Finite Group Databasebeta

← C2×#5696 #86084

Subgroups — C2×#5696 #86084

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#125C211yesmaximal, characteristic, complemented
2#5696C2116yesmaximal, direct product factor
3C2×#898C211yesmaximal, characteristic
4C2×#898C211yesmaximal, characteristic
5C2×#898C211yesmaximal, characteristic
6C2×#896C211yesmaximal, characteristic
7C2×#896C211yesmaximal, characteristic
8C2×#892C211yesmaximal, characteristic
9C2×#890C211yesmaximal, characteristic, complemented
10C2×#889C211yesmaximal, characteristic, complemented
11C2×#889C211yesmaximal, characteristic, complemented
12C2×#889C211yesmaximal, characteristic, complemented
13C2×#888C211yesmaximal, characteristic, complemented
14C2×#887C211yesmaximal, characteristic, complemented
15C2×#876C211yesmaximal, characteristic
16C2×C4×Dic6C211yesmaximal, characteristic, complemented
17C2×#541—31nomaximal, Sylow
18C22×C4×D3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×#125C2218yescomplemented
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C42C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
51C2×C4×Dic3C2211yescharacteristic
52C2×C4×Dic3C2211yescharacteristic
53C2×C4×Dic3C2211yescharacteristic
54#898C2218yes—
55#898C2218yes—
56#898C2218yes—
57#896C2218yes—
58#896C2218yes—
59#892C2218yes—
60#890C2218yescomplemented
61#889C2218yescomplemented
62#889C2218yescomplemented
63#889C2218yescomplemented
64#888C2218yescomplemented
65#887C2218yescomplemented
66#876C2218yes—
67C4×Dic6C2218yescomplemented
68#541—316no—
69C2×C4⋊Q8—31no—
70C2×#126—31no—
71C2×#126—31no—
72C2×#126—31no—
73C2×#126—31no—
74C2×#125—31no—
75C2×#125—31no—
76C2×#125—31no—
77C2×#125—31no—
78C2×#125D311yescharacteristic, complemented
79C2×C22⋊Q8—31no—
80C2×C22⋊Q8—31no—
81C2×C4×Q8—31no—
82C2×C4×Q8—31no—
83C2×#117—31no—
84C23×D3C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C2×C4×D3—24no—
100C2×C4×D3C2314yes—
101C2×Dic6—24no—
102C2×Dic6—24no—
103C2×Dic6C2314yes—
104C2×Dic6C2314yes—
105C3×C4⋊C4C2314yes—
106C3×C4⋊C4C2314yes—
107C3×C4⋊C4C2314yes—
108C3×C4⋊C4C2314yes—
109C3×C4⋊C4C2314yes—
110C3×C4⋊C4C2314yes—
111C3×C42C2314yes—
112(C2×D3)⋊C4C2314yes—
113(C2×D3)⋊C4C2314yes—
114(C2×D3)⋊C4C2314yes—
115(C2×D3)⋊C4C2314yes—
116(C2×D3)⋊C4C2314yes—
117(C2×D3)⋊C4C2314yes—
118C4⋊Dic3C2314yes—
119C4⋊Dic3C2314yes—
120C4⋊Dic3C2314yes—
121C4⋊Dic3C2314yes—
122Dic3⋊C4C2314yes—
123Dic3⋊C4C2314yes—
124Dic3⋊C4C2314yes—
125Dic3⋊C4C2314yes—
126Dic3⋊C4C2314yes—
127Dic3⋊C4C2314yes—
128Dic3⋊C4C2314yes—
129Dic3⋊C4C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132C4×Dic3C2314yes—
133C4×Dic3C2314yes—
134C4×Dic3C2314yes—
135C4×Dic3C2314yes—
136C4×Dic3C2314yes—
137C22×Q8—31no—
138C22×Q8—31no—
139C23×C4—31no—
140C4⋊Q8—38no—
141#126—38no—
142#126—38no—
143#126—38no—
144#126—38no—
145#125—38no—
146#125—38no—
147#125—38no—
148#125—38no—
149#125C2×D318yescomplemented
150C22⋊Q8—38no—
151C22⋊Q8—38no—
152C4×Q8—38no—
153C4×Q8—38no—
154#117—38no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4C2×D311yescharacteristic
170C2×C4⋊C4C2×D311yescharacteristic
171C2×C4⋊C4C2×D311yescharacteristic
172C2×C4⋊C4C2×D311yescharacteristic
173C2×C4⋊C4C2×D311yescharacteristic
174C2×C4⋊C4C2×D311yescharacteristic
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C42—31no—
182C2×C42—31no—
183C2×C42—31no—
184C2×C42—31no—
185C2×C42—31no—
186C2×C42C2×D311yescharacteristic
187C23×C3C2411yescharacteristic
188C22×D3—22no—
189C22×D3—22no—
190C22×D3—22no—
191C22×D3C2412yes—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4C2412yes—
200C2×C3×C4C2412yes—
201C2×C3×C4C2412yes—
202C2×C3×C4C2412yes—
203C2×C3×C4C2412yes—
204C2×C3×C4C2412yes—
205C2×C3×C4C2412yes—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3#4114yes—
213C2×Dic3C2412yes—
214C2×Dic3C2412yes—
215C2×Dic3C2412yes—
216C2×Dic3C2412yes—
217C2×Dic3C2412yes—
218C2×Dic3C2412yes—
219C2×Dic3C2412yes—
220C4×D3—28no—
221Dic6—28no—
222Dic6—28no—
223C24—31no—
224C2×Q8—34no—
225C2×Q8—34no—
226C2×Q8—64no—
227C2×Q8—64no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—34no—
236C22×C4—64no—
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C4⋊C4—34no—
245C4⋊C4—34no—
246C4⋊C4—34no—
247C4⋊C4—34no—
248C4⋊C4—34no—
249C4⋊C4—34no—
250C4⋊C4—34no—
251C4⋊C4—34no—
252C4⋊C4—34no—
253C4⋊C4—34no—
254C4⋊C4—34no—
255C4⋊C4—34no—
256C4⋊C4—34no—
257C4⋊C4—34no—
258C4⋊C4C22×D314yes—
259C4⋊C4C22×D314yes—
260C4⋊C4C22×D314yes—
261C4⋊C4C22×D314yes—
262C4⋊C4C22×D314yes—
263C4⋊C4C22×D314yes—
264C22⋊C4—34no—
265C22⋊C4—34no—
266C22⋊C4—34no—
267C22⋊C4—34no—
268C22⋊C4—34no—
269C22⋊C4—34no—
270C42—34no—
271C42—34no—
272C42—34no—
273C42—34no—
274C42—34no—
275C42C22×D314yes—
276C22×C3C2×#4112yes—
277C22×C32−1+412yes—
278C22×C32−1+412yes—
279C22×C3C2511yescharacteristic
280C2×D3—22no—
281C2×D3—22no—
282C2×D3—22no—
283C2×D3—44no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—22no—
291Dic3—22no—
292Dic3—22no—
293Dic3—22no—
294Dic3—22no—
295Dic3—22no—
296Dic3—22no—
297Dic3C2×#4114yes—
298C23—32no—
299C23—62no—
300C23—62no—
301C23—62no—
302C23C23×D311yescharacteristic
303Q8—68no—
304Q8—68no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—34no—
313C2×C4—32no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—68no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4C23×D312yes—
328C2×C4C23×D312yes—
329C2×C4C23×D312yes—
330C2×C4C23×D312yes—
331C2×C4C23×D312yes—
332C2×C4C23×D312yes—
333C2×C4C23×D312yes—
334C2×C3#54114yescomplemented
335C2×C3C22×#4111yescharacteristic
336C2×C3C2×2−1+411yescharacteristic
337C2×C3C2×2−1+411yescharacteristic
338D3—42no—
339C22—62no—
340C22—62no—
341C22—62no—
342C22—124no—
343C22#100712yes—
344C22D3×#4112yes—
345C22#101012yes—
346C22C24×D311yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—34no—
355C4—62no—
356C4—62no—
357C4—62no—
358C4—62no—
359C4—62no—
360C4—62no—
361C3C2×#54111yescharacteristic, complemented, Sylow
362C2—122no—
363C2#569614yesdirect product factor
364C2C2×#100711yescharacteristic
365C2C2×D3×#4111yescharacteristic
366C2C2×#101011yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#125C2
socleC23×C3C24
prime radicalC23×D3C23
hypercenterC2×#125D3
hypocenterC3C2×#541
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×#125C2semidirect
2. #5696C2direct (N × H)
9. C2×#890C2semidirect
10. C2×#889C2semidirect
11. C2×#889C2semidirect
12. C2×#889C2semidirect
13. C2×#888C2semidirect
14. C2×#887C2semidirect
16. C2×C4×Dic6C2semidirect
21. C3×#125C22semidirect
60. #890C22semidirect
61. #889C22semidirect
62. #889C22semidirect
63. #889C22semidirect
64. #888C22semidirect
65. #887C22semidirect
67. C4×Dic6C22semidirect
78. C2×#125D3semidirect
149. #125C2×D3semidirect
334. C2×C3#541semidirect
361. C3C2×#541semidirect
363. C2#5696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1252
#56962
C2×#8982
C2×#8962
C2×#8922
C2×#8902
C2×#8892
C2×#8882
C2×#8872
C2×#8762
C2×C4×Dic62
C2×#5413