Finite Group Databasebeta

← #10794315

Subgroups — #10794315

Order 768.

4006 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#237C211yesmaximal, characteristic
2C2×#4740C214yesmaximal, complemented
3#73817C212yesmaximal, complemented
4C3×#1781C211yesmaximal, characteristic, complemented
5#72130C211yesmaximal, characteristic, complemented
6#70027C214yesmaximal, complemented
7#69498C212yesmaximal, complemented
8#20584—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67C411yescharacteristic, complemented
11C23×#67C2211yescharacteristic
12C24×Dic3C411yescharacteristic, complemented
13C24×Dic3C2211yescharacteristic
14C22×#237—24no—
15C22×#237—24no—
16C22×#237—216no—
17C22×#237C412yescomplemented
18C22×#237C412yescomplemented
19C22×#237C2212yes—
20C22×#237C2212yes—
21C23×C4×D3C411yescharacteristic, complemented
22C23×Dic6C411yescharacteristic, complemented
23C2×#877C2214yescomplemented
24C2×C3×#99C2214yescomplemented
25C3×#379C2212yes—
26#5247C2212yes—
27C2×#834C2214yescomplemented
28C2×#806C2218yescomplemented
29#4943C2214yescomplemented
30#4942—24no—
31Dic3×C22⋊C4—24no—
32#4740—216no—
33C23×#41—31no—
34C2×#410—34no—
35#2280—32no—
36#1782—34no—
37#1781—32no—
38#1781—31no—
39#1781D311yescharacteristic, complemented
40C25×C3C2×C411yescharacteristic
41C25×C3C2311yescharacteristic
42C24×D3C2×C411yescharacteristic
43C22×C3×D4—24no—
44C22×C3×D4—28no—
45C22×C3×D4C2×C412yes—
46C22×C3×D4C2312yes—
47C23×C3×C4C2×C411yescharacteristic
48C22×#67—24no—
49C22×#67—28no—
50C22×#67—28no—
51C22×#67—24no—
52C22×#67C2×C412yes—
53C22×#67C2×C412yes—
54C22×#67C2×C412yes—
55C22×#67C2312yes—
56C23×Dic3—22no—
57C23×Dic3—22no—
58C23×Dic3—28no—
59C23×Dic3—28no—
60C23×Dic3C2×C411yescharacteristic
61C23×Dic3C2×C412yes—
62C23×Dic3C2×C411yescharacteristic
63C23×Dic3C2×C411yescharacteristic
64C23×Dic3C2×C411yescharacteristic
65C23×Dic3C2×C411yescharacteristic
66C23×Dic3C2×C412yes—
67C23×Dic3D414yes—
68C23×Dic3D414yes—
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C2×#237—264no—
72C2×#237—264no—
73C2×#237—28no—
74C2×#237—28no—
75C2×#237—28no—
76C2×#237—24no—
77C2×#237—24no—
78C2×#237—28no—
79C2×#237—28no—
80C2×#237—28no—
81C2×#237—28no—
82C2×#237—24no—
83C2×#237—24no—
84C2×#237—28no—
85C2×#237—28no—
86C2×#237D4116yes—
87C22×C4×D3—22no—
88C22×C4×D3—22no—
89C22×C4×D3—28no—
90C22×C4×D3C2×C412yes—
91C22×C4×D3C2×C412yes—
92C22×Dic6—24no—
93C22×Dic6—28no—
94C22×Dic6C2×C412yes—
95C22×Dic6C2×C412yes—
96C2×C3×C22⋊C4—24no—
97C2×C3×C22⋊C4C2314yes—
98C2×#217—24no—
99C2×#217C2314yes—
100C2×Dic3⋊C4—24no—
101C2×C4×Dic3—24no—
102#877—216no—
103C3×#99—28no—
104#834—28no—
105C3⋊#95—28no—
106#806—216no—
107C22×#41—32no—
108C22×#41—32no—
109C22×#41—32no—
110C22×#41—32no—
111C22×#41—64no—
112C22×#41—64no—
113C22×#41—616no—
114C23×Q8—31no—
115C23×D4—31no—
116C23×D4—31no—
117C23×D4C2×D311yescharacteristic
118C24×C4—31no—
119C24×C4—31no—
120C24×C4—31no—
121C2×#117—34no—
122#410—616no—
123C2×#99—34no—
124C2×#99—38no—
125C2×#99C2×D314yescomplemented
126#381—34no—
127#380—64no—
128#379—32no—
129#379C2×D312yes—
130C4×C22⋊C4—64no—
131C24×C3—24no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—44no—
135C24×C3—44no—
136C24×C3C22×C411yescharacteristic
137C24×C3C22×C412yes—
138C24×C3C22×C411yescharacteristic
139C24×C3C2×D414yes—
140C24×C3C2411yescharacteristic
141C23×D3—22no—
142C23×D3—24no—
143C23×D3—44no—
144C23×D3C22×C412yes—
145C2×C3×D4—232no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4C2×D418yes—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4C22×C412yes—
156C2×#67—232no—
157C2×#67—232no—
158C2×#67—24no—
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—24no—
162C2×#67—24no—
163C2×#67—24no—
164C2×#67—24no—
165C2×#67—24no—
166C2×#67—24no—
167C2×#67—24no—
168C2×#67C22⋊C418yescomplemented
169C2×#67C2×D418yes—
170C22×Dic3—216no—
171C22×Dic3—216no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—216no—
186C22×Dic3—216no—
187C22×Dic3—24no—
188C22×Dic3C22⋊C414yes—
189C22×Dic3C22⋊C412yes—
190C22×Dic3C22⋊C412yes—
191C22×Dic3C22⋊C412yes—
192C22×Dic3C22⋊C412yes—
193C22×Dic3C22⋊C414yes—
194C22×Dic3C22⋊C418yescomplemented
195C22×Dic3C22×C412yes—
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D418yes—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×D412yes—
204#237—264no—
205#237—232no—
206#237—232no—
207#237—264no—
208#237—232no—
209#237—232no—
210C2×C4×D3—216no—
211C2×C4×D3—216no—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—24no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—24no—
218C2×C4×D3C22⋊C418yescomplemented
219C2×Dic6—232no—
220C2×Dic6—24no—
221C2×Dic6—24no—
222C2×Dic6—24no—
223C2×Dic6—24no—
224C2×Dic6—24no—
225C2×Dic6C22⋊C418yescomplemented
226C3×C22⋊C4—28no—
227C3×C22⋊C4—48no—
228#217—28no—
229#217—48no—
230Dic3⋊C4—48no—
231C4×Dic3—48no—
232C25—31no—
233C25C4×D311yescharacteristic
234C25C22×D311yescharacteristic
235C2×#41—316no—
236C2×#41—64no—
237C2×#41—64no—
238C2×#41—68no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—68no—
242C2×#41—68no—
243C2×#41—68no—
244C2×#41—68no—
245C2×#41—68no—
246C2×#41—68no—
247C2×#41—68no—
248C2×#41—664no—
249C2×#41—664no—
250C2×#41—68no—
251C22×Q8—32no—
252C22×Q8—32no—
253C22×Q8—64no—
254C22×Q8—68no—
255C22×D4—24no—
256C22×D4—28no—
257C22×D4—32no—
258C22×D4—32no—
259C22×D4—32no—
260C22×D4—32no—
261C22×D4—64no—
262C22×D4—68no—
263C22×D4—68no—
264C22×D4—64no—
265C22×D4C4×D312yes—
266C22×D4C22×D312yes—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—32no—
272C23×C4—31no—
273C23×C4—34no—
274C23×C4—34no—
275C23×C4—31no—
276C23×C4—32no—
277C23×C4—31no—
278C23×C4—32no—
279C23×C4—32no—
280C23×C4—62no—
281C23×C4—62no—
282C23×C4—68no—
283C23×C4—68no—
284C23×C4—62no—
285C23×C4—62no—
286C23×C4—68no—
287C23×C4C4×D311yescharacteristic
288#117—616no—
289C2×C4⋊C4—64no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—34no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4C22×D314yes—
294C2×C42—64no—
295#99—28no—
296#99—616no—
297#99—68no—
298#95—68no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—28no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—28no—
306C23×C3—28no—
307C23×C3—42no—
308C23×C3—44no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3—416no—
312C23×C3—416no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3C2×C22⋊C412yes—
316C23×C3C2×C22⋊C412yes—
317C23×C3C2×C22⋊C414yes—
318C23×C3C23×C411yescharacteristic
319C23×C3C22×D412yes—
320C23×C3C22×D414yes—
321C22×D3—28no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—416no—
325C22×D3—44no—
326C22×D3—44no—
327C22×D3—44no—
328C22×D3C2×C22⋊C414yes—
329C3×D4—216no—
330C3×D4—216no—
331C2×C3×C4—28no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—48no—
335C2×C3×C4—44no—
336C2×C3×C4C2×C22⋊C414yes—
337#67—216no—
338#67—216no—
339#67—216no—
340#67—216no—
341C2×Dic3—28no—
342C2×Dic3—216no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—28no—
348C2×Dic3—28no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—28no—
354C2×Dic3—28no—
355C2×Dic3—28no—
356C2×Dic3—48no—
357C2×Dic3—44no—
358C2×Dic3C2×C22⋊C412yes—
359C2×Dic3C2×C22⋊C414yes—
360C2×Dic3C2×C22⋊C412yes—
361C2×Dic3C2×C22⋊C412yes—
362C2×Dic3C2×C22⋊C414yes—
363C2×Dic3C2×C22⋊C411yescharacteristic
364C2×Dic3C2×C22⋊C411yescharacteristic
365C2×Dic3C2×C22⋊C414yes—
366C2×Dic3#12018yes—
367C2×Dic3#12018yes—
368C4×D3—216no—
369C4×D3—28no—
370C4×D3—28no—
371Dic6—216no—
372Dic6—216no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—32no—
377C24—44no—
378C24—44no—
379C24—64no—
380C24—62no—
381C24—124no—
382C24C2×C4×D311yescharacteristic
383C24C2×C4×D311yescharacteristic
384C24C2×C4×D312yes—
385C24D3×D414yes—
386C24C23×D311yescharacteristic
387#41—632no—
388#41—632no—
389#41—632no—
390#41—632no—
391#41—664no—
392#41—664no—
393C2×Q8—38no—
394C2×Q8—64no—
395C2×Q8—64no—
396C2×Q8—64no—
397C2×Q8—64no—
398C2×Q8—632no—
399C2×Q8—64no—
400C2×D4—232no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—38no—
407C2×D4—38no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—632no—
415C2×D4—632no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—64no—
420C2×D4D3×D418yes—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—38no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—34no—
435C22×C4—38no—
436C22×C4—31no—
437C22×C4—32no—
438C22×C4—31no—
439C22×C4—34no—
440C22×C4—38no—
441C22×C4—64no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—616no—
447C22×C4—616no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—616no—
457C22×C4—616no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—616no—
461C22×C4—616no—
462C22×C4—62no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—64no—
467C22×C4C2×C4×D312yes—
468C4⋊C4—128no—
469C22⋊C4—28no—
470C22⋊C4—48no—
471C22⋊C4—68no—
472C22⋊C4—128no—
473C42—128no—
474C22×C3—28no—
475C22×C3—24no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—24no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—44no—
483C22×C3—416no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—416no—
488C22×C3—44no—
489C22×C3#41014yescomplemented
490C22×C3C22×C22⋊C412yes—
491C22×C3C22×C22⋊C411yescharacteristic
492C22×C3C2×#12014yes—
493C2×D3—24no—
494C2×D3—44no—
495C2×D3—416no—
496C2×D3—44no—
497C2×D3—44no—
498C3×C4—24no—
499C3×C4—84no—
500Dic3—22no—
501Dic3—24no—
502Dic3—22no—
503Dic3—84no—
504Dic3#37914yes—
505Dic3#37912yes—
506Dic3#37912yes—
507C23—22no—
508C23—24no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—28no—
515C23—34no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—416no—
521C23—416no—
522C23—44no—
523C23—44no—
524C23—68no—
525C23—62no—
526C23—62no—
527C23—124no—
528C23—1216no—
529C23—124no—
530C23—124no—
531C23D3×C22⋊C412yes—
532C23D3×C22⋊C412yes—
533C23D3×C22⋊C414yes—
534C23C22×C4×D311yescharacteristic
535C23C2×D3×D412yes—
536C23C2×D3×D414yes—
537Q8—616no—
538Q8—616no—
539D4—216no—
540D4—216no—
541D4—616no—
542D4—616no—
543D4—616no—
544D4—616no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—34no—
549C2×C4—34no—
550C2×C4—31no—
551C2×C4—31no—
552C2×C4—32no—
553C2×C4—38no—
554C2×C4—38no—
555C2×C4—34no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—68no—
561C2×C4—68no—
562C2×C4—616no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—68no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—68no—
569C2×C4—68no—
570C2×C4—68no—
571C2×C4—68no—
572C2×C4—616no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—68no—
578C2×C4—128no—
579C2×C4—124no—
580C2×C4D3×C22⋊C414yes—
581C2×C3—24no—
582C2×C3—22no—
583C2×C3—24no—
584C2×C3—42no—
585C2×C3—44no—
586C2×C3C2×#37911yescharacteristic
587C2×C3C2×#41012yes—
588D3—44no—
589C22—28no—
590C22—24no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—22no—
596C22—24no—
597C22—44no—
598C22—416no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—416no—
603C22—44no—
604C22—64no—
605C22—124no—
606C22—124no—
607C22—1216no—
608C22—124no—
609C22#474014yescomplemented
610C22C2×D3×C22⋊C412yes—
611C22C2×D3×C22⋊C411yescharacteristic
612C22D3×#12014yes—
613C4—24no—
614C4—32no—
615C4—32no—
616C4—34no—
617C4—62no—
618C4—64no—
619C4—62no—
620C4—84no—
621C4—244no—
622C3#2058411yescharacteristic, complemented, Sylow
623C2—24no—
624C2—22no—
625C2—24no—
626C2—42no—
627C2—44no—
628C2—124no—
629C2D3×#37911yescharacteristic
630C2C2×#474012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1781C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#237C2
cosocleC24×C3C24
hypercenter#1781D3
hypocenterC3#20584
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. C2×#4740C2semidirect
3. #73817C2semidirect
4. C3×#1781C2semidirect
5. #72130C2semidirect
6. #70027C2semidirect
7. #69498C2semidirect
10. C23×#67C4semidirect
12. C24×Dic3C4semidirect
17. C22×#237C4semidirect
18. C22×#237C4semidirect
21. C23×C4×D3C4semidirect
22. C23×Dic6C4semidirect
23. C2×#877C22semidirect
24. C2×C3×#99C22semidirect
27. C2×#834C22semidirect
28. C2×#806C22semidirect
29. #4943C22semidirect
39. #1781D3semidirect
125. C2×#99C2×D3semidirect
168. C2×#67C22⋊C4semidirect
194. C22×Dic3C22⋊C4semidirect
218. C2×C4×D3C22⋊C4semidirect
225. C2×Dic6C22⋊C4semidirect
489. C22×C3#410semidirect
609. C22#4740semidirect
622. C3#20584semidirect

Maximal subgroups

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

subgroupindex
C23×#2372
C2×#47402
#738172
C3×#17812
#721302
#700272
#694982
#205843