Finite Group Databasebeta

← #34019232

Subgroups — #34019232

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626777C211yesmaximal, characteristic
2#22626726C211yesmaximal, characteristic, complemented
3#22625572C211yesmaximal, characteristic, complemented
4#22625567C211yesmaximal, characteristic, complemented
5#22625560C211yesmaximal, characteristic, complemented
6#22625554C211yesmaximal, characteristic, complemented
7#22625544C211yesmaximal, characteristic, complemented
8#22625541C211yesmaximal, characteristic, complemented
9#22625524C211yesmaximal, characteristic, complemented
10#22625518C211yesmaximal, characteristic, complemented
11#22625504C211yesmaximal, characteristic, complemented
12#22625293C211yesmaximal, characteristic, complemented
13#22625226C211yesmaximal, characteristic
14#22625148C211yesmaximal, characteristic, complemented
15#22619995C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic
18#17989637—21no—
19#17989637C2211yescharacteristic
20#17989636C2211yescharacteristic
21Q8×#10596833C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989622—21no—
25#17989622—21no—
26#17989622C2211yescharacteristic
27#17989622C2211yescharacteristic
28#17989608—21no—
29#17989608C2211yescharacteristic
30#17989603C2211yescharacteristic
31#17989593—21no—
32C2×C4⋊#10596833C2211yescharacteristic
33#17989551—21no—
34C4×#11924037—21no—
35C4×#11924037C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989451C2211yescharacteristic
38#17989448C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989139C2211yescharacteristic
53#41×Dic73C2211yescharacteristic
54#17988929C411yescharacteristic
55#17988925C411yescharacteristic
56#17988870C411yescharacteristic
57#17988865C411yescharacteristic
58#17988795C411yescharacteristic
59#17988790C411yescharacteristic
60#17988749C411yescharacteristic
61Q16×Dic73C2211yescharacteristic
62#17988679C2211yescharacteristic
63#17988678C2211yescharacteristic
64SD16×Dic73C2211yescharacteristic
65#17988637C2211yescharacteristic
66#452×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C2×#11924037D411yescharacteristic
70C2×#11924037D411yescharacteristic
71C22×#10596833—21no—
72#14556658—21no—
73#14556658—21no—
74C4⋊#10596833—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88#14556646—21no—
89#14556646—21no—
90#14556646—21no—
91#14556646D411yescharacteristic
92#14556646D411yescharacteristic
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924018C2×C411yescharacteristic
102C2×#11924016C2×C411yescharacteristic
103#14556590C2×C411yescharacteristic
104#14556589C2×C411yescharacteristic
105#14556581—21no—
106#14556570C2×C411yescharacteristic
107#14556568—21no—
108Q8×Dic73C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110Q8×Dic73C2311yescharacteristic
111#14556556—21no—
112#14556556C2311yescharacteristic
113D4×Dic73—21no—
114D4×Dic73—21no—
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2311yescharacteristic
117#14556550—21no—
118#14556550C2×C411yescharacteristic
119#14556547—21no—
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556524C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556515C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×#41×C73C2×C411yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C2×SD16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—22no—
145#11924037—41no—
146#11924037—42no—
147#11924037—42no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—42no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C22×Dic73—21no—
162C22×Dic73—21no—
163C22×Dic73C2×D411yescharacteristic
164C22×Dic73#4111yescharacteristic
165#11924018—22no—
166#11924016—41no—
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C22×C411yescharacteristic
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73C22×C411yescharacteristic
173C4⋊Dic73C22×C411yescharacteristic
174C4⋊Dic73C22×C411yescharacteristic
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180C22⋊Dic73—21no—
181C22⋊Dic73—21no—
182C22⋊Dic73—21no—
183C22⋊Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73C22×C411yescharacteristic
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73#4111yescharacteristic
192C4×Dic73C2411yescharacteristic
193#41×C73—22no—
194#41×C73—41no—
195C2×Q8×C73C22×C411yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C2×D4×C73—21no—
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200C22×C4×C73C22×C411yescharacteristic
201Q16×C73—22no—
202SD16×C73—41no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C4⋊C4×C73—21no—
206C4⋊C4×C73C22×C411yescharacteristic
207C22⋊C4×C73—21no—
208C23×C73—21no—
209C23×C73C4×D411yescharacteristic
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213Q8×C73#13112yes—
214D4×C73—21no—
215D4×C73—21no—
216D4×C73—42no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—42no—
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C23×C411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—81no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C22×D411yescharacteristic
238C2×Dic73C2×#4111yescharacteristic
239Dic146—21no—
240Dic146—21no—
241Dic146—81no—
242#10596833—21no—
243#10596833—21no—
244#10596833—21no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250#10596833—41no—
251#10596833#13712yes—
252C8×C73—21no—
253C8×C73—21no—
254#10596831—21no—
255#10596831—21no—
256C22×C73—21no—
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73C2×C4×D411yescharacteristic
263C4×C73—21no—
264C4×C73—22no—
265C4×C73—41no—
266C4×C73—41no—
267C4×C73C2×#13111yescharacteristic
268C4×C73#56811yescharacteristic
269Dic73—21no—
270Dic73—21no—
271Dic73—21no—
272Dic73—41no—
273Dic73—41no—
274Dic73—41no—
275Dic73—41no—
276Dic73C2×#13711yescharacteristic
277Dic73#57511yescharacteristic
278#18013—731nomaximal, Sylow
279C2×C73—41no—
280C2×C73—81no—
281C2×C73#293311yescharacteristic
282C2×C73C4×#13711yescharacteristic
283C2×C73#295711yescharacteristic
284#3107—731no—
285#2971—731no—
286#2962—731no—
287#1592—731no—
288#1590—731no—
289#1588—731no—
290#1587—731no—
291#1584—731no—
292#1583—731no—
293#1579—731no—
294C8×Q16—731no—
295#1573—731no—
296#1497—731no—
297#1477—731no—
298#1453—731no—
299C73#1801311yescharacteristic, complemented, Sylow
300#533—731no—
301C4×#41—731no—
302#485—731no—
303#485—731no—
304#475—731no—
305#474—731no—
306#462—731no—
307#461—731no—
308#452—731no—
309#452#6440811yescharacteristic
310#446—731no—
311#446—731no—
312#446—1461no—
313#446—1461no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—731no—
317C8×Q8—731no—
318C4×Q16—731no—
319C4×Q16—731no—
320C4×SD16—731no—
321C4×SD16—731no—
322#436—731no—
323#436—731no—
324#436—1461no—
325#436—1461no—
326#432—731no—
327#432—1461no—
328#431—731no—
329#431—731no—
330#424—1461no—
331C2×C4⋊C8—731no—
332#405—1461no—
333C4×M16—731no—
334C4×M16—1461no—
335#335—731no—
336#335—731no—
337#334—731no—
338#334—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344Q8⋊C8—731no—
345D4⋊C8—731no—
346D4⋊C8—731no—
347#322—731no—
348#322—731no—
349#322—731no—
350C82—731no—
351C2×#41C2×#6440811yescharacteristic
352C2×Q16—731no—
353C2×Q16C2×#6440811yescharacteristic
354C2×SD16—731no—
355C2×SD16C2×#6440811yescharacteristic
356C2×M16—731no—
357C2×M16—731no—
358C2×M16—1461no—
359C2×M16—1461no—
360C22×C8—1461no—
361C4⋊Q8—731no—
362C4⋊Q8—731no—
363#125—1461no—
364C22⋊Q8—731no—
365C22⋊Q8—1461no—
366C22⋊Q8C2×#6440811yescharacteristic
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—1461no—
372C4×D4—731no—
373C4×D4—731no—
374C4×D4—1461no—
375C4×D4—1461no—
376#117—731no—
377#117—1461no—
378#117—1461no—
379C2×C42—731no—
380C2×C42—1461no—
381#107—731no—
382#107—731no—
383#106—731no—
384#106—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—731no—
390C4⋊C8—731no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103—731no—
404#103C2×#6440811yescharacteristic
405#102—731no—
406#102C2×#6440811yescharacteristic
407C22⋊C8—731no—
408C22⋊C8—1461no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—731no—
414#97—731no—
415#97—1461no—
416#97—1461no—
417#97—1461no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427C4×C8—1461no—
428#41—22no—
429#41—41no—
430C2×Q8—731no—
431C2×Q8—1461no—
432C2×Q8C22×#6440811yescharacteristic
433C2×Q8C22×#6440811yescharacteristic
434C2×D4—21no—
435C2×D4C22×#6440811yescharacteristic
436C22×C4—21no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4C22×#6440811yescharacteristic
442Q16—22no—
443Q16—1462no—
444SD16—41no—
445SD16—2921no—
446M16—2922no—
447M16—2922no—
448M16—2921no—
449M16—1462no—
450C2×C8—2922no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—731no—
455C2×C8—731no—
456C2×C8—731no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8C22×#6440811yescharacteristic
464C2×C8C22×#6440811yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—731no—
467C4⋊C4—731no—
468C4⋊C4—731no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4C22×#6440811yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C22⋊C4—1461no—
482C22⋊C4—1461no—
483C42—731no—
484C42—731no—
485C42—731no—
486C42—731no—
487C42—1461no—
488C42—1461no—
489C42—1461no—
490C42—1461no—
491C42—1461no—
492C23—21no—
493C23D4×#6440811yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—5841no—
498Q8—1461no—
499Q8—1461no—
500Q8#1455667812yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—5841no—
514C2×C4—731no—
515C2×C4—731no—
516C2×C4—731no—
517C2×C4—731no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4D4×#6440811yescharacteristic
525C2×C4C23×#6440811yescharacteristic
526C8—21no—
527C8—21no—
528C8—2921no—
529C8—2921no—
530C8—2921no—
531C8—2921no—
532C8—2921no—
533C8—2921no—
534C8—732no—
535C8—1461no—
536C8—1461no—
537C8—1461no—
538C8—1461no—
539C8—1461no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×#6440811yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—2921no—
552C4—2921no—
553C4—2921no—
554C4—2921no—
555C4—731no—
556C4—731no—
557C4—1461no—
558C4—1461no—
559C4—1461no—
560C4C2×#1455667811yescharacteristic
561C4#1798966511yescharacteristic
562C2—41no—
563C2—81no—
564C2#2262666311yescharacteristic
565C2#137×#6440811yescharacteristic
566C2#2262671011yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#452×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C2×C4
cosocleC4×Dic73C24
hypercenter#452#64408
hypocenterC73#18013
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #22626726C2semidirect
3. #22625572C2semidirect
4. #22625567C2semidirect
5. #22625560C2semidirect
6. #22625554C2semidirect
7. #22625544C2semidirect
8. #22625541C2semidirect
9. #22625524C2semidirect
10. #22625518C2semidirect
11. #22625504C2semidirect
12. #22625293C2semidirect
14. #22625148C2semidirect
299. C73#18013semidirect

Maximal subgroups

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

subgroupindex
#226267772
#226267262
#226255722
#226255672
#226255602
#226255542
#226255442
#226255412
#226255242
#226255182
#226255042
#226252932
#226252262
#226251482
#226199952
#1801373