Finite Group Databasebeta

← #10702203

Subgroups — #10702203

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
2C2×#5006C211yesmaximal, characteristic
3#82227C211yesmaximal, characteristic, complemented
4#82213C211yesmaximal, characteristic
5C3×#1733C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7C2×#4532C211yesmaximal, characteristic, complemented
8#11060—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×C4×Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×#825—22no—
17C2×#825—22no—
18C2×#825—22no—
19#5006—22no—
20#5006—22no—
21#5004C412yescomplemented
22#5002—22no—
23#5001C412yescomplemented
24#4984—22no—
25#4966—22no—
26C2×#806—22no—
27#4950—22no—
28C4×C4⋊Dic3—22no—
29C3×#339—22no—
30#4556—22no—
31#4532—22no—
32C22×#104—31no—
33#2896—31no—
34C2×#410—31no—
35#2296—31no—
36C2×#344—31no—
37#1733D311yescharacteristic, complemented
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×M16D412yes—
63C22×C3×C8—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C3×#117C2×C412yes—
70C3×#117D412yes—
71C3×#117D412yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×#217D412yes—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—21no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3D412yes—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208C2×C412yes—
98C22×C3⋊C8—21no—
99C3⋊#95—24no—
100#825—28no—
101#810—24no—
102#809—24no—
103#806—28no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117C2×D311yescharacteristic
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—62no—
125#339—22no—
126#339—62no—
127C24×C3C2×D411yescharacteristic
128C23×D3—21no—
129C22×C3×C4—22no—
130C22×C3×C4—41no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×Q811yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—22no—
146C2×#67—22no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—44no—
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—41no—
194C3×M16—41no—
195C2×C3×C8—22no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4C22⋊C414yes—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—41no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—41no—
210#217—22no—
211#217—22no—
212C4⋊Dic3—48no—
213C4⋊Dic3—41no—
214C4⋊Dic3—41no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—41no—
218C4×Dic3—41no—
219C4×Dic3—41no—
220C4×Dic3—41no—
221#208—22no—
222#208—22no—
223#208—41no—
224#208—41no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—41no—
227C2×C3⋊C8—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—32no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16D1212yes—
258C22×C8—21no—
259C22×C8—61no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117C4×D312yes—
270#117D1212yes—
271#117#6712yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4D1212yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—32no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—32no—
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—24no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—82no—
309C2×C3×C4C2×C22⋊C411yescharacteristic
310C2×C3×C4C2×C4⋊C411yescharacteristic
311C2×C3×C4#11711yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4C4×D411yescharacteristic
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4C22⋊Q811yescharacteristic
323C2×C3×C4C22⋊Q811yescharacteristic
324C2×C3×C4C22⋊Q811yescharacteristic
325C2×C3×C4C22⋊Q811yescharacteristic
326C2×C3×C4C22⋊Q811yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12311yescharacteristic
330C2×C3×C4#12311yescharacteristic
331C2×C3×C4#12311yescharacteristic
332C2×C3×C4#12411yescharacteristic
333C2×C3×C4#12411yescharacteristic
334C2×C3×C4#12611yescharacteristic
335C2×C3×C4#12611yescharacteristic
336C2×C3×C4#12711yescharacteristic
337#67—42no—
338#67—42no—
339#67—42no—
340#67—42no—
341C2×Dic3—24no—
342C2×Dic3—22no—
343C2×Dic3—42no—
344C2×Dic3—44no—
345C2×Dic3—44no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—44no—
350C2×Dic3—42no—
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—
366C24D3×D411yescharacteristic
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—41no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—124no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4C2×C4×D311yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4#23511yescharacteristic
420C22×C4#23511yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4#23711yescharacteristic
424C22×C4#23711yescharacteristic
425C22×C4#23711yescharacteristic
426C22×C4D3×Q811yescharacteristic
427C22×C4#23911yescharacteristic
428C22×C4C2×#6711yescharacteristic
429M16—22no—
430M16—22no—
431M16—41no—
432M16—41no—
433M16—62no—
434M16—62no—
435M16—121no—
436M16—121no—
437C2×C8—22no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—62no—
441C2×C8—121no—
442C2×C8—121no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—128no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C4⋊C4(C2×D3)⋊C414yes—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—41no—
454C22⋊C4—62no—
455C22⋊C4—62no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42—62no—
461C42—62no—
462C42—121no—
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—44no—
485C3×C4—82no—
486C3×C4#38011yescharacteristic
487C3×C4#38511yescharacteristic
488C3×C4#38811yescharacteristic
489C3×C4#39011yescharacteristic
490C3×C4#39411yescharacteristic
491C3×C4#39611yescharacteristic
492C3×C4#39711yescharacteristic
493C3×C4#39911yescharacteristic
494Dic3—44no—
495Dic3—42no—
496Dic3—82no—
497C23—42no—
498C23—62no—
499C23—121no—
500C23—121no—
501C23—242no—
502C23#67⋊C411yescharacteristic
503C23#88211yescharacteristic
504C23#88211yescharacteristic
505C23#88511yescharacteristic
506C23#89511yescharacteristic
507C23#89611yescharacteristic
508C23#93911yescharacteristic
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—24no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—82no—
527C2×C4—122no—
528C2×C4—124no—
529C2×C4—124no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—124no—
538C2×C4—122no—
539C2×C4C4×D1211yescharacteristic
540C2×C4C4×D1211yescharacteristic
541C2×C4#87411yescharacteristic
542C2×C4#87511yescharacteristic
543C2×C4#67⋊C411yescharacteristic
544C2×C4#88211yescharacteristic
545C2×C4#88311yescharacteristic
546C2×C4#88311yescharacteristic
547C2×C4#88511yescharacteristic
548C2×C4#88611yescharacteristic
549C2×C4#88611yescharacteristic
550C2×C4#88611yescharacteristic
551C2×C4D3×C4⋊C411yescharacteristic
552C2×C4#89211yescharacteristic
553C2×C4#89511yescharacteristic
554C2×C4#89511yescharacteristic
555C2×C4#89611yescharacteristic
556C2×C4#89711yescharacteristic
557C2×C4#89711yescharacteristic
558C2×C4#89711yescharacteristic
559C2×C4#89811yescharacteristic
560C2×C4#89811yescharacteristic
561C2×C4C2×(C2×D3)⋊C411yescharacteristic
562C2×C4#93011yescharacteristic
563C2×C4#93011yescharacteristic
564C2×C4#93811yescharacteristic
565C2×C4#93911yescharacteristic
566C2×C4#94611yescharacteristic
567C8—42no—
568C8—122no—
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#466811yescharacteristic
590C22#467011yescharacteristic
591C22#467111yescharacteristic
592C22#467311yescharacteristic
593C22#495811yescharacteristic
594C22#496111yescharacteristic
595C22#499411yescharacteristic
596C4—44no—
597C4—82no—
598C4—124no—
599C4—122no—
600C4—242no—
601C4#466811yescharacteristic
602C4#467011yescharacteristic
603C4#467411yescharacteristic
604C4#467711yescharacteristic
605C4#493911yescharacteristic
606C4#496111yescharacteristic
607C4#498711yescharacteristic
608C4#499411yescharacteristic
609C3#1106011yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—242no—
617C2#6942711yescharacteristic
618C2#7039512yes—
subgroupG/H
centerC2×C4#939
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1733C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1733D3
hypocenterC3#11060
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
3. #82227C2semidirect
5. C3×#1733C2semidirect
6. C2×#4556C2semidirect
7. C2×#4532C2semidirect
21. #5004C4semidirect
23. #5001C4semidirect
37. #1733D3semidirect
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
C2×#50062
#822272
#822132
C3×#17332
C2×#45562
C2×#45322
#110603