Finite Group Databasebeta

← #10868035

Subgroups — #10868035

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
1C3×#2496C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3#69635C211yesmaximal, characteristic, complemented
4#69632C211yesmaximal, characteristic, complemented
5#69619C211yesmaximal, characteristic, complemented
6#69602C211yesmaximal, characteristic, complemented
7#69589C211yesmaximal, characteristic, complemented
8#69567C211yesmaximal, characteristic, complemented
9#69527C211yesmaximal, characteristic
10#69527C211yesmaximal, characteristic
11#69506C211yesmaximal, characteristic
12#69503C211yesmaximal, characteristic
13C4×#810C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69451C211yesmaximal, characteristic
16#27043—31nomaximal, Sylow
17C2×C3×#126—21no—
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C3×#400—21no—
22C3×#397—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#384C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4945—21no—
42#4944—21no—
43C4×C4⋊Dic3C2211yescharacteristic
44#4929C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4652—21no—
49#4649—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2496D311yescharacteristic
61#2362—31no—
62#2072—31no—
63#2068—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1918—31no—
71#1865—31no—
72#1854—31no—
73#1825—31no—
74C4×#103—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C3×#126—44no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#926—22no—
96#926—41no—
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×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C4×Dic6—22no—
113C4×Dic6—41no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125#830—22no—
126#830—41no—
127#810—22no—
128#810—22no—
129#810—22no—
130#810—41no—
131#808—22no—
132#807—22no—
133#807—22no—
134#807—41no—
135C4⋊(C3⋊C8)—22no—
136C4×C3⋊C8—41no—
137C2×#126—21no—
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⋊C8—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×#103—31no—
149C2×#103—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8—31no—
153#400—21no—
154#397—21no—
155#391—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#387—21no—
162#387—61no—
163#384C2×D311yescharacteristic
164#380C2×D311yescharacteristic
165#380C2×D311yescharacteristic
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C4⋊C4C2×D311yescharacteristic
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#336—31no—
179#336—31no—
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×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
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—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—42no—
228#217—24no—
229#217—42no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3#4112yes—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239C4×Dic3—42no—
240C4×Dic3—42no—
241C4×Dic3—42no—
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—21no—
253C2×C3⋊C8#4112yes—
254C22×Q8—31no—
255C23×C4C22×D311yescharacteristic
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260#126—44no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C4×Q8—62no—
264C4×Q8—121no—
265#117—62no—
266#117—121no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4#6711yescharacteristic
286C2×C22⋊C4#6711yescharacteristic
287C2×C42—21no—
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—62no—
298#103—62no—
299#103—62no—
300#103—62no—
301#103—121no—
302C22⋊C8—62no—
303C22⋊C8—121no—
304C4×C8—121no—
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—61no—
314#95#6711yescharacteristic
315#95#6711yescharacteristic
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—41no—
342C2×C3×C4—41no—
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#13512yes—
354C2×C3×C4AGL(1,ℤ/8)12yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
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×#6711yescharacteristic
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—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408C2×C8—32no—
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—61no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—32no—
429C4⋊C4—32no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—64no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—64no—
446C22⋊C4—122no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—42no—
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—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4#52911yescharacteristic
479C3×C4#53911yescharacteristic
480C3×C4#55311yescharacteristic
481C3×C4#55611yescharacteristic
482Dic3—42no—
483Dic3—42no—
484Dic3—42no—
485Dic3—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C22×#6711yescharacteristic
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—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—41no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
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#93212yes—
540C2×C4#95012yes—
541C2×C4D3×#4111yescharacteristic
542C2×C4D3×#4111yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4#100911yescharacteristic
546C2×C4#101011yescharacteristic
547C8—62no—
548C8—121no—
549C8—121no—
550C8—121no—
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—
566C22C2×#93211yescharacteristic
567C22C2×#95011yescharacteristic
568C22#582411yescharacteristic
569C22#582811yescharacteristic
570C22#582911yescharacteristic
571C22#583811yescharacteristic
572C22#583811yescharacteristic
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—122no—
583C4—122no—
584C4—122no—
585C4—241no—
586C4#570211yescharacteristic
587C4#570611yescharacteristic
588C4#570711yescharacteristic
589C4#570911yescharacteristic
590C3#2704311yescharacteristic, complemented, Sylow
591C2—81no—
592C2#8310511yescharacteristic
593C2#8329911yescharacteristic
594C2#8345111yescharacteristic
595C2#8352011yescharacteristic
596C2#8353211yescharacteristic
597C2#8359011yescharacteristic
598C2#8366411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2496C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2496D3
hypocenterC3#27043
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
3. #69635C2semidirect
4. #69632C2semidirect
5. #69619C2semidirect
6. #69602C2semidirect
7. #69589C2semidirect
8. #69567C2semidirect
13. C4×#810C2semidirect
14. #69458C2semidirect
590. C3#27043semidirect

Maximal subgroups

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

subgroupindex
C3×#24962
#734682
#696352
#696322
#696192
#696022
#695892
#695672
#695272
#695062
#695032
C4×#8102
#694582
#694512
#270433