Finite Group Databasebeta

← C2×#80957 #11675598

Subgroups — C2×#80957 #11675598

Order 768.

2286 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×C4×#213C211yesmaximal, characteristic
6#80957C2116yesmaximal, direct product factor
7C2×#4890C212yesmaximal, complemented
8C2×#4874C212yesmaximal, complemented
9C2×#4846C211yesmaximal, characteristic, complemented
10C2×#4830C211yesmaximal, characteristic, complemented
11C2×#4759C212yesmaximal, complemented
12C2×#4756C212yesmaximal
13C2×Q8×D8—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C23×Dic6—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C2×#946—21no—
19C2×D4×Dic3C2212yes—
20C22×#213C2211yescharacteristic
21C2×#924—21no—
22#5727—24no—
23#5727C2218yescomplemented
24D4×Dic6—24no—
25D4×Dic6C2218yescomplemented
26C2×#896—22no—
27C2×#893C2212yes—
28C2×#888C2212yes—
29C2×#888C2212yes—
30C2×#878—22no—
31C2×C4×D12C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#474C2218yescomplemented
35C2×C3×#107C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#102C2212yes—
38C2×#832C2211yescharacteristic
39C2×#821C2212yes—
40C2×#818C2212yes—
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#5078C2218yescomplemented
44C4×#213C2218yes—
45C2×#809C2211yescharacteristic
46C2×#807C2212yes—
47C2×#807C2212yes—
48C2×#807C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4890C22116yescomplemented
51#4874C22116yescomplemented
52#4846C2218yescomplemented
53#4830C2218yescomplemented
54#4759C22116yescomplemented
55#4756C22116yes—
56C2×D4×Q8—31no—
57C2×D4×Q8—31no—
58Q8×D8—316no—
59C2×#500—32no—
60C2×#500—31no—
61C2×#474—32no—
62C2×#474—31no—
63C2×#474—32no—
64C2×#474D311yescharacteristic, complemented
65C2×C8×Q8—31no—
66C2×C4×D8—32no—
67C2×C4×D8—31no—
68C22×C3×Q8—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×Dic3—22no—
72C2×D3×Q8—28no—
73C2×D3×Q8—48no—
74C22×D12C2311yescharacteristic
75C22×C4×D3—21no—
76C22×C4×D3—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6—22no—
80C22×Dic6—28no—
81C22×Dic6—48no—
82C22×Dic6D411yescharacteristic
83C22×C3×C8C2312yes—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8C2314yes—
86C3×C4×D4—22no—
87C3×C4×D4C2314yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—22no—
94#946—28no—
95D4×Dic3—24no—
96D4×Dic3C2318yes—
97C2×#213Q818yescomplemented
98C2×#213C2314yes—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105#924—28no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4D412yes—
109C2×C4×Dic3C2312yes—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#896—216no—
113#893—24no—
114#893C2318yes—
115#888—24no—
116#888—24no—
117#888C2318yes—
118#888C2318yes—
119#878—216no—
120C4×D12—22no—
121C4×D12C2314yes—
122#869—22no—
123#869C2314yes—
124C4×Dic6D414yes—
125C4×Dic6C2314yes—
126C3×#107C2318yes—
127C3×C4⋊C8C2314yes—
128C3×#102C2318yes—
129#832C2314yes—
130#821C2318yes—
131#818C2318yes—
132Dic3⋊C8C2318yes—
133C8×Dic3C2318yes—
134#809C2314yes—
135#807C2314yes—
136#807C2318yes—
137#807C2318yes—
138C4×C3⋊C8C2314yes—
139C23×Q8—61no—
140C23×Q8—61no—
141C22×D8—31no—
142D4×Q8—38no—
143D4×Q8—38no—
144D4×Q8—64no—
145D4×Q8—64no—
146C2×C4⋊Q8—32no—
147C2×C4⋊Q8—32no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8C2×D311yescharacteristic
150C2×C22⋊Q8—62no—
151C2×C22⋊Q8—62no—
152C2×C22⋊Q8—61no—
153C2×C22⋊Q8—61no—
154C2×C4×Q8—31no—
155C2×C4×D4—32no—
156C2×C4×D4—32no—
157C2×C4×D4—31no—
158C2×C4×D4C2×D311yescharacteristic
159#500—38no—
160#500—316no—
161#474—316no—
162#474—38no—
163#474—316no—
164#474C2×D318yescomplemented
165C8×Q8—38no—
166C4×D8—38no—
167C4×D8—316no—
168C2×#107—32no—
169C2×#107—32no—
170C2×#107—31no—
171C2×#107—32no—
172C2×#107C2×D312yes—
173C2×C4⋊C8—32no—
174C2×C4⋊C8C2×D311yescharacteristic
175C2×#102—31no—
176C2×#102—32no—
177C2×#102—31no—
178C2×#102C2×D312yes—
179C2×C4×C8—32no—
180C2×C4×C8—31no—
181C24×C3—21no—
182C23×D3—21no—
183C2×C3×Q8—24no—
184C2×C3×Q8—44no—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4C2×Q814yescomplemented
188C2×C3×D4C2412yes—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—44no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—28no—
199C22×Dic3—48no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202D3×Q8—432no—
203C2×D12—21no—
204C2×D12—22no—
205C2×D12C2×Q814yescomplemented
206C2×D12C2412yes—
207C2×C4×D3—24no—
208C2×C4×D3—28no—
209C2×C4×D3—44no—
210C2×C4×D3—48no—
211C2×Dic6—24no—
212C2×Dic6—28no—
213C2×Dic6—28no—
214C2×Dic6—44no—
215C2×Dic6—48no—
216C2×Dic6—48no—
217C2×Dic6—432no—
218C2×Dic6D818yescomplemented
219C2×Dic6C2×D414yes—
220C2×C3×C8—24no—
221C2×C3×C8C2414yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4C2414yes—
226C3×C4⋊C4C2412yes—
227C3×C22⋊C4—24no—
228C3×C42C2×D412yes—
229C3×C42C2412yes—
230#217—28no—
231#213C2×Q8116yescomplemented
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—28no—
234C4⋊Dic3—22no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3C2×D414yes—
238C4⋊Dic3C2414yes—
239C4⋊Dic3C2412yes—
240Dic3⋊C4—28no—
241Dic3⋊C4—28no—
242Dic3⋊C4C2×D418yes—
243C4×Dic3C2×D414yes—
244C4×Dic3C2414yes—
245C2×C3⋊C8—22no—
246C2×C3⋊C8C2×Q814yescomplemented
247C2×C3⋊C8C2412yes—
248C2×C3⋊C8C2412yes—
249C22×Q8—21no—
250C22×Q8—31no—
251C22×Q8—62no—
252C22×Q8—62no—
253C22×Q8—68no—
254C22×Q8—68no—
255C22×Q8—61no—
256C22×Q8—128no—
257C22×Q8—128no—
258C22×D4—31no—
259C22×D4C22×D311yescharacteristic
260C23×C4—21no—
261C23×C4—62no—
262C23×C4—62no—
263C23×C4—61no—
264C2×D8—38no—
265C2×D8—34no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8C22×D312yes—
269C4⋊Q8—22no—
270C4⋊Q8—38no—
271C4⋊Q8—38no—
272C4⋊Q8—34no—
273C4⋊Q8—64no—
274C4⋊Q8—64no—
275C4⋊Q8—62no—
276C4⋊Q8C22×D314yes—
277C22⋊Q8—616no—
278C22⋊Q8—616no—
279C22⋊Q8—68no—
280C22⋊Q8—68no—
281C4×Q8—34no—
282C4×Q8—34no—
283C4×D4—22no—
284C4×D4—38no—
285C4×D4—38no—
286C4×D4—34no—
287C4×D4—64no—
288C4×D4—64no—
289C4×D4—62no—
290C4×D4C22×D314yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4C22×D312yes—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4—62no—
304C2×C22⋊C4—61no—
305C2×C42—32no—
306C2×C42C22×D311yescharacteristic
307#107—38no—
308#107—34no—
309#107—38no—
310#107—38no—
311#107C22×D318yes—
312C4⋊C8—38no—
313C4⋊C8C22×D314yes—
314#102—34no—
315#102—38no—
316#102—34no—
317#102C22×D318yes—
318C4×C8—38no—
319C4×C8—34no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C22×D411yescharacteristic
325C22×D3—24no—
326C22×D3—22no—
327C22×D3—42no—
328C22×D3—42no—
329C3×Q8—48no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4C22×Q814yescomplemented
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—42no—
337C2×C3×C4—44no—
338C2×C3×C4—48no—
339C2×C3×C4C2×D814yes—
340C2×C3×C4C22×D412yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D412yes—
343C2×C3×C4C22×Q812yes—
344C2×C3×C42−1+412yes—
345C2×C3×C4C2511yescharacteristic
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—42no—
350C2×Dic3—44no—
351C2×Dic3—416no—
352C2×Dic3C2×D818yes—
353C2×Dic3C22×D414yes—
354C2×Dic3C22×D414yes—
355D12—24no—
356D12—22no—
357D12C22×Q814yescomplemented
358C4×D3—48no—
359C4×D3—416no—
360Dic6—48no—
361Dic6—416no—
362Dic6—416no—
363Dic6C2×D8116yescomplemented
364C3×C8—24no—
365C3⋊C8—22no—
366C3⋊C8C22×Q814yescomplemented
367C24—21no—
368C24—61no—
369C2×Q8—24no—
370C2×Q8—38no—
371C2×Q8—34no—
372C2×Q8—44no—
373C2×Q8—68no—
374C2×Q8—68no—
375C2×Q8—64no—
376C2×Q8—128no—
377C2×Q8—128no—
378C2×Q8—1232no—
379C2×Q8—1232no—
380C2×Q8—124no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—34no—
384C2×D4—32no—
385C2×D4—61no—
386C2×D4—62no—
387C2×D4D3×Q814yescomplemented
388C2×D4C23×D312yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—44no—
395C22×C4—68no—
396C22×C4—68no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—64no—
400C22×C4—128no—
401C22×C4—128no—
402C22×C4—124no—
403C22×C4D3×D411yescharacteristic
404C22×C4D3×D411yescharacteristic
405C22×C4C23×D311yescharacteristic
406D8—316no—
407C2×C8—24no—
408C2×C8—34no—
409C2×C8—32no—
410C2×C8—32no—
411C2×C8—62no—
412C2×C8C23×D314yes—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—38no—
417C4⋊C4—34no—
418C4⋊C4—32no—
419C4⋊C4—34no—
420C4⋊C4—68no—
421C4⋊C4—68no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—64no—
425C4⋊C4C23×D312yes—
426C4⋊C4C23×D314yes—
427C22⋊C4—24no—
428C22⋊C4—68no—
429C22⋊C4—68no—
430C22⋊C4—64no—
431C42—34no—
432C42—34no—
433C42D3×D412yes—
434C42C23×D312yes—
435C22×C3—24no—
436C22×C3—48no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3D4×Q812yes—
440C22×C3C22×D812yes—
441C22×C3#57812yes—
442C22×C3C23×D411yescharacteristic
443C2×D3—24no—
444C2×D3—48no—
445C2×D3—42no—
446C2×D3—42no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—44no—
450C3×C4D4×Q812yes—
451C3×C4C22×D814yes—
452C3×C4C23×Q811yescharacteristic
453C3×C4C2×2−1+411yescharacteristic
454Dic3—24no—
455Dic3—42no—
456Dic3—44no—
457Dic3C22×D818yes—
458C23—24no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23—64no—
463C23—62no—
464C23—122no—
465C23—122no—
466C23C2×D3×D411yescharacteristic
467Q8—316no—
468Q8—48no—
469Q8—1216no—
470Q8—1216no—
471Q8—128no—
472D4—24no—
473D4—22no—
474D4—34no—
475D4—64no—
476D4—62no—
477D4C2×D3×Q814yescomplemented
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—34no—
482C2×C4—38no—
483C2×C4—34no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—62no—
490C2×C4—1216no—
491C2×C4—1216no—
492C2×C4—124no—
493C2×C4—122no—
494C2×C4—128no—
495C2×C4D3×D814yes—
496C2×C4C2×D3×D412yes—
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×D3×D412yes—
499C2×C4C2×D3×Q812yes—
500C2×C4#101012yes—
501C2×C4C24×D311yescharacteristic
502C8—24no—
503C8—34no—
504C8—62no—
505C2×C3—44no—
506C2×C3Q8×D814yescomplemented
507C2×C3C2×D4×Q811yescharacteristic
508C2×C3C23×D811yescharacteristic
509C2×C3C2×#57811yescharacteristic
510D3—44no—
511C22—24no—
512C22—48no—
513C22—42no—
514C22—42no—
515C22—64no—
516C22—128no—
517C22—122no—
518C22—122no—
519C22#563212yes—
520C22C2×D3×D812yes—
521C22#577912yes—
522C22C22×D3×D411yescharacteristic
523C4—22no—
524C4—38no—
525C4—42no—
526C4—44no—
527C4—64no—
528C4—124no—
529C4—122no—
530C4#563212yes—
531C4C2×D3×D814yes—
532C4C22×D3×Q811yescharacteristic
533C4C2×#101011yescharacteristic
534C3C2×Q8×D811yescharacteristic, complemented, Sylow
535C2—44no—
536C2—124no—
537C2#8095714yesdirect product factor
538C2C2×#563211yescharacteristic
539C2C22×D3×D811yescharacteristic
540C2C2×#577911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#474D3
hypocenterC3C2×Q8×D8
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#5727C2semidirect
2. C2×D4×Dic6C2semidirect
3. C2×C3×#474C2semidirect
4. C2×#5078C2semidirect
6. #80957C2direct (N × H)
7. C2×#4890C2semidirect
8. C2×#4874C2semidirect
9. C2×#4846C2semidirect
10. C2×#4830C2semidirect
11. C2×#4759C2semidirect
23. #5727C22semidirect
25. D4×Dic6C22semidirect
34. C3×#474C22semidirect
43. #5078C22semidirect
50. #4890C22semidirect
51. #4874C22semidirect
52. #4846C22semidirect
53. #4830C22semidirect
54. #4759C22semidirect
64. C2×#474D3semidirect
97. C2×#213Q8semidirect
164. #474C2×D3semidirect
187. C2×C3×D4C2×Q8semidirect
205. C2×D12C2×Q8semidirect
218. C2×Dic6D8semidirect
231. #213C2×Q8semidirect
246. C2×C3⋊C8C2×Q8semidirect
332. C3×D4C22×Q8semidirect
357. D12C22×Q8semidirect
363. Dic6C2×D8semidirect
366. C3⋊C8C22×Q8semidirect
387. C2×D4D3×Q8semidirect
477. D4C2×D3×Q8semidirect
506. C2×C3Q8×D8semidirect
534. C3C2×Q8×D8semidirect
537. C2#80957direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57272
C2×D4×Dic62
C2×C3×#4742
C2×#50782
C2×C4×#2132
#809572
C2×#48902
C2×#48742
C2×#48462
C2×#48302
C2×#47592
C2×#47562
C2×Q8×D83