Finite Group Databasebeta

← C2×#81912 #11675614

Subgroups — C2×#81912 #11675614

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5690C212yesmaximal, complemented
2C2×C3×#499C211yesmaximal, characteristic, complemented
3C2×#5066C212yesmaximal, complemented
4#81912C2116yesmaximal, direct product factor
5C2×#4878C211yesmaximal, characteristic, complemented
6C2×#4874C211yesmaximal, characteristic, complemented
7C2×#4872C211yesmaximal, characteristic
8C2×#4872C211yesmaximal, characteristic
9C2×#4871C212yesmaximal, complemented
10C2×#4870C212yesmaximal, complemented
11C2×C4×D24C211yesmaximal, characteristic
12C2×C8×Dic6C211yesmaximal, characteristic, complemented
13C2×#3380—31nomaximal, Sylow
14C2×C3×#125C2212yes—
15C22×D24C2211yescharacteristic
16#5690—28no—
17#5690C22116yescomplemented
18C2×#897—22no—
19C2×#896—22no—
20C2×#896—22no—
21C2×#893C2212yes—
22C2×#893C2212yes—
23C2×D3×C4⋊C4—22no—
24C2×#890C2212yes—
25C2×#888C2212yes—
26C2×C4×D12C2212yes—
27C2×C4×Dic6C2211yescharacteristic
28C3×#499C2218yescomplemented
29C2×C3×#107C2211yescharacteristic
30C2×C3×#107C2211yescharacteristic
31C2×C3×#106C2212yes—
32C2×C3×C4×C8C2211yescharacteristic
33C2×#821C2212yes—
34C2×#818C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5066C22116yescomplemented
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×#807C2212yes—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4878C2218yescomplemented
46#4874C2218yescomplemented
47#4872C2218yes—
48#4872C2218yes—
49#4871C22116yescomplemented
50#4870C22116yescomplemented
51C4×D24C2218yes—
52C8×Dic6C2218yescomplemented
53C2×#554—32no—
54#3380—316no—
55C2×#500—31no—
56C2×#499—31no—
57C2×#499D311yescharacteristic, complemented
58C2×#478—32no—
59C2×#478—32no—
60C2×#476—32no—
61C2×C8×Q8—31no—
62C2×C4×D8—31no—
63C2×C4×D8—31no—
64C2×C4×D8—31no—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×Dic6—22no—
70C22×Dic6D411yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#125—24no—
74C3×#125C2318yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C42C2311yescharacteristic
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D411yescharacteristic
89C2×Dic3⋊C4D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×D24Q818yescomplemented
95C2×D24C2314yes—
96#897—216no—
97#896—216no—
98#896—216no—
99#893—24no—
100#893—24no—
101#893C2318yes—
102#893C2318yes—
103D3×C4⋊C4—216no—
104#890—24no—
105#890C2318yes—
106#888—24no—
107#888C2318yes—
108C4×D12—24no—
109C4×D12C2318yes—
110C4×Dic6D414yes—
111C4×Dic6C2314yes—
112C3×#107C2314yes—
113C3×#107C2314yes—
114C3×#106C2318yes—
115C3×C4×C8C2314yes—
116#821C2318yes—
117#818C2314yes—
118Dic3⋊C8C2314yes—
119Dic3⋊C8C2314yes—
120C8×Dic3C2314yes—
121C8×Dic3C2314yes—
122#809C2318yes—
123#809C2318yes—
124#808C2318yes—
125#807C2318yes—
126C4⋊(C3⋊C8)C2314yes—
127C22×D8—31no—
128#554—316no—
129#554—68no—
130C2×C4⋊Q8—32no—
131C2×#125—32no—
132C2×#125C2×D312yes—
133C2×C22⋊Q8—62no—
134C2×C22⋊Q8—62no—
135C2×C22⋊Q8—62no—
136C2×C4×Q8—31no—
137C2×C4×D4—32no—
138C2×C4×D4—32no—
139C2×C4×D4—32no—
140C22×C4⋊C4—62no—
141#500—38no—
142#499—38no—
143#499C2×D318yescomplemented
144#478—316no—
145#478—316no—
146#476—316no—
147C8×Q8—38no—
148C4×D8—38no—
149C4×D8—38no—
150C4×D8—38no—
151C2×#107—31no—
152C2×#107—32no—
153C2×#107C2×D311yescharacteristic
154C2×#107C2×D311yescharacteristic
155C2×#106—32no—
156C2×#106C2×D312yes—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#102—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×C4×C8—31no—
164C2×C4×C8—31no—
165C2×C4×C8C2×D311yescharacteristic
166C23×D3—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C2×D12—22no—
178C2×D12—24no—
179C2×D12C2×Q818yes—
180C2×D12C2414yes—
181C2×C4×D3—28no—
182C2×C4×D3—28no—
183C2×C4×D3—28no—
184C2×C4×D3—48no—
185C2×C4×D3—84no—
186C2×C4×D3—84no—
187C2×Dic6—24no—
188C2×Dic6—28no—
189C2×Dic6—84no—
190C2×Dic6C2×D414yes—
191C2×C3×C8—22no—
192C2×C3×C8C2×Q814yes—
193C2×C3×C8C2412yes—
194C2×C3×C8C2412yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4C2414yes—
200C3×C4⋊C4C2414yes—
201C3×C42C2×D412yes—
202C3×C42C2412yes—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206C4⋊Dic3—22no—
207C4⋊Dic3—28no—
208C4⋊Dic3C2×D414yes—
209C4⋊Dic3C2414yes—
210Dic3⋊C4—28no—
211Dic3⋊C4—28no—
212Dic3⋊C4C2×D414yes—
213Dic3⋊C4C2×D414yes—
214C4×Dic3C2×D412yes—
215C4×Dic3C2×D412yes—
216C4×Dic3C2412yes—
217C4×Dic3C2412yes—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C2412yes—
221C2×C3⋊C8C2412yes—
222D24C2×Q8116yescomplemented
223C22×Q8—31no—
224C22×Q8—62no—
225C22×D4—32no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4—62no—
229C2×D8—38no—
230C2×D8—34no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D311yescharacteristic
234C22×C8C22×D311yescharacteristic
235C4⋊Q8—38no—
236C4⋊Q8—64no—
237#125—24no—
238#125—38no—
239#125—64no—
240#125C22×D318yes—
241C22⋊Q8—616no—
242C22⋊Q8—616no—
243C22⋊Q8—616no—
244C4×Q8—34no—
245C4×Q8—34no—
246C4×D4—38no—
247C4×D4—38no—
248C4×D4—38no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4—64no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—616no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4C22×D312yes—
263C2×C4⋊C4C22×D312yes—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42C22×D311yescharacteristic
270#107—34no—
271#107—38no—
272#107C22×D314yes—
273#107C22×D314yes—
274#106—38no—
275#106C22×D318yes—
276C4⋊C8—34no—
277C4⋊C8—34no—
278C4⋊C8—34no—
279#102—38no—
280#102—38no—
281#102—38no—
282C4×C8—34no—
283C4×C8—34no—
284C4×C8C22×D314yes—
285C23×C3C22×D411yescharacteristic
286C22×D3—28no—
287C22×D3—24no—
288C22×D3—44no—
289C22×D3—44no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—44no—
295C2×C3×C4—44no—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×Q812yes—
300C2×C3×C42−1+412yes—
301C2×C3×C4C2511yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—44no—
307C2×Dic3#13514yes—
308C2×Dic3C22×D412yes—
309C2×Dic3C22×D412yes—
310C2×Dic3C22×D412yes—
311C2×Dic3C22×D412yes—
312D12—28no—
313D12—24no—
314D12C22×Q818yes—
315C4×D3—416no—
316C4×D3—88no—
317C4×D3—88no—
318Dic6—28no—
319Dic6—88no—
320C3×C8—22no—
321C3×C8C22×Q814yes—
322C3⋊C8—22no—
323C3⋊C8—22no—
324C24—62no—
325C2×Q8—34no—
326C2×Q8—64no—
327C2×Q8—68no—
328C2×Q8—244no—
329C2×D4—38no—
330C2×D4—34no—
331C2×D4—62no—
332C2×D4—64no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—68no—
340C22×C4—68no—
341C22×C4—62no—
342C22×C4—68no—
343C22×C4—128no—
344C22×C4—244no—
345C22×C4—244no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4C23×D311yescharacteristic
349D8—316no—
350C2×C8—22no—
351C2×C8—32no—
352C2×C8—32no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8D3×Q814yes—
356C2×C8C23×D312yes—
357C2×C8C23×D312yes—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—34no—
363C4⋊C4—34no—
364C4⋊C4—34no—
365C4⋊C4—34no—
366C4⋊C4—68no—
367C4⋊C4—68no—
368C4⋊C4—68no—
369C4⋊C4—62no—
370C4⋊C4C23×D314yes—
371C4⋊C4C23×D314yes—
372C22⋊C4—68no—
373C22⋊C4—68no—
374C22⋊C4—68no—
375C42—32no—
376C42—32no—
377C42—32no—
378C42—32no—
379C42D3×D412yes—
380C42C23×D312yes—
381C22×C3D4×Q812yes—
382C22×C3C2×#13512yes—
383C22×C3#57612yes—
384C22×C3C23×D411yescharacteristic
385C2×D3—28no—
386C2×D3—416no—
387C2×D3—44no—
388C2×D3—44no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—44no—
392C3×C4—44no—
393C3×C4D4×Q812yes—
394C3×C4C23×Q811yescharacteristic
395C3×C4C2×2−1+411yescharacteristic
396Dic3—22no—
397Dic3—22no—
398Dic3—22no—
399Dic3—44no—
400Dic3C2×#13514yes—
401C23—64no—
402C23—68no—
403C23—124no—
404C23—124no—
405C23C2×D3×D411yescharacteristic
406Q8—68no—
407Q8—248no—
408D4—38no—
409D4—64no—
410D4—68no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—32no—
416C2×C4—34no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—64no—
426C2×C4—1216no—
427C2×C4—124no—
428C2×C4—248no—
429C2×C4—248no—
430C2×C4C2×D3×D412yes—
431C2×C4C2×D3×D412yes—
432C2×C4C2×D3×D411yescharacteristic
433C2×C4C2×D3×Q812yes—
434C2×C4#100712yes—
435C2×C4C24×D311yescharacteristic
436C8—22no—
437C8—62no—
438C8—62no—
439C8C2×D3×Q814yes—
440C2×C3#338014yescomplemented
441C2×C3C2×D4×Q811yescharacteristic
442C2×C3C22×#13511yescharacteristic
443C2×C3C2×#57611yescharacteristic
444D3—48no—
445C22—68no—
446C22—124no—
447C22—124no—
448C22—1216no—
449C22#572712yes—
450C22D3×#13512yes—
451C22#576912yes—
452C22C22×D3×D411yescharacteristic
453C4—22no—
454C4—22no—
455C4—34no—
456C4—44no—
457C4—44no—
458C4—62no—
459C4—62no—
460C4—62no—
461C4—124no—
462C4#572712yes—
463C4C22×D3×Q811yescharacteristic
464C4C2×#100711yescharacteristic
465C3C2×#338011yescharacteristic, complemented, Sylow
466C2—128no—
467C2#8191214yesdirect product factor
468C2C2×#572711yescharacteristic
469C2C2×D3×#13511yescharacteristic
470C2C2×#576911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#499C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenterC2×#499D3
hypocenterC3C2×#3380
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
1. C2×#5690C2semidirect
2. C2×C3×#499C2semidirect
3. C2×#5066C2semidirect
4. #81912C2direct (N × H)
5. C2×#4878C2semidirect
6. C2×#4874C2semidirect
9. C2×#4871C2semidirect
10. C2×#4870C2semidirect
12. C2×C8×Dic6C2semidirect
17. #5690C22semidirect
28. C3×#499C22semidirect
39. #5066C22semidirect
45. #4878C22semidirect
46. #4874C22semidirect
49. #4871C22semidirect
50. #4870C22semidirect
52. C8×Dic6C22semidirect
57. C2×#499D3semidirect
94. C2×D24Q8semidirect
143. #499C2×D3semidirect
222. D24C2×Q8semidirect
440. C2×C3#3380semidirect
465. C3C2×#3380semidirect
467. C2#81912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56902
C2×C3×#4992
C2×#50662
#819122
C2×#48782
C2×#48742
C2×#48722
C2×#48712
C2×#48702
C2×C4×D242
C2×C8×Dic62
C2×#33803