Finite Group Databasebeta

← C2×#81805 #11670332

Subgroups — C2×#81805 #11670332

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#918C212yesmaximal, complemented
2C2×#5727C211yesmaximal, characteristic, complemented
3C2×#5726C211yesmaximal, characteristic
4C2×C3×#494C211yesmaximal, characteristic, complemented
5C2×#5189C212yesmaximal, complemented
6C2×#5187C212yesmaximal, complemented
7C2×#5186C212yesmaximal, complemented
8C2×#5090C211yesmaximal, characteristic, complemented
9C2×#5083C211yesmaximal, characteristic, complemented
10#81805C2116yesmaximal, direct product factor
11C2×C4×#204C211yesmaximal, characteristic, complemented
12C2×#4688C211yesmaximal, characteristic, complemented
13C2×#3276—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×D3×Q8C2212yes—
16C22×C3×Q16C2212yes—
17C22×C3×Q16C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×#946C2212yes—
21C2×#946C2212yes—
22C22×#216C2212yes—
23C22×#215C2212yes—
24C2×#918—216no—
25C2×#918C22116yescomplemented
26C22×#204C2211yescharacteristic
27C22×ΓL(1,𝔽25)C2212yes—
28#5727—24no—
29#5727C2218yescomplemented
30#5726—24no—
31#5726C2218yes—
32C2×#896—22no—
33C2×#895—21no—
34C2×#894—22no—
35C2×#893—21no—
36C2×D3×C4⋊C4—21no—
37C2×#888—21no—
38C2×C4×D12C2211yescharacteristic
39C2×C4×D12C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#494C2218yescomplemented
42C2×C3×C4×C8C2211yescharacteristic
43C2×#835C2212yes—
44C2×#835C2212yes—
45#5189C22116yescomplemented
46#5187C22116yescomplemented
47#5186C22116yescomplemented
48C2×#821C2211yescharacteristic
49C2×(C2×D3)⋊C8C2212yes—
50C2×#817C2211yescharacteristic
51C2×#816C2211yescharacteristic
52#5090C2218yescomplemented
53#5083C2218yescomplemented
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C4×#204C2218yescomplemented
56#4688C2218yescomplemented
57C22×#137—32no—
58C2×D4×Q8—31no—
59C2×#547—31no—
60#3276—316no—
61C2×#494D311yescharacteristic, complemented
62C2×#470—32no—
63C2×#462—31no—
64C2×#461—31no—
65C2×#452—32no—
66C2×#451—32no—
67C2×C4×SD16—31no—
68C2×#436—31no—
69C22×C3×Q8C2312yes—
70C22×C3×Q8C2312yes—
71C2×#239—24no—
72C2×#239—28no—
73C2×#239—44no—
74C2×#239C2318yes—
75C2×D3×Q8—24no—
76C2×D3×Q8—28no—
77C2×D3×Q8—44no—
78C2×D3×Q8C2318yes—
79C22×D12—22no—
80C22×D12D411yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—22no—
83C22×C4×D3—21no—
84C22×C4×D3—22no—
85C22×C4×D3C2312yes—
86C22×Dic6—22no—
87C22×Dic6C2311yescharacteristic
88C2×C3×Q16—28no—
89C2×C3×Q16—44no—
90C2×C3×Q16C2318yes—
91C2×C3×Q16C2318yes—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C3×C4⋊Q8—22no—
95C3×C4⋊Q8—22no—
96C3×C4⋊Q8C2314yes—
97C3×C4⋊Q8C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C42C2311yescharacteristic
103#946—24no—
104#946—24no—
105#946C2318yes—
106#946C2318yes—
107C2×#216—28no—
108C2×#216C2318yes—
109C2×#215—28no—
110C2×#215C2318yes—
111C2×(C2×D3)⋊C4—22no—
112C2×(C2×D3)⋊C4—22no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4D412yes—
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—22no—
120C2×Dic3⋊C4—22no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8C2312yes—
123#918—264no—
124C2×#204D418yescomplemented
125C2×#204C2314yes—
126C2×ΓL(1,𝔽25)—28no—
127C2×ΓL(1,𝔽25)C2318yes—
128#896—216no—
129#895—28no—
130#894—216no—
131#893—28no—
132D3×C4⋊C4—28no—
133#888—28no—
134C4×D12—22no—
135C4×D12D414yes—
136C4×D12C2314yes—
137C4×D12C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140C3×C4×C8C2314yes—
141#835C2318yes—
142#835C2318yes—
143#821C2314yes—
144(C2×D3)⋊C8C2318yes—
145#817C2314yes—
146#816C2314yes—
147C4⋊(C3⋊C8)C2314yes—
148C22×#41—32no—
149C23×Q8—32no—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—32no—
153C22×Q16C2×D312yes—
154C22×Q16C2×D312yes—
155C22×SD16—32no—
156C22×SD16—31no—
157C22×M16—32no—
158D4×Q8—38no—
159D4×Q8—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8—61no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—62no—
169C2×#121—61no—
170C2×C4×Q8—31no—
171C2×C4×D4—31no—
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174C22×C4⋊C4—61no—
175#494C2×D318yescomplemented
176#470—316no—
177#462—38no—
178#461—38no—
179#452—316no—
180#451—316no—
181C4×SD16—38no—
182#436—38no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—32no—
187C2×#103—31no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8C2×D311yescharacteristic
191C23×D3—21no—
192C23×D3C2×D412yes—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8—24no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8C2414yes—
200C2×C3×Q8C2414yes—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D412yes—
210#239—216no—
211#239—48no—
212#239—416no—
213D3×Q8—216no—
214D3×Q8—48no—
215D3×Q8—416no—
216C2×D12—24no—
217C2×D12—21no—
218C2×D12—22no—
219C2×D12—28no—
220C2×D12—48no—
221C2×D12C2×D414yes—
222C2×D12C2×D414yes—
223C2×D12C2412yes—
224C2×C4×D3—24no—
225C2×C4×D3—24no—
226C2×C4×D3—28no—
227C2×C4×D3—24no—
228C2×C4×D3—28no—
229C2×C4×D3—48no—
230C2×C4×D3—48no—
231C2×C4×D3—44no—
232C2×C4×D3C2×D414yes—
233C2×C4×D3C2414yes—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6—28no—
237C2×Dic6—48no—
238C2×Dic6C2×D414yes—
239C2×Dic6C2412yes—
240C3×Q16—216no—
241C3×Q16—48no—
242C2×C3×C8—22no—
243C2×C3×C8C2×D414yes—
244C2×C3×C8C2412yes—
245C2×C3×C8C2412yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C42C2×D412yes—
251C3×C42C2412yes—
252#216—216no—
253#215—216no—
254(C2×D3)⋊C4—28no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4C2×D418yes—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D414yes—
261C4⋊Dic3C2412yes—
262C4⋊Dic3C2412yes—
263Dic3⋊C4—24no—
264Dic3⋊C4—28no—
265Dic3⋊C4—28no—
266C4×Dic3—24no—
267C2×C3⋊C8—24no—
268C2×C3⋊C8C2414yes—
269#204C2×D4116yescomplemented
270ΓL(1,𝔽25)—216no—
271C2×#41—38no—
272C2×#41—64no—
273C2×#41—68no—
274C2×#41—124no—
275C22×Q8—38no—
276C22×Q8—31no—
277C22×Q8—62no—
278C22×Q8—64no—
279C22×Q8—68no—
280C22×Q8—124no—
281C22×Q8C22×D312yes—
282C22×Q8C22×D312yes—
283C22×D4—31no—
284C22×D4—31no—
285C22×D4—62no—
286C23×C4—32no—
287C23×C4—62no—
288C23×C4—62no—
289C23×C4—61no—
290#137—664no—
291C2×Q16—28no—
292C2×Q16—38no—
293C2×Q16—44no—
294C2×Q16—68no—
295C2×Q16C22×D318yes—
296C2×Q16C22×D318yes—
297C2×SD16—38no—
298C2×SD16—38no—
299C2×SD16—34no—
300C2×SD16—68no—
301C2×M16—38no—
302C2×M16—68no—
303C22×C8—32no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—68no—
309C4⋊Q8C22×D314yes—
310C4⋊Q8C22×D314yes—
311#123—616no—
312C22⋊Q8—38no—
313C22⋊Q8—38no—
314C22⋊Q8—616no—
315C22⋊Q8—64no—
316C22⋊Q8—64no—
317#121—68no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×D4—34no—
321C4×D4—34no—
322C4×D4—34no—
323C4×D4—68no—
324C4×D4—62no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—68no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—61no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—62no—
338C2×C22⋊C4—62no—
339C2×C22⋊C4—61no—
340C2×C42—61no—
341C2×C42C22×D311yescharacteristic
342#106—34no—
343C4⋊C8—34no—
344#103—38no—
345#103—38no—
346#103—34no—
347#102—34no—
348C22⋊C8—38no—
349C4×C8C22×D314yes—
350C23×C3C22×D411yescharacteristic
351C22×D3—24no—
352C22×D3—24no—
353C22×D3—24no—
354C22×D3—24no—
355C22×D3—22no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3C22×D414yes—
359C3×Q8—24no—
360C3×Q8—24no—
361C3×Q8—24no—
362C3×Q8—24no—
363C3×Q8—44no—
364C3×Q8—44no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—44no—
369C2×C3×C4—44no—
370C2×C3×C4#13714yescomplemented
371C2×C3×C4C22×D412yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D412yes—
375C2×C3×C4H(5,𝔽2)12yes—
376C2×C3×C4C2511yescharacteristic
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3C22×D414yes—
385D12—24no—
386D12—28no—
387D12—22no—
388D12—416no—
389D12C22×D414yes—
390C4×D3—28no—
391C4×D3—24no—
392C4×D3—24no—
393C4×D3—416no—
394C4×D3—416no—
395C4×D3—48no—
396Dic6—24no—
397Dic6—22no—
398Dic6—416no—
399Dic6C22×D414yes—
400C3×C8—22no—
401C3×C8C22×D414yes—
402C3⋊C8—24no—
403C24—32no—
404C24—61no—
405#41—616no—
406#41—128no—
407#41—1216no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—34no—
415C2×Q8—32no—
416C2×Q8—68no—
417C2×Q8—61no—
418C2×Q8—62no—
419C2×Q8—616no—
420C2×Q8—128no—
421C2×Q8—128no—
422C2×Q8—1216no—
423C2×Q8C23×D314yes—
424C2×Q8C23×D314yes—
425C2×D4—34no—
426C2×D4—34no—
427C2×D4—32no—
428C2×D4—68no—
429C2×D4—64no—
430C2×D4—61no—
431C2×D4—62no—
432C2×D4—128no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—68no—
439C22×C4—68no—
440C22×C4—61no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—64no—
446C22×C4—128no—
447C22×C4—128no—
448C22×C4—124no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—216no—
453Q16—48no—
454Q16—616no—
455SD16—316no—
456SD16—616no—
457M16—616no—
458C2×C8—22no—
459C2×C8—34no—
460C2×C8—64no—
461C2×C8D3×D414yes—
462C2×C8C23×D312yes—
463C2×C8C23×D312yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—34no—
469C4⋊C4—32no—
470C4⋊C4—32no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—64no—
476C22⋊C4—38no—
477C22⋊C4—68no—
478C22⋊C4—68no—
479C22⋊C4—64no—
480C42—64no—
481C42D3×D412yes—
482C42C23×D312yes—
483C22×C3D4212yes—
484C22×C3C2×#13712yes—
485C22×C3#57812yes—
486C22×C3C23×D411yescharacteristic
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—48no—
492C2×D3—48no—
493C2×D3—42no—
494C2×D3—42no—
495C3×C4—22no—
496C3×C4—44no—
497C3×C4—44no—
498C3×C4D4212yes—
499C3×C4C2×#13714yescomplemented
500C3×C4C23×D411yescharacteristic
501C3×C4C2×H(5,𝔽2)11yescharacteristic
502Dic3—24no—
503Dic3—24no—
504Dic3—42no—
505Dic3—42no—
506C23—34no—
507C23—62no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23—122no—
513C23—122no—
514C23C2×D3×D411yescharacteristic
515Q8—24no—
516Q8—24no—
517Q8—24no—
518Q8—24no—
519Q8—34no—
520Q8—44no—
521Q8—44no—
522Q8—62no—
523Q8—64no—
524Q8—1216no—
525D4—34no—
526D4—62no—
527D4—64no—
528D4—68no—
529D4—1216no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—34no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—62no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—68no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—1216no—
545C2×C4—1216no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—128no—
549C2×C4#91814yescomplemented
550C2×C4C2×D3×D412yes—
551C2×C4C2×D3×D412yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D412yes—
554C2×C4#100412yes—
555C2×C4C24×D311yescharacteristic
556C8—22no—
557C8—64no—
558C8C2×D3×D414yes—
559C2×C3#327614yescomplemented
560C2×C3C2×D4211yescharacteristic
561C2×C3C22×#13711yescharacteristic
562C2×C3C2×#57811yescharacteristic
563D3—44no—
564D3—44no—
565C22—64no—
566C22—64no—
567C22—64no—
568C22—64no—
569C22—122no—
570C22—128no—
571C22—128no—
572C22—122no—
573C22#571712yes—
574C22C2×#91812yes—
575C22#576612yes—
576C22C22×D3×D411yescharacteristic
577C4—22no—
578C4—44no—
579C4—44no—
580C4—64no—
581C4—64no—
582C4—122no—
583C4—122no—
584C4#571712yes—
585C4C2×#91814yescomplemented
586C4C22×D3×D411yescharacteristic
587C4C2×#100411yescharacteristic
588C3C2×#327611yescharacteristic, complemented, Sylow
589C2—124no—
590C2—124no—
591C2#8180514yesdirect product factor
592C2C2×#571711yescharacteristic
593C2C22×#91811yescharacteristic
594C2C2×#576611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#494C2
socleC23×C3C22×D4
prime radicalC2×#5726C2
hypercenterC2×#494D3
hypocenterC3C2×#3276
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C22×#918C2semidirect
2. C2×#5727C2semidirect
4. C2×C3×#494C2semidirect
5. C2×#5189C2semidirect
6. C2×#5187C2semidirect
7. C2×#5186C2semidirect
8. C2×#5090C2semidirect
9. C2×#5083C2semidirect
10. #81805C2direct (N × H)
11. C2×C4×#204C2semidirect
12. C2×#4688C2semidirect
25. C2×#918C22semidirect
29. #5727C22semidirect
41. C3×#494C22semidirect
45. #5189C22semidirect
46. #5187C22semidirect
47. #5186C22semidirect
52. #5090C22semidirect
53. #5083C22semidirect
55. C4×#204C22semidirect
56. #4688C22semidirect
61. C2×#494D3semidirect
124. C2×#204D4semidirect
175. #494C2×D3semidirect
269. #204C2×D4semidirect
370. C2×C3×C4#137semidirect
499. C3×C4C2×#137semidirect
549. C2×C4#918semidirect
559. C2×C3#3276semidirect
585. C4C2×#918semidirect
588. C3C2×#3276semidirect
591. C2#81805direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9182
C2×#57272
C2×#57262
C2×C3×#4942
C2×#51892
C2×#51872
C2×#51862
C2×#50902
C2×#50832
#818052
C2×C4×#2042
C2×#46882
C2×#32763