Finite Group Databasebeta

← #10700708

Subgroups — #10700708

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5745C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3C2×#5113C211yesmaximal, characteristic
4C2×C3×#341C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#10782—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5745—22no—
17#5745—22no—
18#5745—22no—
19#5745C414yescomplemented
20C2×#907—22no—
21C2×D3×M16—22no—
22C2×#901—22no—
23#5133C414yescomplemented
24#5131—21no—
25#5120—21no—
26#5113—22no—
27#5113—22no—
28#5113C414yescomplemented
29C2×#821—21no—
30C2×#816—21no—
31C3×#341—22no—
32#4559—22no—
33#4555—22no—
34#4550—22no—
35C2×#568—31no—
36#2896—31no—
37C2×#416—31no—
38C2×#344—31no—
39C2×#341D311yescharacteristic, complemented
40#1733—31no—
41#1729—31no—
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×#235C2×C412yes—
51C2×#235C2×C412yes—
52C2×#235C2×C412yes—
53C2×#235D412yes—
54C2×#235D412yes—
55C2×#235Q812yes—
56C22×D12D411yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6D411yescharacteristic
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×M16C2×C412yes—
68C2×C3×M16C2×C412yes—
69C2×C3×M16D412yes—
70C2×C3×M16D412yes—
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8—21no—
79C22×C3×C8C2×C412yes—
80C2×#217—21no—
81C2×#217—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926—21no—
88#926—21no—
89#926C2×C412yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2×C412yes—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208C2×C412yes—
99C2×#208D412yes—
100C22×C3⋊C8D412yes—
101#907—28no—
102#907—28no—
103D3×M16—28no—
104D3×M16—28no—
105#901—28no—
106#901—28no—
107C2×ΓL(1,𝔽25)—24no—
108C2×D3×C8—24no—
109#821—24no—
110#821—24no—
111#816—24no—
112#816—24no—
113C22×#41—31no—
114#568—34no—
115#568—62no—
116#568—62no—
117#568—62no—
118C2×#131—62no—
119C2×#131—62no—
120C22×M16—31no—
121C22×M16—62no—
122C22×M16C2×D311yescharacteristic
123C22×M16C2×D311yescharacteristic
124C23×C8C2×D311yescharacteristic
125C2×#117—31no—
126C2×#117—31no—
127#419—61no—
128#418—61no—
129#417—34no—
130#416—34no—
131#416—62no—
132#416—62no—
133C2×#103—61no—
134C2×#102—61no—
135#344—62no—
136#341—22no—
137#339—62no—
138#336—62no—
139C24×C3#4111yescharacteristic
140C23×D3—21no—
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×C4C2×D411yescharacteristic
151C22×C3×C4C2×Q811yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C2×#67—22no—
156C2×#67—22no—
157C2×#67—22no—
158C2×#67—22no—
159C2×#67—22no—
160C2×#67—22no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—41no—
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—22no—
173#235—22no—
174#235—22no—
175#235—22no—
176#235C4214yes—
177#235C22⋊C414yes—
178#235C4⋊C414yes—
179#235C4⋊C414yes—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12C22⋊C412yes—
184C2×D12C22⋊C412yes—
185C2×D12C4⋊C412yes—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6C22⋊C412yes—
196C2×Dic6C22⋊C412yes—
197C2×Dic6C4⋊C412yes—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—22no—
203C3×M16—41no—
204C3×M16—41no—
205C3×M16C22⋊C414yes—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—41no—
214C2×C3×C8—41no—
215C2×C3×C8—41no—
216C2×C3×C8—41no—
217C2×C3×C8—41no—
218C2×C3×C8—41no—
219#217—22no—
220#217—41no—
221#217—41no—
222#217—41no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—41no—
227C4×Dic3—22no—
228C4×Dic3—41no—
229C4×Dic3—41no—
230C4×Dic3—41no—
231#208—22no—
232#208—22no—
233#208—22no—
234#208—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8C22⋊C414yes—
238ΓL(1,𝔽25)—216no—
239D3×C8—216no—
240C2×#41—32no—
241C2×#41—32no—
242C2×#41—32no—
243C2×#41—32no—
244C2×#41—32no—
245C2×#41—32no—
246C2×#41—61no—
247C2×#41—61no—
248C2×#41—61no—
249C2×#41—61no—
250C2×#41—61no—
251C2×#41—61no—
252C22×Q8—31no—
253C22×D4—31no—
254C22×D4—61no—
255C23×C4—61no—
256C23×C4C22×D311yescharacteristic
257#131—68no—
258#131—68no—
259#131—68no—
260#131—68no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—32no—
270C2×M16—32no—
271C2×M16—64no—
272C2×M16—68no—
273C2×M16—61no—
274C2×M16—61no—
275C2×M16—61no—
276C2×M16—61no—
277C2×M16—68no—
278C2×M16C4×D312yes—
279C2×M16C4×D312yes—
280C2×M16D1212yes—
281C2×M16#6712yes—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—32no—
291C22×C8—64no—
292C22×C8C4×D312yes—
293#117—32no—
294#117—61no—
295#117—61no—
296#117—61no—
297#117—61no—
298#117—61no—
299#117—61no—
300#117—61no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C42—61no—
306C2×C42—61no—
307#103—64no—
308#103—64no—
309#102—64no—
310#102—64no—
311C23×C3—42no—
312C23×C3#11711yescharacteristic
313C23×C3C4×D411yescharacteristic
314C23×C3C4×D411yescharacteristic
315C23×C3#12111yescharacteristic
316C23×C3C22⋊Q811yescharacteristic
317C23×C3#12311yescharacteristic
318C23×C3#12311yescharacteristic
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—82no—
323C2×C3×C4C2×C4211yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4C2×C22⋊C411yescharacteristic
327C2×C3×C4C2×C22⋊C411yescharacteristic
328C2×C3×C4C2×C22⋊C411yescharacteristic
329C2×C3×C4C2×C4⋊C411yescharacteristic
330C2×C3×C4C2×C4⋊C411yescharacteristic
331C2×C3×C4C2×C4⋊C411yescharacteristic
332C2×C3×C4#11711yescharacteristic
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#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4C22⋊Q811yescharacteristic
347C2×C3×C4C22⋊Q811yescharacteristic
348C2×C3×C4C22⋊Q811yescharacteristic
349C2×C3×C4#12311yescharacteristic
350C2×C3×C4#12311yescharacteristic
351#67—24no—
352#67—24no—
353#67—24no—
354#67—24no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—44no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—82no—
362D12—22no—
363D12—22no—
364D12—22no—
365D12#9514yes—
366C4×D3—24no—
367C4×D3—24no—
368C4×D3—24no—
369C4×D3—24no—
370Dic6—22no—
371Dic6—22no—
372Dic6—22no—
373Dic6#9514yes—
374C3×C8—24no—
375C3×C8—42no—
376C3×C8—42no—
377C3⋊C8—24no—
378C24—61no—
379C24#23911yescharacteristic
380#41—34no—
381#41—34no—
382#41—34no—
383#41—34no—
384#41—62no—
385#41—62no—
386#41—62no—
387#41—62no—
388#41—62no—
389#41—62no—
390#41—62no—
391#41—62no—
392#41—62no—
393#41—62no—
394#41—62no—
395#41—62no—
396C2×Q8—32no—
397C2×Q8—32no—
398C2×Q8—32no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×D4—32no—
403C2×D4—32no—
404C2×D4—32no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
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×C4D3×D411yescharacteristic
434C22×C4#23711yescharacteristic
435C22×C4D3×Q811yescharacteristic
436C22×C4C2×#6711yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—41no—
443M16—41no—
444M16—616no—
445M16—62no—
446M16—62no—
447M16—62no—
448M16—62no—
449M16(C2×D3)⋊C414yes—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—34no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—616no—
465C2×C8—62no—
466C2×C8—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—121no—
471C22⋊C4—62no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C22⋊C4—121no—
475C42—62no—
476C42—121no—
477C42—121no—
478C42—121no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C4×C22⋊C411yescharacteristic
487C22×C3#38011yescharacteristic
488C22×C3#38111yescharacteristic
489C22×C3#38511yescharacteristic
490C22×C3#38511yescharacteristic
491C22×C3#38611yescharacteristic
492C22×C3#38611yescharacteristic
493C2×D3—24no—
494C2×D3—44no—
495C2×D3—82no—
496C2×D3—82no—
497C3×C4C2×#9511yescharacteristic
498C3×C4C4×C22⋊C411yescharacteristic
499C3×C4#37911yescharacteristic
500C3×C4#38011yescharacteristic
501C3×C4#38511yescharacteristic
502C3×C4#38511yescharacteristic
503C3×C4#38611yescharacteristic
504C3×C4#38611yescharacteristic
505Dic3—24no—
506Dic3—82no—
507Dic3—82no—
508C23—42no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—242no—
513C23#89211yescharacteristic
514C23#89311yescharacteristic
515C23#89311yescharacteristic
516C23#89411yescharacteristic
517C23#89411yescharacteristic
518C23#89511yescharacteristic
519C23#94611yescharacteristic
520Q8—34no—
521Q8—62no—
522Q8—62no—
523Q8—62no—
524D4—34no—
525D4—64no—
526D4—64no—
527D4—64no—
528D4—64no—
529D4—62no—
530D4—62no—
531D4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—124no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—242no—
543C2×C4C42×D311yescharacteristic
544C2×C4#87211yescharacteristic
545C2×C4C4×D1211yescharacteristic
546C2×C4C4×D1211yescharacteristic
547C2×C4D3×C22⋊C411yescharacteristic
548C2×C4D3×C22⋊C411yescharacteristic
549C2×C4D3×C22⋊C411yescharacteristic
550C2×C4D3×C22⋊C411yescharacteristic
551C2×C4#67⋊C411yescharacteristic
552C2×C4#67⋊C411yescharacteristic
553C2×C4#88211yescharacteristic
554C2×C4#88211yescharacteristic
555C2×C4#88311yescharacteristic
556C2×C4#88311yescharacteristic
557C2×C4#88411yescharacteristic
558C2×C4#88411yescharacteristic
559C2×C4D3×C4⋊C411yescharacteristic
560C2×C4D3×C4⋊C411yescharacteristic
561C2×C4D3×C4⋊C411yescharacteristic
562C2×C4#89611yescharacteristic
563C2×C4#89611yescharacteristic
564C2×C4#89711yescharacteristic
565C2×C4C2×(C2×D3)⋊C411yescharacteristic
566C2×C4C4×#6711yescharacteristic
567C2×C4C4×#6711yescharacteristic
568C2×C4#93711yescharacteristic
569C2×C4#93711yescharacteristic
570C2×C4#93811yescharacteristic
571C8—24no—
572C8—42no—
573C8—42no—
574C8—64no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3C22⋊#9511yescharacteristic
582C2×C3#188112yes—
583D3—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—64no—
592C22—124no—
593C22—242no—
594C22—242no—
595C22#466411yescharacteristic
596C22#466511yescharacteristic
597C22#466511yescharacteristic
598C22#466611yescharacteristic
599C22#498711yescharacteristic
600C22#498911yescharacteristic
601C22#498911yescharacteristic
602C4—64no—
603C4—242no—
604C4—242no—
605C4D3×#9511yescharacteristic
606C4#466711yescharacteristic
607C4#466811yescharacteristic
608C4#466811yescharacteristic
609C4C4×(C2×D3)⋊C411yescharacteristic
610C4#495511yescharacteristic
611C4#495611yescharacteristic
612C4#495611yescharacteristic
613C3#1078211yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—124no—
621C2#6894712yes—
622C2(C2×D3)⋊#9511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#341C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#341D3
hypocenterC3#10782
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
4. C2×C3×#341C2semidirect
5. C2×#4559C2semidirect
6. #71982C2semidirect
7. #71965C2semidirect
19. #5745C4semidirect
23. #5133C4semidirect
28. #5113C4semidirect
39. C2×#341D3semidirect
613. C3#10782semidirect

Maximal subgroups

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

subgroupindex
C2×#57452
#837682
C2×#51132
C2×C3×#3412
C2×#45592
#719822
#719652
#107823