Finite Group Databasebeta

← #10701570

Subgroups — #10701570

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5132C211yesmaximal, characteristic
2C2×#5005C211yesmaximal, characteristic
3C22×#805C211yesmaximal, characteristic
4C2×C4×#208C211yesmaximal, characteristic, complemented
5C2×C3×#339C211yesmaximal, characteristic, complemented
6#71998C211yesmaximal, characteristic, complemented
7C2×#4533C211yesmaximal, characteristic, complemented
8#10946—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×#208C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×#824C412yescomplemented
17#5132—22no—
18#5132—22no—
19C2×#823C412yescomplemented
20#5005—22no—
21#5005C414yescomplemented
22C2×#805—21no—
23C2×#805—21no—
24C2×#805—22no—
25C2×#805—22no—
26C2×#805C414yescomplemented
27C4×#208—22no—
28C4×#208—22no—
29C2×#803—21no—
30C2×C4×C3⋊C8—21no—
31C3×#339—22no—
32#4552—22no—
33#4533—22no—
34C2×#421—31no—
35C22×#104—31no—
36C2×#413—31no—
37C2×C4×M16—31no—
38C2×#344—31no—
39#1738—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×#208—21no—
81C2×#208—21no—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208C2×C412yes—
92C2×#208C2×C412yes—
93C2×#208C2×C412yes—
94C2×#208D412yes—
95C2×#208D412yes—
96C2×#208D412yes—
97C2×#208D412yes—
98C22×C3⋊C8—21no—
99C22×C3⋊C8—21no—
100C22×C3⋊C8—21no—
101#824—24no—
102#823—24no—
103#805—24no—
104#805—24no—
105#805—28no—
106#803—24no—
107C4×C3⋊C8—24no—
108C22×#41—31no—
109C22×M16—31no—
110C22×M16—31no—
111C22×M16—31no—
112C22×M16C2×D311yescharacteristic
113C2×#117C2×D311yescharacteristic
114C22×C42C2×D311yescharacteristic
115#421—34no—
116#421—62no—
117C2×#104—34no—
118C2×#104—62no—
119C2×#104—61no—
120C2×#104—61no—
121C2×#104—62no—
122#413—62no—
123#413—62no—
124C2×#101—32no—
125C2×#100—32no—
126C4×M16—62no—
127C4×M16—62no—
128C2×#97—61no—
129C2×C4×C8—61no—
130#344—62no—
131#341—62no—
132#339—22no—
133C24×C3#4111yescharacteristic
134C23×D3—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
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—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—22no—
166#235—22no—
167#235—22no—
168#235—22no—
169#235—22no—
170#235—41no—
171#235—41no—
172#235—41no—
173#235—41no—
174#235—41no—
175#235—41no—
176#235—41no—
177#235—41no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12C4212yes—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—41no—
187C2×C4×D3—41no—
188C2×C4×D3—41no—
189C2×C4×D3—41no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6C4212yes—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—41no—
199C3×M16—41no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—41no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—41no—
207C3×C42—22no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—41no—
213C3×C42C22⋊C412yes—
214C3×C42C22⋊C412yes—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208—41no—
224#208—41no—
225#208C22⋊C414yes—
226#208C22⋊C414yes—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—41no—
235C2×C3⋊C8—41no—
236C2×#41—32no—
237C2×#41—32no—
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—
247C2×#41—61no—
248C22×Q8—31no—
249C22×D4—31no—
250C22×D4—61no—
251C23×C4—21no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—32no—
260C2×M16—32no—
261C2×M16—32no—
262C2×M16—32no—
263C2×M16—32no—
264C2×M16—32no—
265C2×M16—32no—
266C2×M16—61no—
267C2×M16—61no—
268C2×M16—61no—
269C2×M16—61no—
270C2×M16—61no—
271C2×M16—61no—
272C2×M16—61no—
273C2×M16—61no—
274C2×M16—61no—
275C2×M16—61no—
276C2×M16—61no—
277C2×M16C4×D312yes—
278C22×C8—21no—
279C22×C8—61no—
280C22×C8—61no—
281C22×C8—61no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117C4×D312yes—
286C2×C4⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42C4×D312yes—
293C2×C42D1211yescharacteristic
294C2×C42D1211yescharacteristic
295C2×C42#6712yes—
296#104—68no—
297#104—64no—
298#104—64no—
299#101—64no—
300#100—64no—
301#97—64no—
302C4×C8—64no—
303C23×C3—42no—
304C23×C3#11711yescharacteristic
305C23×C3C4×D411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3#12111yescharacteristic
308C23×C3#12311yescharacteristic
309C23×C3#12411yescharacteristic
310C23×C3#12611yescharacteristic
311C22×D3—22no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—82no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—82no—
323C2×C3×C4C2×C4211yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4C2×C22⋊C411yescharacteristic
327C2×C3×C4#11711yescharacteristic
328C2×C3×C4#11711yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
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×C4#12011yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12311yescharacteristic
347C2×C3×C4#12411yescharacteristic
348C2×C3×C4#12411yescharacteristic
349C2×C3×C4#12611yescharacteristic
350C2×C3×C4#12711yescharacteristic
351#67—42no—
352#67—42no—
353#67—42no—
354#67—42no—
355C2×Dic3—22no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358D12—22no—
359D12—22no—
360D12—41no—
361D12—41no—
362C4×D3—42no—
363C4×D3—42no—
364C4×D3—42no—
365C4×D3—42no—
366Dic6—22no—
367Dic6—22no—
368Dic6—41no—
369Dic6—41no—
370C3×C8—42no—
371C3⋊C8—24no—
372C3⋊C8—24no—
373C3⋊C8—42no—
374C24—61no—
375C24#23711yescharacteristic
376#41—62no—
377#41—62no—
378#41—62no—
379#41—62no—
380#41—62no—
381#41—62no—
382#41—62no—
383#41—62no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388#41—121no—
389#41—121no—
390#41—121no—
391#41—121no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—32no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—121no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4—121no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—41no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4C2×C4×D311yescharacteristic
424C22×C4C2×C4×D311yescharacteristic
425C22×C4C2×C4×D311yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4#23511yescharacteristic
428C22×C4#23511yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4#23911yescharacteristic
435C22×C4#23911yescharacteristic
436C22×C4C2×#6711yescharacteristic
437M16—22no—
438M16—22no—
439M16—34no—
440M16—34no—
441M16—41no—
442M16—41no—
443M16—62no—
444M16—62no—
445M16—62no—
446M16—62no—
447M16—62no—
448M16—62no—
449M16—62no—
450M16—62no—
451M16—121no—
452M16—121no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—62no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—121no—
464C2×C8—121no—
465C4⋊C4—22no—
466C4⋊C4—41no—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42(C2×D3)⋊C412yes—
476C42(C2×D3)⋊C412yes—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3C4×C22⋊C411yescharacteristic
485C22×C3#38611yescharacteristic
486C22×C3#38811yescharacteristic
487C22×C3#38811yescharacteristic
488C22×C3#38811yescharacteristic
489C22×C3#38811yescharacteristic
490C22×C3#39011yescharacteristic
491C2×D3—42no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—82no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—82no—
498C3×C4C4×C22⋊C411yescharacteristic
499C3×C4C4×C22⋊C411yescharacteristic
500C3×C4#38611yescharacteristic
501C3×C4#38611yescharacteristic
502C3×C4#38811yescharacteristic
503C3×C4#38811yescharacteristic
504C3×C4#39011yescharacteristic
505C3×C4#39011yescharacteristic
506Dic3—42no—
507C23—42no—
508C23—62no—
509C23—121no—
510C23—121no—
511C23—242no—
512C23#67⋊C411yescharacteristic
513C23#67⋊C411yescharacteristic
514C23#88511yescharacteristic
515C23#88611yescharacteristic
516C23#89211yescharacteristic
517C23#89811yescharacteristic
518C23#93911yescharacteristic
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523D4—62no—
524D4—62no—
525D4—122no—
526D4—122no—
527D4—122no—
528D4—122no—
529D4—121no—
530D4—121no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—62no—
539C2×C4—82no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4C42×D311yescharacteristic
547C2×C4#87211yescharacteristic
548C2×C4C4×D1211yescharacteristic
549C2×C4C4×D1211yescharacteristic
550C2×C4D3×C22⋊C411yescharacteristic
551C2×C4D3×C22⋊C411yescharacteristic
552C2×C4#67⋊C411yescharacteristic
553C2×C4#67⋊C411yescharacteristic
554C2×C4#88211yescharacteristic
555C2×C4#88411yescharacteristic
556C2×C4#88411yescharacteristic
557C2×C4#88511yescharacteristic
558C2×C4#89211yescharacteristic
559C2×C4#89311yescharacteristic
560C2×C4#89311yescharacteristic
561C2×C4#89311yescharacteristic
562C2×C4#89311yescharacteristic
563C2×C4#89411yescharacteristic
564C2×C4#89411yescharacteristic
565C2×C4#89511yescharacteristic
566C2×C4#89511yescharacteristic
567C2×C4#89811yescharacteristic
568C2×C4C2×(C2×D3)⋊C411yescharacteristic
569C2×C4C4×#6711yescharacteristic
570C2×C4C4×#6711yescharacteristic
571C2×C4#93911yescharacteristic
572C2×C4#94011yescharacteristic
573C2×C4#94711yescharacteristic
574C8—42no—
575C8—64no—
576C8—64no—
577C8—122no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3#143711yescharacteristic
585C2×C3#193212yes—
586D3—82no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—122no—
595C22—242no—
596C22—242no—
597C22—242no—
598C22#466611yescharacteristic
599C22#466911yescharacteristic
600C22#467011yescharacteristic
601C22#467011yescharacteristic
602C22#495711yescharacteristic
603C22#495811yescharacteristic
604C22#499011yescharacteristic
605C4—24no—
606C4—42no—
607C4—82no—
608C4—122no—
609C4#466511yescharacteristic
610C4#466511yescharacteristic
611C4#466611yescharacteristic
612C4#466911yescharacteristic
613C4C4×(C2×D3)⋊C411yescharacteristic
614C4#495711yescharacteristic
615C4#498811yescharacteristic
616C4#499011yescharacteristic
617C3#1094611yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—242no—
625C2#6875012yes—
626C2#6940811yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC22×C3#388
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#10946
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
4. C2×C4×#208C2semidirect
5. C2×C3×#339C2semidirect
6. #71998C2semidirect
7. C2×#4533C2semidirect
16. C2×#824C4semidirect
19. C2×#823C4semidirect
21. #5005C4semidirect
26. C2×#805C4semidirect
40. C2×#339D3semidirect
617. C3#10946semidirect

Maximal subgroups

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

subgroupindex
C2×#51322
C2×#50052
C22×#8052
C2×C4×#2082
C2×C3×#3392
#719982
C2×#45332
#109463