Finite Group Databasebeta

← C2×#76210 #10963600

Subgroups — C2×#76210 #10963600

Order 768.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#897C211yesmaximal, characteristic, complemented
2C22×#897C211yesmaximal, characteristic, complemented
3C22×#892C211yesmaximal, characteristic, complemented
4C22×C4×D12C211yesmaximal, characteristic
5C2×C3×#389C211yesmaximal, characteristic, complemented
6C2×#4991C212yesmaximal
7C2×#4987C211yesmaximal, characteristic, complemented
8C2×#4984C212yesmaximal, complemented
9C2×#4929C211yesmaximal, characteristic, complemented
10#76210C2116yesmaximal, direct product factor
11C2×#4674C212yesmaximal, complemented
12C2×#4670C212yesmaximal, complemented
13C2×#2415—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C23×Dic6C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×C4⋊Dic3C2212yes—
31C22×C4⋊Dic3C2211yescharacteristic
32C22×C4×Dic3C2212yes—
33C2×#897—28no—
34C2×#897—28no—
35C2×#897C2218yescomplemented
36C2×#897C2218yescomplemented
37C2×#892—28no—
38C2×#892C2218yescomplemented
39C2×C4×D12—28no—
40C2×C4×D12C2218yes—
41C3×#389C2218yescomplemented
42C2×C3×#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46#4991C22116yes—
47#4987C2218yescomplemented
48#4984C22116yescomplemented
49#4929C2218yescomplemented
50#4674C22116yescomplemented
51#4670C22116yescomplemented
52C22×C22⋊Q8—31no—
53C22×C22⋊Q8—31no—
54C22×C4×D4—31no—
55C22×#117—31no—
56#2415—316no—
57C2×#397—32no—
58C2×#394—32no—
59C2×#391—31no—
60C2×#389—32no—
61C2×#389D311yescharacteristic, complemented
62C2×#388—32no—
63C2×#380—31no—
64C24×D3C2311yescharacteristic
65C24×D3C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C23×Dic3C2311yescharacteristic
75C22×D12—24no—
76C22×D12—24no—
77C22×D12—24no—
78C22×D12C2314yes—
79C22×C4×D3—24no—
80C22×C4×D3—24no—
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3—24no—
84C22×C4×D3D418yes—
85C22×C4×D3C2314yes—
86C22×C4×D3C2314yes—
87C22×Dic6—24no—
88C22×Dic6—24no—
89C22×Dic6—24no—
90C22×Dic6C2314yes—
91C2×C3×C4⋊C4—24no—
92C2×C3×C4⋊C4—28no—
93C2×C3×C4⋊C4D418yescomplemented
94C2×C3×C4⋊C4C2318yes—
95C2×C3×C4⋊C4C2314yes—
96C2×C3×C4⋊C4C2314yes—
97C2×C3×C42—24no—
98C2×C3×C42C2314yes—
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4—28no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4—24no—
105C2×(C2×D3)⋊C4C2314yes—
106C2×(C2×D3)⋊C4C2318yes—
107C2×(C2×D3)⋊C4C2318yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×C4⋊Dic3—24no—
112C2×C4⋊Dic3—28no—
113C2×C4⋊Dic3—24no—
114C2×C4⋊Dic3—28no—
115C2×C4⋊Dic3C2318yes—
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2318yes—
118C2×C4⋊Dic3C2314yes—
119C2×C4×Dic3—28no—
120C2×C4×Dic3C2318yes—
121#897—232no—
122#897—232no—
123#892—232no—
124C4×D12—232no—
125C3×#95C2318yes—
126C3⋊#95C2318yes—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C23×Q8—31no—
130C23×D4—31no—
131C24×C4—31no—
132C24×C4—31no—
133C2×C22⋊Q8—38no—
134C2×C22⋊Q8—38no—
135C2×C22⋊Q8—68no—
136C2×C22⋊Q8—68no—
137C2×C4×D4—38no—
138C2×C4×D4—68no—
139C2×#117—38no—
140C2×#117—68no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—31no—
143C22×C4⋊C4—31no—
144C22×C4⋊C4—32no—
145C22×C4⋊C4C2×D312yes—
146C22×C4⋊C4C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic
148C22×C22⋊C4—32no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4—31no—
153C22×C22⋊C4—31no—
154C22×C42—32no—
155C22×C42C2×D311yescharacteristic
156#397—316no—
157#394—316no—
158#391—38no—
159#389—316no—
160#389C2×D318yescomplemented
161#388—316no—
162#380—38no—
163C2×#95—32no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95C2×D312yes—
167C24×C3C2411yescharacteristic
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—22no—
173C23×D3—22no—
174C23×D3—22no—
175C23×D3—22no—
176C23×D3—22no—
177C23×D3—22no—
178C23×D3—22no—
179C23×D3—22no—
180C23×D3C2×D414yes—
181C23×D3#4114yes—
182C23×D3C2412yes—
183C23×D3C2412yes—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4C2×D418yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4114yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2414yes—
201C22×C3×C4C2414yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2412yes—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—22no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—22no—
214C22×Dic3—24no—
215C22×Dic3C2×D414yes—
216C22×Dic3C2414yes—
217C22×Dic3C2412yes—
218C22×Dic3C2414yes—
219C22×Dic3C2412yes—
220C2×D12—232no—
221C2×D12—28no—
222C2×D12—28no—
223C2×D12—28no—
224C2×C4×D3—232no—
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C2×C4×D3—28no—
229C2×C4×D3—28no—
230C2×C4×D3—416no—
231C2×C4×D3C2×D4116yes—
232C2×Dic6—28no—
233C2×Dic6—28no—
234C2×Dic6—28no—
235C2×Dic6—416no—
236C3×C4⋊C4—28no—
237C3×C4⋊C4—216no—
238C3×C4⋊C4C2×D4116yescomplemented
239C3×C42—28no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—216no—
242(C2×D3)⋊C4—216no—
243(C2×D3)⋊C4—216no—
244(C2×D3)⋊C4—216no—
245(C2×D3)⋊C4—28no—
246C4⋊Dic3—28no—
247C4⋊Dic3—216no—
248C4⋊Dic3—28no—
249C4⋊Dic3—216no—
250C4×Dic3—216no—
251C25—31no—
252C25—31no—
253C22×Q8—34no—
254C22×Q8—64no—
255C22×Q8—64no—
256C22×Q8—64no—
257C22×D4—34no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—64no—
261C23×C4—31no—
262C23×C4—38no—
263C23×C4—34no—
264C23×C4—32no—
265C23×C4—32no—
266C23×C4—31no—
267C23×C4—34no—
268C23×C4—64no—
269C23×C4—64no—
270C23×C4—64no—
271C23×C4—64no—
272C23×C4—64no—
273C23×C4C22×D312yes—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D312yes—
276C23×C4C22×D311yescharacteristic
277C23×C4C22×D311yescharacteristic
278C22⋊Q8—632no—
279C22⋊Q8—632no—
280C4×D4—632no—
281#117—632no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—34no—
285C2×C4⋊C4—34no—
286C2×C4⋊C4—38no—
287C2×C4⋊C4—38no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—68no—
290C2×C4⋊C4—64no—
291C2×C4⋊C4—68no—
292C2×C4⋊C4D1218yescomplemented
293C2×C4⋊C4C22×D314yes—
294C2×C4⋊C4C22×D318yes—
295C2×C4⋊C4C22×D314yes—
296C2×C22⋊C4—38no—
297C2×C22⋊C4—34no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—38no—
302C2×C22⋊C4—64no—
303C2×C22⋊C4—64no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—68no—
307C2×C22⋊C4—68no—
308C2×C42—24no—
309C2×C42—38no—
310C2×C42—68no—
311C2×C42C22×D314yes—
312#95—38no—
313#95—38no—
314#95—38no—
315#95C22×D318yes—
316C23×C3C22×D412yes—
317C23×C3C22×D412yes—
318C23×C3C2×#4112yes—
319C23×C3C2×#4112yes—
320C23×C3C2×#4112yes—
321C23×C32−1+412yes—
322C23×C32−1+412yes—
323C23×C3C2511yescharacteristic
324C22×D3—28no—
325C22×D3—28no—
326C22×D3—28no—
327C22×D3—28no—
328C22×D3—28no—
329C22×D3—28no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—44no—
343C22×D3—44no—
344C22×D3—44no—
345C22×D3—44no—
346C22×D3—44no—
347C22×D3—44no—
348C22×D3—44no—
349C22×D3—44no—
350C22×D3C22×D414yes—
351C22×D3C2×#4114yes—
352C2×C3×C4—28no—
353C2×C3×C4—216no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—48no—
365C2×C3×C4—44no—
366C2×C3×C4#121116yes—
367C2×C3×C4C22×D418yes—
368C2×C3×C4C22×D414yes—
369C2×C3×C4C22×D414yes—
370C2×C3×C4C2×#4114yes—
371C2×C3×C4C2×#4114yes—
372C2×Dic3—28no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—24no—
384C2×Dic3—48no—
385C2×Dic3—44no—
386C2×Dic3—48no—
387C2×Dic3C22×D414yes—
388D12—232no—
389C4×D3—232no—
390C4×D3—416no—
391Dic6—416no—
392C24—34no—
393C24—34no—
394C24—32no—
395C24—32no—
396C24—62no—
397C24—62no—
398C24—62no—
399C24—62no—
400C24—62no—
401C24—62no—
402C24—62no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24—62no—
407C24—62no—
408C24C23×D311yescharacteristic
409C2×Q8—68no—
410C2×Q8—68no—
411C2×Q8—68no—
412C2×Q8—1216no—
413C2×D4—68no—
414C2×D4—68no—
415C2×D4—632no—
416C2×D4—68no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—316no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—32no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—632no—
436C22×C4—68no—
437C22×C4—68no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—68no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—1216no—
451C22×C4C2×D1218yes—
452C22×C4C2×D1214yes—
453C22×C4D3×D414yes—
454C22×C4#23714yes—
455C22×C4#23914yes—
456C22×C4C23×D312yes—
457C22×C4C23×D314yes—
458C22×C4C23×D314yes—
459C22×C4C23×D312yes—
460C22×C4C23×D312yes—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—68no—
464C4⋊C4—616no—
465C4⋊C4—68no—
466C4⋊C4—616no—
467C4⋊C4C2×D12116yescomplemented
468C22⋊C4—68no—
469C22⋊C4—68no—
470C22⋊C4—616no—
471C22⋊C4—616no—
472C22⋊C4—616no—
473C22⋊C4—616no—
474C42—28no—
475C42—616no—
476C22×C3C2×#12114yes—
477C22×C3#52514yes—
478C22×C3#53614yes—
479C22×C3#54714yes—
480C22×C3#54814yes—
481C22×C3#55214yes—
482C22×C3#55614yes—
483C22×C3C23×D411yescharacteristic
484C22×C3C23×D411yescharacteristic
485C22×C3C22×#4111yescharacteristic
486C22×C3C22×#4111yescharacteristic
487C22×C3C22×#4111yescharacteristic
488C22×C3C2×2−1+411yescharacteristic
489C22×C3C2×2−1+411yescharacteristic
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—42no—
497C2×D3—416no—
498C2×D3—416no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C3×C4—24no—
507C3×C4—28no—
508C3×C4—44no—
509C3×C4—42no—
510C3×C4C2×#12118yes—
511Dic3—24no—
512Dic3—44no—
513Dic3—42no—
514Dic3—44no—
515C23—34no—
516C23—34no—
517C23—68no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23—68no—
531C23—68no—
532C23—68no—
533C23—68no—
534C23—68no—
535C23—124no—
536C23—124no—
537C23—124no—
538C23—124no—
539C23—124no—
540C23—124no—
541C23—124no—
542C23—124no—
543C23C22×D1212yes—
544C23C2×D3×D412yes—
545C23C2×#23712yes—
546C23C2×#23912yes—
547C23#100712yes—
548C23D3×#4112yes—
549C23#101012yes—
550C23C24×D311yescharacteristic
551Q8—1216no—
552D4—632no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—216no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—34no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—68no—
571C2×C4—632no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—124no—
582C2×C4—128no—
583C2×C4—128no—
584C2×C4—1216no—
585C2×C4#895116yes—
586C2×C4C22×D1218yes—
587C2×C4C22×D1214yes—
588C2×C4C2×D3×D414yes—
589C2×C4C2×#23714yes—
590C2×C4C2×#23914yes—
591C2×C3#241518yescomplemented
592C2×C3C22×#12111yescharacteristic
593C2×C3C2×#52511yescharacteristic
594C2×C3C2×#53611yescharacteristic
595C2×C3C2×#54711yescharacteristic
596C2×C3C2×#54811yescharacteristic
597C2×C3C2×#55211yescharacteristic
598C2×C3C2×#55611yescharacteristic
599D3—44no—
600D3—44no—
601C22—64no—
602C22—64no—
603C22—64no—
604C22—64no—
605C22—64no—
606C22—64no—
607C22—122no—
608C22—122no—
609C22—1216no—
610C22—1216no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22C2×#89514yes—
618C22#555514yes—
619C22#557614yes—
620C22#560614yes—
621C22#562814yes—
622C22#569414yes—
623C22#573214yes—
624C22C23×D1211yescharacteristic
625C22C22×D3×D411yescharacteristic
626C22C22×#23711yescharacteristic
627C22C22×#23911yescharacteristic
628C22C2×#100711yescharacteristic
629C22C2×D3×#4111yescharacteristic
630C22C2×#101011yescharacteristic
631C4—24no—
632C4—28no—
633C4—44no—
634C4—42no—
635C4—64no—
636C4—124no—
637C4—122no—
638C4—124no—
639C4C2×#89518yes—
640C3C2×#241511yescharacteristic, complemented, Sylow
641C2—124no—
642C2—124no—
643C2#7621018yesdirect product factor
644C2C22×#89511yescharacteristic
645C2C2×#555511yescharacteristic
646C2C2×#557611yescharacteristic
647C2C2×#560611yescharacteristic
648C2C2×#562811yescharacteristic
649C2C2×#569411yescharacteristic
650C2C2×#573211yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3C2×#2415
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#897C2semidirect
2. C22×#897C2semidirect
3. C22×#892C2semidirect
5. C2×C3×#389C2semidirect
7. C2×#4987C2semidirect
8. C2×#4984C2semidirect
9. C2×#4929C2semidirect
10. #76210C2direct (N × H)
11. C2×#4674C2semidirect
12. C2×#4670C2semidirect
35. C2×#897C22semidirect
36. C2×#897C22semidirect
38. C2×#892C22semidirect
41. C3×#389C22semidirect
47. #4987C22semidirect
48. #4984C22semidirect
49. #4929C22semidirect
50. #4674C22semidirect
51. #4670C22semidirect
61. C2×#389D3semidirect
93. C2×C3×C4⋊C4D4semidirect
160. #389C2×D3semidirect
238. C3×C4⋊C4C2×D4semidirect
292. C2×C4⋊C4D12semidirect
467. C4⋊C4C2×D12semidirect
591. C2×C3#2415semidirect
640. C3C2×#2415semidirect
643. C2#76210direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8972
C22×#8922
C22×C4×D122
C2×C3×#3892
C2×#49912
C2×#49872
C2×#49842
C2×#49292
#762102
C2×#46742
C2×#46702
C2×#24153