Finite Group Databasebeta

← C2×#5702 #86108

Subgroups — C2×#5702 #86108

Order 384.

1112 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2#5702C2116yesmaximal, direct product factor
3C2×#898C211yesmaximal, characteristic, complemented
4C2×#898C211yesmaximal, characteristic, complemented
5C2×#896C211yesmaximal, characteristic, complemented
6C2×#888C211yesmaximal, characteristic, complemented
7C2×#887C211yesmaximal, characteristic, complemented
8C2×#887C211yesmaximal, characteristic, complemented
9C2×#885C211yesmaximal, characteristic
10C2×#885C211yesmaximal, characteristic
11C2×#67⋊C4C211yesmaximal, characteristic
12C2×#879C211yesmaximal, characteristic, complemented
13C2×#878C211yesmaximal, characteristic, complemented
14C2×#877C211yesmaximal, characteristic, complemented
15C2×#872C211yesmaximal, characteristic, complemented
16C2×C4×Dic6C211yesmaximal, characteristic, complemented
17C2×#556—31nomaximal, Sylow
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C3×#126C2218yescomplemented
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C42C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
51C2×C4×Dic3C2211yescharacteristic
52C2×C4×Dic3C2211yescharacteristic
53C2×C4×Dic3C2211yescharacteristic
54#898C2218yescomplemented
55#898C2218yescomplemented
56#896C2218yescomplemented
57#888C2218yescomplemented
58#887C2218yescomplemented
59#887C2218yescomplemented
60#885C2218yes—
61#885C2218yes—
62#67⋊C4C2218yes—
63#879C2218yescomplemented
64#878C2218yescomplemented
65#877C2218yescomplemented
66#872C2218yescomplemented
67C4×Dic6C2218yescomplemented
68#556—316no—
69C2×C4⋊Q8—31no—
70C2×#126—31no—
71C2×#126—31no—
72C2×#126—31no—
73C2×#126D311yescharacteristic, complemented
74C2×#124—31no—
75C2×#124—31no—
76C2×C22⋊Q8—31no—
77C2×C22⋊Q8—31no—
78C2×C4×Q8—31no—
79C2×C4×Q8—31no—
80C2×C4×Q8—31no—
81C2×C4×D4—31no—
82C2×#117—31no—
83C2×#117—31no—
84C24×C3C2311yescharacteristic
85C23×D3C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C2×#67—24no—
93C2×#67C2314yes—
94C22×Dic3—24no—
95C22×Dic3C2314yes—
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C2×C4×D3—24no—
104C2×C4×D3C2314yes—
105C2×Dic6—24no—
106C2×Dic6—24no—
107C2×Dic6—24no—
108C2×Dic6C2314yes—
109C2×Dic6C2314yes—
110C2×Dic6C2314yes—
111C3×C4⋊C4C2314yes—
112C3×C4⋊C4C2314yes—
113C3×C4⋊C4C2314yes—
114C3×C22⋊C4C2314yes—
115C3×C22⋊C4C2314yes—
116C3×C22⋊C4C2314yes—
117C3×C42C2314yes—
118#217C2314yes—
119#217C2314yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2314yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125C4⋊Dic3C2314yes—
126C4⋊Dic3C2314yes—
127Dic3⋊C4C2314yes—
128Dic3⋊C4C2314yes—
129Dic3⋊C4C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134C4×Dic3C2314yes—
135C4×Dic3C2314yes—
136C4×Dic3C2314yes—
137C4×Dic3C2314yes—
138C4×Dic3C2314yes—
139C4×Dic3C2314yes—
140C22×Q8—31no—
141C22×Q8—31no—
142C22×Q8—31no—
143C22×D4—31no—
144C23×C4—31no—
145C23×C4—31no—
146C4⋊Q8—38no—
147#126—38no—
148#126—38no—
149#126—38no—
150#126C2×D318yescomplemented
151#124—38no—
152#124—38no—
153C22⋊Q8—38no—
154C22⋊Q8—38no—
155C4×Q8—38no—
156C4×Q8—38no—
157C4×Q8—38no—
158C4×D4—38no—
159#117—38no—
160#117—38no—
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⋊C4—31no—
170C2×C4⋊C4C2×D311yescharacteristic
171C2×C4⋊C4C2×D311yescharacteristic
172C2×C4⋊C4C2×D311yescharacteristic
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4C2×D311yescharacteristic
181C2×C22⋊C4C2×D311yescharacteristic
182C2×C22⋊C4C2×D311yescharacteristic
183C2×C42—31no—
184C2×C42—31no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—31no—
189C2×C42C2×D311yescharacteristic
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3C2412yes—
194C23×C3C2411yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3C2412yes—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4C2412yes—
206C2×C3×C4C2412yes—
207C2×C3×C4C2412yes—
208C2×C3×C4C2412yes—
209C2×C3×C4C2412yes—
210C2×C3×C4C2412yes—
211#67—28no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—28no—
218C2×Dic3#4114yes—
219C2×Dic3#4114yes—
220C2×Dic3C2412yes—
221C2×Dic3C2412yes—
222C2×Dic3C2412yes—
223C2×Dic3C2412yes—
224C2×Dic3C2412yes—
225C2×Dic3C2412yes—
226C2×Dic3C2412yes—
227C4×D3—28no—
228Dic6—28no—
229Dic6—28no—
230Dic6—28no—
231C24—31no—
232C24C22×D311yescharacteristic
233C2×Q8—34no—
234C2×Q8—34no—
235C2×Q8—34no—
236C2×Q8—64no—
237C2×Q8—64no—
238C2×Q8—64no—
239C2×D4—34no—
240C2×D4—64no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—34no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—34no—
250C22×C4—64no—
251C22×C4—64no—
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C22×C4C22×D311yescharacteristic
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4—34no—
263C4⋊C4—34no—
264C4⋊C4—34no—
265C4⋊C4—34no—
266C4⋊C4—34no—
267C4⋊C4C22×D314yes—
268C4⋊C4C22×D314yes—
269C4⋊C4C22×D314yes—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4—34no—
276C22⋊C4—34no—
277C22⋊C4C22×D314yes—
278C22⋊C4C22×D314yes—
279C22⋊C4C22×D314yes—
280C42—34no—
281C42—34no—
282C42—34no—
283C42—34no—
284C42—34no—
285C42—34no—
286C42C22×D314yes—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—44no—
291C22×C3C2×#4112yes—
292C22×C3C2×#4112yes—
293C22×C32−1+412yes—
294C22×C3C2511yescharacteristic
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—22no—
298C2×D3—44no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305Dic3—22no—
306Dic3—22no—
307Dic3—22no—
308Dic3—22no—
309Dic3—22no—
310Dic3C2×#4114yes—
311Dic3C2×#4114yes—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—32no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23C23×D312yes—
320C23C23×D311yescharacteristic
321Q8—68no—
322Q8—68no—
323Q8—68no—
324D4—68no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—34no—
332C2×C4—34no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—68no—
341C2×C4—68no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4C23×D312yes—
348C2×C4C23×D312yes—
349C2×C4C23×D312yes—
350C2×C4C23×D312yes—
351C2×C4C23×D312yes—
352C2×C4C23×D312yes—
353C2×C3—42no—
354C2×C3#55614yescomplemented
355C2×C3C22×#4111yescharacteristic
356C2×C3C22×#4111yescharacteristic
357C2×C3C2×2−1+411yescharacteristic
358D3—42no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—44no—
363C22—62no—
364C22—62no—
365C22—62no—
366C22—124no—
367C22D3×#4112yes—
368C22D3×#4112yes—
369C22#101012yes—
370C22C24×D311yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—34no—
378C4—34no—
379C4—62no—
380C4—62no—
381C4—62no—
382C4—62no—
383C4—62no—
384C3C2×#55611yescharacteristic, complemented, Sylow
385C2—42no—
386C2—122no—
387C2#570214yesdirect product factor
388C2C2×D3×#4111yescharacteristic
389C2C2×D3×#4111yescharacteristic
390C2C2×#101011yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
prime radicalC22×#67C22
hypercenterC2×#126D3
hypocenterC3C2×#556
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
2. #5702C2direct (N × H)
3. C2×#898C2semidirect
4. C2×#898C2semidirect
5. C2×#896C2semidirect
6. C2×#888C2semidirect
7. C2×#887C2semidirect
8. C2×#887C2semidirect
12. C2×#879C2semidirect
13. C2×#878C2semidirect
14. C2×#877C2semidirect
15. C2×#872C2semidirect
16. C2×C4×Dic6C2semidirect
24. C3×#126C22semidirect
54. #898C22semidirect
55. #898C22semidirect
56. #896C22semidirect
57. #888C22semidirect
58. #887C22semidirect
59. #887C22semidirect
63. #879C22semidirect
64. #878C22semidirect
65. #877C22semidirect
66. #872C22semidirect
67. C4×Dic6C22semidirect
73. C2×#126D3semidirect
150. #126C2×D3semidirect
354. C2×C3#556semidirect
384. C3C2×#556semidirect
387. C2#5702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
#57022
C2×#8982
C2×#8962
C2×#8882
C2×#8872
C2×#8852
C2×#67⋊C42
C2×#8792
C2×#8782
C2×#8772
C2×#8722
C2×C4×Dic62
C2×#5563