Finite Group Databasebeta

← #10877472

Subgroups — #10877472

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2499C211yesmaximal, characteristic
2#73494C211yesmaximal, characteristic
3#71696C211yesmaximal, characteristic
4#71693C211yesmaximal, characteristic
5#71580C211yesmaximal, characteristic
6#71575C211yesmaximal, characteristic
7#69674C211yesmaximal, characteristic, complemented
8#69674C211yesmaximal, characteristic, complemented
9#69670C211yesmaximal, characteristic, complemented
10#69669C211yesmaximal, characteristic, complemented
11#69574C211yesmaximal, characteristic, complemented
12#69571C211yesmaximal, characteristic, complemented
13#69554C211yesmaximal, characteristic, complemented
14#69542C211yesmaximal, characteristic, complemented
15C22⋊C4×C3⋊C8C211yesmaximal, characteristic
16#27700—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C3×#397—21no—
23C3×#397—21no—
24C3×#391C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#387—21no—
27C3×#387—21no—
28C3×#384C2211yescharacteristic
29C3×#382C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#4949—21no—
38#4948—21no—
39#4935C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4654—21no—
45#4647—21no—
46#4529C2211yescharacteristic
47#4529C2211yescharacteristic
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2499D311yescharacteristic
61#2441—31no—
62#2101—31no—
63#2099—31no—
64#2072—31no—
65#2026—31no—
66#1991—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1873—31no—
71#1834—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96#926—22no—
97#926—41no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—41no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#870—22no—
117#870—41no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126#835—22no—
127#835—22no—
128#835—41no—
129#835—41no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133#830—22no—
134C4×C3⋊C8—22no—
135C4×C3⋊C8—41no—
136C23×C8—31no—
137C2×#125—31no—
138C2×C22⋊Q8C2×D311yescharacteristic
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#399—61no—
149#397—21no—
150#397—21no—
151#396—61no—
152#391C2×D311yescharacteristic
153#391C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—61no—
157#387—21no—
158#387—21no—
159#387—61no—
160#384—31no—
161#384C2×D311yescharacteristic
162#382C2×D311yescharacteristic
163#380—31no—
164C4×C22⋊C4C2×D311yescharacteristic
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—42no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224#217—42no—
225#217—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—42no—
234C4⋊Dic3—81no—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—42no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—81no—
240C4×Dic3—42no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—44no—
253C2×C3⋊C8#4112yes—
254C22×Q8C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—62no—
261C22×C8—62no—
262C22×C8—121no—
263#125—62no—
264#125—121no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#117—22no—
268#117—22no—
269#117—62no—
270#117—121no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4#6711yescharacteristic
292C2×C22⋊C4#6711yescharacteristic
293C2×C42—21no—
294C2×C42—61no—
295C2×C42C22×D311yescharacteristic
296C2×C42C22×D311yescharacteristic
297#103—62no—
298#103—62no—
299#103—121no—
300#103—121no—
301C22⋊C8—62no—
302C4×C8—62no—
303C4×C8—121no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95#6711yescharacteristic
314#95#6711yescharacteristic
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3#12412yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4#12412yes—
356C2×C3×C4#13512yes—
357C2×C3×C4#13512yes—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×Dic3—22no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C3⋊C8—22no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C24C2×#6711yescharacteristic
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4#23712yes—
400C22×C4#23712yes—
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C23×D311yescharacteristic
407C2×C8—32no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—124no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—32no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—241no—
442C4⋊C4—241no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—122no—
448C22⋊C4—241no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—122no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3C2×#12411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3#54711yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4#52911yescharacteristic
479C3×C4#53911yescharacteristic
480C3×C4#55111yescharacteristic
481C3×C4#55211yescharacteristic
482Dic3—42no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#93612yes—
493C23C22×#6711yescharacteristic
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—62no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4#93612yes—
539C2×C4#95012yes—
540C2×C4#95012yes—
541C2×C4C2×#23711yescharacteristic
542C2×C4C2×#23711yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4#100911yescharacteristic
546C2×C4#101011yescharacteristic
547C8—62no—
548C8—121no—
549C8—121no—
550C8—121no—
551C2×C3—42no—
552C2×C3#248811yescharacteristic
553C2×C3#313111yescharacteristic
554C2×C3#313711yescharacteristic
555C2×C3#330211yescharacteristic
556C2×C3#330411yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#333711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×#93611yescharacteristic
567C22C2×#95011yescharacteristic
568C22C2×#95011yescharacteristic
569C22#582411yescharacteristic
570C22#582911yescharacteristic
571C22#583711yescharacteristic
572C22#583811yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—122no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#564911yescharacteristic
587C4#565011yescharacteristic
588C4#565811yescharacteristic
589C4#566511yescharacteristic
590C3#2770011yescharacteristic, complemented, Sylow
591C2—42no—
592C2#8285711yescharacteristic
593C2#8328111yescharacteristic
594C2#8340111yescharacteristic
595C2#8359011yescharacteristic
596C2#8359111yescharacteristic
597C2#8366411yescharacteristic
598C2#8367311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2499C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2499D3
hypocenterC3#27700
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
7. #69674C2semidirect
8. #69674C2semidirect
9. #69670C2semidirect
10. #69669C2semidirect
11. #69574C2semidirect
12. #69571C2semidirect
13. #69554C2semidirect
14. #69542C2semidirect
590. C3#27700semidirect

Maximal subgroups

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

subgroupindex
C3×#24992
#734942
#716962
#716932
#715802
#715752
#696742
#696702
#696692
#695742
#695712
#695542
#695422
C22⋊C4×C3⋊C82
#277003