Finite Group Databasebeta

← #10779128

Subgroups — #10779128

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3116C211yesmaximal, characteristic
2#83339C211yesmaximal, characteristic, complemented
3#82535C211yesmaximal, characteristic
4#41×C3⋊C8C211yesmaximal, characteristic
5#71832C211yesmaximal, characteristic, complemented
6#71149C211yesmaximal, characteristic, complemented
7#71148C211yesmaximal, characteristic, complemented
8#71133C211yesmaximal, characteristic, complemented
9#71132C211yesmaximal, characteristic, complemented
10#71053C211yesmaximal, characteristic, complemented
11#71052C211yesmaximal, characteristic, complemented
12#71037C211yesmaximal, characteristic, complemented
13#71036C211yesmaximal, characteristic, complemented
14#70821C211yesmaximal, characteristic, complemented
15#70794C211yesmaximal, characteristic, complemented
16#19273—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#485—21no—
21C3×#477C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×#461C2211yescharacteristic
25C3×#452—21no—
26C3×#441—21no—
27C3×#440—21no—
28C3×#432C2211yescharacteristic
29#5239—21no—
30#5080C2211yescharacteristic
31#5080C2211yescharacteristic
32#5078C2211yescharacteristic
33#5078C2211yescharacteristic
34#5065—21no—
35#5065—21no—
36#5031C2211yescharacteristic
37#5030—21no—
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40#5021C2211yescharacteristic
41#5019C2211yescharacteristic
42#5018—21no—
43D4×C3⋊C8—21no—
44D4×C3⋊C8—21no—
45D4×C3⋊C8C2211yescharacteristic
46D4×C3⋊C8C2211yescharacteristic
47#5008C2211yescharacteristic
48#4998—21no—
49#4998—21no—
50#4996—21no—
51#4926C2211yescharacteristic
52#4925—21no—
53#4925C2211yescharacteristic
54C2×C4×C3⋊C8—21no—
55C2×C4×C3⋊C8C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3160—31no—
69#3116—31no—
70#3116D311yescharacteristic
71C8×#41—31no—
72#1676—31no—
73#1675—31no—
74#1674—31no—
75#1673—31no—
76#1672—31no—
77#1672—31no—
78#1671—31no—
79#1671—31no—
80#1568—31no—
81#1554—31no—
82#1554—31no—
83C2×C3×#41D411yescharacteristic
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#125—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4D411yescharacteristic
98C3×C4×D4C2311yescharacteristic
99C3×#117—21no—
100C3×#117—21no—
101C3×#117D411yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#216—21no—
105C2×#214—21no—
106#926—21no—
107#924—21no—
108#924—21no—
109C22×C3⋊C8—21no—
110C22×C3⋊C8—21no—
111C22×C3⋊C8D411yescharacteristic
112C22×C3⋊C8D411yescharacteristic
113#870—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6—21no—
118C3×#107—21no—
119C3×#106—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103—21no—
123C3×#103—21no—
124C3×#103—21no—
125C3×#102—21no—
126C3×C22⋊C8—21no—
127C3×#97—21no—
128#835—21no—
129#832—21no—
130#830—21no—
131#830—21no—
132#830—21no—
133#830—21no—
134#810—21no—
135#810—21no—
136#808—21no—
137#808—21no—
138#808—21no—
139#808—21no—
140#808—21no—
141#807—21no—
142#807—21no—
143#807—21no—
144#807—21no—
145#807—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—
153C4×C3⋊C8—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533C2×D311yescharacteristic
162C4×#41C2×D311yescharacteristic
163#500—31no—
164#500—31no—
165#499—61no—
166#499—61no—
167#498—31no—
168#498—31no—
169#485—21no—
170#485—61no—
171#477—31no—
172#477C2×D311yescharacteristic
173#476—31no—
174#476C2×D311yescharacteristic
175#462—31no—
176#462C2×D311yescharacteristic
177#461—31no—
178#461C2×D311yescharacteristic
179#452—21no—
180#452—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#441—21no—
184#441—61no—
185#440—21no—
186#440—61no—
187C8×D4—31no—
188C8×D4—31no—
189C8×D4—61no—
190C8×D4—61no—
191#432—31no—
192#432C2×D311yescharacteristic
193#431—31no—
194#431—61no—
195#427—61no—
196#427—61no—
197C2×C4×C8—31no—
198C2×C4×C8—61no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C3×#41—22no—
212C3×#41—22no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4#4111yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C3×Q16—42no—
227C3×SD16—42no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—41no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—41no—
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42#4111yescharacteristic
251C3×C42C2411yescharacteristic
252#217—41no—
253#216—42no—
254#214—42no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—41no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—42no—
269C2×C3⋊C8—42no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×#41#6711yescharacteristic
275C2×Q16—21no—
276C2×Q16—61no—
277C2×SD16—21no—
278C2×SD16—61no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8C22×D311yescharacteristic
286C4⋊Q8C22×D311yescharacteristic
287#125—21no—
288#125—61no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—61no—
296C4×Q8—61no—
297C4×Q8#6711yescharacteristic
298C4×Q8C22×D311yescharacteristic
299C4×D4—21no—
300C4×D4—21no—
301C4×D4#6711yescharacteristic
302C4×D4C22×D311yescharacteristic
303#117—21no—
304#117—21no—
305#117—61no—
306#117#6711yescharacteristic
307C2×C42—21no—
308C2×C42C22×D311yescharacteristic
309#107—21no—
310#107—61no—
311#107—61no—
312#107—61no—
313#107—61no—
314#107—61no—
315#106—21no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—21no—
331#103—21no—
332#103—61no—
333#103—61no—
334#103—61no—
335#102—21no—
336#102—61no—
337C22⋊C8—21no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#97—21no—
343#97—61no—
344C4×C8—31no—
345C4×C8—31no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8—61no—
351C23×C3—21no—
352C23×C3#12111yescharacteristic
353C3×Q8—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—81no—
357C3×Q8#13512yes—
358C3×D4—24no—
359C3×D4—21no—
360C3×D4#13512yes—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×Dic3—41no—
382C2×Dic3—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—81no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8#13712yes—
395C3⋊C8#13712yes—
396#41—22no—
397#41—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8C2×#6711yescharacteristic
403C2×D4—21no—
404C2×D4C2×#6711yescharacteristic
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4D3×D411yescharacteristic
409C22×C4#23711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411Q16—42no—
412Q16—122no—
413SD16—42no—
414SD16—122no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—122no—
428C2×C8—122no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—121no—
442C4⋊C4—121no—
443C4⋊C4C2×#6711yescharacteristic
444C4⋊C4C2×#6711yescharacteristic
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—41no—
449C22⋊C4—121no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—41no—
454C42—121no—
455C42D3×D411yescharacteristic
456C42#23711yescharacteristic
457C42C2×#6711yescharacteristic
458C42C23×D311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3C2×#12111yescharacteristic
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4C2×#13511yescharacteristic
476C3×C4C2×#13511yescharacteristic
477C3×C4C2×#13711yescharacteristic
478C3×C4C2×#13711yescharacteristic
479Dic3—81no—
480Dic3—81no—
481C23—21no—
482C23#93911yescharacteristic
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—121no—
488Q8—121no—
489Q8—241no—
490Q8#95012yes—
491D4—24no—
492D4—21no—
493D4#95012yes—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4#93911yescharacteristic
511C2×C4#93911yescharacteristic
512C2×C4#93911yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23711yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C8—32no—
517C8—32no—
518C8—41no—
519C8—41no—
520C8—61no—
521C8—61no—
522C8—121no—
523C8—121no—
524C8—121no—
525C8—121no—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3#303911yescharacteristic
529C2×C3#305011yescharacteristic
530C2×C3#306711yescharacteristic
531C22—21no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22C2×#93911yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—22no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—241no—
548C4—241no—
549C4D3×#13711yescharacteristic
550C4D3×#13711yescharacteristic
551C4C2×#95011yescharacteristic
552C4C2×#95011yescharacteristic
553C3#1927311yescharacteristic, complemented, Sylow
554C2—41no—
555C2—42no—
556C2#8292211yescharacteristic
557C2#8396711yescharacteristic
558C2#8400111yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3116C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenter#3116D3
hypocenterC3#19273
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
2. #83339C2semidirect
5. #71832C2semidirect
6. #71149C2semidirect
7. #71148C2semidirect
8. #71133C2semidirect
9. #71132C2semidirect
10. #71053C2semidirect
11. #71052C2semidirect
12. #71037C2semidirect
13. #71036C2semidirect
14. #70821C2semidirect
15. #70794C2semidirect
553. C3#19273semidirect

Maximal subgroups

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

subgroupindex
C3×#31162
#833392
#825352
#41×C3⋊C82
#718322
#711492
#711482
#711332
#711322
#710532
#710522
#710372
#710362
#708212
#707942
#192733