Finite Group Databasebeta

← #10702201

Subgroups — #10702201

Order 768.

938 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
1C22×#825C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3C2×#5006C211yesmaximal, characteristic
4#82226C211yesmaximal, characteristic, complemented
5C2×C3×#344C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7C2×#4555C211yesmaximal, characteristic, complemented
8#11060—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16C2×#825—22no—
17C2×#825—22no—
18C2×#825—22no—
19#5133—22no—
20#5131C412yescomplemented
21#5120C412yescomplemented
22#5113—22no—
23#5006—22no—
24#5006—22no—
25#4972—22no—
26C2×#806—22no—
27#4946—22no—
28C4×Dic3⋊C4—22no—
29C3×#344—22no—
30#4555—22no—
31#4555—22no—
32C22×#104—31no—
33#2896—31no—
34C2×#410—31no—
35#2296—31no—
36C2×#344D311yescharacteristic, complemented
37#1733—31no—
38C2×#339—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C2×#235—21no—
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×#235D412yes—
53C2×#235Q812yes—
54C22×D12C2×C411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×M16—21no—
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×M16C2×C412yes—
68C2×C3×M16D412yes—
69C22×C3×C8—21no—
70C22×C3×C8—21no—
71C3×#117—21no—
72C3×#117—21no—
73C3×#117—21no—
74C3×#117—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42—21no—
78C2×#217—21no—
79C2×#217—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926C2×C412yes—
86#926D412yes—
87#926D412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3D412yes—
90C2×Dic3⋊C4—24no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3D412yes—
99C3⋊#95—24no—
100#825—28no—
101#821—24no—
102#816—24no—
103#806—28no—
104C22×#41—31no—
105C22×M16C2×D311yescharacteristic
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117—31no—
109C2×#117C2×D311yescharacteristic
110C22×C42—31no—
111C2×#104—62no—
112C2×#104—62no—
113C2×#104—62no—
114#419—32no—
115#418—32no—
116#417—62no—
117#416—62no—
118#410—62no—
119#410—62no—
120C2×#99—62no—
121#389—62no—
122#385—62no—
123C4×C4⋊C4—62no—
124#344—22no—
125#339—62no—
126#339—62no—
127C24×C3C2×D411yescharacteristic
128C23×D3—21no—
129C22×C3×C4—22no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×Q811yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C2×#67—22no—
145C2×#67—22no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—44no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C4⋊C412yescomplemented
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C4⋊C412yescomplemented
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16—41no—
196C3×M16—41no—
197C3×M16—41no—
198C3×M16—41no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C42—41no—
210C3×C42—41no—
211#217—22no—
212#217—22no—
213#217—22no—
214#217—41no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—41no—
218C4⋊Dic3C22⋊C414yes—
219Dic3⋊C4—48no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225C4×Dic3—41no—
226C4×Dic3—41no—
227C4×Dic3—41no—
228C2×#41—32no—
229C2×#41—32no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C2×#41—61no—
240C22×Q8—31no—
241C22×D4—31no—
242C22×D4—61no—
243C23×C4—61no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16C4×D312yes—
257C2×M16#6712yes—
258C22×C8—21no—
259C22×C8—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—32no—
265#117—32no—
266#117—32no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117—61no—
271#117—61no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—64no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4D1212yes—
279C2×C42—21no—
280C2×C42—32no—
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288#104—68no—
289#103—64no—
290#102—64no—
291#99—68no—
292#95—64no—
293C23×C3—42no—
294C23×C3C4×D411yescharacteristic
295C23×C3#12011yescharacteristic
296C23×C3#12011yescharacteristic
297C23×C3#12111yescharacteristic
298C23×C3#12111yescharacteristic
299C23×C3C22⋊Q811yescharacteristic
300C23×C3#12411yescharacteristic
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—82no—
305C2×C3×C4—44no—
306C2×C3×C4—42no—
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4C2×C4⋊C411yescharacteristic
309C2×C3×C4#11711yescharacteristic
310C2×C3×C4C4×D411yescharacteristic
311C2×C3×C4C4×D411yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4C22⋊Q811yescharacteristic
321C2×C3×C4C22⋊Q811yescharacteristic
322C2×C3×C4C22⋊Q811yescharacteristic
323C2×C3×C4C22⋊Q811yescharacteristic
324C2×C3×C4C22⋊Q811yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12311yescharacteristic
330C2×C3×C4#12411yescharacteristic
331C2×C3×C4#12411yescharacteristic
332C2×C3×C4#12611yescharacteristic
333C2×C3×C4#12611yescharacteristic
334C2×C3×C4#12711yescharacteristic
335#67—42no—
336#67—42no—
337#67—42no—
338#67—42no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—44no—
343C2×Dic3—42no—
344C2×Dic3—44no—
345C2×Dic3—44no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—82no—
351D12—22no—
352D12—22no—
353D12—41no—
354D12—41no—
355C4×D3—42no—
356C4×D3—42no—
357C4×D3—42no—
358C4×D3—42no—
359Dic6—22no—
360Dic6—22no—
361Dic6—41no—
362Dic6—41no—
363C3×C8—42no—
364C3×C8—42no—
365C24—61no—
366C24C2×D1211yescharacteristic
367#41—62no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372#41—62no—
373#41—62no—
374#41—62no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379#41—121no—
380#41—121no—
381#41—121no—
382#41—121no—
383C2×Q8—32no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×D4—32no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—121no—
398C2×D4—121no—
399C2×D4—121no—
400C2×D4—121no—
401C22×C4—22no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—124no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4C2×C4×D311yescharacteristic
416C22×C4#23511yescharacteristic
417C22×C4#23511yescharacteristic
418C22×C4#23511yescharacteristic
419C22×C4#23511yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4#23711yescharacteristic
426C22×C4D3×Q811yescharacteristic
427C22×C4#23911yescharacteristic
428C22×C4C2×#6711yescharacteristic
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C4⋊C4—34no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—62no—
447C4⋊C4—62no—
448C4⋊C4—128no—
449C4⋊C4—121no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—62no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—121no—
456C42—41no—
457C42—41no—
458C42—62no—
459C42—62no—
460C42—62no—
461C42—62no—
462C42—62no—
463C42—121no—
464C42—121no—
465C42—121no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3#38511yescharacteristic
474C22×C3#38611yescharacteristic
475C22×C3#38811yescharacteristic
476C22×C3#39211yescharacteristic
477C22×C3#39311yescharacteristic
478C22×C3#39411yescharacteristic
479C22×C3#39611yescharacteristic
480C2×D3—42no—
481C2×D3—82no—
482C2×D3—82no—
483C2×D3—82no—
484C3×C4—82no—
485C3×C4#38011yescharacteristic
486C3×C4#38511yescharacteristic
487C3×C4#38811yescharacteristic
488C3×C4#39011yescharacteristic
489C3×C4#39411yescharacteristic
490C3×C4#39611yescharacteristic
491C3×C4#39711yescharacteristic
492C3×C4#39911yescharacteristic
493Dic3—44no—
494Dic3—44no—
495Dic3—42no—
496Dic3—82no—
497C23—42no—
498C23—62no—
499C23—121no—
500C23—121no—
501C23—242no—
502C23C4×D1211yescharacteristic
503C23#87511yescharacteristic
504C23#88211yescharacteristic
505C23#88211yescharacteristic
506C23#89511yescharacteristic
507C23#89711yescharacteristic
508C23#93011yescharacteristic
509Q8—62no—
510Q8—62no—
511Q8—121no—
512Q8—121no—
513D4—62no—
514D4—62no—
515D4—122no—
516D4—122no—
517D4—122no—
518D4—122no—
519D4—121no—
520D4—121no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—124no—
527C2×C4—122no—
528C2×C4—124no—
529C2×C4—124no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—242no—
539C2×C4#87211yescharacteristic
540C2×C4#87611yescharacteristic
541C2×C4D3×C22⋊C411yescharacteristic
542C2×C4#67⋊C411yescharacteristic
543C2×C4#88311yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4#88411yescharacteristic
546C2×C4#88411yescharacteristic
547C2×C4#88411yescharacteristic
548C2×C4#88411yescharacteristic
549C2×C4#88511yescharacteristic
550C2×C4#88511yescharacteristic
551C2×C4D3×C4⋊C411yescharacteristic
552C2×C4#89311yescharacteristic
553C2×C4#89411yescharacteristic
554C2×C4#89411yescharacteristic
555C2×C4#89411yescharacteristic
556C2×C4#89611yescharacteristic
557C2×C4#89611yescharacteristic
558C2×C4#89611yescharacteristic
559C2×C4#89611yescharacteristic
560C2×C4#89811yescharacteristic
561C2×C4C4×#6711yescharacteristic
562C2×C4#92911yescharacteristic
563C2×C4#93711yescharacteristic
564C2×C4#93911yescharacteristic
565C2×C4#94011yescharacteristic
566C2×C4#94611yescharacteristic
567C8—42no—
568C8—42no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3#143911yescharacteristic
576C2×C3#190612yes—
577D3—82no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—122no—
586C22—242no—
587C22—242no—
588C22—242no—
589C22#466511yescharacteristic
590C22#466811yescharacteristic
591C22#467211yescharacteristic
592C22#467311yescharacteristic
593C22#494011yescharacteristic
594C22#496011yescharacteristic
595C22#499411yescharacteristic
596C4—82no—
597C4—124no—
598C4—122no—
599C4—124no—
600C4—242no—
601C4#466711yescharacteristic
602C4#466911yescharacteristic
603C4#467511yescharacteristic
604C4#467611yescharacteristic
605C4#495711yescharacteristic
606C4#495911yescharacteristic
607C4#498911yescharacteristic
608C4#499311yescharacteristic
609C3#1106011yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—242no—
617C2#6903912yes—
618C2#6942811yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#11060
LCSG ▷ C22×C3×C4 ▷ 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. #82226C2semidirect
5. C2×C3×#344C2semidirect
6. #71982C2semidirect
7. C2×#4555C2semidirect
20. #5131C4semidirect
21. #5120C4semidirect
36. C2×#344D3semidirect
178. C2×D12C4⋊C4semidirect
190. C2×Dic6C4⋊C4semidirect
609. C3#11060semidirect

Maximal subgroups

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

subgroupindex
C22×#8252
#837682
C2×#50062
#822262
C2×C3×#3442
#719822
C2×#45552
#110603