Finite Group Databasebeta

← #34018281

Subgroups — #34018281

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626736C211yesmaximal, characteristic
2Q8×#11924036C211yesmaximal, characteristic, complemented
3#22625526C211yesmaximal, characteristic, complemented
4#22625525C211yesmaximal, characteristic, complemented
5#22625522C211yesmaximal, characteristic, complemented
6SD16⋊#10596833C211yesmaximal, characteristic, complemented
7#22625519C211yesmaximal, characteristic, complemented
8#22625518C211yesmaximal, characteristic, complemented
9#22625514C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625483C211yesmaximal, characteristic, complemented
12#22625234C211yesmaximal, characteristic, complemented
13#22625227C211yesmaximal, characteristic
14#22625129C211yesmaximal, characteristic, complemented
15#22618753C211yesmaximal, characteristic
16#17989636—21no—
17#17989636—21no—
18#17989636C2211yescharacteristic
19#17989636C2211yescharacteristic
20#17989636C2211yescharacteristic
21#17989636C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989614—21no—
25#17989614—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989606C2211yescharacteristic
29#17989605C2211yescharacteristic
30#17989604—21no—
31#17989588—21no—
32#17989588C2211yescharacteristic
33C4×#11924036—21no—
34C4×#11924036C2211yescharacteristic
35C2×C4×#10596833—21no—
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989428C2211yescharacteristic
49#17989428C2211yescharacteristic
50C8×#10596833C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52C2×#14556620C411yescharacteristic
53C4×#11924027C2211yescharacteristic
54C4×Q8×D73C2211yescharacteristic
55#17988695C411yescharacteristic
56#17988692C411yescharacteristic
57#17988687C411yescharacteristic
58#17988681C2211yescharacteristic
59#17988679C2211yescharacteristic
60#17988678C2211yescharacteristic
61SD16×Dic73C2211yescharacteristic
62#17988628C411yescharacteristic
63#17988594C411yescharacteristic
64#17988585C411yescharacteristic
65#17988539C2211yescharacteristic
66#440×C73C411yescharacteristic
67C2×#11924036—21no—
68C2×#11924036D411yescharacteristic
69C2×#11924036D411yescharacteristic
70C22×#10596833—21no—
71C22×#10596833—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658—21no—
77#14556658—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C4×#10596833—21no—
89C4×#10596833—21no—
90C4×#10596833—21no—
91C4×#10596833—21no—
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833D411yescharacteristic
96C4×#10596833D411yescharacteristic
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924027C2×C411yescharacteristic
102C2×Q8×D73C2×C411yescharacteristic
103#14556620C818yescomplemented
104C2×Dic292C2×C411yescharacteristic
105C2×#11924018C2×C411yescharacteristic
106C2×#11924016C2×C411yescharacteristic
107C2×#11924011C2×C411yescharacteristic
108C2×#11924011C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110C4×Dic146—21no—
111C4×Dic146C2×C411yescharacteristic
112C4×Dic146C2×C411yescharacteristic
113#14556556—21no—
114#14556556C2311yescharacteristic
115#14556556C2311yescharacteristic
116D4×Dic73C2311yescharacteristic
117C4×C22⋊D73—21no—
118#14556551—21no—
119#14556549C2×C411yescharacteristic
120#14556547—21no—
121C4⋊C4×D73—21no—
122C4⋊C4×D73C2×C411yescharacteristic
123C2×C4×Dic73—21no—
124C42×D73—21no—
125C42×D73C2311yescharacteristic
126#14556524C2×C411yescharacteristic
127#14556523C2×C411yescharacteristic
128#14556522C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556516C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556514C2×C411yescharacteristic
133#14556514C2×C411yescharacteristic
134#14556512C2×C411yescharacteristic
135#14556502C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×SD16×C73C2×C411yescharacteristic
139C4×Q8×C73C2×C411yescharacteristic
140C4×D4×C73C2×C411yescharacteristic
141#107×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#97×C73C2×C411yescharacteristic
145#11924036—22no—
146#11924036—41no—
147#11924036—42no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—42no—
155C2×#10596833—42no—
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160#11924027—22no—
161#11924027C2×C814yes—
162C2×Dic146—21no—
163C2×Dic146C22×C411yescharacteristic
164C2×Dic146C22×C411yescharacteristic
165Q8×D73—22no—
166Q8×D73C2×C814yescomplemented
167C2×C22⋊D73—21no—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C2×C4×D73—21no—
171C2×C4×D73—21no—
172C2×C4×D73C22×C411yescharacteristic
173C2×C4×D73C2×D411yescharacteristic
174C2×C4×D73#4111yescharacteristic
175Dic292C2×C814yescomplemented
176#11924018C2×C814yescomplemented
177#11924016C2×C814yescomplemented
178#11924011—22no—
179#11924011C2×C814yescomplemented
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C22×C411yescharacteristic
182C4⋊Dic73—21no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189#11924007—21no—
190C22⋊Dic73—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73C2×D411yescharacteristic
199C4×Dic73#4111yescharacteristic
200C4×Dic73C2411yescharacteristic
201C2×Q8×C73C22×C411yescharacteristic
202C2×D4×C73C22×C411yescharacteristic
203C22×C4×C73—21no—
204SD16×C73C2×C814yescomplemented
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C22×C411yescharacteristic
207C4⋊C4×C73—21no—
208C4⋊C4×C73C22×C411yescharacteristic
209C4⋊C4×C73C22×C411yescharacteristic
210C22⋊C4×C73—21no—
211C42×C73—21no—
212C42×C73C22×C411yescharacteristic
213C23×C73—21no—
214C22×D73C4×D411yescharacteristic
215Q8×C73—21no—
216Q8×C73C22×C812yes—
217D4×C73—21no—
218D4×C73C22×C812yes—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—42no—
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C4×D411yescharacteristic
225C2×C4×C73C23×C411yescharacteristic
226C22⋊D73—24no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—24no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73C4×D411yescharacteristic
235C2×Dic73C22×D411yescharacteristic
236C2×Dic73C2×#4111yescharacteristic
237C4×D73—24no—
238C4×D73—21no—
239C4×D73—21no—
240C4×D73—41no—
241C4×D73—42no—
242C4×D73C22×C812yes—
243C4×D73#13112yes—
244Dic146—21no—
245Dic146—21no—
246Dic146—24no—
247Dic146C22×C812yes—
248Dic146C22×C812yes—
249#10596833—21no—
250#10596833—21no—
251#10596833—21no—
252#10596833—41no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258#10596833#13712yescomplemented
259C8×C73—21no—
260C8×C73C22×C812yes—
261#10596831—21no—
262#10596831C22×C812yes—
263C22×C73—22no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73C2×C4×D411yescharacteristic
267C2×D73—21no—
268C2×D73—21no—
269C2×D73C8×D412yes—
270C4×C73—22no—
271C4×C73—21no—
272C4×C73—21no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73C23×C811yescharacteristic
276C4×C73C2×#13111yescharacteristic
277Dic73—22no—
278Dic73—22no—
279Dic73—21no—
280Dic73—41no—
281Dic73—41no—
282Dic73C8×D412yes—
283Dic73C2×#13711yescharacteristic
284Dic73#57511yescharacteristic
285C8×#137—731nomaximal, Sylow
286C2×C73—42no—
287C2×C73C2×C8×D411yescharacteristic
288C2×C73C4×#13711yescharacteristic
289C2×C73#295511yescharacteristic
290D73—22no—
291C8×#41—731no—
292C2×C8×Q8—731no—
293C4×#137—731no—
294#1579—731no—
295#1579—731no—
296SD16⋊C8—731no—
297SD16⋊C8—731no—
298C8×Q16—731no—
299C8×Q16—731no—
300C8×SD16—731no—
301C8×SD16—731no—
302#1565—731no—
303#1479—731no—
304#1477—731no—
305C8×M16—731no—
306C73C8×#13711yescharacteristic, complemented, Sylow
307C2×#137—731no—
308C4×#41—731no—
309C2×C4×Q8—731no—
310C8×Q8—731no—
311C8×Q8—731no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—1461no—
317C8×Q8—1461no—
318#441—731no—
319#441—731no—
320#440—731no—
321#440#6440811yescharacteristic
322C4×Q16—731no—
323C4×Q16—731no—
324C4×SD16—731no—
325C4×SD16—731no—
326C8×D4—731no—
327C8×D4—731no—
328C8×D4—1461no—
329C8×D4—1461no—
330#431—731no—
331#431—731no—
332#431—1461no—
333#428—731no—
334C2×C4⋊C8—731no—
335C2×C4⋊C8—1461no—
336#419—731no—
337#417—731no—
338C4×M16—731no—
339C2×C4×C8—731no—
340C2×C4×C8—1461no—
341C2×C4×C8—1461no—
342#335—731no—
343#335—731no—
344#334—731no—
345#334—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352D4⋊C8—731no—
353D4⋊C8—731no—
354#322—731no—
355#322—731no—
356C82—731no—
357C82—731no—
358C2×#41—731no—
359C22×Q8—731no—
360#137—738no—
361C2×Q16—731no—
362C2×Q16—731no—
363C2×SD16—731no—
364C2×SD16C2×#6440811yescharacteristic
365C2×M16—731no—
366C2×M16—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—1461no—
370C22×C8—1461no—
371C22×C8—1461no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—731no—
375C4×Q8—731no—
376C4×Q8—731no—
377C4×Q8—1461no—
378C4×Q8—1461no—
379C4×Q8C2×#6440811yescharacteristic
380C4×D4—731no—
381C4×D4—1461no—
382C4×D4—1461no—
383C4×D4C2×#6440811yescharacteristic
384#117—731no—
385#117—1461no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—1461no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391#107—731no—
392#107C2×#6440811yescharacteristic
393#106—731no—
394#106—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8—1461no—
407#103—731no—
408#103—731no—
409#103—731no—
410#103—731no—
411#103—731no—
412#103C2×#6440811yescharacteristic
413#102—731no—
414#102C2×#6440811yescharacteristic
415C22⋊C8—1461no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—731no—
420#97C2×#6440811yescharacteristic
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—1461no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8—1461no—
433C4×C8—1461no—
434C4×C8—1461no—
435C4×C8—1461no—
436#41—734no—
437#41—1462no—
438C2×Q8—734no—
439C2×Q8—731no—
440C2×Q8—731no—
441C2×Q8—1461no—
442C2×Q8—1462no—
443C2×Q8C22×#6440811yescharacteristic
444C2×D4—1461no—
445C2×D4C22×#6440811yescharacteristic
446C22×C4—21no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454Q16—734no—
455Q16—734no—
456SD16—734no—
457SD16#1192403614yescomplemented
458M16—734no—
459M16—1462no—
460C2×C8—2922no—
461C2×C8—2922no—
462C2×C8—2922no—
463C2×C8—2921no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—731no—
469C2×C8—731no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1462no—
477C2×C8C22×#6440811yescharacteristic
478C2×C8C22×#6440811yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—731no—
481C4⋊C4—731no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4C22×#6440811yescharacteristic
489C4⋊C4C22×#6440811yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—1461no—
492C22⋊C4—1461no—
493C42—21no—
494C42—731no—
495C42—731no—
496C42—731no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42—1461no—
503C42C22×#6440811yescharacteristic
504C23—21no—
505C23—731no—
506Q8—21no—
507Q8—732no—
508Q8—732no—
509Q8—1464no—
510Q8—1461no—
511Q8—1461no—
512Q8C2×#1192403612yes—
513D4—21no—
514D4—1464no—
515D4C2×#1192403612yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—2921no—
521C2×C4—2922no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—731no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1464no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1464no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4D4×#6440811yescharacteristic
538C2×C4D4×#6440811yescharacteristic
539C2×C4C23×#6440811yescharacteristic
540C8—21no—
541C8—2921no—
542C8—2921no—
543C8—2921no—
544C8—2921no—
545C8—2921no—
546C8—2921no—
547C8—732no—
548C8—732no—
549C8—1461no—
550C8—1461no—
551C8—1461no—
552C8—1461no—
553C8C2×#1192403612yes—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—732no—
558C22—1461no—
559C22—1461no—
560C22C2×D4×#6440811yescharacteristic
561C4—22no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—2921no—
567C4—2921no—
568C4—732no—
569C4—731no—
570C4—731no—
571C4—1462no—
572C4—1462no—
573C4—1461no—
574C4C22×#1192403611yescharacteristic
575C4D73⋊#13111yescharacteristic
576C2—42no—
577C2—1462no—
578C2D4×#1192403611yescharacteristic
579C2#137×#6440811yescharacteristic
580C2#2262670311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC4×C73C23×C8
FrattiniC2×C4C23×#64408
Fitting#440×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#11924027C2×C4
cosocleC4×Dic73C24
hypercenter#440#64408
hypocenterC73C8×#137
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. Q8×#11924036C2semidirect
3. #22625526C2semidirect
4. #22625525C2semidirect
5. #22625522C2semidirect
6. SD16⋊#10596833C2semidirect
7. #22625519C2semidirect
8. #22625518C2semidirect
9. #22625514C2semidirect
10. SD16×#10596833C2semidirect
11. #22625483C2semidirect
12. #22625234C2semidirect
14. #22625129C2semidirect
103. #14556620C8semidirect
166. Q8×D73C2×C8semidirect
175. Dic292C2×C8semidirect
176. #11924018C2×C8semidirect
177. #11924016C2×C8semidirect
179. #11924011C2×C8semidirect
204. SD16×C73C2×C8semidirect
258. #10596833#137semidirect
306. C73C8×#137semidirect
457. SD16#11924036semidirect

Maximal subgroups

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

subgroupindex
#226267362
Q8×#119240362
#226255262
#226255252
#226255222
SD16⋊#105968332
#226255192
#226255182
#226255142
SD16×#105968332
#226254832
#226252342
#226252272
#226251292
#226187532
C8×#13773