Finite Group Databasebeta

← #10765618

Subgroups — #10765618

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×#3107C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3#82521C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic, complemented
5#71640C211yesmaximal, characteristic
6#70906C211yesmaximal, characteristic, complemented
7#70905C211yesmaximal, characteristic, complemented
8#70904C211yesmaximal, characteristic, complemented
9#70903C211yesmaximal, characteristic, complemented
10#70900C211yesmaximal, characteristic, complemented
11#70899C211yesmaximal, characteristic, complemented
12#70898C211yesmaximal, characteristic, complemented
13#70896C211yesmaximal, characteristic, complemented
14#70752C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#17949—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C3×#485C411yescharacteristic
20C3×#485C411yescharacteristic
21C3×#475C411yescharacteristic
22C3×#474C411yescharacteristic
23C3×#462C411yescharacteristic
24C3×#461C411yescharacteristic
25C3×#452C411yescharacteristic
26C3×#452C411yescharacteristic
27C3×C4×Q16C2211yescharacteristic
28C3×C4×Q16C2211yescharacteristic
29C3×C4×SD16C2211yescharacteristic
30C3×C4×SD16C2211yescharacteristic
31C3×#431C2211yescharacteristic
32#5024—21no—
33#5024—21no—
34#5024C2211yescharacteristic
35#5024C2211yescharacteristic
36#5024C2211yescharacteristic
37#5024C2211yescharacteristic
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40#5012C2211yescharacteristic
41#5012C2211yescharacteristic
42#5011—21no—
43#5011—21no—
44#5000—21no—
45#5000—21no—
46#4926—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49#4925C2211yescharacteristic
50#4924—21no—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4535C2211yescharacteristic
53#4535C2211yescharacteristic
54#4535C2211yescharacteristic
55#4535C2211yescharacteristic
56#4535C2211yescharacteristic
57#4535C2211yescharacteristic
58D4⋊(C3⋊C8)C2211yescharacteristic
59D4⋊(C3⋊C8)C2211yescharacteristic
60#4458C2211yescharacteristic
61#4458C2211yescharacteristic
62#4457C2211yescharacteristic
63#4457C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107D311yescharacteristic
69#2989—31no—
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×Q16C2×C411yescharacteristic
85C2×C3×Q16C2×C411yescharacteristic
86C2×C3×SD16C2×C411yescharacteristic
87C2×C3×SD16C2×C411yescharacteristic
88C3×C4⋊Q8C2×C411yescharacteristic
89C3×C4⋊Q8C2×C411yescharacteristic
90C3×#125—21no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8C2×C411yescharacteristic
93C3×C22⋊Q8C2×C411yescharacteristic
94C3×C4×Q8—21no—
95C3×C4×Q8C2×C411yescharacteristic
96C3×C4×Q8C2311yescharacteristic
97C3×C4×Q8C2311yescharacteristic
98C3×C4×Q8C2311yescharacteristic
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4C2×C411yescharacteristic
102C3×C4×D4C2311yescharacteristic
103C3×#117—21no—
104C3×#117—21no—
105C3×#117C2×C411yescharacteristic
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#208—21no—
109C2×#208—21no—
110C22×C3⋊C8—21no—
111C3×#107C2×C411yescharacteristic
112C3×#107C2×C411yescharacteristic
113C3×#106C2×C411yescharacteristic
114C3×#106C2×C411yescharacteristic
115C3×C4⋊C8C2×C411yescharacteristic
116C3×C4⋊C8C2×C411yescharacteristic
117C3×#103C2×C411yescharacteristic
118C3×#103C2×C411yescharacteristic
119C3×#103C2×C411yescharacteristic
120C3×#103C2×C411yescharacteristic
121C3×#103C2×C411yescharacteristic
122C3×#103C2×C411yescharacteristic
123C3×#102C2×C411yescharacteristic
124C3×#102C2×C411yescharacteristic
125C3×C22⋊C8C2×C411yescharacteristic
126C3×C22⋊C8C2×C411yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#830—21no—
130#830—21no—
131#830—21no—
132#830—21no—
133#830D411yescharacteristic
134#830D411yescharacteristic
135C4⋊(C3⋊C8)—21no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138C4⋊(C3⋊C8)—21no—
139C4⋊(C3⋊C8)—21no—
140C4⋊(C3⋊C8)—21no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)D411yescharacteristic
144C4⋊(C3⋊C8)D411yescharacteristic
145C4⋊(C3⋊C8)C2311yescharacteristic
146C4⋊(C3⋊C8)C2311yescharacteristic
147C4⋊(C3⋊C8)C2311yescharacteristic
148C4⋊(C3⋊C8)C2311yescharacteristic
149#803—21no—
150#803—21no—
151#803—21no—
152#803—21no—
153#803—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161C4×#41C2×D311yescharacteristic
162#485Dic311yescharacteristic
163#485Dic311yescharacteristic
164#475Dic311yescharacteristic
165#474Dic311yescharacteristic
166#462Dic311yescharacteristic
167#461Dic311yescharacteristic
168#452Dic311yescharacteristic
169#452Dic311yescharacteristic
170#446—31no—
171#446—31no—
172#446—31no—
173#446—31no—
174#446—61no—
175#446—61no—
176C8×Q8—31no—
177C8×Q8—31no—
178C4×Q16C2×D311yescharacteristic
179C4×Q16C2×D311yescharacteristic
180C4×SD16C2×D311yescharacteristic
181C4×SD16C2×D311yescharacteristic
182#436—31no—
183#436—31no—
184#435—61no—
185#435—61no—
186#433—61no—
187#433—61no—
188#432—31no—
189#432—61no—
190#431—31no—
191#431—31no—
192#431C2×D311yescharacteristic
193#423—61no—
194C2×C4⋊C8—31no—
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×Q8—21no—
214C2×C3×Q8C22×C411yescharacteristic
215C2×C3×Q8C22×C411yescharacteristic
216C2×C3×Q8C22×C411yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4C22×C411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C22×C411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4#4111yescharacteristic
225C3×Q16—41no—
226C3×Q16—41no—
227C3×SD16—41no—
228C3×SD16—41no—
229C2×C3×C8C22×C411yescharacteristic
230C2×C3×C8C22×C411yescharacteristic
231C2×C3×C8C22×C411yescharacteristic
232C2×C3×C8C22×C411yescharacteristic
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C22×C411yescharacteristic
243C3×C4⋊C4C22×C411yescharacteristic
244C3×C4⋊C4C22×C411yescharacteristic
245C3×C4⋊C4C22×C411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42C22×C411yescharacteristic
257C3×C42C2×D411yescharacteristic
258C3×C42#4111yescharacteristic
259C3×C42C2411yescharacteristic
260#208—81no—
261#208—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—81no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×#41C2×Dic311yescharacteristic
274C2×Q16C2×Dic311yescharacteristic
275C2×Q16C2×Dic311yescharacteristic
276C2×SD16C2×Dic311yescharacteristic
277C2×SD16C2×Dic311yescharacteristic
278C2×M16—61no—
279C2×M16—61no—
280C22×C8—61no—
281C4⋊Q8C2×Dic311yescharacteristic
282C4⋊Q8C2×Dic311yescharacteristic
283#125—21no—
284C22⋊Q8—21no—
285C22⋊Q8C2×Dic311yescharacteristic
286C22⋊Q8C2×Dic311yescharacteristic
287C4×Q8—21no—
288C4×Q8C2×Dic311yescharacteristic
289C4×Q8C22×D311yescharacteristic
290C4×Q8C22×D311yescharacteristic
291C4×Q8C22×D311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C2×Dic311yescharacteristic
295C4×D4C22×D311yescharacteristic
296#117—21no—
297#117—21no—
298#117C2×Dic311yescharacteristic
299C2×C42—21no—
300C2×C42C22×D311yescharacteristic
301#107C2×Dic311yescharacteristic
302#107C2×Dic311yescharacteristic
303#106C2×Dic311yescharacteristic
304#106C2×Dic311yescharacteristic
305C4⋊C8—31no—
306C4⋊C8—31no—
307C4⋊C8—31no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—61no—
312C4⋊C8—61no—
313C4⋊C8—61no—
314C4⋊C8—61no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C2×Dic311yescharacteristic
320C4⋊C8C2×Dic311yescharacteristic
321#103C2×Dic311yescharacteristic
322#103C2×Dic311yescharacteristic
323#103C2×Dic311yescharacteristic
324#103C2×Dic311yescharacteristic
325#103C2×Dic311yescharacteristic
326#103C2×Dic311yescharacteristic
327#102C2×Dic311yescharacteristic
328#102C2×Dic311yescharacteristic
329C22⋊C8—31no—
330C22⋊C8—31no—
331C22⋊C8—61no—
332C22⋊C8—61no—
333C22⋊C8—61no—
334C22⋊C8—61no—
335C22⋊C8C2×Dic311yescharacteristic
336C22⋊C8C2×Dic311yescharacteristic
337#97—61no—
338#97—61no—
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×Q8—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—81no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
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—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—81no—
378C2×C3×C4C4×D411yescharacteristic
379C2×C3×C4C4×D411yescharacteristic
380C2×C3×C4C4×D411yescharacteristic
381C2×C3×C4C23×C411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—21no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—41no—
399#41—41no—
400C2×Q8—21no—
401C2×Q8C22×Dic311yescharacteristic
402C2×Q8C22×Dic311yescharacteristic
403C2×Q8C22×Dic311yescharacteristic
404C2×D4—21no—
405C2×D4C22×Dic311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4D3×D411yescharacteristic
410C22×C4#23711yescharacteristic
411C22×C4C22×Dic311yescharacteristic
412Q16—41no—
413Q16—41no—
414SD16—41no—
415SD16—41no—
416M16—241no—
417M16—241no—
418C2×C8—31no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—241no—
429C2×C8C22×Dic311yescharacteristic
430C2×C8C22×Dic311yescharacteristic
431C2×C8C22×Dic311yescharacteristic
432C2×C8C22×Dic311yescharacteristic
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4C22×Dic311yescharacteristic
443C4⋊C4C22×Dic311yescharacteristic
444C4⋊C4C22×Dic311yescharacteristic
445C4⋊C4C22×Dic311yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42D3×D411yescharacteristic
457C42#23711yescharacteristic
458C42C22×Dic311yescharacteristic
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—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4#56811yescharacteristic
479C3×C4#56811yescharacteristic
480C3×C4#57711yescharacteristic
481C3×C4#57811yescharacteristic
482C23—21no—
483C23D4×Dic311yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—81no—
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—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—81no—
510C2×C4D4×Dic311yescharacteristic
511C2×C4D4×Dic311yescharacteristic
512C2×C4D4×Dic311yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23711yescharacteristic
515C2×C4C23×Dic311yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—61no—
521C8—61no—
522C8—61no—
523C8—61no—
524C8—121no—
525C8—121no—
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×D4×Dic311yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4#577811yescharacteristic
554C4#577911yescharacteristic
555C4#581911yescharacteristic
556C4#581911yescharacteristic
557C3#1794911yescharacteristic, complemented, Sylow
558C2—41no—
559C2—81no—
560C2#8291711yescharacteristic
561C2#8391011yescharacteristic
562C2#8396511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#3107C2
socleC22×C3C2×C4×D4
prime radicalC2×C3×#41C2×C4
cosocleC3×C42C24
hypercenter#3107D3
hypocenterC3#17949
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
4. #71824C2semidirect
6. #70906C2semidirect
7. #70905C2semidirect
8. #70904C2semidirect
9. #70903C2semidirect
10. #70900C2semidirect
11. #70899C2semidirect
12. #70898C2semidirect
13. #70896C2semidirect
14. #70752C2semidirect
15. #70744C2semidirect
557. C3#17949semidirect

Maximal subgroups

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

subgroupindex
C3×#31072
#833312
#825212
#718242
#716402
#709062
#709052
#709042
#709032
#709002
#708992
#708982
#708962
#707522
#707442
#179493