Finite Group Databasebeta

← #10765616

Subgroups — #10765616

Order 768.

568 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
1C3×#2989C211yesmaximal, characteristic
2#82534C211yesmaximal, characteristic
3#79266C211yesmaximal, characteristic, complemented
4#71640C211yesmaximal, characteristic
5#70957C211yesmaximal, characteristic, complemented
6#70956C211yesmaximal, characteristic, complemented
7#70952C211yesmaximal, characteristic, complemented
8#70951C211yesmaximal, characteristic, complemented
9#70866C211yesmaximal, characteristic, complemented
10#70865C211yesmaximal, characteristic, complemented
11#70864C211yesmaximal, characteristic, complemented
12#70860C211yesmaximal, characteristic, complemented
13#70744C211yesmaximal, characteristic, complemented
14#70429C211yesmaximal, characteristic, complemented
15#70423C211yesmaximal, characteristic, complemented
16#17949—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5470C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×#436C2211yescharacteristic
22C3×#436C2211yescharacteristic
23C3×#435—21no—
24C3×#435—21no—
25C3×#432—21no—
26C3×#431C2211yescharacteristic
27C3×#423—21no—
28C2×C3×C4⋊C8C2211yescharacteristic
29#5239C411yescharacteristic
30#5239C411yescharacteristic
31#5031C411yescharacteristic
32C4×#216C2211yescharacteristic
33C4×#216C2211yescharacteristic
34#5020C411yescharacteristic
35#5019C411yescharacteristic
36C4×#214C2211yescharacteristic
37C4×#214C2211yescharacteristic
38#5007C411yescharacteristic
39#4996C411yescharacteristic
40#4996C411yescharacteristic
41#4926C2211yescharacteristic
42#4925C2211yescharacteristic
43#4925C2211yescharacteristic
44#4828—21no—
45#4828—21no—
46#4720—21no—
47#4720—21no—
48#4679C2211yescharacteristic
49#4679C2211yescharacteristic
50C8×Dic6C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52#4535C2211yescharacteristic
53#4535C2211yescharacteristic
54D4⋊(C3⋊C8)C2211yescharacteristic
55D4⋊(C3⋊C8)C2211yescharacteristic
56#4484C2211yescharacteristic
57#4484C2211yescharacteristic
58#4484C2211yescharacteristic
59#4484C2211yescharacteristic
60#4480C2211yescharacteristic
61#4480C2211yescharacteristic
62#4479C2211yescharacteristic
63#4479C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107—31no—
69#2989D311yescharacteristic
70#2985—31no—
71#1593—31no—
72#1592—31no—
73#1591—31no—
74#1590—31no—
75#1588—31no—
76#1587—31no—
77#1584—31no—
78#1583—31no—
79#1573—31no—
80#1488—31no—
81#1481—31no—
82#1454—31no—
83C2×C3×#41C2×C411yescharacteristic
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C22×C3×C8—21no—
87C3×C4×Q8C2×C411yescharacteristic
88C3×C4×Q8C2311yescharacteristic
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4C2×C411yescharacteristic
92C3×C4×D4C2311yescharacteristic
93C3×#117—21no—
94C3×#117C2×C411yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216C2×C411yescharacteristic
98C2×#216C2×C411yescharacteristic
99C2×#214C2×C411yescharacteristic
100C2×#214C2×C411yescharacteristic
101#926—21no—
102#924—21no—
103#924C2×C411yescharacteristic
104#924C2×C411yescharacteristic
105#870—21no—
106#869C2×C411yescharacteristic
107#869C2×C411yescharacteristic
108C4×Dic6—21no—
109C4×Dic6C2311yescharacteristic
110C4×Dic6C2311yescharacteristic
111C3×C4⋊C8—21no—
112C3×C4⋊C8—21no—
113C3×C4⋊C8—21no—
114C3×C4⋊C8D411yescharacteristic
115C3×C4⋊C8D411yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C22⋊C8—21no—
120C3×C22⋊C8D411yescharacteristic
121C3×C22⋊C8D411yescharacteristic
122C3×#97—21no—
123C3×#97—21no—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126#835C2×C411yescharacteristic
127#835C2×C411yescharacteristic
128#832C2×C411yescharacteristic
129#832C2×C411yescharacteristic
130#830—21no—
131#830—21no—
132#830C2×C411yescharacteristic
133#830C2×C411yescharacteristic
134#815—21no—
135#815—21no—
136Dic3⋊C8—21no—
137Dic3⋊C8—21no—
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140C8×Dic3—21no—
141C8×Dic3—21no—
142#810C2×C411yescharacteristic
143#810C2×C411yescharacteristic
144#810C2×C411yescharacteristic
145#810C2×C411yescharacteristic
146#808C2×C411yescharacteristic
147#808C2×C411yescharacteristic
148#807C2×C411yescharacteristic
149#807C2×C411yescharacteristic
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)C2×C411yescharacteristic
152C4⋊(C3⋊C8)C2×C411yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161C4×#41C2×D311yescharacteristic
162#485—31no—
163#485—31no—
164#475—31no—
165#474—31no—
166#462—31no—
167#461—31no—
168#452—31no—
169#452—31no—
170#446—31no—
171#446—31no—
172#446—61no—
173#446—61no—
174#446C2×D311yescharacteristic
175#446C2×D311yescharacteristic
176C8×Q8—31no—
177C8×Q8—31no—
178C4×Q16—31no—
179C4×Q16—31no—
180C4×SD16—31no—
181C4×SD16—31no—
182#436C2×D311yescharacteristic
183#436C2×D311yescharacteristic
184#435—21no—
185#435—21no—
186#433—61no—
187#433—61no—
188#432—21no—
189#432—31no—
190#431—31no—
191#431—31no—
192#431C2×D311yescharacteristic
193#423—21no—
194C2×C4⋊C8C2×D311yescharacteristic
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203Q8⋊C8—31no—
204Q8⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8C22×C411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C22×C411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C22×C411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×Dic6—21no—
223C2×Dic6C22×C411yescharacteristic
224C2×Dic6C22×C411yescharacteristic
225C3×M16—81no—
226C3×M16—81no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—81no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42C22×C411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42#4111yescharacteristic
247C3×C42C2411yescharacteristic
248#217—21no—
249#217—21no—
250#216—41no—
251#216—41no—
252#214—41no—
253#214—41no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C22×C411yescharacteristic
258C4⋊Dic3C22×C411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×#41C4×D311yescharacteristic
274C2×Q16—31no—
275C2×Q16—31no—
276C2×SD16—31no—
277C2×SD16—31no—
278C2×M16—21no—
279C2×M16—21no—
280C22×C8—21no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283#125—61no—
284C22⋊Q8—31no—
285C22⋊Q8—31no—
286C22⋊Q8—61no—
287C4×Q8—31no—
288C4×Q8—31no—
289C4×Q8—61no—
290C4×Q8C4×D311yescharacteristic
291C4×Q8C22×D311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C4×D311yescharacteristic
295C4×D4C22×D311yescharacteristic
296#117—21no—
297#117—61no—
298#117C4×D311yescharacteristic
299C2×C42—21no—
300C2×C42C22×D311yescharacteristic
301#107—31no—
302#107—31no—
303#106—31no—
304#106—31no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—31no—
312C4⋊C8—61no—
313C4⋊C8—61no—
314C4⋊C8—61no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8#6711yescharacteristic
318C4⋊C8#6711yescharacteristic
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321#103—31no—
322#103—31no—
323#103—31no—
324#103—31no—
325#103—31no—
326#103—31no—
327#102—31no—
328#102—31no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—61no—
334C22⋊C8—61no—
335C22⋊C8#6711yescharacteristic
336C22⋊C8#6711yescharacteristic
337#97—21no—
338#97—21no—
339#97—61no—
340#97—61no—
341#97—61no—
342C4×C8—31no—
343C4×C8—31no—
344C4×C8—31no—
345C4×C8—31no—
346C4×C8—61no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C23×C3—21no—
351C23×C3C4×D411yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—42no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—81no—
369C2×C3×C4C4×D411yescharacteristic
370C2×C3×C4C4×D411yescharacteristic
371C2×C3×C4C4×D411yescharacteristic
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—21no—
383Dic6—81no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—41no—
399#41—41no—
400C2×Q8—31no—
401C2×Q8—31no—
402C2×Q8—61no—
403C2×Q8C2×C4×D311yescharacteristic
404C2×D4—21no—
405C2×D4C2×C4×D311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4C2×C4×D311yescharacteristic
410C22×C4#23511yescharacteristic
411C22×C4C2×#6711yescharacteristic
412Q16—121no—
413Q16—121no—
414SD16—121no—
415SD16—121no—
416M16—81no—
417M16—81no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—81no—
429C2×C8C2×#6711yescharacteristic
430C2×C8C2×#6711yescharacteristic
431C2×C8C2×#6711yescharacteristic
432C2×C8C2×#6711yescharacteristic
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4C2×C4×D311yescharacteristic
445C4⋊C4C2×C4×D311yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—61no—
455C42—61no—
456C42C2×C4×D311yescharacteristic
457C42#23511yescharacteristic
458C42C2×#6711yescharacteristic
459C42C23×D311yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3C2×C4×D411yescharacteristic
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4#56811yescharacteristic
475C3×C4#56811yescharacteristic
476C3×C4#57711yescharacteristic
477C3×C4#57811yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482C23—21no—
483C23C4×#6711yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488Q8—61no—
489Q8—61no—
490Q8—241no—
491D4—21no—
492D4—21no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—81no—
510C2×C4C4×#6711yescharacteristic
511C2×C4C4×#6711yescharacteristic
512C2×C4C4×#6711yescharacteristic
513C2×C4C22×C4×D311yescharacteristic
514C2×C4C2×#23511yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—61no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—121no—
527C8—121no—
528C8—121no—
529C8—121no—
530C2×C3—41no—
531C2×C3—81no—
532C2×C3#293011yescharacteristic
533C2×C3#294711yescharacteristic
534C2×C3#295711yescharacteristic
535C22—21no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22C2×C4×#6711yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—121no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4#574511yescharacteristic
554C4#574511yescharacteristic
555C4#583611yescharacteristic
556C4#583811yescharacteristic
557C3#1794911yescharacteristic, complemented, Sylow
558C2—41no—
559C2—81no—
560C2#8257111yescharacteristic
561C2#8375611yescharacteristic
562C2#8381211yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2989C2
socleC22×C3C2×C4×D4
prime radicalC2×C3×#41C2×C4
cosocleC3×C42C24
hypercenter#2989D3
hypocenterC3#17949
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2989

Complements

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

NH ≅ G/Ntype
3. #79266C2semidirect
5. #70957C2semidirect
6. #70956C2semidirect
7. #70952C2semidirect
8. #70951C2semidirect
9. #70866C2semidirect
10. #70865C2semidirect
11. #70864C2semidirect
12. #70860C2semidirect
13. #70744C2semidirect
14. #70429C2semidirect
15. #70423C2semidirect
557. C3#17949semidirect

Maximal subgroups

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

subgroupindex
C3×#29892
#825342
#792662
#716402
#709572
#709562
#709522
#709512
#708662
#708652
#708642
#708602
#707442
#704292
#704232
#179493