Finite Group Databasebeta

← #10810639

Subgroups — #10810639

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5551C211yesmaximal, characteristic
2C3×#2354C211yesmaximal, characteristic, complemented
3C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#5002C212yesmaximal, complemented
5#71585C212yesmaximal, complemented
6#69584C211yesmaximal, characteristic, complemented
7#69579C211yesmaximal, characteristic, complemented
8#69522C212yesmaximal, complemented
9#69514C212yesmaximal, complemented
10#69486C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic, complemented
12#22417—31nomaximal, Sylow
13C22×#235C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×#117C2212yes—
16C2×#938—21no—
17C2×#924C2212yes—
18C23×C3⋊C8C2212yes—
19#5551—24no—
20#5551C418yescomplemented
21C2×D3×C4⋊C4—21no—
22C2×#883—22no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#390—21no—
26C3×#388—22no—
27C3×#384C2211yescharacteristic
28C3×#380C2212yes—
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#830C2212yes—
32D4×C3⋊C8—24no—
33#5002—28no—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45#4529C2212yes—
46C2×#547—31no—
47C2×C8×D4—31no—
48C2×#416—32no—
49#2354D311yescharacteristic, complemented
50#1981—31no—
51#1956—31no—
52#1892—32no—
53#1873—32no—
54#1802—31no—
55#1788—32no—
56C4×#102—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C2×#235—24no—
61C2×#235—24no—
62C2×#235—24no—
63C2×#235—42no—
64C2×#235—42no—
65C2×#235C2×C418yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—21no—
69C22×Dic6C2312yes—
70C3×C4×D4C2×C414yes—
71C3×C4×D4D414yes—
72C3×#117—24no—
73C3×#117—42no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4D412yes—
83C2×C3×C42—21no—
84C2×C3×C42—22no—
85C2×C3×C42C2311yescharacteristic
86C2×#217—22no—
87#938—28no—
88C2×(C2×D3)⋊C4—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92#924—24no—
93#924C2×C418yes—
94C2×Dic3⋊C4—22no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8—24no—
97C22×C3⋊C8—42no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101D3×C4⋊C4—28no—
102#883—216no—
103C4×D12—22no—
104C4×D12C2×C414yes—
105#869—22no—
106#869C2×C414yes—
107C3×#95—22no—
108C3×#95—21no—
109#830—24no—
110#810—24no—
111#810—24no—
112#810—42no—
113#810—42no—
114#809—24no—
115#809—22no—
116#809D414yescomplemented
117C4⋊(C3⋊C8)—22no—
118C4×C3⋊C8—22no—
119C22×#41—32no—
120C23×C8—32no—
121#547—38no—
122#547—64no—
123C2×C4⋊Q8—31no—
124C2×#123—62no—
125C2×C22⋊Q8—32no—
126C2×#121—61no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117C2×D312yes—
130C22×C4⋊C4—61no—
131C8×D4—64no—
132C2×C4⋊C8—31no—
133#416—68no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—31no—
141C2×C22⋊C8—32no—
142C2×C4×C8—31no—
143#390—21no—
144#388—22no—
145#384C2×D311yescharacteristic
146#380C2×D312yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2×D311yescharacteristic
149#340—32no—
150#340—31no—
151#340—31no—
152C24×C3C2×D412yes—
153C23×D3—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4C22⋊C414yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4C22×C414yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C2×#67—28no—
171C2×#67—44no—
172C2×#67—44no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175#235—216no—
176#235—48no—
177#235—48no—
178#235—48no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C22×C412yes—
183C2×D12C2×D412yes—
184C2×D12#4112yes—
185C2×C4×D3—24no—
186C2×C4×D3—28no—
187C2×C4×D3—44no—
188C2×C4×D3—44no—
189C2×C4×D3—42no—
190C2×C4×D3—42no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6C22×C414yes—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—81no—
204C3×C4⋊C4C22⋊C414yes—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—82no—
208C3×C22⋊C4C22⋊C418yes—
209C3×C42—44no—
210C3×C42—42no—
211C3×C42C22×C412yes—
212C3×C42C2×D412yes—
213#217—28no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—24no—
216C4⋊Dic3—24no—
217C4⋊Dic3—21no—
218C4⋊Dic3C22×C412yes—
219Dic3⋊C4—28no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—48no—
229C2×C3⋊C8C2×D412yes—
230C2×#41—38no—
231C2×#41—64no—
232C2×#41—64no—
233C2×#41—64no—
234C2×#41—122no—
235C2×#41—122no—
236C22×Q8—32no—
237C22×D4—31no—
238C22×D4—62no—
239C22×D4#6711yescharacteristic
240C23×C4—61no—
241C23×C4—62no—
242C23×C4C22×D312yes—
243C22×C8—31no—
244C22×C8—32no—
245C22×C8—31no—
246C22×C8—64no—
247C22×C8—64no—
248C22×C8—122no—
249C4⋊Q8—34no—
250C4⋊Q8—62no—
251#123—616no—
252C22⋊Q8—38no—
253C22⋊Q8—64no—
254#121—68no—
255C4×D4—34no—
256C4×D4—62no—
257C4×D4C4×D314yes—
258C4×D4D1214yes—
259#117—24no—
260#117—42no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4#6712yes—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42C22×D311yescharacteristic
280C4⋊C8—62no—
281#103—64no—
282#103—64no—
283#103—122no—
284#103—122no—
285#102—34no—
286#102—62no—
287#102—64no—
288C22⋊C8—64no—
289C4×C8—62no—
290#95—22no—
291#95—21no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3C2×C22⋊C414yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C3×D4—28no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4C2×C22⋊C414yes—
326C2×C3×C4C2×C22⋊C412yes—
327C2×C3×C4C2×C22⋊C412yes—
328C2×C3×C4#13512yes—
329C2×C3×C4#13512yes—
330C2×C3×C4C23×C411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336#67—416no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343D12—22no—
344D12—22no—
345D12—22no—
346D12C4×D414yes—
347C4×D3—416no—
348C4×D3—48no—
349Dic6—24no—
350Dic6—24no—
351Dic6—42no—
352Dic6—42no—
353C3⋊C8—22no—
354C3⋊C8—42no—
355C3⋊C8—41no—
356C24—61no—
357C24C2×#6712yes—
358#41—616no—
359#41—128no—
360#41—128no—
361#41—128no—
362C2×Q8—34no—
363C2×Q8—62no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—32no—
371C2×D4—32no—
372C2×D4—32no—
373C2×D4—68no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—124no—
378C2×D4—124no—
379C2×D4(C2×D3)⋊C414yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—64no—
390C22×C4—68no—
391C22×C4—124no—
392C22×C4—124no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4C2×C4×D314yes—
396C22×C4C2×D1214yes—
397C22×C4C2×#6712yes—
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400C22×C4C23×D311yescharacteristic
401C2×C8—32no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—62no—
410C2×C8—128no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—32no—
417C4⋊C4—42no—
418C4⋊C4—64no—
419C4⋊C4—68no—
420C4⋊C4—64no—
421C4⋊C4—61no—
422C4⋊C4—81no—
423C4⋊C4(C2×D3)⋊C414yes—
424C4⋊C4D3×D412yes—
425C22⋊C4—44no—
426C22⋊C4—68no—
427C22⋊C4—64no—
428C22⋊C4—82no—
429C22⋊C4(C2×D3)⋊C418yes—
430C42—44no—
431C42—42no—
432C42C2×C4×D312yes—
433C42C2×D1212yes—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3C22×C22⋊C411yescharacteristic
442C22×C3D4211yescharacteristic
443C22×C3#54611yescharacteristic
444C22×C3C2×#13511yescharacteristic
445C22×C3C2×#13511yescharacteristic
446C22×C3#57711yescharacteristic
447C22×C3#57811yescharacteristic
448C2×D3—24no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—82no—
459C3×C4—81no—
460C3×C4#41614yescomplemented
461C3×C4C2×C4×D411yescharacteristic
462C3×C4#51811yescharacteristic
463C3×C4D4211yescharacteristic
464C3×C4#54611yescharacteristic
465Dic3—44no—
466Dic3—42no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23C2×(C2×D3)⋊C414yes—
481C23C22×#6711yescharacteristic
482Q8—64no—
483Q8—64no—
484Q8—122no—
485Q8—122no—
486D4—28no—
487D4—34no—
488D4—62no—
489D4—62no—
490D4—62no—
491D4—1216no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—1216no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—128no—
517C2×C4C2×(C2×D3)⋊C412yes—
518C2×C4C2×(C2×D3)⋊C414yes—
519C2×C4C2×(C2×D3)⋊C412yes—
520C2×C4#95012yes—
521C2×C4#95012yes—
522C2×C4C22×C4×D311yescharacteristic
523C2×C4C22×D1211yescharacteristic
524C2×C4C2×D3×D411yescharacteristic
525C2×C4#100411yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4#100911yescharacteristic
528C8—62no—
529C8—122no—
530C8—121no—
531C2×C3—44no—
532C2×C3D4×C22⋊C411yescharacteristic
533C2×C3C2×#41611yescharacteristic
534C2×C3#289811yescharacteristic
535C2×C3#328311yescharacteristic
536C2×C3#328411yescharacteristic
537C2×C3#330211yescharacteristic
538C2×C3#330311yescharacteristic
539D3—44no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—64no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—122no—
554C22C22×(C2×D3)⋊C411yescharacteristic
555C22D4×#6711yescharacteristic
556C22C2×#95011yescharacteristic
557C22C2×#95011yescharacteristic
558C22#582911yescharacteristic
559C22#583611yescharacteristic
560C22#583811yescharacteristic
561C4—22no—
562C4—24no—
563C4—42no—
564C4—82no—
565C4—81no—
566C4—122no—
567C4—124no—
568C4#500214yescomplemented
569C4C4×D3×D411yescharacteristic
570C4#554511yescharacteristic
571C4D4×D1211yescharacteristic
572C4#555411yescharacteristic
573C3#2241711yescharacteristic, complemented, Sylow
574C2—44no—
575C2—124no—
576C2C2×#500211yescharacteristic
577C2D4×(C2×D3)⋊C411yescharacteristic
578C2#8257011yescharacteristic
579C2#8352411yescharacteristic
580C2#8357911yescharacteristic
581C2#8365211yescharacteristic
582C2#8366411yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC2×#5551C2
cosocleC22×C3×C4C24
hypercenter#2354D3
hypocenterC3#22417
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. C3×#2354C2semidirect
3. C2×D4×C3⋊C8C2semidirect
4. C2×#5002C2semidirect
5. #71585C2semidirect
6. #69584C2semidirect
7. #69579C2semidirect
8. #69522C2semidirect
9. #69514C2semidirect
10. #69486C2semidirect
11. C4×#809C2semidirect
20. #5551C4semidirect
49. #2354D3semidirect
116. #809D4semidirect
460. C3×C4#416semidirect
568. C4#5002semidirect
573. C3#22417semidirect

Maximal subgroups

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

subgroupindex
C2×#55512
C3×#23542
C2×D4×C3⋊C82
C2×#50022
#715852
#695842
#695792
#695222
#695142
#694862
C4×#8092
#224173