Finite Group Databasebeta

← #30826985

Subgroups — #30826985

Order 17152.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570258C211yesmaximal, characteristic
2#22569365C211yesmaximal, characteristic, complemented
3#22569246C211yesmaximal, characteristic, complemented
4#22559098C211yesmaximal, characteristic, complemented
5#22559093C211yesmaximal, characteristic, complemented
6#22559086C211yesmaximal, characteristic, complemented
7#22559072C211yesmaximal, characteristic, complemented
8#22559072C211yesmaximal, characteristic, complemented
9#22559065C211yesmaximal, characteristic, complemented
10#22559008C211yesmaximal, characteristic, complemented
11#22559003C211yesmaximal, characteristic, complemented
12#22558667C211yesmaximal, characteristic, complemented
13C2×#16863123C211yesmaximal, characteristic, complemented
14#22558493C211yesmaximal, characteristic, complemented
15C67×#1573C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18#16863732C411yescharacteristic
19#16863715C411yescharacteristic
20#16863697C411yescharacteristic
21#16863697C411yescharacteristic
22#16863612C411yescharacteristic
23#16863612C411yescharacteristic
24#16863549C411yescharacteristic
25#16863532C411yescharacteristic
26#16863523C2211yescharacteristic, complemented
27#16863520C2211yescharacteristic, complemented
28C4×#11761617C2211yescharacteristic
29C4×#11761617C2211yescharacteristic
30C4×D536C2211yescharacteristic
31C4×D536C2211yescharacteristic
32#16863473C2211yescharacteristic
33#16863473C2211yescharacteristic
34#16863473C2211yescharacteristic, complemented
35#16863469—21no—
36#16863469—21no—
37#16863467—21no—
38#16863461C2211yescharacteristic
39#16863461C2211yescharacteristic
40#16863461C2211yescharacteristic, complemented
41C8×C22⋊D67—21no—
42#16863456—21no—
43#16863449C2211yescharacteristic
44#16863421—21no—
45#16863358—21no—
46C22⋊C8×D67—21no—
47C2×C4×C67⋊C8C2211yescharacteristic
48#16863126C2211yescharacteristic, complemented
49#16863126C2211yescharacteristic, complemented
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863123C2211yescharacteristic
53#16863123C2211yescharacteristic
54#16863123C2211yescharacteristic, complemented
55#16863123C2211yescharacteristic, complemented
56#16863118C2211yescharacteristic, complemented
57#16863118C2211yescharacteristic, complemented
58C8×C67⋊C8C2211yescharacteristic, complemented
59C8×C67⋊C8C2211yescharacteristic, complemented
60#432×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62C2×C4⋊C8×C67C2211yescharacteristic
63#335×C67C2211yescharacteristic, complemented
64#335×C67C2211yescharacteristic, complemented
65#334×C67C2211yescharacteristic, complemented
66#334×C67C2211yescharacteristic, complemented
67C22×D268C2×C411yescharacteristic
68C22×C4×D67—21no—
69C2×#11761617C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×D536C2×C411yescharacteristic
72C2×D536C2×C411yescharacteristic
73C2×#11761611—21no—
74C2×#11761611—21no—
75C2×#11761611—21no—
76C22×C67⋊C8—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)D411yescharacteristic
103C4⋊(C67⋊C8)D411yescharacteristic
104Dic67⋊C8—21no—
105Dic67⋊C8—21no—
106#14523968C2×C411yescharacteristic
107#14523968C2×C411yescharacteristic
108#14523965C2×C411yescharacteristic
109#14523965C2×C411yescharacteristic
110#14523965C2×C411yescharacteristic
111#14523965C2×C411yescharacteristic
112#14523965C2×C411yescharacteristic
113#14523965C2×C411yescharacteristic
114C22⋊(C67⋊C8)D411yescharacteristic
115C22⋊(C67⋊C8)D411yescharacteristic
116(C2×D67)⋊C8—21no—
117(C2×D67)⋊C8—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120(C2×D67)⋊C8—21no—
121(C2×D67)⋊C8—21no—
122#14523953—21no—
123C4×C67⋊C8—21no—
124C4×C67⋊C8C2311yescharacteristic
125C4×C67⋊C8C2311yescharacteristic
126C4×C67⋊C8C2311yescharacteristic
127C4×C67⋊C8C2311yescharacteristic
128C8×Dic67—21no—
129C22×C8×C67—21no—
130C2×C42×C67C2311yescharacteristic
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2×C411yescharacteristic
133C4⋊C8×C67C2×C411yescharacteristic
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136C4⋊C8×C67C2311yescharacteristic
137C4⋊C8×C67C2311yescharacteristic
138C22⋊C8×C67—21no—
139C22⋊C8×C67C2×C411yescharacteristic
140C22⋊C8×C67C2×C411yescharacteristic
141#97×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C23×D67—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×D268—22no—
147C2×D268—21no—
148C2×D268—41no—
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—41no—
163D536—22no—
164D536—41no—
165#11761611—81no—
166#11761611—81no—
167#11761611—81no—
168C2×C67⋊C8—24no—
169C2×C67⋊C8—21no—
170C2×C67⋊C8—21no—
171C2×C67⋊C8C2×D411yescharacteristic
172C2×C67⋊C8C2×D411yescharacteristic
173C2×C67⋊C8C2×D411yescharacteristic
174C2×C67⋊C8C2×D411yescharacteristic
175C8×D67—81no—
176C8×D67—81no—
177C8×D67—81no—
178C4⋊Dic67—21no—
179C4⋊Dic67C22×C411yescharacteristic
180C4⋊Dic67C22×C411yescharacteristic
181C4⋊Dic67C22×C411yescharacteristic
182C4⋊Dic67C22×C411yescharacteristic
183Dic67⋊C4—21no—
184Dic67⋊C4—21no—
185Dic67⋊C4—21no—
186Dic67⋊C4—21no—
187C22⋊Dic67—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—
194(C2×D67)⋊C4—21no—
195(C2×D67)⋊C4—21no—
196C4×Dic67—21no—
197C22×C4×C67C22×C411yescharacteristic
198C22×C4×C67C2×D411yescharacteristic
199C22×C4×C67#4111yescharacteristic
200C2×C8×C67—21no—
201C2×C8×C67—21no—
202C2×C8×C67—21no—
203C2×C8×C67—21no—
204C2×C8×C67—81no—
205C2×C8×C67C22×C411yescharacteristic
206C2×C8×C67C22×C411yescharacteristic
207C2×C8×C67C22×C411yescharacteristic
208C2×C8×C67C22×C411yescharacteristic
209C42×C67C22×C411yescharacteristic
210C42×C67C2×D411yescharacteristic
211C42×C67#4111yescharacteristic
212C42×C67C2411yescharacteristic
213C23×C67C4×D411yescharacteristic
214C22×D67—21no—
215C22×D67—21no—
216C22×D67—21no—
217C22×D67—42no—
218C22×D67—42no—
219C22×D67—81no—
220C2×C4×C67—22no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C23×C411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C22⋊D67—42no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236C2×Dic67—21no—
237C2×Dic67—81no—
238D268—21no—
239D268—21no—
240D268—21no—
241D268—21no—
242D268—21no—
243D268—42no—
244D268#13112yes—
245C4×D67—41no—
246C4×D67—41no—
247C4×D67—41no—
248C4×D67—41no—
249C4×D67—41no—
250C4×D67—41no—
251C4×D67—81no—
252C4×D67—81no—
253Dic134—21no—
254Dic134—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—21no—
258C8×C67—21no—
259C8×C67—41no—
260C8×C67—41no—
261C8×C67—41no—
262C8×C67—41no—
263C67⋊C8—22no—
264C67⋊C8—22no—
265C67⋊C8—21no—
266C67⋊C8—21no—
267C67⋊C8—21no—
268C67⋊C8—21no—
269C22×C67—21no—
270C22×C67—21no—
271C22×C67#44312yes—
272C22×C67C2×C4×D411yescharacteristic
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—41no—
282C2×D67—84no—
283C4×C67—21no—
284C4×C67—21no—
285C4×C67—21no—
286C4×C67—21no—
287C4×C67C2×#13111yescharacteristic
288C4×C67#56811yescharacteristic
289C4×C67#57611yescharacteristic
290C4×C67#57711yescharacteristic
291Dic67—41no—
292Dic67—41no—
293Dic67—41no—
294Dic67—41no—
295Dic67—41no—
296#18014—671nomaximal, Sylow
297C2×C67—22no—
298C2×C67#293311yescharacteristic
299C2×C67#294611yescharacteristic
300C2×C67C2×#44311yescharacteristic
301D67—42no—
302D67—81no—
303D67—81no—
304#3105—671no—
305#2994—671no—
306#2974—671no—
307#1590—671no—
308#1589—671no—
309#1588—671no—
310#1586—671no—
311#1586—671no—
312#1585—671no—
313C8×SD16—671no—
314C8×D8—671no—
315#1573D6711yescharacteristic, complemented
316#1494—671no—
317C2×D4⋊C8—671no—
318#1450—671no—
319C67#1801411yescharacteristic, complemented, Sylow
320#529—671no—
321C2×C4×D4—671no—
322#484—671no—
323#481—671no—
324#478—671no—
325#478—671no—
326#463—671no—
327#463—671no—
328#450—671no—
329#447—671no—
330#446—671no—
331C8×Q8—671no—
332C4×SD16—671no—
333C4×SD16—671no—
334C4×D8—671no—
335C4×D8—671no—
336#436—671no—
337#436—671no—
338#436—671no—
339#435—1341no—
340#435—1341no—
341#435—1341no—
342C8×D4—671no—
343C8×D4—671no—
344C8×D4—671no—
345C8×D4—1341no—
346#433—1341no—
347#432C2×D6711yescharacteristic
348#431—671no—
349#431C2×D6711yescharacteristic
350#424—1341no—
351C2×C4⋊C8C2×D6711yescharacteristic
352#407—1341no—
353C2×C22⋊C8—1341no—
354C2×C4×C8—671no—
355#335C2×D6711yescharacteristic, complemented
356#335C2×D6711yescharacteristic, complemented
357#334C2×D6711yescharacteristic, complemented
358#334C2×D6711yescharacteristic, complemented
359Q8⋊C8—671no—
360Q8⋊C8—671no—
361D4⋊C8—671no—
362D4⋊C8—671no—
363D4⋊C8—671no—
364D4⋊C8—671no—
365D4⋊C8—671no—
366D4⋊C8—671no—
367#322—671no—
368#322—671no—
369C82—671no—
370C82—671no—
371C22×D4—671no—
372C23×C4—1341no—
373C2×SD16—671no—
374C2×SD16—671no—
375C2×D8—671no—
376C2×D8—671no—
377C2×M16—1341no—
378C2×M16—1341no—
379C2×M16—1341no—
380C22×C8—21no—
381C22×C8—1341no—
382C22×C8—1341no—
383C22×C8—1341no—
384C22×C8—1341no—
385#126—1341no—
386#125—671no—
387#124—671no—
388#123—1341no—
389C22⋊Q8—671no—
390#121—671no—
391C4×Q8—671no—
392C4×D4—671no—
393C4×D4—671no—
394C4×D4—671no—
395C4×D4—671no—
396C4×D4—671no—
397C4×D4—1341no—
398C4×D4—1341no—
399C4×D4—1341no—
400#117—1341no—
401C2×C4⋊C4—671no—
402C2×C22⋊C4—1341no—
403C2×C42C22×D6711yescharacteristic
404#107—671no—
405#107—671no—
406#106—671no—
407#106—671no—
408C4⋊C8—21no—
409C4⋊C8—671no—
410C4⋊C8—671no—
411C4⋊C8—1341no—
412C4⋊C8—1341no—
413C4⋊C8C4×D6711yescharacteristic
414C4⋊C8C4×D6711yescharacteristic
415C4⋊C8C22×D6711yescharacteristic
416C4⋊C8C22×D6711yescharacteristic
417C4⋊C8C22×D6711yescharacteristic
418C4⋊C8C22×D6711yescharacteristic
419#103—671no—
420#103—671no—
421#102—671no—
422#102—671no—
423#102—671no—
424#102—671no—
425#102—671no—
426#102—671no—
427C22⋊C8—21no—
428C22⋊C8—671no—
429C22⋊C8—671no—
430C22⋊C8—1341no—
431C22⋊C8—1341no—
432C22⋊C8—1341no—
433C22⋊C8—1341no—
434C22⋊C8—1341no—
435C22⋊C8—1341no—
436C22⋊C8C4×D6711yescharacteristic
437C22⋊C8C4×D6711yescharacteristic
438#97—21no—
439#97—1341no—
440C4×C8—671no—
441C4×C8—671no—
442C4×C8—671no—
443C4×C8—671no—
444C4×C8—1341no—
445C4×C8—1341no—
446C4×C8C22×D6711yescharacteristic
447C4×C8C22×D6711yescharacteristic
448C24—1341no—
449C2×Q8—671no—
450C2×D4—2681no—
451C2×D4—671no—
452C2×D4—671no—
453C2×D4—671no—
454C2×D4—1342no—
455C2×D4—1341no—
456C2×D4—1341no—
457C22×C4—5361no—
458C22×C4—5361no—
459C22×C4—1341no—
460C22×C4—1341no—
461C22×C4—1341no—
462C22×C4—1341no—
463C22×C4—1341no—
464C22×C4—1341no—
465C22×C4C2×C4×D6711yescharacteristic
466C22×C4D4×D6711yescharacteristic
467C22×C4#1176162911yescharacteristic
468SD16—2681no—
469SD16—1342no—
470D8—2681no—
471D8—1342no—
472M16—5361no—
473M16—5361no—
474M16—5361no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—81no—
480C2×C8—5361no—
481C2×C8—5361no—
482C2×C8—5361no—
483C2×C8—671no—
484C2×C8—671no—
485C2×C8—671no—
486C2×C8—671no—
487C2×C8—1344no—
488C2×C8—1341no—
489C2×C8—1341no—
490C2×C8C2×C4×D6711yescharacteristic
491C2×C8C2×C4×D6711yescharacteristic
492C2×C8C2×C4×D6711yescharacteristic
493C2×C8C2×C4×D6711yescharacteristic
494C4⋊C4—671no—
495C4⋊C4—671no—
496C4⋊C4—671no—
497C4⋊C4—671no—
498C4⋊C4—1341no—
499C4⋊C4—1341no—
500C4⋊C4—1341no—
501C4⋊C4—1341no—
502C4⋊C4—1341no—
503C22⋊C4—1341no—
504C22⋊C4—1341no—
505C22⋊C4—1341no—
506C22⋊C4—1341no—
507C22⋊C4—1341no—
508C22⋊C4—1341no—
509C22⋊C4—1341no—
510C22⋊C4—1341no—
511C22⋊C4—1341no—
512C42—1341no—
513C42C2×C4×D6711yescharacteristic
514C42D4×D6711yescharacteristic
515C42#1176162911yescharacteristic
516C42C23×D6711yescharacteristic
517C23—2682no—
518C23—2682no—
519C23—5361no—
520C23—1341no—
521C23—1341no—
522C23—1341no—
523C23#1452400511yescharacteristic
524Q8—1341no—
525Q8—1341no—
526D4—2682no—
527D4—2682no—
528D4—672no—
529D4—1341no—
530D4—1341no—
531D4—1341no—
532D4—1341no—
533D4—1341no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—2681no—
540C2×C4—2681no—
541C2×C4—2681no—
542C2×C4—2681no—
543C2×C4—2681no—
544C2×C4—2681no—
545C2×C4—5361no—
546C2×C4—5361no—
547C2×C4—5361no—
548C2×C4—1341no—
549C2×C4—1341no—
550C2×C4—1341no—
551C2×C4—1341no—
552C2×C4—1341no—
553C2×C4#1452400511yescharacteristic
554C2×C4#1452400511yescharacteristic
555C2×C4#1452400511yescharacteristic
556C2×C4C22×C4×D6711yescharacteristic
557C2×C4C2×D4×D6711yescharacteristic
558C2×C4C2×#1176162911yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—1342no—
568C8—1342no—
569C8—1341no—
570C8—1341no—
571C8—1341no—
572C8—1341no—
573C22—21no—
574C22—21no—
575C22—2681no—
576C22—2681no—
577C22—2681no—
578C22—2681no—
579C22—2681no—
580C22—2681no—
581C22—2681no—
582C22—2681no—
583C22—5364no—
584C22—1342no—
585C22#1686351212yes—
586C22C2×#1452400511yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—2681no—
592C4—2681no—
593C4—2681no—
594C4—2681no—
595C4—2681no—
596C4#131×D6711yescharacteristic
597C4#1686416111yescharacteristic
598C4#1686420011yescharacteristic
599C4#1686420511yescharacteristic
600C2—22no—
601C2—2682no—
602C2—5361no—
603C2—5361no—
604C2#2256893311yescharacteristic
605C2#2256903911yescharacteristic
606C2C2×#1686351211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1573C2
socleC22×C67C2×C4×D4
prime radical#16863532C4
cosocleC42×C67C24
hypercenter#1573D67
hypocenterC67#18014
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
2. #22569365C2semidirect
3. #22569246C2semidirect
4. #22559098C2semidirect
5. #22559093C2semidirect
6. #22559086C2semidirect
7. #22559072C2semidirect
8. #22559072C2semidirect
9. #22559065C2semidirect
10. #22559008C2semidirect
11. #22559003C2semidirect
12. #22558667C2semidirect
13. C2×#16863123C2semidirect
14. #22558493C2semidirect
15. C67×#1573C2semidirect
26. #16863523C22semidirect
27. #16863520C22semidirect
34. #16863473C22semidirect
40. #16863461C22semidirect
48. #16863126C22semidirect
49. #16863126C22semidirect
54. #16863123C22semidirect
55. #16863123C22semidirect
56. #16863118C22semidirect
57. #16863118C22semidirect
58. C8×C67⋊C8C22semidirect
59. C8×C67⋊C8C22semidirect
63. #335×C67C22semidirect
64. #335×C67C22semidirect
65. #334×C67C22semidirect
66. #334×C67C22semidirect
315. #1573D67semidirect
319. C67#18014semidirect
355. #335C2×D67semidirect
356. #335C2×D67semidirect
357. #334C2×D67semidirect
358. #334C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702582
#225693652
#225692462
#225590982
#225590932
#225590862
#225590722
#225590652
#225590082
#225590032
#225586672
C2×#168631232
#225584932
C67×#15732
#1801467