Finite Group Databasebeta

← #10868018

Subgroups — #10868018

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2362C211yesmaximal, characteristic
2#73511C211yesmaximal, characteristic
3#71695C211yesmaximal, characteristic
4#71694C211yesmaximal, characteristic
5C4×#835C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#71576C211yesmaximal, characteristic
8#71575C211yesmaximal, characteristic
9#69682C211yesmaximal, characteristic, complemented
10#69671C211yesmaximal, characteristic, complemented
11#69618C211yesmaximal, characteristic, complemented
12#69600C211yesmaximal, characteristic, complemented
13#69468C211yesmaximal, characteristic, complemented
14#69461C211yesmaximal, characteristic, complemented
15#69451C211yesmaximal, characteristic
16#27042—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C3×#391C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#388—21no—
23C3×#388—21no—
24C3×#387—21no—
25C3×C4×C4⋊C4—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4952—21no—
41#4944—21no—
42#4936C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4658—21no—
49#4651—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2495—31no—
61#2362D311yescharacteristic
62#2074—31no—
63#2066—31no—
64#2036—31no—
65#2035—31no—
66#1949—31no—
67#1941—31no—
68#1923—31no—
69#1921—31no—
70#1918—31no—
71#1863—31no—
72#1854—31no—
73#1825—31no—
74C4×#103—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C3×#117—22no—
82C3×#117—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95D411yescharacteristic
116C3×#95D411yescharacteristic
117#835—22no—
118#835—22no—
119#835—22no—
120#835—41no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—41no—
130#808—22no—
131#808—41no—
132#807—22no—
133#807—22no—
134C4⋊(C3⋊C8)—22no—
135C4×C3⋊C8—41no—
136C2×C22⋊Q8C2×D311yescharacteristic
137C2×C4×Q8C2×D311yescharacteristic
138C2×#117C2×D311yescharacteristic
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#397—61no—
153#391C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389C2×D311yescharacteristic
157#388—21no—
158#388—21no—
159#388—61no—
160#387—21no—
161#383—61no—
162#382—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168C4×C22⋊C4C2×D311yescharacteristic
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#4112yes—
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×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4#4112yes—
214C3×C4⋊C4#4112yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—41no—
218C3×C22⋊C4—42no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224C3×C42—42no—
225C3×C42—42no—
226#217—42no—
227#217—81no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236Dic3⋊C4—81no—
237Dic3⋊C4—81no—
238C4×Dic3—42no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
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#4112yes—
251C22×Q8C22×D311yescharacteristic
252C23×C4C22×D311yescharacteristic
253C22×C8—31no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8—31no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C4×Q8—22no—
260C4×Q8—41no—
261#117—22no—
262#117—41no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4#6711yescharacteristic
280C2×C22⋊C4#6711yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—61no—
285C2×C42C22×D311yescharacteristic
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#107—62no—
289#106—62no—
290#106—121no—
291C4⋊C8—62no—
292#103—62no—
293#103—62no—
294#103—62no—
295#103—121no—
296C22⋊C8—62no—
297C22⋊C8—121no—
298C4×C8—121no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95#6711yescharacteristic
311#95#6711yescharacteristic
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
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—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
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—41no—
354C2×C3×C4#13512yes—
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—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—
373C3⋊C8—22no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376C3⋊C8—41no—
377C24C2×#6711yescharacteristic
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8#23512yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
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×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C23×D311yescharacteristic
404C2×C8—32no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—42no—
422C4⋊C4—42no—
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—241no—
432C4⋊C4—241no—
433C4⋊C4#23512yes—
434C4⋊C4#23712yes—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—122no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3#52511yescharacteristic
457C22×C3#54611yescharacteristic
458C22×C3#54711yescharacteristic
459C22×C3C2×#13511yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57811yescharacteristic
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—81no—
472C3×C4#52911yescharacteristic
473C3×C4#53911yescharacteristic
474C3×C4#55311yescharacteristic
475C3×C4#55611yescharacteristic
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23C22×#6711yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
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—41no—
510C2×C4—41no—
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—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#95012yes—
535C2×C4C2×#23511yescharacteristic
536C2×C4C2×#23511yescharacteristic
537C2×C4C2×#23711yescharacteristic
538C2×C4#100411yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4#101011yescharacteristic
541C8—62no—
542C8—121no—
543C8—121no—
544C8—121no—
545C2×C3—81no—
546C2×C3#246611yescharacteristic
547C2×C3#310211yescharacteristic
548C2×C3#312111yescharacteristic
549C2×C3#330011yescharacteristic
550C2×C3#330711yescharacteristic
551C2×C3#334011yescharacteristic
552C2×C3#335211yescharacteristic
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#579111yescharacteristic
561C22#580611yescharacteristic
562C22C2×#95011yescharacteristic
563C22#582211yescharacteristic
564C22#582411yescharacteristic
565C22#583711yescharacteristic
566C22#583811yescharacteristic
567C4—22no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—81no—
576C4—241no—
577C4—241no—
578C4—241no—
579C4—241no—
580C4#553811yescharacteristic
581C4#554211yescharacteristic
582C4#555811yescharacteristic
583C4#556411yescharacteristic
584C3#2704211yescharacteristic, complemented, Sylow
585C2—81no—
586C2#8236911yescharacteristic
587C2#8252811yescharacteristic
588C2#8255211yescharacteristic
589C2#8351211yescharacteristic
590C2#8353411yescharacteristic
591C2#8359711yescharacteristic
592C2#8365811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2362C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2362D3
hypocenterC3#27042
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
5. C4×#835C2semidirect
6. #71628C2semidirect
9. #69682C2semidirect
10. #69671C2semidirect
11. #69618C2semidirect
12. #69600C2semidirect
13. #69468C2semidirect
14. #69461C2semidirect
584. C3#27042semidirect

Maximal subgroups

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

subgroupindex
C3×#23622
#735112
#716952
#716942
C4×#8352
#716282
#715762
#715752
#696822
#696712
#696182
#696002
#694682
#694612
#694512
#270423