Finite Group Databasebeta

← C2×#81054 #11675616

Subgroups — C2×#81054 #11675616

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic, complemented
2C2×#5543C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×C4×#213C211yesmaximal, characteristic
6#81054C2116yesmaximal, direct product factor
7C2×#4871C212yesmaximal, complemented
8C2×#4858C212yesmaximal, complemented
9C2×#4842C211yesmaximal, characteristic, complemented
10C2×#4830C211yesmaximal, characteristic, complemented
11C2×#4763C212yesmaximal, complemented
12C2×#4756C212yesmaximal
13C2×#3380—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946—22no—
17C2×D4×Dic3C2212yes—
18C22×#213C2211yescharacteristic
19C22×C4⋊Dic3—21no—
20C2×#924—21no—
21#5730—24no—
22#5730C2218yescomplemented
23#5543—24no—
24#5543C2218yescomplemented
25C2×#897—21no—
26C2×#893C2212yes—
27C2×D3×C4⋊C4—21no—
28C2×#890C2212yes—
29C2×#890C2212yes—
30C2×#878—22no—
31C2×C4×D12C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#476C2218yescomplemented
35C2×C3×#106C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#102C2212yes—
38C2×#832C2211yescharacteristic
39C2×#821C2212yes—
40C2×#817C2212yes—
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#5078C2218yescomplemented
44C4×#213C2218yes—
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2212yes—
48C2×#807C2212yes—
49C2×C4×C3⋊C8C2211yescharacteristic
50#4871C22116yescomplemented
51#4858C22116yescomplemented
52#4842C2218yescomplemented
53#4830C2218yescomplemented
54#4763C22116yescomplemented
55#4756C22116yes—
56C2×#554—31no—
57C2×#554—31no—
58#3380—316no—
59C2×#500—31no—
60C2×#499—32no—
61C2×#478—32no—
62C2×#478—32no—
63C2×#476—31no—
64C2×#476D311yescharacteristic, 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—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—22no—
75C22×Dic6—21no—
76C22×Dic6D411yescharacteristic
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2314yes—
80C3×C4×D4—22no—
81C3×C4×D4C2314yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88#946—216no—
89D4×Dic3—24no—
90D4×Dic3C2318yes—
91C2×#213Q818yescomplemented
92C2×#213C2314yes—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—28no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102#924—28no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4D412yes—
106C2×C4×Dic3C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#897—28no—
110#893—24no—
111#893C2318yes—
112D3×C4⋊C4—28no—
113#890—24no—
114#890—24no—
115#890C2318yes—
116#890C2318yes—
117#878—216no—
118C4×D12—22no—
119C4×D12C2314yes—
120#869—22no—
121#869C2314yes—
122C4×Dic6D414yes—
123C4×Dic6C2314yes—
124C3×#106C2318yes—
125C3×C4⋊C8C2314yes—
126C3×#102C2318yes—
127#832C2314yes—
128#821C2318yes—
129#817C2318yes—
130Dic3⋊C8C2318yes—
131C8×Dic3C2318yes—
132#809C2314yes—
133#807C2318yes—
134#807C2318yes—
135#807C2314yes—
136C4×C3⋊C8C2314yes—
137C22×D8—31no—
138#554—38no—
139#554—38no—
140#554—64no—
141#554—64no—
142C2×C4⋊Q8—31no—
143C2×C4⋊Q8C2×D311yescharacteristic
144C2×#125—32no—
145C2×#125—32no—
146C2×C22⋊Q8—62no—
147C2×C22⋊Q8—61no—
148C2×C22⋊Q8—62no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×D4—32no—
152C2×C4×D4—32no—
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155C22×C4⋊C4—61no—
156C22×C4⋊C4—61no—
157#500—38no—
158#499—316no—
159#478—316no—
160#478—316no—
161#476—38no—
162#476C2×D318yescomplemented
163C8×Q8—38no—
164C4×D8—38no—
165C4×D8—316no—
166C2×#107—32no—
167C2×#107—31no—
168C2×#107—32no—
169C2×#106—32no—
170C2×#106C2×D312yes—
171C2×C4⋊C8—32no—
172C2×C4⋊C8C2×D311yescharacteristic
173C2×#102—31no—
174C2×#102—32no—
175C2×#102—31no—
176C2×#102C2×D312yes—
177C2×C4×C8—32no—
178C2×C4×C8—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—82no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4C2×Q814yes—
186C2×C3×D4C2412yes—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—44no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—28no—
197C22×Dic3—84no—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×D412yes—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12C2×Q814yes—
203C2×D12C2412yes—
204C2×C4×D3—24no—
205C2×C4×D3—28no—
206C2×C4×D3—44no—
207C2×C4×D3—84no—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—82no—
211C2×Dic6C2×D414yes—
212C2×C3×C8—24no—
213C2×C3×C8C2414yes—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4C2414yes—
218C3×C4⋊C4C2412yes—
219C3×C22⋊C4—24no—
220C3×C42C2×D412yes—
221C3×C42C2412yes—
222#217—28no—
223#213C2×Q8116yescomplemented
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—28no—
226C4⋊Dic3—22no—
227C4⋊Dic3—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—28no—
230C4⋊Dic3—28no—
231C4⋊Dic3C2×D414yes—
232C4⋊Dic3C2414yes—
233C4⋊Dic3C2412yes—
234Dic3⋊C4—28no—
235Dic3⋊C4—28no—
236Dic3⋊C4C2×D418yes—
237C4×Dic3C2×D414yes—
238C4×Dic3C2414yes—
239C2×C3⋊C8—22no—
240C2×C3⋊C8C2×Q814yes—
241C2×C3⋊C8C2412yes—
242C2×C3⋊C8C2412yes—
243C22×Q8—21no—
244C22×Q8—31no—
245C22×Q8—61no—
246C22×D4—31no—
247C22×D4C22×D311yescharacteristic
248C23×C4—21no—
249C23×C4—62no—
250C23×C4—62no—
251C23×C4—61no—
252C2×D8—38no—
253C2×D8—34no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8C22×D312yes—
257C4⋊Q8—22no—
258C4⋊Q8—34no—
259C4⋊Q8—62no—
260C4⋊Q8C22×D314yes—
261#125—38no—
262#125—38no—
263#125—64no—
264#125—64no—
265C22⋊Q8—616no—
266C22⋊Q8—68no—
267C22⋊Q8—616no—
268C22⋊Q8—68no—
269C4×Q8—34no—
270C4×Q8—34no—
271C4×D4—22no—
272C4×D4—38no—
273C4×D4—38no—
274C4×D4—34no—
275C4×D4—64no—
276C4×D4—64no—
277C4×D4—62no—
278C4×D4C22×D314yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—68no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D312yes—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—61no—
297C2×C42—32no—
298C2×C42C22×D311yescharacteristic
299#107—34no—
300#107—38no—
301#107—38no—
302#106—38no—
303#106C22×D318yes—
304C4⋊C8—38no—
305C4⋊C8C22×D314yes—
306#102—34no—
307#102—38no—
308#102—34no—
309#102C22×D318yes—
310C4×C8—38no—
311C4×C8—34no—
312C23×C3—24no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C22×D3—24no—
318C22×D3—22no—
319C22×D3—42no—
320C22×D3—42no—
321C3×Q8—84no—
322C3×D4—24no—
323C3×D4—22no—
324C3×D4C22×Q814yes—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—42no—
329C2×C3×C4—44no—
330C2×C3×C4—48no—
331C2×C3×C4#13514yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×Q812yes—
336C2×C3×C42−1+412yes—
337C2×C3×C4C2511yescharacteristic
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344C2×Dic3—88no—
345C2×Dic3C22×D414yes—
346C2×Dic3C22×D414yes—
347D12—24no—
348D12—22no—
349D12C22×Q814yes—
350C4×D3—48no—
351C4×D3—88no—
352Dic6—28no—
353Dic6—84no—
354C3×C8—24no—
355C3⋊C8—22no—
356C3⋊C8C22×Q814yes—
357C24—21no—
358C24—61no—
359C2×Q8—24no—
360C2×Q8—34no—
361C2×Q8—64no—
362C2×Q8—64no—
363C2×Q8—82no—
364C2×Q8—242no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—34no—
368C2×D4—32no—
369C2×D4—61no—
370C2×D4—62no—
371C2×D4D3×Q814yes—
372C2×D4C23×D312yes—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—44no—
379C22×C4—68no—
380C22×C4—68no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—64no—
384C22×C4—124no—
385C22×C4—244no—
386C22×C4—244no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390D8—316no—
391C2×C8—24no—
392C2×C8—34no—
393C2×C8—32no—
394C2×C8—32no—
395C2×C8—62no—
396C2×C8C23×D314yes—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—34no—
401C4⋊C4—38no—
402C4⋊C4—32no—
403C4⋊C4—34no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—68no—
407C4⋊C4—68no—
408C4⋊C4—61no—
409C4⋊C4—62no—
410C4⋊C4—64no—
411C4⋊C4C23×D312yes—
412C4⋊C4C23×D314yes—
413C22⋊C4—24no—
414C22⋊C4—68no—
415C22⋊C4—68no—
416C22⋊C4—64no—
417C42—34no—
418C42—34no—
419C42D3×D412yes—
420C42C23×D312yes—
421C22×C3—24no—
422C22×C3—48no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3D4×Q812yes—
426C22×C3C2×#13512yes—
427C22×C3#57612yes—
428C22×C3C23×D411yescharacteristic
429C2×D3—24no—
430C2×D3—48no—
431C2×D3—42no—
432C2×D3—42no—
433C3×C4—22no—
434C3×C4—42no—
435C3×C4—44no—
436C3×C4D4×Q812yes—
437C3×C4C2×#13514yes—
438C3×C4C23×Q811yescharacteristic
439C3×C4C2×2−1+411yescharacteristic
440Dic3—24no—
441Dic3—24no—
442Dic3—42no—
443Dic3—44no—
444C23—24no—
445C23—22no—
446C23—42no—
447C23—42no—
448C23—64no—
449C23—62no—
450C23—122no—
451C23—122no—
452C23C2×D3×D411yescharacteristic
453Q8—68no—
454Q8—84no—
455Q8—244no—
456D4—24no—
457D4—22no—
458D4—34no—
459D4—64no—
460D4—62no—
461D4C2×D3×Q814yes—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—34no—
466C2×C4—34no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—62no—
473C2×C4—64no—
474C2×C4—122no—
475C2×C4—124no—
476C2×C4—128no—
477C2×C4—248no—
478C2×C4—248no—
479C2×C4#91614yes—
480C2×C4C2×D3×D412yes—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D412yes—
483C2×C4C2×D3×Q812yes—
484C2×C4#101012yes—
485C2×C4C24×D311yescharacteristic
486C8—24no—
487C8—34no—
488C8—62no—
489C2×C3—44no—
490C2×C3#338014yescomplemented
491C2×C3C2×D4×Q811yescharacteristic
492C2×C3C22×#13511yescharacteristic
493C2×C3C2×#57611yescharacteristic
494D3—44no—
495C22—24no—
496C22—42no—
497C22—42no—
498C22—48no—
499C22—64no—
500C22—122no—
501C22—122no—
502C22—128no—
503C22#563212yes—
504C22C2×#91612yes—
505C22#577412yes—
506C22C22×D3×D411yescharacteristic
507C4—22no—
508C4—42no—
509C4—44no—
510C4—64no—
511C4—64no—
512C4—122no—
513C4—124no—
514C4#563212yes—
515C4C2×#91614yes—
516C4C22×D3×Q811yescharacteristic
517C4C2×#101011yescharacteristic
518C3C2×#338011yescharacteristic, complemented, Sylow
519C2—44no—
520C2—124no—
521C2#8105414yesdirect product factor
522C2C2×#563211yescharacteristic
523C2C22×#91611yescharacteristic
524C2C2×#577411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#476C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#476D3
hypocenterC3C2×#3380
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×#5543C2semidirect
3. C2×C3×#476C2semidirect
4. C2×#5078C2semidirect
6. #81054C2direct (N × H)
7. C2×#4871C2semidirect
8. C2×#4858C2semidirect
9. C2×#4842C2semidirect
10. C2×#4830C2semidirect
11. C2×#4763C2semidirect
22. #5730C22semidirect
24. #5543C22semidirect
34. C3×#476C22semidirect
43. #5078C22semidirect
50. #4871C22semidirect
51. #4858C22semidirect
52. #4842C22semidirect
53. #4830C22semidirect
54. #4763C22semidirect
64. C2×#476D3semidirect
91. C2×#213Q8semidirect
162. #476C2×D3semidirect
223. #213C2×Q8semidirect
490. C2×C3#3380semidirect
518. C3C2×#3380semidirect
521. C2#81054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#55432
C2×C3×#4762
C2×#50782
C2×C4×#2132
#810542
C2×#48712
C2×#48582
C2×#48422
C2×#48302
C2×#47632
C2×#47562
C2×#33803