Finite Group Databasebeta

← #10780330

Subgroups — #10780330

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3251C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#83154C211yesmaximal, characteristic, complemented
4#83129C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71122C211yesmaximal, characteristic, complemented
7#71118C211yesmaximal, characteristic, complemented
8#71106C211yesmaximal, characteristic, complemented
9#71102C211yesmaximal, characteristic, complemented
10#71027C211yesmaximal, characteristic, complemented
11#71023C211yesmaximal, characteristic, complemented
12#71011C211yesmaximal, characteristic, complemented
13#71007C211yesmaximal, characteristic, complemented
14#70506C211yesmaximal, characteristic, complemented
15#70492C211yesmaximal, characteristic, complemented
16#19383—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C3×#529C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#478—21no—
21C3×#475C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#469—21no—
24C3×#468—21no—
25C3×#463—21no—
26C3×#460C2211yescharacteristic
27C3×#459C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5232—21no—
30#5232—21no—
31#5079C2211yescharacteristic
32#5079C2211yescharacteristic
33#5070C2211yescharacteristic
34#5070C2211yescharacteristic
35#5069—21no—
36#5060—21no—
37#5055—21no—
38#5055—21no—
39#5050—21no—
40#5040—21no—
41#5031C2211yescharacteristic
42#5031C2211yescharacteristic
43#5024C2211yescharacteristic
44Q8×C3⋊C8C2211yescharacteristic
45#5019C2211yescharacteristic
46#5019C2211yescharacteristic
47#5012—21no—
48#5012C2211yescharacteristic
49#5011—21no—
50D4×C3⋊C8C2211yescharacteristic
51#5000—21no—
52#4997—21no—
53#4926C2211yescharacteristic
54#4925C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3251D311yescharacteristic
69#3218—31no—
70#3141—31no—
71#2969—31no—
72#1702—31no—
73#1701—31no—
74#1700—31no—
75#1699—31no—
76#1682—31no—
77#1681—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1553—31no—
82#1546—31no—
83C2×C3×#41—21no—
84C22×Dic6—21no—
85C2×C3×SD16—21no—
86C2×C3×D8—21no—
87C2×C3×M16—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#126—21no—
91C3×#125D411yescharacteristic
92C3×#124—21no—
93C3×#124D411yescharacteristic
94C3×#123—21no—
95C3×C22⋊Q8D411yescharacteristic
96C3×#121—21no—
97C3×#121—21no—
98C3×#121D411yescharacteristic
99C3×C4×Q8C2311yescharacteristic
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4C2311yescharacteristic
104C3×#117—21no—
105C2×C3×C42C2311yescharacteristic
106C2×#216—21no—
107C2×#216—21no—
108C2×#214—21no—
109C2×#214—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×#208—21no—
113C2×#208D411yescharacteristic
114C2×#208D411yescharacteristic
115C22×C3⋊C8—21no—
116#869—21no—
117#869—21no—
118#869—21no—
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C3×#107—21no—
122C3×#106—21no—
123C3×C4⋊C8—21no—
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#103—21no—
127C3×#102—21no—
128C3×#102—21no—
129C3×#102—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#835—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#830—21no—
139#830—21no—
140#830—21no—
141#830—21no—
142#810—21no—
143#810—21no—
144#810—21no—
145#810—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532C2×D311yescharacteristic
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#497—61no—
164#491—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#478—21no—
172#475C2×D311yescharacteristic
173#474C2×D311yescharacteristic
174#469—21no—
175#468—21no—
176#463—21no—
177#462—31no—
178#462—31no—
179#461—31no—
180#461—31no—
181#460C2×D311yescharacteristic
182#459C2×D311yescharacteristic
183#451—61no—
184#448—61no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423C2×D311yescharacteristic
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—42no—
212C3×#41—42no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—41no—
220C2×C3×D4C2×D411yescharacteristic
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4111yescharacteristic
226C22×Dic3—41no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C3×SD16—81no—
233C3×D8—81no—
234C3×M16—81no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—41no—
252C3×C42—21no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42#4111yescharacteristic
256C3×C42C2411yescharacteristic
257#216—81no—
258#216—81no—
259#214—81no—
260#214—81no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—41no—
266C4⋊Dic3—41no—
267#208—41no—
268#208—41no—
269#208—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—42no—
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277C2×C3⋊C8C2×D411yescharacteristic
278C2×C3⋊C8C2×D411yescharacteristic
279C2×#41—21no—
280C22×Q8—61no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×SD16—21no—
284C2×SD16—61no—
285C2×SD16—61no—
286C2×D8—21no—
287C2×M16—21no—
288C2×M16—31no—
289C2×M16—31no—
290C2×M16—61no—
291C22×C8—61no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—61no—
295C4⋊Q8—61no—
296C4⋊Q8—61no—
297C4⋊Q8C22×D311yescharacteristic
298#127C22×D311yescharacteristic
299#126—21no—
300#125#6711yescharacteristic
301#124—21no—
302#124#6711yescharacteristic
303#123—21no—
304C22⋊Q8#6711yescharacteristic
305#121—21no—
306#121—21no—
307#121#6711yescharacteristic
308C4×Q8C22×D311yescharacteristic
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C22×D311yescharacteristic
313#117—21no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C42C22×D311yescharacteristic
317#107—21no—
318#106—21no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—21no—
339#102—21no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—31no—
349#97—31no—
350#97—61no—
351#97—61no—
352C4×C8—31no—
353C4×C8—31no—
354C4×C8—61no—
355C4×C8—61no—
356C23×C3—21no—
357C23×C3—41no—
358C23×C3#12111yescharacteristic
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—81no—
370C3×D4—81no—
371C3×D4—82no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—82no—
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4#12111yescharacteristic
386C2×C3×C4#12111yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—82no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—82no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—21no—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—42no—
409#41—42no—
410C2×Q8—21no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—121no—
414C2×Q8—241no—
415C2×Q8—241no—
416C2×Q8C2×#6711yescharacteristic
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—41no—
422C2×D4C2×#6711yescharacteristic
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—121no—
426C22×C4D3×D411yescharacteristic
427C22×C4#23711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429Q16—241no—
430Q16—241no—
431SD16—81no—
432SD16—241no—
433SD16—241no—
434D8—81no—
435M16—81no—
436M16—121no—
437M16—121no—
438M16—122no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—31no—
444C2×C8—31no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—122no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—121no—
461C4⋊C4—121no—
462C4⋊C4C2×#6711yescharacteristic
463C4⋊C4C2×#6711yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C42—21no—
472C42D3×D411yescharacteristic
473C42#23711yescharacteristic
474C42C2×#6711yescharacteristic
475C42C23×D311yescharacteristic
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12111yescharacteristic
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—81no—
494C3×C4#57511yescharacteristic
495C3×C4#57511yescharacteristic
496C3×C4#57611yescharacteristic
497C3×C4#57711yescharacteristic
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—41no—
502C23#93911yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—41no—
506Q8—41no—
507Q8—121no—
508Q8—121no—
509Q8—121no—
510Q8—121no—
511Q8—242no—
512D4—21no—
513D4—21no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—81no—
519D4—81no—
520D4—82no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—242no—
536C2×C4#93911yescharacteristic
537C2×C4#93911yescharacteristic
538C2×C4#93911yescharacteristic
539C2×C4C2×D3×D411yescharacteristic
540C2×C4C2×#23711yescharacteristic
541C2×C4C22×#6711yescharacteristic
542C8—41no—
543C8—41no—
544C8—61no—
545C8—61no—
546C8—61no—
547C8—61no—
548C8—121no—
549C8—121no—
550C8—121no—
551C8—121no—
552C2×C3—41no—
553C2×C3—81no—
554C2×C3—161no—
555C2×C3#304011yescharacteristic
556C2×C3#306211yescharacteristic
557C2×C3#306911yescharacteristic
558C22—21no—
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#93911yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—81no—
576C4—241no—
577C4—241no—
578C4#577311yescharacteristic
579C4#577311yescharacteristic
580C4#583511yescharacteristic
581C4#583711yescharacteristic
582C3#1938311yescharacteristic, complemented, Sylow
583C2—41no—
584C2—81no—
585C2—161no—
586C2#8342211yescharacteristic
587C2#8393311yescharacteristic
588C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3251C2
socleC22×C3C2×#121
hypercenter#3251D3
hypocenterC3#19383
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3251

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #83154C2semidirect
4. #83129C2semidirect
5. #71808C2semidirect
6. #71122C2semidirect
7. #71118C2semidirect
8. #71106C2semidirect
9. #71102C2semidirect
10. #71027C2semidirect
11. #71023C2semidirect
12. #71011C2semidirect
13. #71007C2semidirect
14. #70506C2semidirect
15. #70492C2semidirect
582. C3#19383semidirect

Maximal subgroups

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

subgroupindex
C3×#32512
#832832
#831542
#831292
#718082
#711222
#711182
#711062
#711022
#710272
#710232
#710112
#710072
#705062
#704922
#193833