811yescharacteristic
542Q8—1341no—
543Q8—1341no—
544D4—2682no—
545D4—2682no—
546D4—672no—
547D4—672no—
548D4—1341no—
549D4—1341no—
550D4—1341no—
551D4—1341no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—2682no—
558C2×C4—2682no—
559C2×C4—2682no—
560C2×C4—2682no—
561C2×C4—2681no—
562C2×C4—2681no—
563C2×C4—2681no—
564C2×C4—2681no—
565C2×C4—2681no—
566C2×C4—2681no—
567C2×C4—5361no—
568C2×C4—1341no—
569C2×C4—1341no—
570C2×C4—1341no—
571C2×C4—1341no—
572C2×C4—1341no—
573C2×C4C4×D26811yescharacteristic
574C2×C4C4×D26811yescharacteristic
575C2×C4C4×D26811yescharacteristic
576C2×C4C22×C4×D6711yescharacteristic
577C2×C4C22×D26811yescharacteristic
578C2×C4C2×#1176162811yescharacteristic
579C8—22no—
580C8—21no—
581C8—21no—
582C8—21no—
583C8—21no—
584C8—21no—
585C8—21no—
586C8—41no—
587C8—2681no—
588C8—2681no—
589C8—2681no—
590C8—2681no—
591C8M16×D6712yescomplemented
592C8#1452407112yescomplemented
593C22—21no—
594C22—21no—
595C22—21no—
596C22—2681no—
597C22—2681no—
598C22—2681no—
599C22—2681no—
600C22—2681no—
601C22—2681no—
602C22—2681no—
603C22—5364no—
604C22—1342no—
605C22—1342no—
606C22C2×C4×D26811yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—2681no—
612C4—2681no—
613C4—2681no—
614C4—2681no—
615C4—2681no—
616C4C2×M16×D6711yescharacteristic
617C4#1686416211yescharacteristic
618C4C2×#1452407111yescharacteristic
619C4#1686419611yescharacteristic
620C2—41no—
621C2—2682no—
622C2—2682no—
623C2—5361no—
624C2M16×D26811yescharacteristic
625C2C4×#1452407111yescharacteristic
626C2#2256912011yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1455C2
socleC22×C67C2×C4×D4
prime radicalC2×#14524071C4
cosocleC42×C67C24
hypercenter#1455D67
hypocenterC67#18046
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22569224C2semidirect
3. M16×D268C2semidirect
4. #22559113C2semidirect
5. #22559101C2semidirect
6. #22559075C2semidirect
7. #22559068C2semidirect
8. #22559047C2semidirect
9. #22559040C2semidirect
10. #22559034C2semidirect
11. C8×D536C2semidirect
12. #22558976C2semidirect
13. #22558653C2semidirect
14. #22558649C2semidirect
15. C67×#1455C2semidirect
20. #16863526C22semidirect
21. #16863526C22semidirect
38. C8×D268C22semidirect
39. C8×D268C22semidirect
48. #16863149C22semidirect
49. #16863149C22semidirect
50. #16863147C22semidirect
51. #16863147C22semidirect
52. #16863128C22semidirect
53. #16863128C22semidirect
58. #16863125C22semidirect
59. #16863125C22semidirect
63. #322×C67C22semidirect
64. #322×C67C22semidirect
65. #322×C67C22semidirect
66. C82×C67C22semidirect
170. D536M16semidirect
252. D268C2×M16semidirect
273. C8×C67C2×M16semidirect
274. C8×C67AGL(1,ℤ/8)semidirect
328. #1455D67semidirect
329. C67#18046semidirect
377. #322C2×D67semidirect
378. #322C2×D67semidirect
379. #322C2×D67semidirect
380. C82C2×D67semidirect
591. C8M16×D67semidirect
592. C8#14524071semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225692242
C4×#145240712
M16×D2682
#225591132
#225591012
#225590752
#225590682
#225590472
#225590402
#225590342
C8×D5362
#225589762
#225586532
#225586492
C67×#14552
#1804667