Finite Group Databasebeta

← #10868037

Subgroups — #10868037

Order 768.

706 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
1#73512C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3C3×#1923C211yesmaximal, characteristic
4#70489C211yesmaximal, characteristic, complemented
5#70453C211yesmaximal, characteristic, complemented
6#69937C211yesmaximal, characteristic
7#69937C211yesmaximal, characteristic
8#69929C211yesmaximal, characteristic
9#69926C211yesmaximal, characteristic
10#69770C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#68684C211yesmaximal, characteristic, complemented
13#68636C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#27043—31nomaximal, Sylow
17C2×#926C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#879—21no—
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4953—21no—
37#4945—21no—
38#4944—21no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4659—21no—
47#4653—21no—
48#4652—21no—
49#4649—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2496—31no—
61#2362—31no—
62#2072—31no—
63#2068—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1942—31no—
68#1923D311yescharacteristic
69#1920—31no—
70#1918—31no—
71#1865—31no—
72#1854—31no—
73#1825—31no—
74C4×#103—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89#926—22no—
90#926—41no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100#924—22no—
101#924—22no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111#879—44no—
112C4×Dic6—22no—
113C4×Dic6—41no—
114C3×C4⋊C8—22no—
115C3×C22⋊C8—22no—
116C3×C22⋊C8—41no—
117C3×C4×C8—41no—
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#818—22no—
130#818—22no—
131#818—41no—
132#817—22no—
133#816—22no—
134#816—22no—
135#816—22no—
136#816—41no—
137C2×#126—61no—
138C2×C22⋊Q8—31no—
139C2×C4×Q8—31no—
140C2×#117—31no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#103—31no—
147C2×#103—31no—
148C2×#103—31no—
149C2×#103—31no—
150C2×C22⋊C8C2×D311yescharacteristic
151C2×C22⋊C8C2×D311yescharacteristic
152C2×C4×C8C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#388—61no—
161#387—61no—
162#387—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—61no—
169C4×C22⋊C4C2×D311yescharacteristic
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C24×C3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6#4112yes—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8#4112yes—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225#217—24no—
226#217—42no—
227#217—42no—
228#217—42no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—42no—
242C4⋊Dic3—42no—
243C4⋊Dic3#4112yes—
244C4⋊Dic3#4112yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C22×Q8—31no—
255C23×C4C22×D311yescharacteristic
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#126—124no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C4×Q8—62no—
264C4×Q8—121no—
265#117—62no—
266#117—121no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4D1211yescharacteristic
286C2×C22⋊C4D1211yescharacteristic
287C2×C42—61no—
288C2×C42—61no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42C22×D311yescharacteristic
292C2×C42C22×D311yescharacteristic
293#107—62no—
294#107—62no—
295#107—121no—
296#106—62no—
297C4⋊C8—22no—
298#103—62no—
299#103—62no—
300#103—62no—
301#103—121no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C4×C8—41no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95D1211yescharacteristic
315#95D1211yescharacteristic
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3C22×D411yescharacteristic
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
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—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4#13512yes—
342C2×C3×C4AGL(1,ℤ/8)12yes—
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C42−1+411yescharacteristic
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373Dic6—22no—
374Dic6—22no—
375Dic6—41no—
376Dic6—41no—
377C3×C8—22no—
378C3×C8—41no—
379C3×C8—41no—
380C3×C8—41no—
381C24C2×D1211yescharacteristic
382C2×Q8—32no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4C2×D1211yescharacteristic
403C22×C4C2×D1211yescharacteristic
404C22×C4C2×D1211yescharacteristic
405C22×C4C2×D1211yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4C23×D311yescharacteristic
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8#23712yes—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—64no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—64no—
444C22⋊C4—122no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—122no—
452C42—122no—
453C42—122no—
454C42—122no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3#52511yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3#54711yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
467C22×C3#57811yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4#52911yescharacteristic
475C3×C4#53911yescharacteristic
476C3×C4#55311yescharacteristic
477C3×C4#55611yescharacteristic
478Dic3—42no—
479Dic3—42no—
480Dic3—42no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C22×D1211yescharacteristic
494Q8—62no—
495Q8—62no—
496Q8—121no—
497Q8—121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
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—121no—
539C2×C4#90412yes—
540C2×C4#90812yes—
541C2×C4C2×#23711yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4#101011yescharacteristic
547C8—22no—
548C8—41no—
549C8—41no—
550C8—41no—
551C2×C3—81no—
552C2×C3#246611yescharacteristic
553C2×C3#310611yescharacteristic
554C2×C3#312711yescharacteristic
555C2×C3#330211yescharacteristic
556C2×C3#330811yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#335311yescharacteristic
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22#552411yescharacteristic
567C22#555411yescharacteristic
568C22#555511yescharacteristic
569C22C2×#90411yescharacteristic
570C22C2×#90811yescharacteristic
571C22#575311yescharacteristic
572C22#575311yescharacteristic
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—122no—
579C4—122no—
580C4—122no—
581C4—241no—
582C4—241no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#560111yescharacteristic
587C4#566011yescharacteristic
588C4#568011yescharacteristic
589C4#570211yescharacteristic
590C3#2704311yescharacteristic, complemented, Sylow
591C2—81no—
592C2#7551611yescharacteristic
593C2#7930111yescharacteristic
594C2#7939611yescharacteristic
595C2#7941311yescharacteristic
596C2#7944211yescharacteristic
597C2#7950911yescharacteristic
598C2#7953111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1923C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1923D3
hypocenterC3#27043
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
4. #70489C2semidirect
5. #70453C2semidirect
10. #69770C2semidirect
11. C4×#816C2semidirect
12. #68684C2semidirect
13. #68636C2semidirect
14. #68623C2semidirect
15. #68621C2semidirect
590. C3#27043semidirect

Maximal subgroups

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

subgroupindex
#735122
#734682
C3×#19232
#704892
#704532
#699372
#699292
#699262
#697702
C4×#8162
#686842
#686362
#686232
#686212
#270433