Finite Group Databasebeta

← C2×#76632 #10964413

Subgroups — C2×#76632 #10964413

Order 768.

2854 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C22×#888C211yesmaximal, characteristic, complemented
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C2×#5229C211yesmaximal, characteristic, complemented
5C2×#4987C211yesmaximal, characteristic
6C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
7C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
8#76632C2116yesmaximal, direct product factor
9C2×#4673C212yesmaximal
10C2×#4658C212yesmaximal, complemented
11C2×#4646C212yesmaximal, complemented
12C2×#4669C212yesmaximal
13C2×#2492—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×Dic6C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×#946—28no—
20C2×#946C2218yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3C2211yescharacteristic
30C22×C4×Dic3C2212yes—
31C2×#888—28no—
32C2×#888C2218yescomplemented
33C3×#391C2218yescomplemented
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36#5229C2218yescomplemented
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2211yescharacteristic
42C2×C3⋊#95C2212yes—
43#4987C2218yes—
44C4×(C2×D3)⋊C4C2218yes—
45C4×C4⋊Dic3C2218yescomplemented
46#4673C22116yes—
47#4669C22116yes—
48#4658C22116yescomplemented
49#4646C22116yescomplemented
50C22×C4⋊Q8—31no—
51C22×C22⋊Q8—31no—
52#2492—316no—
53C2×#400—32no—
54C2×#393—32no—
55C2×#391—32no—
56C2×#391—31no—
57C2×#391D311yescharacteristic, complemented
58C2×#388—32no—
59C2×#380—31no—
60C2×C4×C4⋊C4—31no—
61C2×C4×C22⋊C4—31no—
62C24×D3C2311yescharacteristic
63C22×C3×Q8—24no—
64C22×C3×Q8—24no—
65C22×C3×Q8—24no—
66C22×C3×Q8C2314yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C23×Dic3C2312yes—
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C22×C4×D3—24no—
78C22×C4×D3—24no—
79C22×C4×D3—24no—
80C22×C4×D3C2314yes—
81C22×Dic6—28no—
82C22×Dic6—28no—
83C22×Dic6—44no—
84C22×Dic6C2318yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C42—24no—
88C2×C3×C42C2314yes—
89#946—232no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—44no—
93C2×(C2×D3)⋊C4Q818yescomplemented
94C2×(C2×D3)⋊C4C2318yes—
95C2×(C2×D3)⋊C4C2314yes—
96C2×(C2×D3)⋊C4C2318yes—
97C2×(C2×D3)⋊C4C2314yes—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3D418yes—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2314yes—
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—28no—
104C2×Dic3⋊C4C2318yes—
105C2×Dic3⋊C4C2318yes—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—44no—
108C2×C4×Dic3C2314yes—
109C2×C4×Dic3C2318yes—
110#888—232no—
111C3×#95C2318yes—
112C3×#95C2318yes—
113C3⋊#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2318yes—
116C3⋊#95C2314yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C23×Q8—32no—
120C23×Q8C2×D311yescharacteristic
121C24×C4—31no—
122C2×C4⋊Q8—38no—
123C2×C4⋊Q8—68no—
124C2×C22⋊Q8—38no—
125C2×C22⋊Q8—68no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4C2×D311yescharacteristic
131C22×C22⋊C4—31no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C42—31no—
136C22×C42—32no—
137C22×C42C2×D311yescharacteristic
138#400—316no—
139#393—316no—
140#391—38no—
141#391—316no—
142#391C2×D318yescomplemented
143#388—316no—
144#380—38no—
145C4×C4⋊C4—38no—
146C4×C22⋊C4—38no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—31no—
153C2×#95C2×D312yes—
154C2×#95C2×D312yes—
155C24×C3C2411yescharacteristic
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3—22no—
162C23×D3C2×Q814yes—
163C23×D3C2412yes—
164C2×C3×Q8—28no—
165C2×C3×Q8—28no—
166C2×C3×Q8—28no—
167C2×C3×Q8—416no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×Q814yes—
181C22×C3×C4#4114yes—
182C22×C3×C4#4114yes—
183C22×C3×C4C2412yes—
184C22×C3×C4C2414yes—
185C22×C3×C4C2414yes—
186C22×C3×C4C2412yes—
187C22×C3×C4C2412yes—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3C2×D418yes—
201C22×Dic3C2×Q814yes—
202C22×Dic3#4114yes—
203C22×Dic3C2414yes—
204C22×Dic3C2412yes—
205C22×Dic3C2414yes—
206C22×Dic3C2412yes—
207C22×Dic3C2412yes—
208C2×C4×D3—28no—
209C2×C4×D3—28no—
210C2×C4×D3—28no—
211C2×C4×D3—416no—
212C2×Dic6—216no—
213C2×Dic6—216no—
214C2×Dic6—48no—
215C2×Dic6—432no—
216C3×C4⋊C4—28no—
217C3×C42—28no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—48no—
221(C2×D3)⋊C4C2×Q8116yescomplemented
222C4⋊Dic3—28no—
223C4⋊Dic3C2×D4116yes—
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226C4×Dic3—28no—
227C4×Dic3—48no—
228C25—31no—
229C22×Q8—24no—
230C22×Q8—24no—
231C22×Q8—24no—
232C22×Q8—38no—
233C22×Q8—68no—
234C22×Q8—68no—
235C22×Q8—124no—
236C22×Q8C22×D314yes—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—32no—
241C23×C4—31no—
242C23×C4—34no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—64no—
246C23×C4C22×D312yes—
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251C4⋊Q8—632no—
252C22⋊Q8—632no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—38no—
257C2×C4⋊C4—38no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—68no—
260C2×C4⋊C4—68no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4C22×D314yes—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—34no—
265C2×C22⋊C4—38no—
266C2×C22⋊C4—34no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—68no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—124no—
271C2×C42—24no—
272C2×C42—34no—
273C2×C42—38no—
274C2×C42—64no—
275C2×C42—124no—
276C2×C42C22×D314yes—
277#95—34no—
278#95—34no—
279#95—38no—
280#95—38no—
281#95—38no—
282#95—38no—
283#95C22×D318yes—
284#95C22×D318yes—
285C23×C3C22×D412yes—
286C23×C3C22×Q812yes—
287C23×C3C2×#4112yes—
288C23×C3C2×#4112yes—
289C23×C3C2×#4112yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C32−1+412yes—
292C23×C3C2511yescharacteristic
293C22×D3—28no—
294C22×D3—28no—
295C22×D3—28no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3C22×Q814yes—
307C3×Q8—416no—
308C2×C3×C4—28no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—48no—
321C2×C3×C4—48no—
322C2×C3×C4—44no—
323C2×C3×C4#124116yes—
324C2×C3×C4C22×D414yes—
325C2×C3×C4C22×Q814yes—
326C2×C3×C4C2×#4114yes—
327C2×C3×C4C2×#4114yes—
328C2×Dic3—28no—
329C2×Dic3—216no—
330C2×Dic3—28no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—44no—
344C2×Dic3—48no—
345C2×Dic3C22×D418yes—
346C2×Dic3C22×Q814yes—
347C2×Dic3C2×#4114yes—
348C4×D3—416no—
349Dic6—432no—
350C24—34no—
351C24—32no—
352C24—62no—
353C24—62no—
354C24—62no—
355C24—62no—
356C24—62no—
357C24—62no—
358C24C23×D311yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—416no—
363C2×Q8—616no—
364C2×Q8—616no—
365C2×Q8—128no—
366C2×Q8—1232no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—38no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—34no—
384C22×C4—34no—
385C22×C4—32no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—62no—
392C22×C4—64no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—68no—
399C22×C4—68no—
400C22×C4—68no—
401C22×C4—1216no—
402C22×C4#23514yes—
403C22×C4D3×D414yes—
404C22×C4#23714yes—
405C22×C4D3×Q814yes—
406C22×C4C23×D312yes—
407C22×C4C23×D314yes—
408C22×C4C23×D314yes—
409C22×C4C23×D312yes—
410C22×C4C23×D312yes—
411C4⋊C4—28no—
412C4⋊C4—316no—
413C4⋊C4—616no—
414C4⋊C4—616no—
415C4⋊C4—68no—
416C22⋊C4—316no—
417C22⋊C4—616no—
418C22⋊C4—68no—
419C22⋊C4—128no—
420C42—28no—
421C42—68no—
422C42—128no—
423C22×C3C2×#12414yes—
424C22×C3#53314yes—
425C22×C3#54214yes—
426C22×C3#54614yes—
427C22×C3D4×Q814yes—
428C22×C3#55414yes—
429C22×C3#55614yes—
430C22×C3C23×D411yescharacteristic
431C22×C3C23×Q811yescharacteristic
432C22×C3C22×#4111yescharacteristic
433C22×C3C22×#4111yescharacteristic
434C22×C3C22×#4111yescharacteristic
435C22×C3C2×H(5,𝔽2)11yescharacteristic
436C22×C3C2×2−1+411yescharacteristic
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—416no—
445C3×C4—24no—
446C3×C4—44no—
447C3×C4—44no—
448C3×C4—42no—
449C3×C4C2×#12418yes—
450Dic3—24no—
451Dic3—28no—
452Dic3—24no—
453Dic3—42no—
454Dic3—44no—
455C23—34no—
456C23—68no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—68no—
464C23—68no—
465C23—124no—
466C23—124no—
467C23—124no—
468C23—124no—
469C23C2×#23512yes—
470C23C2×D3×D412yes—
471C23C2×#23712yes—
472C23C2×D3×Q812yes—
473C23#100712yes—
474C23D3×#4112yes—
475C23#100912yes—
476C23C24×D311yescharacteristic
477Q8—416no—
478Q8—1232no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—38no—
492C2×C4—34no—
493C2×C4—34no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—68no—
498C2×C4—616no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—68no—
506C2×C4—64no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—124no—
513C2×C4—128no—
514C2×C4—1216no—
515C2×C4#885116yes—
516C2×C4C2×#23514yes—
517C2×C4C2×D3×D414yes—
518C2×C4C2×#23714yes—
519C2×C4C2×D3×Q814yes—
520C2×C3#249218yescomplemented
521C2×C3C22×#12411yescharacteristic
522C2×C3C2×#53311yescharacteristic
523C2×C3C2×#54211yescharacteristic
524C2×C3C2×#54611yescharacteristic
525C2×C3C2×D4×Q811yescharacteristic
526C2×C3C2×#55411yescharacteristic
527C2×C3C2×#55611yescharacteristic
528D3—44no—
529C22—64no—
530C22—64no—
531C22—64no—
532C22—122no—
533C22—122no—
534C22—1216no—
535C22—122no—
536C22—122no—
537C22C2×#88514yes—
538C22#557814yes—
539C22#558214yes—
540C22#563514yes—
541C22#564114yes—
542C22#567614yes—
543C22#572714yes—
544C22C22×#23511yescharacteristic
545C22C22×D3×D411yescharacteristic
546C22C22×#23711yescharacteristic
547C22C22×D3×Q811yescharacteristic
548C22C2×#100711yescharacteristic
549C22C2×D3×#4111yescharacteristic
550C22C2×#100911yescharacteristic
551C4—24no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—64no—
556C4—68no—
557C4—64no—
558C4—124no—
559C4—122no—
560C4C2×#88518yes—
561C3C2×#249211yescharacteristic, complemented, Sylow
562C2—124no—
563C2#7663218yesdirect product factor
564C2C22×#88511yescharacteristic
565C2C2×#557811yescharacteristic
566C2C2×#558211yescharacteristic
567C2C2×#563511yescharacteristic
568C2C2×#564111yescharacteristic
569C2C2×#567611yescharacteristic
570C2C2×#572711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#391C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#391D3
hypocenterC3C2×#2492
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×#888C2semidirect
3. C2×C3×#391C2semidirect
4. C2×#5229C2semidirect
6. C2×C4×C4⋊Dic3C2semidirect
8. #76632C2direct (N × H)
10. C2×#4658C2semidirect
11. C2×#4646C2semidirect
32. C2×#888C22semidirect
33. C3×#391C22semidirect
36. #5229C22semidirect
45. C4×C4⋊Dic3C22semidirect
48. #4658C22semidirect
49. #4646C22semidirect
57. C2×#391D3semidirect
93. C2×(C2×D3)⋊C4Q8semidirect
142. #391C2×D3semidirect
221. (C2×D3)⋊C4C2×Q8semidirect
520. C2×C3#2492semidirect
561. C3C2×#2492semidirect
563. C2#76632direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#8882
C2×C3×#3912
C2×#52292
C2×#49872
C2×C4×C4⋊Dic32
C2×C4×(C2×D3)⋊C42
#766322
C2×#46732
C2×#46582
C2×#46462
C2×#46692
C2×#24923