Finite Group Databasebeta

← #10853322

Subgroups — #10853322

Order 768.

734 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
1#83037C211yesmaximal, characteristic, complemented
2#73639C211yesmaximal, characteristic
3C3×#1989C211yesmaximal, characteristic, complemented
4#71743C211yesmaximal, characteristic
5#70484C211yesmaximal, characteristic
6#70451C211yesmaximal, characteristic, complemented
7#69461C211yesmaximal, characteristic, complemented
8#68901C211yesmaximal, characteristic
9#68899C211yesmaximal, characteristic
10#68892C211yesmaximal, characteristic
11#68847C211yesmaximal, characteristic, complemented
12#68820C211yesmaximal, characteristic, complemented
13#68801C211yesmaximal, characteristic, complemented
14#68795C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic
16#26025—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#890C2211yescharacteristic
20C2×#872—21no—
21C2×C3×C4⋊C8C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×Dic3⋊C8C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#4988C2211yescharacteristic
30#4983—21no—
31#4981—21no—
32#4976—21no—
33#4975—21no—
34#4975C2211yescharacteristic
35#4975C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4940—21no—
43#4939C2211yescharacteristic
44C4×Dic3⋊C4C2211yescharacteristic
45#4928C2211yescharacteristic
46#4669—21no—
47#4667—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2552—31no—
61#2354—31no—
62#2067—31no—
63#2067—31no—
64#2025—31no—
65#2025—31no—
66#1989—31no—
67#1989D311yescharacteristic, complemented
68#1947—31no—
69#1939—31no—
70#1921—31no—
71#1921—31no—
72#1834—31no—
73#1802—31no—
74C4×#102—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#125—22no—
80C3×#125—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×Dic3⋊C4D411yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#893—22no—
114#893Q814yes—
115#890—22no—
116#890—41no—
117#872—44no—
118C3×C4⋊C8—41no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95D411yescharacteristic
125C3⋊#95D411yescharacteristic
126#821—22no—
127#821—41no—
128#818—22no—
129Dic3⋊C8—22no—
130Dic3⋊C8—41no—
131C8×Dic3—22no—
132#809—22no—
133#809—41no—
134#807—22no—
135C2×#125—31no—
136C2×#125C2×D311yescharacteristic
137C2×C4×D4—31no—
138C2×#117—61no—
139C2×#107—31no—
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×C4⋊C8—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×C4×C8—31no—
149#398—61no—
150#398—61no—
151#390—31no—
152#390—31no—
153#389—31no—
154#389—31no—
155#389—61no—
156#388—61no—
157#388—61no—
158#384—31no—
159#384—31no—
160#384—61no—
161#380—61no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178C23×D3—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12C2×Q812yes—
199C2×C4×D3—24no—
200C2×C4×D3—42no—
201C2×C4×D3—42no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—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—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×Q812yes—
217C3×C4⋊C4#4112yes—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—42no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—81no—
225(C2×D3)⋊C4—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—42no—
232C4⋊Dic3#4112yes—
233Dic3⋊C4—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—42no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240Dic3⋊C4—81no—
241Dic3⋊C4—81no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C4×Dic3C2×Q812yes—
247C4×Dic3#4112yes—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C22×D4—31no—
255C23×C4—61no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#125—22no—
261#125—41no—
262#125—62no—
263#125—121no—
264C4×D4—34no—
265C4×D4—62no—
266#117—124no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
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×C4⋊C4—61no—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—21no—
292C2×C42—31no—
293C2×C42—31no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#107—62no—
298#107—62no—
299C4⋊C8—41no—
300C4⋊C8—62no—
301C4⋊C8—121no—
302#102—62no—
303#102—62no—
304#102—121no—
305#102—121no—
306C4×C8—62no—
307#95—21no—
308#95—31no—
309#95—31no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314C23×C3C22×D411yescharacteristic
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
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—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4C22×Q811yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C42−1+411yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—22no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
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×Dic3C22⋊Q812yes—
364C2×Dic3C22⋊Q812yes—
365C2×Dic3#13512yes—
366C2×Dic3AGL(1,ℤ/8)12yes—
367D12—22no—
368D12—22no—
369D12—41no—
370D12—41no—
371C4×D3—48no—
372C3×C8—41no—
373C3×C8—41no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376C24—61no—
377C2×D4—32no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—64no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C23×D311yescharacteristic
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—32no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4—241no—
435C4⋊C4—241no—
436C4⋊C4#23512yes—
437C4⋊C4D3×Q812yes—
438C22⋊C4—64no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C22⋊C4—241no—
442C22⋊C4—241no—
443C42—21no—
444C42—21no—
445C42—32no—
446C42—32no—
447C42—42no—
448C42—61no—
449C42—61no—
450C42—122no—
451C42—122no—
452C22×C3C2×C22⋊Q811yescharacteristic
453C22×C3#54611yescharacteristic
454C22×C3#54811yescharacteristic
455C22×C3C2×#13511yescharacteristic
456C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
457C22×C3#57611yescharacteristic
458C22×C3#57811yescharacteristic
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—81no—
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—81no—
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4#53311yescharacteristic
473C3×C4#54411yescharacteristic
474C3×C4#55511yescharacteristic
475C3×C4#55611yescharacteristic
476Dic3—22no—
477Dic3—41no—
478Dic3—42no—
479Dic3—41no—
480Dic3—81no—
481Dic3—81no—
482Dic3#47814yes—
483C23—62no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23C2×D3×D411yescharacteristic
491D4—62no—
492D4—62no—
493D4—121no—
494D4—121no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—41no—
508C2×C4—41no—
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—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—128no—
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×C4C2×#23511yescharacteristic
536C2×C4#100411yescharacteristic
537C2×C4C2×D3×Q811yescharacteristic
538C2×C4#100711yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C8—41no—
542C8—41no—
543C8—121no—
544C8—121no—
545C2×C3#243911yescharacteristic
546C2×C3C2×#47811yescharacteristic
547C2×C3#308311yescharacteristic
548C2×C3#330511yescharacteristic
549C2×C3#333011yescharacteristic
550C2×C3#336411yescharacteristic
551C2×C3#336611yescharacteristic
552D3—82no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22#549011yescharacteristic
561C22D3×C22⋊Q811yescharacteristic
562C22#563311yescharacteristic
563C22D3×#13511yescharacteristic
564C22#576911yescharacteristic
565C22D3×AGL(1,ℤ/8)11yescharacteristic
566C22#577911yescharacteristic
567C4—34no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—62no—
573C4—81no—
574C4—81no—
575C4—122no—
576C4—121no—
577C4—121no—
578C4—241no—
579C4—241no—
580C4#556811yescharacteristic
581C4#557811yescharacteristic
582C4#567011yescharacteristic
583C4#572911yescharacteristic
584C3#2602511yescharacteristic, complemented, Sylow
585C2—242no—
586C2#7650111yescharacteristic
587C2#7885211yescharacteristic
588C2#7890611yescharacteristic
589C2#8115211yescharacteristic
590C2D3×#47811yescharacteristic
591C2#8121611yescharacteristic
592C2#8123411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1989C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1989D3
hypocenterC3#26025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
1. #83037C2semidirect
3. C3×#1989C2semidirect
6. #70451C2semidirect
7. #69461C2semidirect
11. #68847C2semidirect
12. #68820C2semidirect
13. #68801C2semidirect
14. #68795C2semidirect
67. #1989D3semidirect
584. C3#26025semidirect

Maximal subgroups

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

subgroupindex
#830372
#736392
C3×#19892
#717432
#704842
#704512
#694612
#689012
#688992
#688922
#688472
#688202
#688012
#687952
#687802
#260253