Finite Group Databasebeta

← #10701563

Subgroups — #10701563

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83768C211yesmaximal, characteristic
2C2×#5006C211yesmaximal, characteristic
3C4×#926C211yesmaximal, characteristic, complemented
4C22×#805C211yesmaximal, characteristic
5C2×C3×#339C211yesmaximal, characteristic, complemented
6C2×#4559C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#10945—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C4×#217—22no—
17#5133—22no—
18#5131C412yescomplemented
19#5120C412yescomplemented
20#5113—22no—
21#5006—22no—
22#5006C414yescomplemented
23C2×#806—22no—
24C4×C4⋊Dic3—22no—
25#4932—21no—
26C42×Dic3—21no—
27C2×#805—21no—
28C2×#805—21no—
29C2×#805—22no—
30C2×#805C414yescomplemented
31C3×#339—22no—
32#4559—22no—
33#4523—22no—
34C22×#104—31no—
35#2896—31no—
36C2×#410—31no—
37C4×#117—31no—
38C2×#344—31no—
39#1733—31no—
40C2×#339D311yescharacteristic, complemented
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235C2×C412yes—
55C2×#235C2×C412yes—
56C22×D12C2×C411yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6C2×C411yescharacteristic
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16C2×C412yes—
65C22×C3×C8—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C3×#117C2×C412yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2×C412yes—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D412yes—
80C2×#217—21no—
81C2×#217D412yes—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926C2×C412yes—
87#926C2×C412yes—
88#926D412yes—
89#926D412yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D412yes—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—24no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#208C2×C412yes—
101C22×C3⋊C8—21no—
102C3⋊#95—24no—
103#821—24no—
104#816—24no—
105#806—28no—
106#805—24no—
107#805—24no—
108C22×#41—31no—
109C22×M16—31no—
110C22×M16C2×D311yescharacteristic
111C2×#117—31no—
112C2×#117—31no—
113C2×#117C2×D311yescharacteristic
114C22×C42C2×D311yescharacteristic
115C2×#104—34no—
116C2×#104—61no—
117C2×#104—61no—
118C2×#104—62no—
119#419—32no—
120#418—32no—
121#417—62no—
122#416—62no—
123#410—34no—
124#410—62no—
125C2×#99—62no—
126C4×C4⋊C4—62no—
127C4×C22⋊C4—62no—
128#376—61no—
129C43—61no—
130#344—62no—
131#339—22no—
132#339—62no—
133C24×C3C2×D411yescharacteristic
134C23×D3—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—44no—
140C22×C3×C4—41no—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C2×#67—22no—
156C2×#67—22no—
157C2×#67—41no—
158C2×#67—41no—
159C2×#67—41no—
160C2×#67—41no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164#235—22no—
165#235—22no—
166#235—22no—
167#235—22no—
168#235—22no—
169#235—22no—
170#235—22no—
171#235—22no—
172#235—41no—
173#235—41no—
174#235—41no—
175#235—41no—
176#235—41no—
177#235—41no—
178#235—41no—
179#235—41no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12C4212yescomplemented
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—41no—
189C2×C4×D3—41no—
190C2×C4×D3—41no—
191C2×C4×D3—41no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6C4212yescomplemented
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—41no—
201C3×M16—41no—
202C2×C3×C8—22no—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—41no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—41no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—22no—
214C3×C42—41no—
215C3×C42C22⋊C412yes—
216C3×C42C22⋊C412yes—
217#217—22no—
218#217—22no—
219#217—22no—
220#217C22⋊C414yes—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3C22⋊C414yes—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C4×Dic3—22no—
229C4×Dic3—48no—
230#208—22no—
231#208—22no—
232#208—41no—
233#208—41no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—41no—
236C2×C3⋊C8—41no—
237C2×#41—32no—
238C2×#41—32no—
239C2×#41—61no—
240C2×#41—61no—
241C2×#41—61no—
242C2×#41—61no—
243C2×#41—61no—
244C2×#41—61no—
245C2×#41—61no—
246C2×#41—61no—
247C2×#41—61no—
248C2×#41—61no—
249C22×Q8—31no—
250C22×D4—31no—
251C22×D4—61no—
252C23×C4—21no—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—32no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16—61no—
264C2×M16—61no—
265C2×M16—61no—
266C2×M16C4×D312yes—
267C22×C8—21no—
268C22×C8—61no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—32no—
273#117—32no—
274#117—32no—
275#117—32no—
276#117—61no—
277#117—61no—
278#117—61no—
279#117—61no—
280#117C4×D312yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—61no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—32no—
286C2×C22⋊C4—61no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—64no—
294C2×C42C4×D312yes—
295C2×C42D1211yescharacteristic
296C2×C42D1211yescharacteristic
297C2×C42#6712yes—
298#104—64no—
299#104—64no—
300#103—64no—
301#102—64no—
302#99—68no—
303#95—64no—
304C23×C3—42no—
305C23×C3C2×C22⋊C411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3C4×D411yescharacteristic
308C23×C3#12011yescharacteristic
309C23×C3#12111yescharacteristic
310C23×C3#12111yescharacteristic
311C23×C3#12311yescharacteristic
312C22×D3—22no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—82no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—44no—
320C2×C3×C4—44no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—82no—
324C2×C3×C4C2×C4211yescharacteristic
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4C2×C22⋊C411yescharacteristic
327C2×C3×C4#11711yescharacteristic
328C2×C3×C4#11711yescharacteristic
329C2×C3×C4#11711yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4C4×D411yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12311yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12411yescharacteristic
347C2×C3×C4#12411yescharacteristic
348C2×C3×C4#12411yescharacteristic
349C2×C3×C4#12611yescharacteristic
350C2×C3×C4#12611yescharacteristic
351C2×C3×C4#12711yescharacteristic
352#67—42no—
353#67—42no—
354#67—42no—
355#67—42no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—44no—
360C2×Dic3—44no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363D12—22no—
364D12—22no—
365D12—41no—
366D12—41no—
367C4×D3—42no—
368C4×D3—42no—
369C4×D3—42no—
370C4×D3—42no—
371Dic6—22no—
372Dic6—22no—
373Dic6—41no—
374Dic6—41no—
375C3×C8—42no—
376C3⋊C8—42no—
377C24—61no—
378C24D3×D411yescharacteristic
379#41—62no—
380#41—62no—
381#41—62no—
382#41—62no—
383#41—62no—
384#41—62no—
385#41—62no—
386#41—62no—
387#41—121no—
388#41—121no—
389#41—121no—
390#41—121no—
391#41—121no—
392#41—121no—
393#41—121no—
394#41—121no—
395C2×Q8—32no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×D4—32no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—121no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—121no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—41no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—121no—
425C22×C4—121no—
426C22×C4—121no—
427C22×C4—121no—
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×C4×D311yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×D1211yescharacteristic
432C22×C4#23511yescharacteristic
433C22×C4#23511yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4#23711yescharacteristic
438C22×C4#23711yescharacteristic
439C22×C4#23911yescharacteristic
440C22×C4#23911yescharacteristic
441C22×C4C2×#6711yescharacteristic
442M16—22no—
443M16—22no—
444M16—41no—
445M16—41no—
446M16—62no—
447M16—62no—
448M16—121no—
449M16—121no—
450C2×C8—22no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—62no—
454C2×C8—121no—
455C2×C8—121no—
456C4⋊C4—22no—
457C4⋊C4—34no—
458C4⋊C4—41no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C22⋊C4—22no—
463C22⋊C4—34no—
464C22⋊C4—41no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—62no—
475C42—62no—
476C42—62no—
477C42—62no—
478C42—128no—
479C42(C2×D3)⋊C412yes—
480C42(C2×D3)⋊C412yes—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3C4×C22⋊C411yescharacteristic
489C22×C3#38611yescharacteristic
490C22×C3#38611yescharacteristic
491C22×C3#38611yescharacteristic
492C22×C3#38811yescharacteristic
493C22×C3#38811yescharacteristic
494C22×C3#39011yescharacteristic
495C2×D3—42no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—24no—
500C3×C4—42no—
501C3×C4—82no—
502C3×C4C4×C22⋊C411yescharacteristic
503C3×C4C4×C22⋊C411yescharacteristic
504C3×C4#38811yescharacteristic
505C3×C4#38811yescharacteristic
506C3×C4#38811yescharacteristic
507C3×C4#38811yescharacteristic
508C3×C4#39011yescharacteristic
509C3×C4#39011yescharacteristic
510Dic3—44no—
511Dic3—44no—
512Dic3—42no—
513C23—42no—
514C23—62no—
515C23—121no—
516C23—121no—
517C23—242no—
518C23D3×C22⋊C411yescharacteristic
519C23#67⋊C411yescharacteristic
520C23#88211yescharacteristic
521C23#88411yescharacteristic
522C23#89311yescharacteristic
523C23#89411yescharacteristic
524C23#93911yescharacteristic
525Q8—62no—
526Q8—62no—
527Q8—121no—
528Q8—121no—
529D4—62no—
530D4—62no—
531D4—122no—
532D4—122no—
533D4—122no—
534D4—122no—
535D4—121no—
536D4—121no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—82no—
548C2×C4—124no—
549C2×C4—124no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4C42×D311yescharacteristic
557C2×C4#87211yescharacteristic
558C2×C4C4×D1211yescharacteristic
559C2×C4C4×D1211yescharacteristic
560C2×C4D3×C22⋊C411yescharacteristic
561C2×C4#67⋊C411yescharacteristic
562C2×C4#67⋊C411yescharacteristic
563C2×C4#67⋊C411yescharacteristic
564C2×C4#88411yescharacteristic
565C2×C4#88511yescharacteristic
566C2×C4#88511yescharacteristic
567C2×C4#88611yescharacteristic
568C2×C4#89211yescharacteristic
569C2×C4#89211yescharacteristic
570C2×C4#89311yescharacteristic
571C2×C4#89311yescharacteristic
572C2×C4#89311yescharacteristic
573C2×C4#89411yescharacteristic
574C2×C4#89511yescharacteristic
575C2×C4#89511yescharacteristic
576C2×C4#89811yescharacteristic
577C2×C4#89811yescharacteristic
578C2×C4C2×(C2×D3)⋊C411yescharacteristic
579C2×C4C4×#6711yescharacteristic
580C2×C4C4×#6711yescharacteristic
581C2×C4#93911yescharacteristic
582C2×C4#94011yescharacteristic
583C2×C4#94711yescharacteristic
584C8—42no—
585C8—122no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3#143711yescharacteristic
593C2×C3#190612yes—
594D3—82no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—122no—
603C22—242no—
604C22—242no—
605C22—242no—
606C22#466511yescharacteristic
607C22#466511yescharacteristic
608C22#466611yescharacteristic
609C22#466911yescharacteristic
610C22#495711yescharacteristic
611C22#495811yescharacteristic
612C22#499011yescharacteristic
613C4—24no—
614C4—42no—
615C4—82no—
616C4—124no—
617C4—122no—
618C4—124no—
619C4#466611yescharacteristic
620C4#466911yescharacteristic
621C4#467011yescharacteristic
622C4#467011yescharacteristic
623C4C4×(C2×D3)⋊C411yescharacteristic
624C4#495711yescharacteristic
625C4#498811yescharacteristic
626C4#499011yescharacteristic
627C3#1094511yescharacteristic, complemented, Sylow
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—242no—
635C2#6877812yes—
636C2#6940811yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#10945
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. C4×#926C2semidirect
5. C2×C3×#339C2semidirect
6. C2×#4559C2semidirect
7. #71578C2semidirect
18. #5131C4semidirect
19. #5120C4semidirect
22. #5006C4semidirect
30. C2×#805C4semidirect
40. C2×#339D3semidirect
185. C2×D12C42semidirect
197. C2×Dic6C42semidirect
627. C3#10945semidirect

Maximal subgroups

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

subgroupindex
#837682
C2×#50062
C4×#9262
C22×#8052
C2×C3×#3392
C2×#45592
#715782
#109453