Finite Group Databasebeta

← #10701656

Subgroups — #10701656

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5132C211yesmaximal, characteristic
2C2×#5129C211yesmaximal, characteristic
3C2×#5006C211yesmaximal, characteristic
4C2×#4997C211yesmaximal, characteristic, complemented
5C2×C3×#344C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7#71980C211yesmaximal, characteristic, complemented
8#10961—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16C2×#824C412yescomplemented
17#5132—22no—
18#5132—22no—
19C2×#823C412yescomplemented
20#5129—21no—
21#5129—21no—
22#5129—22no—
23C2×(C2×D3)⋊C8C414yescomplemented
24#5006—22no—
25#5006—22no—
26#4997—22no—
27#4997—22no—
28C2×#806C414yes—
29C2×#803—21no—
30C2×C4×C3⋊C8—21no—
31C3×#344—22no—
32#4556—22no—
33#4550—22no—
34C2×#413—31no—
35C2×#410—31no—
36C2×#408—31no—
37C2×#405—31no—
38C2×#344—31no—
39C2×#344D311yescharacteristic, complemented
40#1729—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67C2×C412yes—
43C2×#235—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×#235—21no—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3C2×C412yes—
57C22×Dic6C2×C411yescharacteristic
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16C2×C412yes—
70C22×C3×C8—21no—
71C22×C3×C8C2×C412yes—
72C3×#117—21no—
73C3×#117—21no—
74C3×#117D412yes—
75C3×#117D412yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C22⋊C4C2×C412yes—
78C2×C3×C42—21no—
79C2×#217C2×C412yes—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84C2×C4⋊Dic3—21no—
85C2×C4×Dic3—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208C2×C412yes—
90C2×#208D412yes—
91C2×#208D412yes—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2×C412yes—
98C22×C3⋊C8D411yescharacteristic
99C22×C3⋊C8D411yescharacteristic
100C22×C3⋊C8D412yes—
101#824—24no—
102#823—24no—
103(C2×D3)⋊C8—24no—
104(C2×D3)⋊C8—24no—
105#806—28no—
106#803—24no—
107C4×C3⋊C8—24no—
108C22×#41—31no—
109C22×M16—31no—
110C22×M16C2×D311yescharacteristic
111C22×M16C2×D311yescharacteristic
112C23×C8—31no—
113C2×#117—31no—
114C2×#117C2×D311yescharacteristic
115#413—62no—
116#413—62no—
117C2×#101—32no—
118C2×#100—32no—
119#410—62no—
120#410—62no—
121C2×#99—34no—
122#408—62no—
123#408—61no—
124#408—61no—
125C2×C22⋊C8—34no—
126#405—62no—
127#405—62no—
128C2×#97—61no—
129C2×C4×C8—61no—
130#344—22no—
131#344—62no—
132#336—62no—
133C24×C3C22×C411yescharacteristic
134C23×D3C4212yes—
135C22×C3×C4—22no—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C2×#67—22no—
151C2×#67—22no—
152C2×#67—22no—
153C2×#67—22no—
154C2×#67—22no—
155C2×#67—22no—
156C22×Dic3—22no—
157C22×Dic3C4212yes—
158#235—41no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173#235—41no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—41no—
195C3×M16—41no—
196C3×M16—41no—
197C3×M16—41no—
198C3×M16—41no—
199C3×M16—41no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4C22⋊C414yes—
210C3×C42—22no—
211C3×C42—22no—
212#217—22no—
213#217—22no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216C4×Dic3—41no—
217C4×Dic3—41no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208C22⋊C414yes—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8C22⋊C412yes—
234C2×C3⋊C8C22⋊C412yes—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C2×#41—61no—
240C2×#41—61no—
241C2×#41—61no—
242C2×#41—61no—
243C2×#41—61no—
244C2×#41—61no—
245C2×#41—61no—
246C2×#41—61no—
247C22×Q8—31no—
248C22×D4—32no—
249C22×D4—31no—
250C23×C4—32no—
251C23×C4C22×D311yescharacteristic
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—32no—
264C2×M16—32no—
265C2×M16—32no—
266C2×M16—61no—
267C2×M16—61no—
268C2×M16—61no—
269C2×M16C4×D312yes—
270C22×C8—21no—
271C22×C8—32no—
272C22×C8—32no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—61no—
276C22×C8—61no—
277C22×C8—61no—
278C22×C8—61no—
279C22×C8—61no—
280C22×C8C4×D312yes—
281#117—21no—
282#117—21no—
283#117—61no—
284#117—61no—
285#117—61no—
286#117—61no—
287#117D1212yes—
288#117#6712yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—61no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4C4×D312yes—
293C2×C42—21no—
294C2×C42—61no—
295#101—64no—
296#100—64no—
297#99—68no—
298C22⋊C8—64no—
299C22⋊C8—64no—
300#97—64no—
301C4×C8—64no—
302C23×C3—42no—
303C23×C3C2×C4211yescharacteristic
304C23×C3C2×C22⋊C411yescharacteristic
305C23×C3#11711yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3C4×D411yescharacteristic
308C23×C3C4×D411yescharacteristic
309C23×C3C4×D411yescharacteristic
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—44no—
314C2×C3×C4—24no—
315C2×C3×C4—44no—
316C2×C3×C4C2×C22⋊C411yescharacteristic
317C2×C3×C4C2×C22⋊C411yescharacteristic
318C2×C3×C4#11711yescharacteristic
319C2×C3×C4#11711yescharacteristic
320C2×C3×C4C4×D411yescharacteristic
321C2×C3×C4C4×D411yescharacteristic
322C2×C3×C4C4×D411yescharacteristic
323C2×C3×C4C4×D411yescharacteristic
324C2×C3×C4C4×D411yescharacteristic
325C2×C3×C4C4×D411yescharacteristic
326C2×C3×C4C4×D411yescharacteristic
327C2×C3×C4C4×D411yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12111yescharacteristic
330C2×C3×C4#12111yescharacteristic
331C2×C3×C4#12111yescharacteristic
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12111yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12311yescharacteristic
337C2×C3×C4#12311yescharacteristic
338C2×C3×C4#12411yescharacteristic
339C2×C3×C4#12411yescharacteristic
340C2×C3×C4#12411yescharacteristic
341C2×C3×C4#12611yescharacteristic
342C2×C3×C4#12611yescharacteristic
343C2×C3×C4#12711yescharacteristic
344#67—42no—
345#67—42no—
346#67—42no—
347#67—42no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—44no—
352C2×Dic3—42no—
353D12—41no—
354D12—41no—
355D12—41no—
356D12—41no—
357C4×D3—42no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—42no—
361Dic6—41no—
362Dic6—41no—
363Dic6—41no—
364Dic6—41no—
365C3×C8—42no—
366C3×C8—42no—
367C3⋊C8—22no—
368C3⋊C8—22no—
369C3⋊C8—24no—
370C24—32no—
371C24C2×C4×D311yescharacteristic
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379#41—121no—
380#41—121no—
381#41—121no—
382#41—121no—
383#41—121no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C22×C4—22no—
407C22×C4—32no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4C2×C4×D311yescharacteristic
416C22×C4C2×C4×D311yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4#23511yescharacteristic
419C22×C4#23511yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4#23711yescharacteristic
425C22×C4#23711yescharacteristic
426C22×C4#23911yescharacteristic
427C22×C4#23911yescharacteristic
428C22×C4C2×#6711yescharacteristic
429M16—22no—
430M16—22no—
431M16—34no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—62no—
439M16—62no—
440M16—62no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—62no—
450C2×C8—61no—
451C2×C8—62no—
452C2×C8—61no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—62no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—121no—
463C4⋊C4—121no—
464C22⋊C4—22no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4(C2×D3)⋊C414yes—
468C42—22no—
469C42—22no—
470C42—121no—
471C42—121no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3C4×C22⋊C411yescharacteristic
480C22×C3C4×C22⋊C411yescharacteristic
481C22×C3C4×C22⋊C411yescharacteristic
482C22×C3#38611yescharacteristic
483C22×C3#38811yescharacteristic
484C22×C3#38811yescharacteristic
485C22×C3#39011yescharacteristic
486C2×D3—42no—
487C2×D3—44no—
488C2×D3—82no—
489C2×D3—82no—
490C3×C4—44no—
491C3×C4#38611yescharacteristic
492C3×C4#38611yescharacteristic
493C3×C4#38811yescharacteristic
494C3×C4#38811yescharacteristic
495C3×C4#38811yescharacteristic
496C3×C4#38811yescharacteristic
497C3×C4#39011yescharacteristic
498C3×C4#39011yescharacteristic
499Dic3—42no—
500Dic3—82no—
501C23—42no—
502C23—62no—
503C23—62no—
504C23—62no—
505C23—124no—
506C23C42×D311yescharacteristic
507C23#87211yescharacteristic
508C23#67⋊C411yescharacteristic
509C23#67⋊C411yescharacteristic
510C23#89311yescharacteristic
511C23#89311yescharacteristic
512C23C2×(C2×D3)⋊C411yescharacteristic
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8—121no—
517D4—122no—
518D4—122no—
519D4—122no—
520D4—122no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—121no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—124no—
535C2×C4—122no—
536C2×C4C4×D1211yescharacteristic
537C2×C4C4×D1211yescharacteristic
538C2×C4D3×C22⋊C411yescharacteristic
539C2×C4D3×C22⋊C411yescharacteristic
540C2×C4#67⋊C411yescharacteristic
541C2×C4#67⋊C411yescharacteristic
542C2×C4#88211yescharacteristic
543C2×C4#88411yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4#88511yescharacteristic
546C2×C4#88511yescharacteristic
547C2×C4#88611yescharacteristic
548C2×C4#89211yescharacteristic
549C2×C4#89211yescharacteristic
550C2×C4#89311yescharacteristic
551C2×C4#89311yescharacteristic
552C2×C4#89411yescharacteristic
553C2×C4#89411yescharacteristic
554C2×C4#89511yescharacteristic
555C2×C4#89511yescharacteristic
556C2×C4#89811yescharacteristic
557C2×C4#89811yescharacteristic
558C2×C4C4×#6711yescharacteristic
559C2×C4C4×#6711yescharacteristic
560C2×C4#93911yescharacteristic
561C2×C4#93911yescharacteristic
562C2×C4#94011yescharacteristic
563C2×C4#94711yescharacteristic
564C8—42no—
565C8—42no—
566C8—62no—
567C8—62no—
568C8—64no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3#143711yescharacteristic
576C2×C3#175712yes—
577D3—82no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—122no—
586C22—124no—
587C22—242no—
588C22—242no—
589C22#466611yescharacteristic
590C22#466611yescharacteristic
591C22#466911yescharacteristic
592C22#466911yescharacteristic
593C22C4×(C2×D3)⋊C411yescharacteristic
594C22#495811yescharacteristic
595C22#498811yescharacteristic
596C4—44no—
597C4—122no—
598C4—242no—
599C4#466511yescharacteristic
600C4#466511yescharacteristic
601C4#467011yescharacteristic
602C4#467011yescharacteristic
603C4#495711yescharacteristic
604C4#495711yescharacteristic
605C4#499011yescharacteristic
606C4#499011yescharacteristic
607C3#1096111yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—242no—
615C2#6901412yes—
616C2#6940811yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC22×C3C4×C22⋊C4
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#10961
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×#4997C2semidirect
5. C2×C3×#344C2semidirect
6. C2×#4556C2semidirect
7. #71980C2semidirect
16. C2×#824C4semidirect
19. C2×#823C4semidirect
23. C2×(C2×D3)⋊C8C4semidirect
39. C2×#344D3semidirect
607. C3#10961semidirect

Maximal subgroups

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

subgroupindex
C2×#51322
C2×#51292
C2×#50062
C2×#49972
C2×C3×#3442
C2×#45562
#719802
#109613