Finite Group Databasebeta

← #10827630

Subgroups — #10827630

Order 768.

2810 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 2810 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3490C211yesmaximal, characteristic, complemented
2#82612C212yesmaximal, complemented
3C2×#4955C211yesmaximal, characteristic, complemented
4#71897C212yesmaximal, complemented
5#71851C212yesmaximal, complemented
6#71670C212yesmaximal, complemented
7#71654C214yesmaximal, complemented
8#71542C211yesmaximal, characteristic, complemented
9#23984—31nomaximal, Sylow
10C25×D3C2211yescharacteristic
11C3×#551—24no—
12C3×#546—24no—
13C3×#543C2211yescharacteristic, complemented
14C3×#538—22no—
15C3×#534C2211yescharacteristic, complemented
16C3×#525—22no—
17C2×C3×#120C2212yes—
18C3×#518C2212yescomplemented
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C2×#937—24no—
22C2×#937C2214yes—
23C22×(C2×D3)⋊C4—22no—
24#5247C2212yes—
25C2×#834C2214yescomplemented
26#4955—24no—
27#4955—22no—
28#4955C2212yes—
29#4955C2214yes—
30#4955C2212yes—
31#4526C2212yescomplemented
32#4526C2212yescomplemented
33#4526C2214yescomplemented
34#3490D311yescharacteristic, complemented
35#2846—32no—
36C2×#379—31no—
37#2029—32no—
38#2021—32no—
39#1898—32no—
40#1889—34no—
41#1819—31no—
42C25×C3C2311yescharacteristic
43C24×D3—21no—
44C24×D3—21no—
45C24×D3—24no—
46C24×D3—24no—
47C24×D3—48no—
48C24×D3—42no—
49C24×D3C2312yes—
50C2×C3×#41—22no—
51C22×C3×D4D414yescomplemented
52C22×C3×D4C2312yes—
53C23×C3×C4D412yescomplemented
54C22×#67—24no—
55C23×Dic3—21no—
56C3×#126—24no—
57C3×#124—24no—
58C3×#123—24no—
59C3×#123—24no—
60C3×#123—22no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—24no—
64C3×#121—22no—
65C3×#121—24no—
66C3×#121—22no—
67C3×#120—22no—
68C3×#120—22no—
69C3×#120—24no—
70C3×#120—22no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×#117—22no—
75C3×#117—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D412yescomplemented
79C2×C3×C22⋊C4D414yescomplemented
80C2×C3×C22⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4C2312yes—
83C2×#217—21no—
84C2×#217—24no—
85C2×#217—42no—
86C2×#217C2312yes—
87C2×#217C2314yes—
88#937—28no—
89#937—44no—
90#937—416no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—28no—
95C2×(C2×D3)⋊C4—48no—
96#834—44no—
97C26—31no—
98#551—24no—
99#546—24no—
100#543C2×D311yescharacteristic, complemented
101#538—22no—
102#534C2×D311yescharacteristic, complemented
103#525—22no—
104C2×#120—34no—
105C2×#120—64no—
106C2×#120C2×D312yes—
107#518C2×D312yescomplemented
108C22×C22⋊C4—31no—
109C22×C22⋊C4—62no—
110C22×C22⋊C4C2×D311yescharacteristic
111C2×#99—34no—
112#379—32no—
113#379—34no—
114#379—32no—
115#379—32no—
116#379—64no—
117#379—62no—
118#342—32no—
119#342—32no—
120#342—34no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—44no—
124C24×C3C2×D411yescharacteristic
125C24×C3C2×D414yes—
126C24×C3C2×D412yes—
127C24×C3C2×D411yescharacteristic
128C24×C3C2411yescharacteristic
129C23×D3—28no—
130C23×D3—24no—
131C23×D3—22no—
132C23×D3—24no—
133C23×D3—216no—
134C23×D3—24no—
135C23×D3—22no—
136C23×D3—24no—
137C23×D3—48no—
138C23×D3—416no—
139C23×D3—48no—
140C23×D3—416no—
141C23×D3—432no—
142C23×D3—42no—
143C23×D3—44no—
144C23×D3—42no—
145C23×D3—48no—
146C23×D3—432no—
147C23×D3—42no—
148C23×D3—42no—
149C23×D3#4114yescomplemented
150C3×#41—84no—
151C2×C3×Q8—22no—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—48no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—44no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D412yes—
168C2×#67—48no—
169C2×#67—416no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—44no—
173C22×Dic3—42no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C42—24no—
188#217—24no—
189#217—24no—
190#217—24no—
191#217—48no—
192#217—42no—
193(C2×D3)⋊C4—48no—
194(C2×D3)⋊C4—48no—
195(C2×D3)⋊C4—48no—
196(C2×D3)⋊C4—416no—
197C25—32no—
198C25—64no—
199C25—61no—
200C25—64no—
201C25—61no—
202C25—122no—
203C25—128no—
204C25C22×D311yescharacteristic
205C2×#41—22no—
206C22×D4—64no—
207C22×D4#6714yescomplemented
208C22×D4C22×D312yes—
209C23×C4—61no—
210C23×C4#6712yescomplemented
211#126—24no—
212#124—24no—
213#123—24no—
214#123—24no—
215#123—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—24no—
219#121—22no—
220#121—24no—
221#121—22no—
222#120—22no—
223#120—22no—
224#120—24no—
225#120—22no—
226#120—68no—
227#120—124no—
228#120—1216no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232#117—22no—
233#117—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—32no—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4—68no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—128no—
246C2×C22⋊C4#6712yescomplemented
247C2×C22⋊C4#6714yescomplemented
248C2×C22⋊C4C22×D311yescharacteristic
249C2×C22⋊C4C22×D311yescharacteristic
250C2×C22⋊C4C22×D312yes—
251#99—124no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—24no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—28no—
258C23×C3—28no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—42no—
262C23×C3—44no—
263C23×C3—42no—
264C23×C3—48no—
265C23×C3—48no—
266C23×C3—44no—
267C23×C3#12012yes—
268C23×C3#12012yes—
269C23×C3C22×D411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C2×#4112yes—
272C23×C3H(5,𝔽2)12yes—
273C22×D3—28no—
274C22×D3—216no—
275C22×D3—216no—
276C22×D3—24no—
277C22×D3—28no—
278C22×D3—48no—
279C22×D3—416no—
280C22×D3—464no—
281C22×D3—48no—
282C22×D3—416no—
283C22×D3—416no—
284C22×D3—48no—
285C22×D3—48no—
286C22×D3—44no—
287C22×D3—48no—
288C22×D3—48no—
289C22×D3—48no—
290C22×D3—416no—
291C22×D3—464no—
292C22×D3—416no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—44no—
296C22×D3—48no—
297C3×Q8—82no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—44no—
301C3×D4—48no—
302C3×D4—84no—
303C3×D4—82no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—44no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—44no—
313C2×C3×C4—82no—
314C2×C3×C4—84no—
315#67—816no—
316C2×Dic3—44no—
317C2×Dic3—48no—
318C2×Dic3—42no—
319C2×Dic3—44no—
320C2×Dic3—84no—
321C24—22no—
322C24—22no—
323C24—34no—
324C24—44no—
325C24—68no—
326C24—64no—
327C24—616no—
328C24—64no—
329C24—62no—
330C24—64no—
331C24—62no—
332C24—64no—
333C24—1216no—
334C24—124no—
335C24—122no—
336C24—128no—
337C24—122no—
338C24—128no—
339C24—128no—
340C24—1216no—
341C24—1232no—
342C24—1232no—
343C24—122no—
344C24—122no—
345C24C2×#6711yescharacteristic
346C24C2×#6714yes—
347C24C2×#6712yes—
348C24C2×#6711yescharacteristic
349C24C23×D311yescharacteristic
350#41—84no—
351C2×Q8—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—48no—
360C2×D4—128no—
361C2×D4—1216no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—44no—
366C22×C4—62no—
367C22×C4—64no—
368C22×C4—124no—
369C22×C4—122no—
370C22×C4C2×#6712yes—
371C22×C4C2×#6712yes—
372C22×C4C2×#6714yes—
373C22×C4C2×#6712yes—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—64no—
388C22⋊C4—64no—
389C22⋊C4—64no—
390C22⋊C4—128no—
391C22⋊C4—128no—
392C22⋊C4—128no—
393C22⋊C4—1216no—
394C22⋊C4—122no—
395C22⋊C4—128no—
396C42—24no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—24no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—416no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—84no—
413C22×C3B(4,𝔽2)14yes—
414C22×C3C2×#12011yescharacteristic
415C22×C3#52511yescharacteristic
416C22×C3#53511yescharacteristic
417C22×C3#54614yes—
418C2×D3—216no—
419C2×D3—28no—
420C2×D3—48no—
421C2×D3—416no—
422C2×D3—48no—
423C2×D3—44no—
424C2×D3—432no—
425C2×D3—432no—
426C2×D3—44no—
427C2×D3—48no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—44no—
432Dic3—82no—
433Dic3—84no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—28no—
438C23—21no—
439C23—28no—
440C23—21no—
441C23—22no—
442C23—24no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—48no—
447C23—48no—
448C23—44no—
449C23—68no—
450C23—616no—
451C23—616no—
452C23—64no—
453C23—68no—
454C23—1216no—
455C23—128no—
456C23—128no—
457C23—124no—
458C23—128no—
459C23—128no—
460C23—128no—
461C23—1216no—
462C23—128no—
463C23—128no—
464C23—1216no—
465C23—1264no—
466C23—1216no—
467C23—1264no—
468C23—1216no—
469C23—128no—
470C23—124no—
471C23—122no—
472C23—122no—
473C23#95312yes—
474C23#95312yes—
475C23D3×#4112yes—
476C23#100912yes—
477C23C22×#6711yescharacteristic
478C23C22×#6712yes—
479Q8—82no—
480D4—48no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—84no—
485D4—82no—
486D4—2416no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4—124no—
499C2×C4—128no—
500C2×C4—124no—
501C2×C4—122no—
502C2×C4—244no—
503C2×C3—21no—
504C2×C3—24no—
505C2×C3—21no—
506C2×C3—24no—
507C2×C3—42no—
508C2×C3—82no—
509C2×C3#240411yescharacteristic
510C2×C3C2×B(4,𝔽2)12yes—
511D3—48no—
512C22—24no—
513C22—24no—
514C22—22no—
515C22—22no—
516C22—24no—
517C22—22no—
518C22—22no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—416no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—616no—
528C22—68no—
529C22—84no—
530C22—128no—
531C22—1216no—
532C22—128no—
533C22—124no—
534C22—1232no—
535C22—1232no—
536C22—124no—
537C22—128no—
538C22#524414yes—
539C22#582811yescharacteristic
540C22#582914yes—
541C22#583011yescharacteristic
542C22C2×#95311yescharacteristic
543C4—42no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—242no—
548C4—244no—
549C3#2398411yescharacteristic, complemented, Sylow
550C2—21no—
551C2—24no—
552C2—21no—
553C2—24no—
554C2—42no—
555C2—82no—
556C2—128no—
557C2#8409611yescharacteristic
558C2C2×#524412yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3490C2
socleC22×C3C2×#120
hypercenter#3490D3
hypocenterC3#23984
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #3490

Complements

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

NH ≅ G/Ntype
1. C3×#3490C2semidirect
2. #82612C2semidirect
3. C2×#4955C2semidirect
4. #71897C2semidirect
5. #71851C2semidirect
6. #71670C2semidirect
7. #71654C2semidirect
8. #71542C2semidirect
13. C3×#543C22semidirect
15. C3×#534C22semidirect
18. C3×#518C22semidirect
25. C2×#834C22semidirect
31. #4526C22semidirect
32. #4526C22semidirect
33. #4526C22semidirect
34. #3490D3semidirect
51. C22×C3×D4D4semidirect
53. C23×C3×C4D4semidirect
78. C2×C3×C22⋊C4D4semidirect
79. C2×C3×C22⋊C4D4semidirect
100. #543C2×D3semidirect
102. #534C2×D3semidirect
107. #518C2×D3semidirect
149. C23×D3#41semidirect
207. C22×D4#67semidirect
210. C23×C4#67semidirect
246. C2×C22⋊C4#67semidirect
247. C2×C22⋊C4#67semidirect
549. C3#23984semidirect

Maximal subgroups

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

subgroupindex
C3×#34902
#826122
C2×#49552
#718972
#718512
#716702
#716542
#715422
#239843