Finite Group Databasebeta

← #34018684

Subgroups — #34018684

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626846C211yesmaximal, characteristic
2#22626835C211yesmaximal, characteristic, complemented
3#22625556C211yesmaximal, characteristic, complemented
4#22625546C211yesmaximal, characteristic, complemented
5#22625542C211yesmaximal, characteristic, complemented
6#22625536C211yesmaximal, characteristic, complemented
7#22625525C211yesmaximal, characteristic, complemented
8SD16⋊#10596833C211yesmaximal, characteristic, complemented
9Q16×#10596833C211yesmaximal, characteristic, complemented
10#22625514C211yesmaximal, characteristic, complemented
11#22625506C211yesmaximal, characteristic, complemented
12#22625262C211yesmaximal, characteristic
13#22625239C211yesmaximal, characteristic, complemented
14#22625152C211yesmaximal, characteristic, complemented
15#22619995C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic
18#17989637—21no—
19#17989637C2211yescharacteristic
20#17989636C2211yescharacteristic
21Q8×#10596833C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24D4×#10596833—21no—
25D4×#10596833—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989611—21no—
30#17989608C2211yescharacteristic
31#17989604C2211yescharacteristic
32#17989603—21no—
33#17989603C2211yescharacteristic
34C2×C4⋊#10596833—21no—
35C2×C4⋊#10596833C2211yescharacteristic
36#17989450C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989449C2211yescharacteristic
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
67C22×#10596833—21no—
68C22×#10596833—21no—
69C22×#10596833—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73C4⋊#10596833—21no—
74C4⋊#10596833—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833D411yescharacteristic
79C4⋊#10596833D411yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833D411yescharacteristic
89C22⋊#10596833D411yescharacteristic
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
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
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—81no—
151C2×#10596833—81no—
152C2×#10596833—81no—
153C2×#10596833C2×D411yescharacteristic
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×Dic146—21no—
158C2×Dic146C22×C411yescharacteristic
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73C2×D411yescharacteristic
162C22×Dic73#4111yescharacteristic
163#11924018—22no—
164#11924016—22no—
165C2×#10596831C22×C411yescharacteristic
166C2×#10596831C22×C411yescharacteristic
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73C22×C411yescharacteristic
171C4⋊Dic73C22×C411yescharacteristic
172C4⋊Dic73C22×C411yescharacteristic
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178C22⋊Dic73—21no—
179C22⋊Dic73—21no—
180C22⋊Dic73—21no—
181C22⋊Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73C22×C411yescharacteristic
188C4×Dic73C2×D411yescharacteristic
189C4×Dic73#4111yescharacteristic
190C4×Dic73C2411yescharacteristic
191#41×C73—22no—
192#41×C73—22no—
193C2×Q8×C73C22×C411yescharacteristic
194C2×Q8×C73C22×C411yescharacteristic
195C2×D4×C73—21no—
196C2×D4×C73C22×C411yescharacteristic
197C22×C4×C73—21no—
198C22×C4×C73C22×C411yescharacteristic
199Q16×C73—22no—
200SD16×C73—22no—
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C22×C411yescharacteristic
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C22⋊C4×C73—21no—
206C23×C73—21no—
207C23×C73C4×D411yescharacteristic
208Q8×C73—21no—
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73#13112yes—
212D4×C73—24no—
213D4×C73—21no—
214D4×C73#13112yes—
215C2×C4×C73—24no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73C4×D411yescharacteristic
222C2×C4×C73C23×C411yescharacteristic
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—81no—
233C2×Dic73C4×D411yescharacteristic
234C2×Dic73C4×D411yescharacteristic
235C2×Dic73C22×D411yescharacteristic
236C2×Dic73C2×#4111yescharacteristic
237Dic146—21no—
238Dic146—21no—
239Dic146—81no—
240#10596833—21no—
241#10596833—21no—
242#10596833—21no—
243#10596833—21no—
244#10596833—41no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250C8×C73—21no—
251C8×C73—21no—
252#10596831—21no—
253#10596831—21no—
254C22×C73—21no—
255C22×C73—22no—
256C22×C73—21no—
257C22×C73—21no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73C2×C4×D411yescharacteristic
261C4×C73—21no—
262C4×C73—22no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73C2×#13111yescharacteristic
266C4×C73C2×#13111yescharacteristic
267Dic73—21no—
268Dic73—21no—
269Dic73—21no—
270Dic73—41no—
271Dic73—41no—
272Dic73—41no—
273Dic73—41no—
274Dic73#57711yescharacteristic
275Dic73#57811yescharacteristic
276#17921—731nomaximal, Sylow
277C2×C73—41no—
278C2×C73—42no—
279C2×C73#292911yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#295711yescharacteristic
282#3107—731no—
283#2988—731no—
284#2985—731no—
285#1587—731no—
286#1585—731no—
287#1584—731no—
288#1582—731no—
289#1579—731no—
290SD16⋊C8—731no—
291C8×Q16—731no—
292C8×SD16—731no—
293#1573—731no—
294#1488—731no—
295#1481—731no—
296#1454—731no—
297C73#1792111yescharacteristic, complemented, Sylow
298#533—731no—
299C4×#41—731no—
300#485—731no—
301#485—731no—
302#475—731no—
303#474—731no—
304#462—731no—
305#461—731no—
306#452—731no—
307#452#6440811yescharacteristic
308#446—731no—
309#446—731no—
310#446—1461no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C4×Q16—731no—
317C4×Q16—731no—
318C4×SD16—731no—
319C4×SD16—731no—
320C8×D4—731no—
321C8×D4—731no—
322C8×D4—1461no—
323C8×D4—1461no—
324#433—1461no—
325#433—1461no—
326#432—731no—
327#431—731no—
328#431—731no—
329#431—731no—
330#431—1461no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—1461no—
333#335—731no—
334#335—731no—
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C2×#41C2×#6440811yescharacteristic
350C2×Q16—731no—
351C2×Q16C2×#6440811yescharacteristic
352C2×SD16—731no—
353C2×SD16C2×#6440811yescharacteristic
354C22×C8—1461no—
355C22×C8—1461no—
356C22×C8—1461no—
357C4⋊Q8—731no—
358C4⋊Q8—731no—
359#125—1461no—
360C22⋊Q8—731no—
361C22⋊Q8—1461no—
362C22⋊Q8C2×#6440811yescharacteristic
363C4×Q8—731no—
364C4×Q8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—1461no—
368C4×D4—731no—
369C4×D4—731no—
370C4×D4—1461no—
371C4×D4—1461no—
372#117—731no—
373#117—1461no—
374#117—1461no—
375C2×C42—731no—
376C2×C42—1461no—
377#107—731no—
378#107—731no—
379#106—731no—
380#106—731no—
381C4⋊C8—731no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103C2×#6440811yescharacteristic
403#102—731no—
404#102C2×#6440811yescharacteristic
405C22⋊C8—731no—
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—1461no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—1461no—
414#97—1461no—
415#97—1461no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8—1461no—
425C4×C8—1461no—
426#41—22no—
427#41—22no—
428C2×Q8—731no—
429C2×Q8—1461no—
430C2×Q8C22×#6440811yescharacteristic
431C2×Q8C22×#6440811yescharacteristic
432C2×D4—21no—
433C2×D4C22×#6440811yescharacteristic
434C22×C4—21no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4C22×#6440811yescharacteristic
440Q16—22no—
441Q16—1462no—
442SD16—22no—
443SD16—1462no—
444C2×C8—5841no—
445C2×C8—5841no—
446C2×C8—5841no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8C22×#6440811yescharacteristic
460C2×C8C22×#6440811yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—731no—
463C4⋊C4—731no—
464C4⋊C4—731no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4C22×#6440811yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1461no—
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C42—731no—
480C42—731no—
481C42—731no—
482C42—731no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C23—21no—
489C23D4×#6440811yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—5841no—
494Q8—1461no—
495Q8—1461no—
496Q8#1455667812yes—
497D4—24no—
498D4—21no—
499D4#1455667812yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—5841no—
510C2×C4—731no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—731no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4D4×#6440811yescharacteristic
521C2×C4C23×#6440811yescharacteristic
522C8—21no—
523C8—21no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×#6440811yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—2921no—
548C4—2921no—
549C4—2921no—
550C4—2921no—
551C4—731no—
552C4—731no—
553C4—1461no—
554C4—1461no—
555C4—1461no—
556C4C2×#1455667811yescharacteristic
557C4C2×#1455667811yescharacteristic
558C2—41no—
559C2—42no—
560C2#2262664611yescharacteristic
561C2#2262668911yescharacteristic
562C2#2262670811yescharacteristic
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#17921
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. #22626835C2semidirect
3. #22625556C2semidirect
4. #22625546C2semidirect
5. #22625542C2semidirect
6. #22625536C2semidirect
7. #22625525C2semidirect
8. SD16⋊#10596833C2semidirect
9. Q16×#10596833C2semidirect
10. #22625514C2semidirect
11. #22625506C2semidirect
13. #22625239C2semidirect
14. #22625152C2semidirect
297. C73#17921semidirect

Maximal subgroups

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

subgroupindex
#226268462
#226268352
#226255562
#226255462
#226255422
#226255362
#226255252
SD16⋊#105968332
Q16×#105968332
#226255142
#226255062
#226252622
#226252392
#226251522
#226199952
#1792173