Finite Group Databasebeta

← #11069284

Subgroups — #11069284

Order 768.

2890 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic, complemented
2C2×#5675C212yesmaximal, complemented
3C2×#5555C211yesmaximal, characteristic
4C3×#2547C211yesmaximal, characteristic, complemented
5#83253C211yesmaximal, characteristic, complemented
6#82959C212yesmaximal, complemented
7C4×#936C211yesmaximal, characteristic, complemented
8#77568C218yesmaximal, complemented
9#76641C212yesmaximal, complemented
10#76499C212yesmaximal, complemented
11#73899C214yesmaximal, complemented
12#73893C214yesmaximal, complemented
13#73711C211yesmaximal, characteristic, complemented
14#73617C211yesmaximal, characteristic, complemented
15#39360—31nomaximal, Sylow
16C22×D3×Q8C2212yes—
17C22×#237C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#946C2212yes—
22C2×#944C2212yes—
23C2×#944C2212yes—
24C2×#936C2211yescharacteristic
25C2×#930C2212yes—
26C22×C4⋊Dic3C2212yes—
27#5727—28no—
28#5675—216no—
29#5555—28no—
30C2×#897C2212yes—
31C2×#896C2214yes—
32C2×#893C2212yes—
33C2×#888C2212yes—
34C2×#886C2214yes—
35C2×#883C2218yes—
36C2×#878C2218yes—
37C2×C4×D12C2211yescharacteristic
38C2×#872C2212yes—
39C2×#869C2211yescharacteristic
40C2×#869C2212yes—
41C2×C4×Dic6C2211yescharacteristic
42C3×#397C2214yes—
43C3×#391C2212yes—
44C3×#380C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5221C2211yescharacteristic
47C4×#217C2214yes—
48#4988C2212yes—
49#4984C2214yes—
50#4977C2211yescharacteristic
51#4975C2214yes—
52#4975C2214yes—
53#4973C2211yescharacteristic
54#4972C2212yes—
55#4959C2218yes—
56#4946C2218yes—
57#4945C2214yes—
58C4×(C2×D3)⋊C4C2212yes—
59C42×Dic3C2211yescharacteristic
60#4929C2212yes—
61#4675C2218yes—
62#4669C2214yes—
63#4668C2218yes—
64#4661C2214yes—
65#4657C2218yes—
66#4656C2218yes—
67C2×D4×Q8—31no—
68C2×#547—31no—
69C2×#536—32no—
70#2868—31no—
71#2680—38no—
72#2621—32no—
73#2547D311yescharacteristic, complemented
74#2546—31no—
75#2440—34no—
76#2434—32no—
77#2432—32no—
78#2422—34no—
79C4×#124—31no—
80#2293—31no—
81C22×C3×Q8C2312yes—
82C22×C3×D4C2311yescharacteristic
83C23×C3×C4C2312yes—
84C22×#67C2314yes—
85C23×Dic3C2314yes—
86C2×D3×Q8—28no—
87C2×D3×Q8—28no—
88C2×D3×Q8—44no—
89C2×#237—28no—
90C2×#237—28no—
91C2×#237—44no—
92C22×D12C2311yescharacteristic
93C22×C4×D3C2314yes—
94C22×C4×D3C2312yes—
95C22×Dic6C2312yes—
96C22×Dic6C2314yes—
97C22×Dic6C2311yescharacteristic
98C3×C4⋊Q8—24no—
99C3×#124—28no—
100C3×C4×D4—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2318yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C2×#217C2314yes—
110C2×#217C2314yes—
111#946—28no—
112#944—28no—
113#944—44no—
114#936D418yescomplemented
115#930—28no—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×C4⋊Dic3—28no—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2318yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2318yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2318yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2311yescharacteristic
138#897—28no—
139#896—216no—
140#893—28no—
141#888—28no—
142#886—216no—
143#883—232no—
144#878—232no—
145C4×D12D418yes—
146#872—28no—
147#869—28no—
148#869—24no—
149C4×Dic6—24no—
150C3×#95C2318yes—
151C3×#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2318yes—
156C22×#41—32no—
157C23×Q8—32no—
158D4×Q8—68no—
159#547—68no—
160#536—616no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8—32no—
166C2×C4⋊Q8C2×D311yescharacteristic
167C2×#124—31no—
168C2×#124C2×D312yes—
169C2×#123—38no—
170C2×#123—34no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—34no—
174C2×C22⋊Q8—38no—
175C2×#121—32no—
176C2×C4×Q8—31no—
177C2×C4×D4—32no—
178C2×C4×D4—31no—
179C2×C4×D4C2×D311yescharacteristic
180C2×#117—32no—
181C22×C4⋊C4—32no—
182#398—38no—
183#398—34no—
184#397—38no—
185#397C2×D314yes—
186#395—38no—
187#394—38no—
188#391—31no—
189#391—32no—
190#391C2×D312yes—
191#390—31no—
192#390—32no—
193#389—32no—
194#389—34no—
195#388—34no—
196#388—34no—
197#385—38no—
198#385—38no—
199#384—34no—
200#384—34no—
201#382—31no—
202#380C2×D314yes—
203C4×C4⋊C4C2×D311yescharacteristic
204C4×C22⋊C4—34no—
205C4×C22⋊C4—32no—
206C43—31no—
207C24×C3C2412yes—
208C23×D3C2412yes—
209C2×C3×Q8—24no—
210C2×C3×Q8—24no—
211C2×C3×Q8—24no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4C2×D414yes—
215C22×C3×C4—24no—
216C22×C3×C4—28no—
217C22×C3×C4C2412yes—
218C22×C3×C4C2414yes—
219C22×C3×C4C2414yes—
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C2×#67—28no—
224C2×#67—28no—
225C2×#67—44no—
226C22×Dic3—28no—
227C22×Dic3—28no—
228C22×Dic3—44no—
229C22×Dic3C2412yes—
230C22×Dic3C2414yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235D3×Q8—432no—
236#237—432no—
237C2×D12—24no—
238C2×D12C2×D414yes—
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—28no—
242C2×C4×D3—28no—
243C2×C4×D3—44no—
244C2×C4×D3C2×D418yes—
245C2×Dic6—22no—
246C2×Dic6—22no—
247C2×Dic6—24no—
248C2×Dic6—24no—
249C2×Dic6—24no—
250C2×Dic6—28no—
251C2×Dic6—28no—
252C2×Dic6—44no—
253C2×Dic6C2×D414yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—24no—
258C3×C22⋊C4—216no—
259C3×C22⋊C4—24no—
260C3×C42—24no—
261C3×C42C2×D414yes—
262#217—28no—
263#217C2×D4116yes—
264(C2×D3)⋊C4—28no—
265(C2×D3)⋊C4—28no—
266(C2×D3)⋊C4—28no—
267(C2×D3)⋊C4—28no—
268(C2×D3)⋊C4C2×D418yes—
269C4⋊Dic3—22no—
270C4⋊Dic3—24no—
271C4⋊Dic3—24no—
272C4⋊Dic3—216no—
273C4⋊Dic3—28no—
274C4⋊Dic3C2×D414yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—28no—
277Dic3⋊C4—216no—
278Dic3⋊C4—28no—
279Dic3⋊C4—48no—
280C4×Dic3—24no—
281C4×Dic3—22no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3C2×D414yes—
285C2×#41—68no—
286C2×#41—68no—
287C2×#41—124no—
288C22×Q8—31no—
289C22×Q8—32no—
290C22×Q8—34no—
291C22×Q8—68no—
292C22×Q8—68no—
293C22×Q8—124no—
294C22×Q8C22×D312yes—
295C22×D4—34no—
296C22×D4—31no—
297C22×D4C22×D311yescharacteristic
298C23×C4—32no—
299C23×C4—34no—
300C23×C4—34no—
301C23×C4C22×D312yes—
302C4⋊Q8—24no—
303C4⋊Q8—68no—
304C4⋊Q8—68no—
305C4⋊Q8—68no—
306C4⋊Q8—64no—
307C4⋊Q8—124no—
308#124—28no—
309#124—38no—
310#123—632no—
311#123—616no—
312C22⋊Q8—632no—
313C22⋊Q8—616no—
314C22⋊Q8—68no—
315C22⋊Q8—68no—
316#121—68no—
317C4×Q8—64no—
318C4×D4—24no—
319C4×D4—38no—
320C4×D4—68no—
321#117—68no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—38no—
330C2×C4⋊C4—38no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—38no—
333C2×C4⋊C4—68no—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D312yes—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4C22×D312yes—
346C2×C22⋊C4C22×D318yes—
347C2×C42—31no—
348C2×C42—32no—
349C2×C42—31no—
350C2×C42—32no—
351C2×C42—32no—
352C2×C42C22×D312yes—
353C2×C42C22×D311yescharacteristic
354#95—34no—
355#95—34no—
356#95—34no—
357#95—38no—
358#95C22×D312yes—
359#95C22×D318yes—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—24no—
363C23×C3—24no—
364C23×C3C22×D414yes—
365C23×C3C2511yescharacteristic
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—24no—
369C22×D3—24no—
370C22×D3C22×D414yes—
371C3×Q8—48no—
372C3×D4—28no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—22no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—48no—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D412yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4H(5,𝔽2)12yes—
388C2×C3×C42−1+414yes—
389#67—416no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—416no—
404C2×Dic3C22×D414yes—
405C2×Dic3C22×D414yes—
406C2×Dic3C22×D414yes—
407C2×Dic32−1+414yes—
408D12—28no—
409C4×D3—216no—
410C4×D3—416no—
411Dic6—28no—
412Dic6—416no—
413Dic6—48no—
414C24—32no—
415C24C23×D312yes—
416#41—1232no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—34no—
421C2×Q8—68no—
422C2×Q8—68no—
423C2×Q8—62no—
424C2×Q8—64no—
425C2×Q8—64no—
426C2×Q8—62no—
427C2×Q8—64no—
428C2×Q8—124no—
429C2×Q8—1232no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—34no—
433C2×D4—68no—
434C2×D4—68no—
435C2×D4—64no—
436C2×D4—124no—
437C2×D4D3×D414yes—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—38no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—68no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—68no—
453C22×C4—124no—
454C22×C4—124no—
455C22×C4C23×D312yes—
456C22×C4C23×D314yes—
457C22×C4C23×D314yes—
458C22×C4C23×D311yescharacteristic
459C22×C4C23×D311yescharacteristic
460C22×C4C23×D311yescharacteristic
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—34no—
466C4⋊C4—68no—
467C4⋊C4—68no—
468C4⋊C4—68no—
469C4⋊C4—616no—
470C4⋊C4—616no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—62no—
474C4⋊C4—64no—
475C4⋊C4—128no—
476C22⋊C4—216no—
477C22⋊C4—24no—
478C22⋊C4—316no—
479C22⋊C4—38no—
480C22⋊C4—68no—
481C22⋊C4—68no—
482C22⋊C4—68no—
483C22⋊C4—68no—
484C22⋊C4—68no—
485C42—24no—
486C42—34no—
487C42—64no—
488C42—64no—
489C42—62no—
490C42—64no—
491C42D3×D414yes—
492C22×C3—24no—
493C22×C3—28no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—44no—
497C22×C3C23×D411yescharacteristic
498C22×C3C23×D411yescharacteristic
499C22×C3C2×H(5,𝔽2)11yescharacteristic
500C22×C3C2×2−1+412yes—
501C22×C3C2×2−1+412yes—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3—44no—
506C2×D3—44no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—44no—
510C3×C4—44no—
511C3×C4#53614yescomplemented
512C3×C4D4214yes—
513Dic3—24no—
514Dic3—24no—
515Dic3—24no—
516Dic3—24no—
517Dic3—44no—
518Dic3—42no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—62no—
525C23—62no—
526C23—64no—
527C23—64no—
528C23C2×D3×D414yes—
529C23C24×D311yescharacteristic
530Q8—48no—
531Q8—68no—
532Q8—1216no—
533Q8—128no—
534D4—28no—
535D4—68no—
536D4—1216no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—34no—
547C2×C4—34no—
548C2×C4—34no—
549C2×C4—34no—
550C2×C4—48no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—616no—
560C2×C4—62no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—1216no—
566C2×C4—1216no—
567C2×C4C2×D3×D412yes—
568C2×C4C2×D3×D412yes—
569C2×C4C2×D3×D412yes—
570C2×C4C2×D3×D412yes—
571C2×C4#100912yes—
572C2×C4#101014yes—
573C2×C3—44no—
574C2×C3C2×#53611yescharacteristic
575C2×C3C2×D4211yescharacteristic
576C2×C3#349911yescharacteristic
577C2×C3#349912yes—
578C2×C3#355212yes—
579D3—44no—
580C22—28no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—64no—
586C22—64no—
587C22—64no—
588C22—124no—
589C22—124no—
590C22C22×D3×D411yescharacteristic
591C22C22×D3×D411yescharacteristic
592C22C2×#100911yescharacteristic
593C22C2×#101012yes—
594C22D3×2−1+412yes—
595C4—22no—
596C4—42no—
597C4—44no—
598C4—44no—
599C4—64no—
600C4—64no—
601C4—64no—
602C4—64no—
603C4—122no—
604C4—124no—
605C4#560914yes—
606C4#567514yescomplemented
607C3#3936011yescharacteristic, complemented, Sylow
608C2—44no—
609C2—124no—
610C2C2×#560911yescharacteristic
611C2C2×#567511yescharacteristic
612C2#8634011yescharacteristic
613C2#8667912yes—
614C2#8701012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2547C2
prime radicalC2×#5555C2
hypercenter#2547D3
hypocenterC3#39360
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#5727C2semidirect
2. C2×#5675C2semidirect
4. C3×#2547C2semidirect
5. #83253C2semidirect
6. #82959C2semidirect
7. C4×#936C2semidirect
8. #77568C2semidirect
9. #76641C2semidirect
10. #76499C2semidirect
11. #73899C2semidirect
12. #73893C2semidirect
13. #73711C2semidirect
14. #73617C2semidirect
73. #2547D3semidirect
114. #936D4semidirect
511. C3×C4#536semidirect
606. C4#5675semidirect
607. C3#39360semidirect

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#56752
C2×#55552
C3×#25472
#832532
#829592
C4×#9362
#775682
#766412
#764992
#738992
#738932
#737112
#736172
#393603