Finite Group Databasebeta

← #34018293

Subgroups — #34018293

Order 18688.

702 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626847C211yesmaximal, characteristic
2#22626825C211yesmaximal, 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#22625223C211yesmaximal, characteristic
13#22625220C211yesmaximal, characteristic, complemented
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#17989606—21no—
29#17989606C2211yescharacteristic
30#17989605—21no—
31#17989605C2211yescharacteristic
32#17989604—21no—
33C2×#14556658—21no—
34#17989588C2211yescharacteristic
35C4×#11924036C2211yescharacteristic
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×#11924036—21no—
69C22×#10596833—21no—
70C22×#10596833—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648D411yescharacteristic
86#14556648D411yescharacteristic
87C22⋊#10596833—21no—
88C22⋊#10596833—21no—
89C4×#10596833—21no—
90C4×#10596833—21no—
91C4×#10596833—21no—
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
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—42no—
146#11924036—81no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—81no—
154C2×#10596833—81no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159#11924027—22no—
160#11924027C2×C814yes—
161C2×Dic146—21no—
162C2×Dic146C22×C411yescharacteristic
163C2×Dic146C22×C411yescharacteristic
164Q8×D73—22no—
165Q8×D73C2×C814yescomplemented
166C2×C22⋊D73—21no—
167C22×Dic73—21no—
168C22×Dic73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171C2×C4×D73C22×C411yescharacteristic
172C2×C4×D73C2×D411yescharacteristic
173C2×C4×D73#4111yescharacteristic
174Dic292C2×C814yescomplemented
175#11924018C2×C814yescomplemented
176#11924016C2×C814yescomplemented
177#11924011—22no—
178#11924011C2×C814yescomplemented
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C22×C411yescharacteristic
181C4⋊Dic73—21no—
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189C22⋊Dic73—21no—
190#11924005—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C4×Dic73C2411yescharacteristic
200C2×Q8×C73C22×C411yescharacteristic
201C2×D4×C73C22×C411yescharacteristic
202C22×C4×C73—21no—
203SD16×C73C2×C814yescomplemented
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C4⋊C4×C73C22×C411yescharacteristic
209C22⋊C4×C73—21no—
210C42×C73—21no—
211C42×C73C22×C411yescharacteristic
212C23×C73—21no—
213C22×D73C4×D411yescharacteristic
214Q8×C73—21no—
215Q8×C73C22×C812yes—
216D4×C73—21no—
217D4×C73C22×C812yes—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—81no—
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C23×C411yescharacteristic
225C22⋊D73—24no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—24no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73C4×D411yescharacteristic
234C2×Dic73C22×D411yescharacteristic
235C2×Dic73C2×#4111yescharacteristic
236C4×D73—24no—
237C4×D73—21no—
238C4×D73—21no—
239C4×D73—41no—
240C4×D73—81no—
241C4×D73C22×C812yes—
242C4×D73#13112yes—
243Dic146—21no—
244Dic146—21no—
245Dic146—24no—
246Dic146C22×C812yes—
247Dic146C22×C812yes—
248#10596833—22no—
249#10596833—21no—
250#10596833—21no—
251#10596833—21no—
252#10596833—21no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258C8×C73—21no—
259C8×C73C22×C812yes—
260#10596831—21no—
261#10596831C22×C812yes—
262C22×C73—22no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73C2×C4×D411yescharacteristic
266C2×D73—21no—
267C2×D73—21no—
268C2×D73C8×D412yes—
269C4×C73—22no—
270C4×C73—21no—
271C4×C73—21no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73C23×C811yescharacteristic
275C4×C73C2×#13111yescharacteristic
276Dic73—22no—
277Dic73—22no—
278Dic73—21no—
279Dic73—41no—
280Dic73—41no—
281Dic73C8×D412yes—
282Dic73#57711yescharacteristic
283Dic73#57811yescharacteristic
284#17853—731nomaximal, Sylow
285C2×C73—42no—
286C2×C73C2×C8×D411yescharacteristic
287C2×C73#294711yescharacteristic
288C2×C73#295611yescharacteristic
289D73—22no—
290#2988—731no—
291#2982—731no—
292C4×#137—731no—
293#1579—731no—
294#1579—731no—
295SD16⋊C8—731no—
296SD16⋊C8—731no—
297C8×Q16—731no—
298C8×Q16—731no—
299C8×SD16—731no—
300C8×SD16—731no—
301#1565—731no—
302#1476—731no—
303C2×Q8⋊C8—731no—
304C8×M16—731no—
305C73#1785311yescharacteristic, complemented, Sylow
306C2×#137—731no—
307C4×#41—731no—
308C2×C4×Q8—731no—
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—731no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—1461no—
316C8×Q8—1461no—
317#441—731no—
318#441—731no—
319#440—731no—
320#440#6440811yescharacteristic
321C4×Q16—731no—
322C4×Q16—731no—
323C4×SD16—731no—
324C4×SD16—731no—
325C8×D4—731no—
326C8×D4—731no—
327C8×D4—1461no—
328C8×D4—1461no—
329#431—731no—
330#431—731no—
331#431—1461no—
332#431—1461no—
333#431—1461no—
334#428—731no—
335C2×C4⋊C8—731no—
336C2×C4⋊C8—1461no—
337#419—731no—
338#417—731no—
339C4×M16—731no—
340C2×C4×C8—731no—
341#335—731no—
342#335—731no—
343#334—731no—
344#334—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350Q8⋊C8—731no—
351D4⋊C8—731no—
352D4⋊C8—731no—
353#322—731no—
354#322—731no—
355C82—731no—
356C82—731no—
357C2×#41—731no—
358C22×Q8—731no—
359#137—738no—
360C2×Q16—731no—
361C2×Q16—731no—
362C2×SD16—731no—
363C2×SD16C2×#6440811yescharacteristic
364C2×M16—731no—
365C2×M16—731no—
366C22×C8—1461no—
367C22×C8—1461no—
368C22×C8—1461no—
369C22×C8—1461no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—731no—
375C4×Q8—1461no—
376C4×Q8—1461no—
377C4×Q8C2×#6440811yescharacteristic
378C4×D4—731no—
379C4×D4—1461no—
380C4×D4—1461no—
381C4×D4C2×#6440811yescharacteristic
382#117—731no—
383#117—1461no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1461no—
386C2×C42—731no—
387C2×C42—1461no—
388C2×C42—1461no—
389#107—731no—
390#107C2×#6440811yescharacteristic
391#106—731no—
392#106—731no—
393C4⋊C8—731no—
394C4⋊C8—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—
406#103—731no—
407#103—731no—
408#103—731no—
409#103—731no—
410#103—731no—
411#103C2×#6440811yescharacteristic
412#102—731no—
413#102C2×#6440811yescharacteristic
414C22⋊C8—731no—
415C22⋊C8—731no—
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—1461no—
428C4×C8—1461no—
429C4×C8—1461no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8—1461no—
433C4×C8—1461no—
434C4×C8—1461no—
435#41—734no—
436#41—1462no—
437C2×Q8—734no—
438C2×Q8—731no—
439C2×Q8—731no—
440C2×Q8—1461no—
441C2×Q8—1462no—
442C2×Q8C22×#6440811yescharacteristic
443C2×D4—1461no—
444C2×D4C22×#6440811yescharacteristic
445C22×C4—21no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453Q16—734no—
454Q16—734no—
455SD16—734no—
456SD16#1192403614yescomplemented
457M16—734no—
458M16—1462no—
459C2×C8—2922no—
460C2×C8—5841no—
461C2×C8—5841no—
462C2×C8—5841no—
463C2×C8—731no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—731no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C22×#6440811yescharacteristic
476C2×C8C22×#6440811yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—731no—
479C4⋊C4—731no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4C22×#6440811yescharacteristic
487C4⋊C4C22×#6440811yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C42—21no—
492C42—731no—
493C42—731no—
494C42—731no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42C22×#6440811yescharacteristic
502C23—21no—
503C23—731no—
504Q8—21no—
505Q8—732no—
506Q8—732no—
507Q8—1464no—
508Q8—1461no—
509Q8—1461no—
510Q8C2×#1192403612yes—
511D4—21no—
512D4—1464no—
513D4C2×#1192403612yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—81no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—5841no—
522C2×C4—731no—
523C2×C4—731no—
524C2×C4—732no—
525C2×C4—732no—
526C2×C4—731no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1464no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1464no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4D4×#6440811yescharacteristic
536C2×C4D4×#6440811yescharacteristic
537C2×C4C23×#6440811yescharacteristic
538C8—21no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—2921no—
543C8—2921no—
544C8—732no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8—1461no—
549C8—1462no—
550C8—1461no—
551C8C2×#1192403612yes—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—732no—
556C22—1461no—
557C22—1461no—
558C22C2×D4×#6440811yescharacteristic
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—2921no—
565C4—2921no—
566C4—732no—
567C4—731no—
568C4—731no—
569C4—1462no—
570C4—1462no—
571C4—1461no—
572C4C22×#1192403611yescharacteristic
573C4D73⋊#13111yescharacteristic
574C2—42no—
575C2—1462no—
576C2D4×#1192403611yescharacteristic
577C2#2262668911yescharacteristic
578C2#2262670511yescharacteristic
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
hypocenterC73#17853
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. #22626825C2semidirect
3. #22625526C2semidirect
4. #22625525C2semidirect
5. #22625522C2semidirect
6. SD16⋊#10596833C2semidirect
7. #22625519C2semidirect
8. #22625518C2semidirect
9. #22625514C2semidirect
10. SD16×#10596833C2semidirect
11. #22625483C2semidirect
13. #22625220C2semidirect
14. #22625129C2semidirect
103. #14556620C8semidirect
165. Q8×D73C2×C8semidirect
174. Dic292C2×C8semidirect
175. #11924018C2×C8semidirect
176. #11924016C2×C8semidirect
178. #11924011C2×C8semidirect
203. SD16×C73C2×C8semidirect
305. C73#17853semidirect
456. SD16#11924036semidirect

Maximal subgroups

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

subgroupindex
#226268472
#226268252
#226255262
#226255252
#226255222
SD16⋊#105968332
#226255192
#226255182
#226255142
SD16×#105968332
#226254832
#226252232
#226252202
#226251292
#226187532
#1785373