Finite Group Databasebeta

← #10701891

Subgroups — #10701891

Order 768.

1270 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3C2×#5225C211yesmaximal, characteristic, complemented
4C2×#5139C211yesmaximal, characteristic
5C2×#5005C211yesmaximal, characteristic
6#71987C211yesmaximal, characteristic, complemented
7#71593C211yesmaximal, characteristic, complemented
8#11007—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×#235C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13#5816—22no—
14#5816—22no—
15#5816—22no—
16#5816—22no—
17C2×#947C412yescomplemented
18C2×#946—22no—
19C2×#944C412yescomplemented
20C2×#929—22no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#419—22no—
25C3×#419—22no—
26C2×C3×#103—22no—
27#5225—22no—
28#5225—22no—
29C2×#835—22no—
30#5139—22no—
31#5139—22no—
32C2×#825—22no—
33#5005—22no—
34#5005—22no—
35C2×#805—22no—
36#4554—22no—
37#4527—22no—
38C2×#536—31no—
39C2×#421—31no—
40C2×#421—31no—
41C2×#419—31no—
42C2×#419D311yescharacteristic, complemented
43#1742—31no—
44#1740—31no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—21no—
54C22×C3×Q8—21no—
55C22×C3×Q8—21no—
56C22×C3×Q8C2×C412yes—
57C22×C3×Q8D412yes—
58C22×C3×Q8D412yes—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—21no—
69C2×#235—21no—
70C2×#235—21no—
71C2×#235—21no—
72C2×#235D412yes—
73C2×#235D412yes—
74C22×D12C2×C411yescharacteristic
75C22×C4×D3—21no—
76C22×Dic6C2×C411yescharacteristic
77C2×C3×M16—21no—
78C2×C3×M16—21no—
79C2×C3×M16—21no—
80C2×C3×M16—21no—
81C2×C3×M16—21no—
82C2×C3×M16D412yes—
83C22×C3×C8—21no—
84C3×#117—21no—
85C3×#117—21no—
86C3×#117—21no—
87C3×#117D412yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×#217—21no—
92C2×#217D412yes—
93#947—24no—
94#947—24no—
95#946—28no—
96#946—28no—
97#944—24no—
98#944—24no—
99#929—28no—
100#929—28no—
101C2×(C2×D3)⋊C4—24no—
102#926—21no—
103#926—21no—
104#926—21no—
105#926—21no—
106#926—21no—
107#926—21no—
108#926—21no—
109#926D412yes—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D412yes—
112C2×Dic3⋊C4—24no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2×C412yes—
115C2×#208—21no—
116C2×#208—21no—
117C2×#208—21no—
118C2×#208—21no—
119C2×#208—21no—
120C2×#208D412yes—
121C22×C3⋊C8—21no—
122C3×#103—28no—
123#835—28no—
124#825—28no—
125#805—28no—
126C22×#41—31no—
127C23×Q8C2×D311yescharacteristic
128C22×M16—31no—
129C22×M16C2×D311yescharacteristic
130#536—62no—
131#536—62no—
132#536—62no—
133#536—62no—
134C2×C4⋊Q8—32no—
135C2×#124—32no—
136C2×#123—62no—
137C2×C22⋊Q8—62no—
138C2×#117—31no—
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141#421—62no—
142#421—62no—
143#421—62no—
144#421—62no—
145C2×#104—62no—
146C2×#104—62no—
147#419—22no—
148#419—22no—
149#419—62no—
150#419—62no—
151C2×#103—22no—
152C2×#103—62no—
153#343—62no—
154#342—62no—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—41no—
173C2×C3×Q8—41no—
174C2×C3×Q8—41no—
175C2×C3×Q8—41no—
176C2×C3×Q8—41no—
177C2×C3×Q8—41no—
178C2×C3×Q8—41no—
179C2×C3×Q8—41no—
180C2×C3×Q8—41no—
181C2×C3×Q8—41no—
182C2×C3×Q8—41no—
183C2×C3×Q8—41no—
184C2×C3×Q8C22⋊C414yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—41no—
189C22×C3×C4—41no—
190C22×C3×C4—41no—
191C22×C3×C4—41no—
192C22×C3×C4—41no—
193C22×C3×C4C22×C411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4#4111yescharacteristic
204C22×C3×C4#4111yescharacteristic
205C22×C3×C4#4111yescharacteristic
206C22×C3×C4#4111yescharacteristic
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—41no—
210C2×#67—41no—
211C2×#67—41no—
212C2×#67—41no—
213C22×Dic3—21no—
214C22×Dic3—41no—
215C22×Dic3C22⋊C414yes—
216#235—22no—
217#235—22no—
218#235—22no—
219#235—22no—
220#235—22no—
221#235—22no—
222#235—22no—
223#235—22no—
224#235—41no—
225#235—41no—
226#235—41no—
227#235—41no—
228#235—41no—
229#235—41no—
230#235—41no—
231#235—41no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12C22⋊C412yes—
238C2×C4×D3—22no—
239C2×C4×D3—22no—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×C4×D3—41no—
243C2×C4×D3—41no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6C22⋊C412yes—
250C3×M16—22no—
251C3×M16—22no—
252C3×M16—41no—
253C3×M16—41no—
254C2×C3×C8—22no—
255C2×C3×C8—41no—
256C2×C3×C8—41no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—41no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—41no—
261C3×C42—22no—
262C3×C42—41no—
263#217—22no—
264#217—22no—
265#217—22no—
266#217—41no—
267(C2×D3)⋊C4—216no—
268C4⋊Dic3—22no—
269C4⋊Dic3—22no—
270C4⋊Dic3—22no—
271C4⋊Dic3—41no—
272Dic3⋊C4—216no—
273C4×Dic3—22no—
274C4×Dic3—22no—
275C4×Dic3—22no—
276C4×Dic3—41no—
277#208—22no—
278#208—22no—
279#208—41no—
280#208—41no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—41no—
283C2×C3⋊C8—41no—
284C2×#41—32no—
285C2×#41—32no—
286C2×#41—61no—
287C2×#41—61no—
288C2×#41—61no—
289C2×#41—61no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41—61no—
293C2×#41—61no—
294C2×#41—61no—
295C2×#41—61no—
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8—21no—
302C22×Q8—21no—
303C22×Q8—21no—
304C22×Q8—21no—
305C22×Q8—21no—
306C22×Q8—21no—
307C22×Q8—31no—
308C22×Q8C4×D312yes—
309C22×Q8D1212yes—
310C22×Q8#6712yes—
311C22×D4—31no—
312C22×D4—61no—
313C23×C4—21no—
314C23×C4—61no—
315C23×C4C22×D311yescharacteristic
316C2×M16—21no—
317C2×M16—21no—
318C2×M16—21no—
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—32no—
322C2×M16—61no—
323C2×M16—61no—
324C2×M16—61no—
325C2×M16—61no—
326C2×M16—61no—
327C2×M16#6712yes—
328C22×C8—21no—
329C22×C8—61no—
330C4⋊Q8—64no—
331C4⋊Q8—64no—
332#124—64no—
333#124—64no—
334#123—68no—
335#123—68no—
336C22⋊Q8—68no—
337C22⋊Q8—68no—
338#117—21no—
339#117—21no—
340#117—21no—
341#117—32no—
342#117—61no—
343#117—61no—
344#117—61no—
345#117—61no—
346#117—61no—
347#117—61no—
348#117—61no—
349#117#6712yes—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—64no—
353C2×C4⋊C4—61no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—64no—
357C2×C22⋊C4—61no—
358C2×C42—21no—
359C2×C42—32no—
360C2×C42—61no—
361#104—68no—
362#104—68no—
363#103—28no—
364#103—68no—
365C23×C3—42no—
366C23×C3C2×C22⋊C411yescharacteristic
367C23×C3#12011yescharacteristic
368C23×C3#12011yescharacteristic
369C23×C3#12111yescharacteristic
370C23×C3#12111yescharacteristic
371C23×C3#12411yescharacteristic
372C23×C3#12711yescharacteristic
373C22×D3—22no—
374C22×D3—44no—
375C22×D3—41no—
376C22×D3—41no—
377C3×Q8—24no—
378C3×Q8—24no—
379C3×Q8—42no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×Q8—42no—
383C3×Q8—42no—
384C3×Q8—42no—
385C2×C3×C4—24no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—82no—
394C2×C3×C4C2×C22⋊C411yescharacteristic
395C2×C3×C4C2×C22⋊C411yescharacteristic
396C2×C3×C4C4×D411yescharacteristic
397C2×C3×C4C4×D411yescharacteristic
398C2×C3×C4C4×D411yescharacteristic
399C2×C3×C4C4×D411yescharacteristic
400C2×C3×C4#12011yescharacteristic
401C2×C3×C4#12011yescharacteristic
402C2×C3×C4#12011yescharacteristic
403C2×C3×C4#12011yescharacteristic
404C2×C3×C4#12011yescharacteristic
405C2×C3×C4#12011yescharacteristic
406C2×C3×C4#12111yescharacteristic
407C2×C3×C4#12111yescharacteristic
408C2×C3×C4#12111yescharacteristic
409C2×C3×C4#12111yescharacteristic
410C2×C3×C4#12111yescharacteristic
411C2×C3×C4#12111yescharacteristic
412C2×C3×C4#12111yescharacteristic
413C2×C3×C4#12111yescharacteristic
414C2×C3×C4#12111yescharacteristic
415C2×C3×C4#12111yescharacteristic
416C2×C3×C4#12311yescharacteristic
417C2×C3×C4#12311yescharacteristic
418C2×C3×C4#12311yescharacteristic
419C2×C3×C4#12311yescharacteristic
420C2×C3×C4#12411yescharacteristic
421C2×C3×C4#12711yescharacteristic
422#67—42no—
423#67—42no—
424#67—42no—
425#67—42no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—44no—
431C2×Dic3—42no—
432C2×Dic3—82no—
433D12—22no—
434D12—22no—
435D12—41no—
436D12—41no—
437C4×D3—42no—
438C4×D3—42no—
439C4×D3—42no—
440C4×D3—42no—
441Dic6—22no—
442Dic6—22no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—42no—
446C3⋊C8—42no—
447C24—61no—
448C24D3×D411yescharacteristic
449#41—62no—
450#41—62no—
451#41—62no—
452#41—62no—
453#41—62no—
454#41—62no—
455#41—62no—
456#41—62no—
457#41—121no—
458#41—121no—
459#41—121no—
460#41—121no—
461#41—121no—
462#41—121no—
463#41—121no—
464#41—121no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—32no—
481C2×Q8—41no—
482C2×Q8—41no—
483C2×Q8—41no—
484C2×Q8—41no—
485C2×Q8—41no—
486C2×Q8—41no—
487C2×Q8—41no—
488C2×Q8—41no—
489C2×Q8—41no—
490C2×Q8—41no—
491C2×Q8—41no—
492C2×Q8—41no—
493C2×Q8—61no—
494C2×Q8—61no—
495C2×Q8—61no—
496C2×Q8—61no—
497C2×Q8—61no—
498C2×Q8(C2×D3)⋊C414yes—
499C2×D4—32no—
500C2×D4—62no—
501C2×D4—62no—
502C2×D4—61no—
503C2×D4—61no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—121no—
508C2×D4—121no—
509C2×D4—121no—
510C2×D4—121no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—34no—
515C22×C4—41no—
516C22×C4—41no—
517C22×C4—41no—
518C22×C4—41no—
519C22×C4—41no—
520C22×C4—61no—
521C22×C4—62no—
522C22×C4—62no—
523C22×C4—121no—
524C22×C4—121no—
525C22×C4—121no—
526C22×C4—121no—
527C22×C4—121no—
528C22×C4C2×C4×D311yescharacteristic
529C22×C4C2×D1211yescharacteristic
530C22×C4#23511yescharacteristic
531C22×C4#23511yescharacteristic
532C22×C4D3×D411yescharacteristic
533C22×C4D3×D411yescharacteristic
534C22×C4D3×D411yescharacteristic
535C22×C4D3×D411yescharacteristic
536C22×C4D3×D411yescharacteristic
537C22×C4#23711yescharacteristic
538C22×C4#23711yescharacteristic
539C22×C4C2×#6711yescharacteristic
540C22×C4C2×#6711yescharacteristic
541C22×C4C2×#6711yescharacteristic
542M16—22no—
543M16—22no—
544M16—41no—
545M16—41no—
546M16—62no—
547M16—62no—
548M16—121no—
549M16—121no—
550C2×C8—22no—
551C2×C8—41no—
552C2×C8—41no—
553C2×C8—62no—
554C2×C8—121no—
555C2×C8—121no—
556C4⋊C4—22no—
557C4⋊C4—41no—
558C4⋊C4—616no—
559C4⋊C4—62no—
560C4⋊C4—62no—
561C4⋊C4—62no—
562C4⋊C4—121no—
563C22⋊C4—22no—
564C22⋊C4—41no—
565C22⋊C4—616no—
566C22⋊C4—62no—
567C22⋊C4—62no—
568C22⋊C4—62no—
569C22⋊C4—121no—
570C42—22no—
571C42—41no—
572C42—62no—
573C42—62no—
574C42—62no—
575C42—121no—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—41no—
580C22×C3—41no—
581C22×C3—41no—
582C22×C3—41no—
583C22×C3#37911yescharacteristic
584C22×C3#39011yescharacteristic
585C22×C3#39011yescharacteristic
586C22×C3#39211yescharacteristic
587C22×C3#39211yescharacteristic
588C22×C3#39411yescharacteristic
589C22×C3#39411yescharacteristic
590C2×D3—44no—
591C2×D3—42no—
592C2×D3—82no—
593C2×D3—82no—
594C3×C4—44no—
595C3×C4—82no—
596C3×C4#38611yescharacteristic
597C3×C4#38611yescharacteristic
598C3×C4#38611yescharacteristic
599C3×C4#38611yescharacteristic
600C3×C4#39211yescharacteristic
601C3×C4#39211yescharacteristic
602C3×C4#39411yescharacteristic
603C3×C4#39411yescharacteristic
604Dic3—44no—
605Dic3—42no—
606Dic3—82no—
607C23—42no—
608C23—62no—
609C23—124no—
610C23—121no—
611C23—121no—
612C23D3×C22⋊C411yescharacteristic
613C23#88211yescharacteristic
614C23#88411yescharacteristic
615C23#88511yescharacteristic
616C23#93711yescharacteristic
617C23#93911yescharacteristic
618C23#94011yescharacteristic
619Q8—24no—
620Q8—24no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626Q8—42no—
627Q8—62no—
628Q8—62no—
629Q8—121no—
630Q8—121no—
631D4—62no—
632D4—62no—
633D4—122no—
634D4—122no—
635D4—122no—
636D4—122no—
637D4—121no—
638D4—121no—
639C2×C4—24no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—64no—
648C2×C4—62no—
649C2×C4—82no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—124no—
657C2×C4—122no—
658C2×C4—242no—
659C2×C4D3×C22⋊C411yescharacteristic
660C2×C4#67⋊C411yescharacteristic
661C2×C4#67⋊C411yescharacteristic
662C2×C4#88211yescharacteristic
663C2×C4#88211yescharacteristic
664C2×C4#88311yescharacteristic
665C2×C4#88311yescharacteristic
666C2×C4#88411yescharacteristic
667C2×C4#88411yescharacteristic
668C2×C4#88411yescharacteristic
669C2×C4#88511yescharacteristic
670C2×C4#88611yescharacteristic
671C2×C4C2×(C2×D3)⋊C411yescharacteristic
672C2×C4C4×#6711yescharacteristic
673C2×C4C4×#6711yescharacteristic
674C2×C4#92911yescharacteristic
675C2×C4#93011yescharacteristic
676C2×C4#93011yescharacteristic
677C2×C4#93711yescharacteristic
678C2×C4#93711yescharacteristic
679C2×C4#93711yescharacteristic
680C2×C4#93811yescharacteristic
681C2×C4#93811yescharacteristic
682C2×C4#93911yescharacteristic
683C2×C4#93911yescharacteristic
684C2×C4#93911yescharacteristic
685C2×C4#94011yescharacteristic
686C2×C4#95311yescharacteristic
687C8—42no—
688C8—122no—
689C2×C3—21no—
690C2×C3—21no—
691C2×C3—21no—
692C2×C3—21no—
693C2×C3—21no—
694C2×C3—21no—
695C2×C3#143811yescharacteristic
696C2×C3#179612yes—
697D3—82no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—122no—
706C22—124no—
707C22—242no—
708C22—242no—
709C22#495511yescharacteristic
710C22#495711yescharacteristic
711C22#495711yescharacteristic
712C22#495911yescharacteristic
713C22#496011yescharacteristic
714C22#496111yescharacteristic
715C22#522311yescharacteristic
716C4—44no—
717C4—82no—
718C4—122no—
719C4—124no—
720C4—242no—
721C4#495611yescharacteristic
722C4#495611yescharacteristic
723C4#495811yescharacteristic
724C4#495911yescharacteristic
725C4#496011yescharacteristic
726C4#496111yescharacteristic
727C4#521311yescharacteristic
728C4#522311yescharacteristic
729C3#1100711yescharacteristic, complemented, Sylow
730C2—21no—
731C2—21no—
732C2—21no—
733C2—21no—
734C2—21no—
735C2—21no—
736C2—242no—
737C2#7034712yes—
738C2#7155011yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#419C2
socleC22×C3#379
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#419D3
hypocenterC3#11007
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. C2×#5225C2semidirect
6. #71987C2semidirect
7. #71593C2semidirect
17. C2×#947C4semidirect
19. C2×#944C4semidirect
42. C2×#419D3semidirect
729. C3#11007semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×C3×#4192
C2×#52252
C2×#51392
C2×#50052
#719872
#715932
#110073