Finite Group Databasebeta

← C2×Q8×#676 #12963684

Subgroups — C2×Q8×#676 #12963684

Order 1280.

2154 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D5C211yesmaximal, characteristic
2C2×C5×C8×Q8C211yesmaximal, characteristic, complemented
3C2×Q8×#168C211yesmaximal, characteristic, complemented
4Q8×#676C2116yesmaximal, direct product factor
5C2×#65126C213yesmaximal
6C2×#65120C213yesmaximal, complemented
7C2×C4×#676C213yesmaximal
8C2×#64968C213yesmaximal, complemented
9C22×Q8×D5C411yescharacteristic
10C2×C4×C5×Q8C2211yescharacteristic
11C2×Q8×Dic5C411yescharacteristic
12C22×#676C2213yes—
13C22×#676C2211yescharacteristic
14C4×Q8×D5C418yes—
15C4×Q8×D5C2218yes—
16C2×D5×C4⋊C4C413yes—
17C2×#4033C413yes—
18C2×C42×D5C2213yes—
19C2×C4×Dic10C2213yes—
20C5×C8×Q8C2218yescomplemented
21C2×C5×C4⋊C8C2213yes—
22C2×C4×C5×C8C2213yes—
23C2×#3947C2213yes—
24C2×#3947C2213yes—
25C2×#3946C2216yes—
26Q8×#168C2218yescomplemented
27C2×C4⋊#168C2213yes—
28C2×C4×#168C2213yes—
29#65126C22124yes—
30#65120C22124yescomplemented
31C4×#676C22124yes—
32#64968C22124yescomplemented
33C2×Q8×M16—51nomaximal, Sylow
34C22×C5×Q8C2×C411yescharacteristic
35C2×Q8×D5—28no—
36C2×Q8×D5C2×C418yes—
37C22×C4×D5C2×C413yes—
38C22×C4×D5C2×C413yes—
39C22×C4×D5C2311yescharacteristic
40C22×Dic10C2×C413yes—
41C22×C5×C8C2313yes—
42C22×C5×C8C2311yescharacteristic
43C4×C5×Q8C2×C414yes—
44C4×C5×Q8C2314yes—
45C2×C5×C4⋊C4C2×C413yes—
46C2×C42×C5C2313yes—
47Q8×Dic5C2×C418yes—
48C2×C4⋊Dic5C2×C413yes—
49C2×#683C2×C416yes—
50C2×C4×Dic5C2×C413yes—
51C2×C4×Dic5C2×C413yes—
52C2×C4×Dic5C2313yes—
53C22×#168C2313yes—
54C22×#168C2311yescharacteristic
55C2×#676—212no—
56C2×#676Q818yesdirect product factor
57C2×#676C23112yes—
58C2×#676C2314yes—
59D5×C4⋊C4C2×C4124yes—
60#4033C2×C4124yes—
61C42×D5C2×C4112yes—
62C42×D5C23112yes—
63C4×Dic10C2×C4112yes—
64C4×Dic10C23112yes—
65C5×C4⋊C8C23112yes—
66C4×C5×C8C23112yes—
67#3947C23112yes—
68#3947C23112yes—
69#3946C23124yes—
70C4⋊#168C23112yes—
71C4×#168C23112yes—
72C22×C4×Q8—51no—
73Q8×M16—516no—
74C2×#446—53no—
75C2×#446—53no—
76C2×C8×Q8—51no—
77C2×C8×Q8D511yescharacteristic, complemented
78C2×#423—53no—
79C2×C4×M16—53no—
80C23×D5C22×C411yescharacteristic
81C2×C5×Q8M1618yescomplemented
82C2×C5×Q8C22×C414yes—
83C22×C4×C5C22×C413yes—
84C22×C4×C5C22×C413yes—
85C22×C4×C5C2411yescharacteristic
86C22×Dic5C22×C413yes—
87C22×Dic5C22×C413yes—
88C22×Dic5C22×C411yescharacteristic
89Q8×D5—232no—
90C2×C4×D5—212no—
91C2×C4×D5—46no—
92C2×C4×D5C22×C4112yes—
93C2×C4×D5C22×C4112yes—
94C2×C4×D5C22×C412yes—
95C2×C4×D5C2×Q814yes—
96C2×C4×D5#4114yes—
97C2×C4×D5C2412yes—
98C2×Dic10—212no—
99C2×Dic10C22×C4112yes—
100C2×C5×C8—26no—
101C2×C5×C8C2×Q814yescomplemented
102C2×C5×C8C2416yes—
103C2×C5×C8C2412yes—
104C5×C4⋊C4C22×C4112yes—
105C42×C5C22×C416yes—
106C42×C5C2416yes—
107C4⋊Dic5C22×C4112yes—
108#683C22×C4124yes—
109C4×Dic5C22×C416yes—
110C4×Dic5C22×C4112yes—
111C4×Dic5C22×C4112yes—
112C4×Dic5C2416yes—
113C2×#168—26no—
114C2×#168C2×Q814yescomplemented
115C2×#168C2416yes—
116C2×#168C2412yes—
117#676—224no—
118#676C2×Q8116yesdirect product factor
119C23×Q8—51no—
120C22×M16—51no—
121C22×M16—53no—
122C2×C4×Q8—51no—
123C2×C4×Q8—58no—
124C2×C4×Q8—58no—
125C2×C4×Q8—53no—
126C2×C4×Q8—53no—
127C2×C4×Q8C2×D511yescharacteristic
128C22×C4⋊C4—53no—
129C22×C42—53no—
130#446—524no—
131#446—524no—
132C8×Q8—58no—
133C8×Q8C2×D518yescomplemented
134#423—524no—
135C2×C4⋊C8—56no—
136C2×C4⋊C8—53no—
137C2×C4⋊C8C2×D513yes—
138C4×M16—524no—
139C2×#97—53no—
140C2×#97—53no—
141C2×C4×C8—53no—
142C2×C4×C8C2×D513yes—
143C23×C5C23×C411yescharacteristic
144C22×D5—22no—
145C22×D5—22no—
146C22×D5C4×Q814yes—
147C22×D5C23×C412yes—
148C5×Q8C2×M16116yescomplemented
149C2×C4×C5—26no—
150C2×C4×C5C2×M16112yes—
151C2×C4×C5C23×C416yes—
152C2×C4×C5C23×C411yescharacteristic
153C2×C4×C5C23×C416yes—
154C2×C4×C5C22×Q812yes—
155C2×C4×C5C2×#4112yes—
156C2×C4×C5C2511yescharacteristic
157C2×Dic5—26no—
158C2×Dic5—26no—
159C2×Dic5C4×Q814yes—
160C2×Dic5C23×C416yes—
161C2×Dic5C23×C416yes—
162C2×Dic5C23×C412yes—
163C4×D5—224no—
164C4×D5—412no—
165C4×D5C4×Q818yes—
166C4×D5C22×Q814yes—
167C4×D5C2×#4114yes—
168Dic10—224no—
169C5×C8—26no—
170C5×C8C22×Q814yescomplemented
171#168—26no—
172#168C22×Q814yescomplemented
173C22×Q8—58no—
174C22×Q8—53no—
175C22×Q8—108no—
176C22×Q8C4×D511yescharacteristic
177C23×C4—51no—
178C23×C4—53no—
179C23×C4—53no—
180C2×M16—58no—
181C2×M16—54no—
182C2×M16—512no—
183C2×M16—1012no—
184C22×C8—51no—
185C22×C8—53no—
186C22×C8C22×D513yes—
187C22×C8C22×D511yescharacteristic
188C4×Q8—58no—
189C4×Q8—524no—
190C4×Q8—512no—
191C4×Q8—512no—
192C4×Q8C4×D514yes—
193C4×Q8C22×D514yes—
194C2×C4⋊C4—56no—
195C2×C4⋊C4—53no—
196C2×C4⋊C4—524no—
197C2×C4⋊C4C4×D513yes—
198C2×C42—53no—
199C2×C42—53no—
200C2×C42—53no—
201C2×C42—512no—
202C2×C42—512no—
203C2×C42C22×D513yes—
204C4⋊C8—524no—
205C4⋊C8—512no—
206C4⋊C8C22×D5112yes—
207#97—512no—
208#97—512no—
209C4×C8—512no—
210C4×C8C22×D5112yes—
211C22×C5C2×C4×Q812yes—
212C22×C5C22×M1612yes—
213C22×C5#56812yes—
214C22×C5C24×C411yescharacteristic
215C2×D5—28no—
216C2×D5—22no—
217C2×D5—22no—
218C2×D5C2×C4×Q814yes—
219C4×C5—26no—
220C4×C5C2×C4×Q812yes—
221C4×C5C22×M16112yes—
222C4×C5C23×Q811yescharacteristic
223C4×C5C22×#4111yescharacteristic
224Dic5—26no—
225Dic5—26no—
226Dic5C2×C4×Q814yes—
227C24—51no—
228C2×Q8—512no—
229C2×Q8—1032no—
230C2×Q8—1012no—
231C2×Q8#67618yesdirect product factor
232C2×Q8C2×C4×D514yes—
233C22×C4—51no—
234C22×C4—52no—
235C22×C4—54no—
236C22×C4—54no—
237C22×C4—52no—
238C22×C4—53no—
239C22×C4—53no—
240C22×C4—512no—
241C22×C4—512no—
242C22×C4—1012no—
243C22×C4—206no—
244C22×C4C2×C4×D513yes—
245C22×C4C2×C4×D513yes—
246C22×C4C23×D511yescharacteristic
247M16—516no—
248M16—1024no—
249C2×C8—26no—
250C2×C8—54no—
251C2×C8—52no—
252C2×C8—56no—
253C2×C8—106no—
254C2×C8Q8×D514yescomplemented
255C2×C8C23×D516yes—
256C2×C8C23×D512yes—
257C4⋊C4—524no—
258C4⋊C4—512no—
259C4⋊C4C2×C4×D5112yes—
260C42—512no—
261C42—512no—
262C42—56no—
263C42—56no—
264C42C2×C4×D516yes—
265C42C23×D516yes—
266C2×C5Q8×M1614yescomplemented
267C2×C5C22×C4×Q811yescharacteristic
268C2×C5C23×M1611yescharacteristic
269C2×C5C2×#56811yescharacteristic
270D5—24no—
271C23—54no—
272C23—52no—
273C23—102no—
274C23—102no—
275C23C22×C4×D511yescharacteristic
276Q8—1024no—
277Q8C2×#676116yesdirect product factor
278C2×C4—26no—
279C2×C4—54no—
280C2×C4—52no—
281C2×C4—58no—
282C2×C4—54no—
283C2×C4—54no—
284C2×C4—56no—
285C2×C4—56no—
286C2×C4—106no—
287C2×C4—106no—
288C2×C4—1024no—
289C2×C4—2012no—
290C2×C4C2×#676112yes—
291C2×C4C22×C4×D516yes—
292C2×C4C22×C4×D516yes—
293C2×C4C22×C4×D511yescharacteristic
294C2×C4C2×Q8×D512yes—
295C2×C4D5×#4112yes—
296C2×C4C24×D511yescharacteristic
297C8—26no—
298C8—54no—
299C8—106no—
300C8C2×Q8×D514yescomplemented
301C5C2×Q8×M1611yescharacteristic, complemented, Sylow
302C22—54no—
303C22—102no—
304C22—102no—
305C22—108no—
306C22C4×Q8×D512yes—
307C22C22×#67612yes—
308C22#6608412yes—
309C22C23×C4×D511yescharacteristic
310C4—26no—
311C4—54no—
312C4—106no—
313C4—106no—
314C4C4×Q8×D512yes—
315C4C22×#676112yes—
316C4C22×Q8×D511yescharacteristic
317C4C2×D5×#4111yescharacteristic
318C2—104no—
319C2Q8×#67614yesdirect product factor
320C2C2×C4×Q8×D511yescharacteristic
321C2C23×#67611yescharacteristic
322C2C2×#6608411yescharacteristic
subgroupG/H
centerC22×C4C23×D5
commutatorC22×C5C24×C4
FrattiniC2×C4C24×D5
FittingC2×C5×C8×Q8C2
socleC23×C5C23×C4
prime radicalC23×D5C22×C4
cosocleC2×C4×C5C25
hypercenterC2×C8×Q8D5
hypocenterC5C2×Q8×M16
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C5×C8×Q8C2semidirect
3. C2×Q8×#168C2semidirect
4. Q8×#676C2direct (N × H)
6. C2×#65120C2semidirect
8. C2×#64968C2semidirect
20. C5×C8×Q8C22semidirect
26. Q8×#168C22semidirect
30. #65120C22semidirect
32. #64968C22semidirect
56. C2×#676Q8direct (N × H)
77. C2×C8×Q8D5semidirect
81. C2×C5×Q8M16semidirect
101. C2×C5×C8C2×Q8semidirect
114. C2×#168C2×Q8semidirect
118. #676C2×Q8direct (N × H)
133. C8×Q8C2×D5semidirect
148. C5×Q8C2×M16semidirect
170. C5×C8C22×Q8semidirect
172. #168C22×Q8semidirect
231. C2×Q8#676direct (N × H)
254. C2×C8Q8×D5semidirect
266. C2×C5Q8×M16semidirect
277. Q8C2×#676direct (N × H)
300. C8C2×Q8×D5semidirect
301. C5C2×Q8×M16semidirect
319. C2Q8×#676direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D52
C2×C5×C8×Q82
C2×Q8×#1682
Q8×#6762
C2×#651262
C2×#651202
C2×C4×#6762
C2×#649682
C2×Q8×M165