Finite Group Databasebeta

← #10700686

Subgroups — #10700686

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5745C211yesmaximal, characteristic
2C2×#5139C211yesmaximal, characteristic
3#83768C211yesmaximal, characteristic
4C3×#1738C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6C2×#4559C211yesmaximal, characteristic, complemented
7#71982C211yesmaximal, characteristic, complemented
8#10778—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5745—22no—
17#5745—22no—
18#5745—22no—
19#5745—22no—
20C2×#907C414yescomplemented
21C2×D3×M16—22no—
22C2×#901—22no—
23#5139—22no—
24#5139C414yescomplemented
25C2×#825—22no—
26#5133—22no—
27#5131C412yescomplemented
28#5120C412yescomplemented
29#5113—22no—
30C3×#341—22no—
31#4559—22no—
32#4559—22no—
33#4555—22no—
34C2×#568—31no—
35C2×#421—31no—
36#2896—31no—
37C2×#344—31no—
38C2×#344—31no—
39#1738D311yescharacteristic, complemented
40#1733—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235C2×C412yes—
50C2×#235C2×C412yes—
51C2×#235D412yes—
52C2×#235D412yes—
53C2×#235D412yes—
54C2×#235Q812yes—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3—21no—
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×M16—21no—
70C2×C3×M16—21no—
71C2×C3×M16C2×C412yes—
72C2×C3×M16C2×C412yes—
73C2×C3×M16C2×C412yes—
74C2×C3×M16D412yes—
75C2×C3×M16D412yes—
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8—21no—
79C2×#217—21no—
80C2×#217—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926C2×C412yes—
88#926C2×C412yes—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208D412yes—
98C2×#208D412yes—
99C22×C3⋊C8C2×C412yes—
100#907—28no—
101#907—28no—
102D3×M16—28no—
103D3×M16—28no—
104#901—28no—
105#901—28no—
106C2×ΓL(1,𝔽25)—24no—
107C2×D3×C8—24no—
108#825—28no—
109#821—24no—
110#816—24no—
111C22×#41—31no—
112#568—62no—
113#568—62no—
114#568—62no—
115#568—62no—
116C2×#131—34no—
117C2×#131—62no—
118C22×M16—31no—
119C22×M16—62no—
120C22×M16C2×D311yescharacteristic
121C22×M16C2×D311yescharacteristic
122C22×M16C2×D311yescharacteristic
123C2×#117—31no—
124C2×#117—31no—
125#421—34no—
126#421—62no—
127C2×#104—62no—
128#419—32no—
129#418—32no—
130#417—62no—
131#416—62no—
132#344—62no—
133#344—62no—
134#341—22no—
135#339—62no—
136C24×C3#4111yescharacteristic
137C23×D3—21no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×Q811yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C2×#67—22no—
153C2×#67—22no—
154C2×#67—22no—
155C2×#67—22no—
156C2×#67—22no—
157C2×#67—22no—
158C22×Dic3—21no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161#235—22no—
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—22no—
171#235—22no—
172#235—22no—
173#235C22⋊C414yes—
174#235C22⋊C414yes—
175#235C4⋊C414yes—
176#235C4⋊C414yes—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12C4212yes—
181C2×D12C22⋊C412yes—
182C2×D12C4⋊C412yes—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6C4212yes—
193C2×Dic6C22⋊C412yes—
194C2×Dic6C4⋊C412yes—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—41no—
203C3×M16—41no—
204C3×M16—41no—
205C3×M16—41no—
206C3×M16C22⋊C414yes—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—41no—
213C2×C3×C8—41no—
214C2×C3×C8—41no—
215C2×C3×C8—41no—
216#217—22no—
217#217—22no—
218#217—41no—
219#217—41no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—41no—
223C4⋊Dic3—41no—
224C4×Dic3—22no—
225C4×Dic3—22no—
226C4×Dic3—41no—
227C4×Dic3—41no—
228#208—22no—
229#208—22no—
230#208—22no—
231#208—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8C22⋊C414yes—
235ΓL(1,𝔽25)—216no—
236D3×C8—216no—
237C2×#41—32no—
238C2×#41—32no—
239C2×#41—32no—
240C2×#41—32no—
241C2×#41—32no—
242C2×#41—32no—
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—61no—
253C23×C4C22×D311yescharacteristic
254#131—68no—
255#131—68no—
256#131—68no—
257#131—68no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
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—21no—
270C2×M16—21no—
271C2×M16—32no—
272C2×M16—32no—
273C2×M16—64no—
274C2×M16—68no—
275C2×M16—68no—
276C2×M16—61no—
277C2×M16—61no—
278C2×M16—61no—
279C2×M16—61no—
280C2×M16C4×D312yes—
281C2×M16C4×D312yes—
282C2×M16C4×D312yes—
283C2×M16D1212yes—
284C2×M16#6712yes—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—32no—
289C22×C8—64no—
290#117—32no—
291#117—32no—
292#117—61no—
293#117—61no—
294#117—61no—
295#117—61no—
296#117—61no—
297#117—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—61no—
303C2×C42—61no—
304#104—68no—
305#103—64no—
306#102—64no—
307C23×C3—81no—
308C23×C3#11711yescharacteristic
309C23×C3C4×D411yescharacteristic
310C23×C3C4×D411yescharacteristic
311C23×C3#12111yescharacteristic
312C23×C3C22⋊Q811yescharacteristic
313C23×C3#12311yescharacteristic
314C23×C3#12311yescharacteristic
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—82no—
319C2×C3×C4C2×C4211yescharacteristic
320C2×C3×C4C2×C22⋊C411yescharacteristic
321C2×C3×C4C2×C22⋊C411yescharacteristic
322C2×C3×C4C2×C22⋊C411yescharacteristic
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4C2×C4⋊C411yescharacteristic
326C2×C3×C4C2×C4⋊C411yescharacteristic
327C2×C3×C4C2×C4⋊C411yescharacteristic
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×C4#12011yescharacteristic
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4C22⋊Q811yescharacteristic
343C2×C3×C4C22⋊Q811yescharacteristic
344C2×C3×C4C22⋊Q811yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12311yescharacteristic
347#67—24no—
348#67—24no—
349#67—24no—
350#67—24no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—82no—
357C2×Dic3—82no—
358D12—22no—
359D12—22no—
360D12—22no—
361D12#9514yes—
362C4×D3—24no—
363C4×D3—24no—
364C4×D3—24no—
365C4×D3—24no—
366Dic6—22no—
367Dic6—22no—
368Dic6—22no—
369Dic6#9514yes—
370C3×C8—24no—
371C3×C8—42no—
372C3×C8—42no—
373C3⋊C8—24no—
374C24—61no—
375C24#23911yescharacteristic
376#41—34no—
377#41—34no—
378#41—34no—
379#41—34no—
380#41—62no—
381#41—62no—
382#41—62no—
383#41—62no—
384#41—62no—
385#41—62no—
386#41—62no—
387#41—62no—
388#41—62no—
389#41—62no—
390#41—62no—
391#41—62no—
392C2×Q8—32no—
393C2×Q8—32no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—32no—
399C2×D4—32no—
400C2×D4—32no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4C2×C4×D311yescharacteristic
420C22×C4C2×C4×D311yescharacteristic
421C22×C4C2×C4×D311yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4#23511yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4#23711yescharacteristic
431C22×C4D3×Q811yescharacteristic
432C22×C4C2×#6711yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—41no—
441M16—41no—
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—34no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—616no—
461C2×C8—62no—
462C2×C8—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—121no—
466C4⋊C4—121no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C42—62no—
472C42—62no—
473C42—121no—
474C42—121no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3C4×C22⋊C411yescharacteristic
483C22×C3#38011yescharacteristic
484C22×C3#38111yescharacteristic
485C22×C3#38511yescharacteristic
486C22×C3#38511yescharacteristic
487C22×C3#38611yescharacteristic
488C22×C3#38611yescharacteristic
489C2×D3—24no—
490C2×D3—44no—
491C2×D3—82no—
492C2×D3—82no—
493C3×C4C2×#9511yescharacteristic
494C3×C4C4×C22⋊C411yescharacteristic
495C3×C4#37911yescharacteristic
496C3×C4#38011yescharacteristic
497C3×C4#38511yescharacteristic
498C3×C4#38511yescharacteristic
499C3×C4#38611yescharacteristic
500C3×C4#38611yescharacteristic
501Dic3—24no—
502Dic3—82no—
503Dic3—82no—
504C23—62no—
505C23—62no—
506C23—62no—
507C23—81no—
508C23—242no—
509C23#89211yescharacteristic
510C23#89311yescharacteristic
511C23#89311yescharacteristic
512C23#89411yescharacteristic
513C23#89411yescharacteristic
514C23#89511yescharacteristic
515C23#94611yescharacteristic
516Q8—34no—
517Q8—62no—
518Q8—62no—
519Q8—62no—
520D4—34no—
521D4—64no—
522D4—64no—
523D4—64no—
524D4—64no—
525D4—62no—
526D4—62no—
527D4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—242no—
538C2×C4—242no—
539C2×C4C42×D311yescharacteristic
540C2×C4#87211yescharacteristic
541C2×C4C4×D1211yescharacteristic
542C2×C4C4×D1211yescharacteristic
543C2×C4D3×C22⋊C411yescharacteristic
544C2×C4D3×C22⋊C411yescharacteristic
545C2×C4D3×C22⋊C411yescharacteristic
546C2×C4D3×C22⋊C411yescharacteristic
547C2×C4#67⋊C411yescharacteristic
548C2×C4#67⋊C411yescharacteristic
549C2×C4#88211yescharacteristic
550C2×C4#88211yescharacteristic
551C2×C4#88311yescharacteristic
552C2×C4#88311yescharacteristic
553C2×C4#88411yescharacteristic
554C2×C4#88411yescharacteristic
555C2×C4D3×C4⋊C411yescharacteristic
556C2×C4D3×C4⋊C411yescharacteristic
557C2×C4D3×C4⋊C411yescharacteristic
558C2×C4#89611yescharacteristic
559C2×C4#89611yescharacteristic
560C2×C4#89711yescharacteristic
561C2×C4C2×(C2×D3)⋊C411yescharacteristic
562C2×C4C4×#6711yescharacteristic
563C2×C4C4×#6711yescharacteristic
564C2×C4#93711yescharacteristic
565C2×C4#93711yescharacteristic
566C2×C4#93811yescharacteristic
567C8—24no—
568C8—42no—
569C8—42no—
570C8—64no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3C22⋊#9511yescharacteristic
579D3—44no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—64no—
588C22—124no—
589C22—242no—
590C22—242no—
591C22#466411yescharacteristic
592C22#466511yescharacteristic
593C22#466511yescharacteristic
594C22#466611yescharacteristic
595C22#498711yescharacteristic
596C22#498911yescharacteristic
597C22#498911yescharacteristic
598C4—64no—
599C4—242no—
600C4—242no—
601C4D3×#9511yescharacteristic
602C4#466711yescharacteristic
603C4#466811yescharacteristic
604C4#466811yescharacteristic
605C4C4×(C2×D3)⋊C411yescharacteristic
606C4#495511yescharacteristic
607C4#495611yescharacteristic
608C4#495611yescharacteristic
609C3#1077811yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—124no—
618C2(C2×D3)⋊#9511yescharacteristic
subgroupG/H
centerC4#4955
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1738C2
socleC2×C3C22⋊#95
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1738D3
hypocenterC3#10778
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
4. C3×#1738C2semidirect
5. C2×#4559C2semidirect
6. C2×#4559C2semidirect
7. #71982C2semidirect
20. C2×#907C4semidirect
24. #5139C4semidirect
27. #5131C4semidirect
28. #5120C4semidirect
39. #1738D3semidirect
609. C3#10778semidirect

Maximal subgroups

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

subgroupindex
C2×#57452
C2×#51392
#837682
C3×#17382
C2×#45592
#719822
#107783