Finite Group Databasebeta

← #30825895

Subgroups — #30825895

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570258C211yesmaximal, characteristic
2#22569360C211yesmaximal, characteristic, complemented
3#22569232C211yesmaximal, characteristic, complemented
4#22559075C211yesmaximal, characteristic, complemented
5#22559068C211yesmaximal, characteristic, complemented
6#22559047C211yesmaximal, characteristic, complemented
7#22559040C211yesmaximal, characteristic, complemented
8#22559034C211yesmaximal, characteristic, complemented
9#22559023C211yesmaximal, characteristic, complemented
10C8×#11761617C211yesmaximal, characteristic, complemented
11C8×D536C211yesmaximal, characteristic, complemented
12#22558998C211yesmaximal, characteristic, complemented
13#22558635C211yesmaximal, characteristic, complemented
14#22558618C211yesmaximal, characteristic, complemented
15C67×#1454C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18#16863732C411yescharacteristic
19#16863715C411yescharacteristic
20#16863697C411yescharacteristic
21#16863697C411yescharacteristic
22#16863612C411yescharacteristic
23#16863612C411yescharacteristic
24#16863549C411yescharacteristic
25#16863532C411yescharacteristic
26#16863526C2211yescharacteristic, complemented
27C8×Dic134C2211yescharacteristic, complemented
28C4×#11761617C2211yescharacteristic
29C4×#11761617C2211yescharacteristic
30C4×D536C2211yescharacteristic
31C4×D536C2211yescharacteristic
32#16863475C2211yescharacteristic, complemented
33#16863471—21no—
34#16863470—21no—
35C8×D268C2211yescharacteristic
36C8×D268C2211yescharacteristic
37C8×D268C2211yescharacteristic
38C8×D268C2211yescharacteristic
39C8×D268C2211yescharacteristic, complemented
40#16863461—21no—
41#16863461—21no—
42#16863456—21no—
43#16863449C2211yescharacteristic
44#16863421—21no—
45#16863420C2211yescharacteristic
46C22⋊C8×D67—21no—
47C22⋊C8×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
60#431×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic, complemented
64#322×C67C2211yescharacteristic, complemented
65#322×C67C2211yescharacteristic, complemented
66C82×C67C2211yescharacteristic, complemented
67C22×D268C2×C411yescharacteristic
68C22×C4×D67—21no—
69C2×#11761617C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×D536C2×C411yescharacteristic
72C2×D536C2×C411yescharacteristic
73C2×#11761612—21no—
74C2×#11761611—21no—
75C2×C8×D67—21no—
76C2×C8×D67—21no—
77C2×C8×D67—21no—
78C2×C8×D67—21no—
79C2×C8×D67—21no—
80#14524037—21no—
81#14524035C2×C411yescharacteristic
82#14524031C2×C411yescharacteristic
83#14524026—21no—
84#14524022C2×C411yescharacteristic
85#14524017C2×C411yescharacteristic
86C4×Dic134C2311yescharacteristic
87C4×D268—21no—
88C4×D268—21no—
89C4×D268C2×C411yescharacteristic
90C4×D268C2×C411yescharacteristic
91C4×D268C2311yescharacteristic
92C4×D268C2311yescharacteristic
93C4×D268C2311yescharacteristic
94C4×C22⋊D67—21no—
95#14524000—21no—
96C2×C4⋊Dic67C2×C411yescharacteristic
97C2×(C2×D67)⋊C4—21no—
98#14523977C2×C411yescharacteristic
99#14523977C2×C411yescharacteristic
100#14523975C2×C411yescharacteristic
101#14523975C2×C411yescharacteristic
102C4⋊(C67⋊C8)—21no—
103C4⋊(C67⋊C8)C2311yescharacteristic
104C4⋊(C67⋊C8)C2311yescharacteristic
105C4⋊(C67⋊C8)C2311yescharacteristic
106C4⋊(C67⋊C8)C2311yescharacteristic
107Dic67⋊C8—21no—
108Dic67⋊C8—21no—
109#14523968C2×C411yescharacteristic
110#14523968C2×C411yescharacteristic
111#14523965C2×C411yescharacteristic
112#14523965C2×C411yescharacteristic
113#14523965C2×C411yescharacteristic
114#14523965C2×C411yescharacteristic
115#14523965C2×C411yescharacteristic
116#14523965C2×C411yescharacteristic
117C22⋊(C67⋊C8)—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120(C2×D67)⋊C8—21no—
121(C2×D67)⋊C8—21no—
122(C2×D67)⋊C8—21no—
123(C2×D67)⋊C8—21no—
124#14523953—21no—
125C4×C67⋊C8—21no—
126C8×Dic67—21no—
127C2×C42×C67C2311yescharacteristic
128C4⋊C8×C67—21no—
129C4⋊C8×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67D411yescharacteristic
133C22⋊C8×C67—21no—
134C22⋊C8×C67C2×C411yescharacteristic
135C22⋊C8×C67C2×C411yescharacteristic
136C22⋊C8×C67D411yescharacteristic
137C22⋊C8×C67D411yescharacteristic
138C4×C8×C67C2311yescharacteristic
139C4×C8×C67C2311yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C23×D67—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×D268—21no—
147C2×D268—22no—
148C2×D268—22no—
149C2×D268C22×C411yescharacteristic
150C2×D268C22×C411yescharacteristic
151C2×D268C22×C411yescharacteristic
152C2×C22⋊D67—21no—
153C22×Dic67—21no—
154C2×C4×D67—21no—
155C2×C4×D67—21no—
156C2×C4×D67—21no—
157C2×C4×D67—21no—
158C2×C4×D67—21no—
159C2×C4×D67—81no—
160C2×C4×D67—81no—
161#11761617—22no—
162#11761617—22no—
163D536—22no—
164D536—22no—
165#11761612—81no—
166#11761611—81no—
167C2×C67⋊C8—21no—
168C2×C67⋊C8—21no—
169C2×C67⋊C8—21no—
170C2×C67⋊C8—21no—
171C8×D67—81no—
172C8×D67—81no—
173C8×D67—81no—
174C8×D67—81no—
175C8×D67—81no—
176C4⋊Dic67—21no—
177C4⋊Dic67C22×C411yescharacteristic
178C4⋊Dic67C22×C411yescharacteristic
179C4⋊Dic67C22×C411yescharacteristic
180C4⋊Dic67C22×C411yescharacteristic
181Dic67⋊C4—21no—
182Dic67⋊C4—21no—
183Dic67⋊C4—21no—
184Dic67⋊C4—21no—
185C22⋊Dic67—21no—
186(C2×D67)⋊C4—21no—
187(C2×D67)⋊C4—21no—
188(C2×D67)⋊C4—21no—
189(C2×D67)⋊C4—21no—
190(C2×D67)⋊C4—21no—
191(C2×D67)⋊C4—21no—
192(C2×D67)⋊C4—21no—
193(C2×D67)⋊C4—21no—
194C4×Dic67—21no—
195C22×C4×C67C22×C411yescharacteristic
196C22×C4×C67C2×D411yescharacteristic
197C22×C4×C67#4111yescharacteristic
198C2×C8×C67—21no—
199C2×C8×C67—21no—
200C2×C8×C67C22×C411yescharacteristic
201C2×C8×C67C22×C411yescharacteristic
202C2×C8×C67C22×C411yescharacteristic
203C2×C8×C67C22×C411yescharacteristic
204C2×C8×C67C2×D411yescharacteristic
205C2×C8×C67C2×D411yescharacteristic
206C2×C8×C67C2×D411yescharacteristic
207C2×C8×C67C2×D411yescharacteristic
208C42×C67C22×C411yescharacteristic
209C42×C67C2×D411yescharacteristic
210C42×C67#4111yescharacteristic
211C42×C67C2411yescharacteristic
212C23×C67C4×D411yescharacteristic
213C22×D67—21no—
214C22×D67—21no—
215C22×D67—24no—
216C22×D67—21no—
217C22×D67—81no—
218C22×D67—81no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—41no—
224C2×C4×C67C4×D411yescharacteristic
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C23×C411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C2×#4111yescharacteristic
230C22⋊D67—81no—
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236C2×Dic67—81no—
237D268—24no—
238D268—21no—
239D268—21no—
240D268—21no—
241D268—21no—
242D268#13112yes—
243D268#13112yes—
244C4×D67—41no—
245C4×D67—41no—
246C4×D67—41no—
247C4×D67—41no—
248C4×D67—41no—
249C4×D67—41no—
250C4×D67—81no—
251C4×D67—81no—
252Dic134—21no—
253Dic134—21no—
254C8×C67—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—21no—
258C8×C67—21no—
259C8×C67—21no—
260C8×C67—21no—
261C8×C67—21no—
262C8×C67—41no—
263C8×C67—41no—
264C67⋊C8—41no—
265C67⋊C8—41no—
266C67⋊C8—41no—
267C67⋊C8—41no—
268C22×C67—21no—
269C22×C67—21no—
270C22×C67—21no—
271C22×C67C2×C4×D411yescharacteristic
272C2×D67—22no—
273C2×D67—22no—
274C2×D67—41no—
275C2×D67—41no—
276C2×D67—41no—
277C2×D67—41no—
278C2×D67—41no—
279C2×D67—41no—
280C2×D67—41no—
281C2×D67—84no—
282C4×C67—21no—
283C4×C67—21no—
284C4×C67—21no—
285C4×C67—21no—
286C4×C67C2×#13111yescharacteristic
287C4×C67C2×#13111yescharacteristic
288C4×C67#57611yescharacteristic
289C4×C67#57711yescharacteristic
290Dic67—41no—
291Dic67—41no—
292Dic67—41no—
293Dic67—41no—
294Dic67—41no—
295#17918—671nomaximal, Sylow
296C2×C67—41no—
297C2×C67#292911yescharacteristic
298C2×C67#294611yescharacteristic
299C2×C67#295711yescharacteristic
300D67—42no—
301D67—42no—
302D67—81no—
303#3105—671no—
304#2993—671no—
305#2972—671no—
306#1586—671no—
307#1585—671no—
308#1582—671no—
309#1581—671no—
310#1580—671no—
311SD16⋊C8—671no—
312C8×SD16—671no—
313C8×D8—671no—
314#1574—671no—
315#1486—671no—
316#1482—671no—
317#1454D6711yescharacteristic, complemented
318C67#1791811yescharacteristic, complemented, Sylow
319#529—671no—
320C2×C4×D4—671no—
321#484—671no—
322#481—671no—
323#478—671no—
324#478—671no—
325#463—671no—
326#463—671no—
327#450—671no—
328#447—671no—
329#446—671no—
330C8×Q8—671no—
331C4×SD16—671no—
332C4×SD16—671no—
333C4×D8—671no—
334C4×D8—671no—
335#436—671no—
336#436—1341no—
337#435—1341no—
338C8×D4—671no—
339C8×D4—671no—
340C8×D4—671no—
341C8×D4—671no—
342C8×D4—671no—
343C8×D4—1341no—
344C8×D4—1341no—
345#433—1341no—
346#431—671no—
347#431C2×D6711yescharacteristic
348#431C2×D6711yescharacteristic
349#431C2×D6711yescharacteristic
350#424—1341no—
351#423—671no—
352C2×C22⋊C8—1341no—
353C2×C22⋊C8—1341no—
354#335—671no—
355#335—671no—
356#334—671no—
357#334—671no—
358Q8⋊C8—671no—
359Q8⋊C8—671no—
360D4⋊C8—671no—
361D4⋊C8—671no—
362D4⋊C8—671no—
363D4⋊C8—671no—
364D4⋊C8—671no—
365D4⋊C8—671no—
366#322C2×D6711yescharacteristic, complemented
367#322C2×D6711yescharacteristic, complemented
368#322C2×D6711yescharacteristic, complemented
369C82C2×D6711yescharacteristic, complemented
370C22×D4—671no—
371C23×C4—1341no—
372C2×SD16—671no—
373C2×SD16—671no—
374C2×D8—671no—
375C2×D8—671no—
376C2×M16—1341no—
377C2×M16—1341no—
378C22×C8—1341no—
379C22×C8—1341no—
380C22×C8—1341no—
381C22×C8—1341no—
382C22×C8—1341no—
383#126—1341no—
384#125—671no—
385#124—671no—
386#123—1341no—
387C22⋊Q8—671no—
388#121—671no—
389C4×Q8—671no—
390C4×D4—671no—
391C4×D4—671no—
392C4×D4—671no—
393C4×D4—671no—
394C4×D4—671no—
395C4×D4—1341no—
396C4×D4—1341no—
397C4×D4—1341no—
398#117—1341no—
399C2×C4⋊C4—671no—
400C2×C22⋊C4—1341no—
401C2×C42C22×D6711yescharacteristic
402#107—671no—
403#107—671no—
404#106—671no—
405#106—671no—
406C4⋊C8—21no—
407C4⋊C8—671no—
408C4⋊C8—671no—
409C4⋊C8—671no—
410C4⋊C8—671no—
411C4⋊C8—1341no—
412C4⋊C8—1341no—
413C4⋊C8—1341no—
414C4⋊C8C4×D6711yescharacteristic
415C4⋊C8C4×D6711yescharacteristic
416C4⋊C8D26811yescharacteristic
417C4⋊C8D26811yescharacteristic
418#103—671no—
419#103—671no—
420#102—671no—
421#102—671no—
422#102—671no—
423#102—671no—
424#102—671no—
425#102—671no—
426C22⋊C8—21no—
427C22⋊C8—1341no—
428C22⋊C8—1341no—
429C22⋊C8—1341no—
430C22⋊C8—1341no—
431C22⋊C8—1341no—
432C22⋊C8—1341no—
433C22⋊C8—1341no—
434C22⋊C8C4×D6711yescharacteristic
435C22⋊C8C4×D6711yescharacteristic
436C22⋊C8D26811yescharacteristic
437C22⋊C8D26811yescharacteristic
438#97—1341no—
439C4×C8—1341no—
440C4×C8—1341no—
441C4×C8C22×D6711yescharacteristic
442C4×C8C22×D6711yescharacteristic
443C4×C8C22×D6711yescharacteristic
444C4×C8C22×D6711yescharacteristic
445C4×C8C22×D6711yescharacteristic
446C4×C8C22×D6711yescharacteristic
447C24—1341no—
448C2×Q8—671no—
449C2×D4—671no—
450C2×D4—671no—
451C2×D4—671no—
452C2×D4—1342no—
453C2×D4—1342no—
454C2×D4—1341no—
455C2×D4—1341no—
456C22×C4—5361no—
457C22×C4—5361no—
458C22×C4—1341no—
459C22×C4—1341no—
460C22×C4—1341no—
461C22×C4—1341no—
462C22×C4—1341no—
463C22×C4—1341no—
464C22×C4C2×C4×D6711yescharacteristic
465C22×C4C2×D26811yescharacteristic
466C22×C4#1176162811yescharacteristic
467SD16—1342no—
468SD16—1342no—
469D8—1342no—
470D8—1342no—
471M16—5361no—
472M16—5361no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—5361no—
476C2×C8—5361no—
477C2×C8—5361no—
478C2×C8—5361no—
479C2×C8—5361no—
480C2×C8—1341no—
481C2×C8—1341no—
482C2×C8—1341no—
483C2×C8—1341no—
484C2×C8C2×C4×D6711yescharacteristic
485C2×C8C2×C4×D6711yescharacteristic
486C2×C8C2×C4×D6711yescharacteristic
487C2×C8C2×C4×D6711yescharacteristic
488C2×C8C2×D26811yescharacteristic
489C2×C8C2×D26811yescharacteristic
490C2×C8C2×D26811yescharacteristic
491C2×C8C2×D26811yescharacteristic
492C4⋊C4—671no—
493C4⋊C4—671no—
494C4⋊C4—671no—
495C4⋊C4—671no—
496C4⋊C4—1341no—
497C4⋊C4—1341no—
498C4⋊C4—1341no—
499C4⋊C4—1341no—
500C4⋊C4—1341no—
501C22⋊C4—1341no—
502C22⋊C4—1341no—
503C22⋊C4—1341no—
504C22⋊C4—1341no—
505C22⋊C4—1341no—
506C22⋊C4—1341no—
507C22⋊C4—1341no—
508C22⋊C4—1341no—
509C22⋊C4—1341no—
510C42—1341no—
511C42C2×C4×D6711yescharacteristic
512C42C2×D26811yescharacteristic
513C42#1176162811yescharacteristic
514C42C23×D6711yescharacteristic
515C23—5361no—
516C23—5361no—
517C23—1344no—
518C23—1341no—
519C23—1341no—
520C23—1341no—
521C23C4×D26811yescharacteristic
522Q8—1341no—
523Q8—1341no—
524D4—5361no—
525D4—672no—
526D4—672no—
527D4—1344no—
528D4—1341no—
529D4—1341no—
530D4—1341no—
531D4—1341no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—2681no—
538C2×C4—2681no—
539C2×C4—2681no—
540C2×C4—2681no—
541C2×C4—2681no—
542C2×C4—2681no—
543C2×C4—5361no—
544C2×C4—5361no—
545C2×C4—5361no—
546C2×C4—1341no—
547C2×C4—1341no—
548C2×C4—1341no—
549C2×C4—1341no—
550C2×C4—1341no—
551C2×C4C4×D26811yescharacteristic
552C2×C4C4×D26811yescharacteristic
553C2×C4C4×D26811yescharacteristic
554C2×C4C22×C4×D6711yescharacteristic
555C2×C4C22×D26811yescharacteristic
556C2×C4C2×#1176162811yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—2681no—
568C8—2681no—
569C8—2681no—
570C8—2681no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—2681no—
575C22—2681no—
576C22—2681no—
577C22—2681no—
578C22—2681no—
579C22—2681no—
580C22—2681no—
581C22—5364no—
582C22—1342no—
583C22—1342no—
584C22C2×C4×D26811yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—2681no—
590C4—2681no—
591C4—2681no—
592C4—2681no—
593C4—2681no—
594C4#131×D6711yescharacteristic
595C4#131×D6711yescharacteristic
596C4#1686420111yescharacteristic
597C4#1686420811yescharacteristic
598C2—41no—
599C2—2682no—
600C2—2682no—
601C2—5361no—
602C2#2256890511yescharacteristic
603C2#2256904311yescharacteristic
604C2#2256912711yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1454C2
socleC22×C67C2×C4×D4
prime radical#16863532C4
cosocleC42×C67C24
hypercenter#1454D67
hypocenterC67#17918
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #22569360C2semidirect
3. #22569232C2semidirect
4. #22559075C2semidirect
5. #22559068C2semidirect
6. #22559047C2semidirect
7. #22559040C2semidirect
8. #22559034C2semidirect
9. #22559023C2semidirect
10. C8×#11761617C2semidirect
11. C8×D536C2semidirect
12. #22558998C2semidirect
13. #22558635C2semidirect
14. #22558618C2semidirect
15. C67×#1454C2semidirect
26. #16863526C22semidirect
27. C8×Dic134C22semidirect
32. #16863475C22semidirect
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
317. #1454D67semidirect
318. C67#17918semidirect
366. #322C2×D67semidirect
367. #322C2×D67semidirect
368. #322C2×D67semidirect
369. C82C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702582
#225693602
#225692322
#225590752
#225590682
#225590472
#225590402
#225590342
#225590232
C8×#117616172
C8×D5362
#225589982
#225586352
#225586182
C67×#14542
#1791867