Finite Group Databasebeta

← C2×Q8×#11765849 #32861840

Subgroups — C2×Q8×#11765849 #32861840

Order 18176.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11765849C2116yesmaximal, direct product factor
2C2×#17986130C216yesmaximal, complemented
3C2×Q8×C71⋊C8C212yesmaximal, complemented
4C2×#17986023C213yesmaximal
5C2×C4×#11765849C213yesmaximal
6C22×C4×Q8×C71C211yesmaximal, characteristic
7C22×#11765849C2213yes—
8C22×#11765849C2211yescharacteristic
9#17986130C22148yescomplemented
10Q8×C71⋊C8C22116yescomplemented
11#17986023C22124yes—
12C2×C4⋊(C71⋊C8)C2216yes—
13C2×C4⋊(C71⋊C8)C2216yes—
14C4×#11765849C22124yes—
15C2×#14556039C2216yes—
16C2×C4×C71⋊C8C2216yes—
17C23×Q8×C71C411yescharacteristic
18C2×C4×Q8×C71C413yes—
19C2×C4×Q8×C71C418yes—
20C2×C4×Q8×C71C411yescharacteristic
21C2×C4×Q8×C71C2211yescharacteristic
22C2×C4×Q8×C71C2213yes—
23C2×C4×Q8×C71C2218yes—
24C22×C4⋊C4×C71C413yes—
25C22×C42×C71C2213yes—
26C2×#11765849—212no—
27C2×#11765849Q818yesdirect product factor
28C2×#11765849C23112yes—
29C2×#11765849C2314yes—
30C22×C71⋊C8C2316yes—
31C22×C71⋊C8C2312yes—
32C4⋊(C71⋊C8)C23124yes—
33C4⋊(C71⋊C8)C23124yes—
34#14556039C23124yes—
35C4×C71⋊C8C23124yes—
36C22×Q8×C71—28no—
37C22×Q8×C71C2×C418yes—
38C22×Q8×C71C2×C413yes—
39C22×Q8×C71C2×C411yescharacteristic
40C23×C4×C71C2×C413yes—
41C23×C4×C71C2×C413yes—
42C23×C4×C71C2311yescharacteristic
43C4×Q8×C71C2×C4112yes—
44C4×Q8×C71C2×C4124yes—
45C4×Q8×C71C2×C414yes—
46C4×Q8×C71C2×C418yes—
47C4×Q8×C71C2314yes—
48C4×Q8×C71C23112yes—
49C2×C4⋊C4×C71C2×C416yes—
50C2×C4⋊C4×C71C2×C413yes—
51C2×C4⋊C4×C71C2×C413yes—
52C2×C4⋊C4×C71C2×C4124yes—
53C2×C42×C71C2×C413yes—
54C2×C42×C71C2×C413yes—
55C2×C42×C71C2×C4112yes—
56C2×C42×C71C2313yes—
57C2×C42×C71C2313yes—
58C2×C42×C71C23112yes—
59#11765849—224no—
60#11765849C2×Q8116yesdirect product factor
61C2×C71⋊C8—212no—
62C2×C71⋊C8C2×Q818yescomplemented
63C2×C71⋊C8C24112yes—
64C2×C71⋊C8C2414yes—
65C24×C71C22×C411yescharacteristic
66C2×Q8×C71—232no—
67C2×Q8×C71—212no—
68C2×Q8×C71M1618yescomplemented
69C2×Q8×C71C22×C4112yes—
70C2×Q8×C71C22×C414yes—
71C22×C4×C71—212no—
72C22×C4×C71—46no—
73C22×C4×C71C22×C413yes—
74C22×C4×C71C22×C413yes—
75C22×C4×C71C22×C413yes—
76C22×C4×C71C22×C4112yes—
77C22×C4×C71C22×C4112yes—
78C22×C4×C71C22×C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C22×C413yes—
81C22×C4×C71C2×Q814yes—
82C22×C4×C71#4114yes—
83C22×C4×C71C2412yes—
84C22×C4×C71C2411yescharacteristic
85C4⋊C4×C71C22×C4124yes—
86C4⋊C4×C71C22×C4112yes—
87C4⋊C4×C71C22×C4112yes—
88C42×C71C22×C416yes—
89C42×C71C22×C4112yes—
90C42×C71C22×C416yes—
91C42×C71C22×C4112yes—
92C42×C71C2416yes—
93C42×C71C2416yes—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71C4×Q814yes—
97C23×C71C23×C412yes—
98C23×C71C23×C411yescharacteristic
99Q8×C71—224no—
100Q8×C71C2×M16116yescomplemented
101C2×C4×C71—26no—
102C2×C4×C71—26no—
103C2×C4×C71—26no—
104C2×C4×C71—224no—
105C2×C4×C71—412no—
106C2×C4×C71C4×Q818yes—
107C2×C4×C71C4×Q814yes—
108C2×C4×C71C2×M16112yes—
109C2×C4×C71C23×C416yes—
110C2×C4×C71C23×C416yes—
111C2×C4×C71C23×C416yes—
112C2×C4×C71C23×C412yes—
113C2×C4×C71C23×C411yescharacteristic
114C2×C4×C71C23×C416yes—
115C2×C4×C71C22×Q812yes—
116C2×C4×C71C22×Q814yes—
117C2×C4×C71C2×#4112yes—
118C2×C4×C71C2×#4114yes—
119C2×C4×C71C2511yescharacteristic
120C71⋊C8—212no—
121C71⋊C8C22×Q818yescomplemented
122C22×C71—28no—
123C22×C71—22no—
124C22×C71—22no—
125C22×C71C2×C4×Q814yes—
126C22×C71C2×C4×Q812yes—
127C22×C71C22×M1612yes—
128C22×C71#56812yes—
129C22×C71C24×C411yescharacteristic
130C4×C71—26no—
131C4×C71—26no—
132C4×C71—26no—
133C4×C71C2×C4×Q812yes—
134C4×C71C2×C4×Q814yes—
135C4×C71C22×M16112yes—
136C4×C71C23×Q811yescharacteristic
137C4×C71C22×#4111yescharacteristic
138C2×Q8×M16—711nomaximal, Sylow
139C2×C71—24no—
140C2×C71Q8×M1614yescomplemented
141C2×C71C22×C4×Q811yescharacteristic
142C2×C71C23×M1611yescharacteristic
143C2×C71C2×#56811yescharacteristic
144C22×C4×Q8D7111yescharacteristic
145Q8×M16—7116no—
146C2×#446—716no—
147C2×C8×Q8—712no—
148C2×#423—713no—
149C2×C4×M16—713no—
150C71C2×Q8×M1611yescharacteristic, complemented, Sylow
151C23×Q8Dic7111yescharacteristic
152C22×M16—711no—
153C22×M16—713no—
154C2×C4×Q8Dic7113yes—
155C2×C4×Q8Dic7118yes—
156C2×C4×Q8Dic7111yescharacteristic
157C2×C4×Q8C2×D7111yescharacteristic
158C2×C4×Q8C2×D7113yes—
159C2×C4×Q8C2×D7118yes—
160C22×C4⋊C4Dic7113yes—
161C22×C42C2×D7113yes—
162#446—7148no—
163C8×Q8—7116no—
164#423—7124no—
165C2×C4⋊C8—716no—
166C2×C4⋊C8—716no—
167C4×M16—7124no—
168C2×#97—716no—
169C2×C4×C8—716no—
170C22×Q8—28no—
171C22×Q8C2×Dic7118yes—
172C22×Q8C2×Dic7113yes—
173C22×Q8C2×Dic7111yescharacteristic
174C23×C4C2×Dic7113yes—
175C23×C4C2×Dic7113yes—
176C23×C4C22×D7111yescharacteristic
177C2×M16—718no—
178C2×M16—714no—
179C2×M16—7112no—
180C2×M16—14212no—
181C22×C8—712no—
182C22×C8—716no—
183C4×Q8C2×Dic71112yes—
184C4×Q8C2×Dic71124yes—
185C4×Q8C2×Dic7114yes—
186C4×Q8C2×Dic7118yes—
187C4×Q8C22×D7114yes—
188C4×Q8C22×D71112yes—
189C2×C4⋊C4C2×Dic7116yes—
190C2×C4⋊C4C2×Dic7113yes—
191C2×C4⋊C4C2×Dic7113yes—
192C2×C4⋊C4C2×Dic71124yes—
193C2×C42C2×Dic7113yes—
194C2×C42C2×Dic7113yes—
195C2×C42C2×Dic71112yes—
196C2×C42C22×D71112yes—
197C2×C42C22×D7113yes—
198C2×C42C22×D7113yes—
199C4⋊C8—7124no—
200C4⋊C8—7124no—
201#97—7124no—
202C4×C8—7124no—
203C24C22×Dic7111yescharacteristic
204C2×Q8—232no—
205C2×Q8—212no—
206C2×Q8#1176584918yesdirect product factor
207C2×Q8C22×Dic71112yes—
208C2×Q8C22×Dic7114yes—
209C22×C4—212no—
210C22×C4—46no—
211C22×C4C22×Dic7113yes—
212C22×C4C22×Dic7113yes—
213C22×C4C22×Dic7113yes—
214C22×C4C22×Dic71112yes—
215C22×C4C22×Dic71112yes—
216C22×C4C22×Dic7111yescharacteristic
217C22×C4C22×Dic7112yes—
218C22×C4C22×Dic7113yes—
219C22×C4Q8×D7114yes—
220C22×C4#1176586614yes—
221C22×C4C23×D7112yes—
222C22×C4C23×D7111yescharacteristic
223M16—7116no—
224M16—14224no—
225C2×C8—718no—
226C2×C8—714no—
227C2×C8—7112no—
228C2×C8—14212no—
229C4⋊C4C22×Dic71124yes—
230C4⋊C4C22×Dic71112yes—
231C4⋊C4C22×Dic71112yes—
232C42C22×Dic71112yes—
233C42C22×Dic7116yes—
234C42C22×Dic71112yes—
235C42C22×Dic7116yes—
236C42C23×D7116yes—
237C42C23×D7116yes—
238C23—22no—
239C23—22no—
240C23Q8×Dic7114yes—
241C23C23×Dic7112yes—
242C23C23×Dic7111yescharacteristic
243Q8—224no—
244Q8C2×#11765849116yesdirect product factor
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—224no—
249C2×C4—412no—
250C2×C4Q8×Dic7118yes—
251C2×C4Q8×Dic7114yes—
252C2×C4C2×#11765849112yes—
253C2×C4C23×Dic7116yes—
254C2×C4C23×Dic7116yes—
255C2×C4C23×Dic7116yes—
256C2×C4C23×Dic7112yes—
257C2×C4C23×Dic7111yescharacteristic
258C2×C4C23×Dic7116yes—
259C2×C4C2×Q8×D7112yes—
260C2×C4C2×Q8×D7114yes—
261C2×C4C2×#1176586612yes—
262C2×C4C2×#1176586614yes—
263C2×C4C24×D7111yescharacteristic
264C8—718no—
265C8—14212no—
266C22—22no—
267C22—22no—
268C22—28no—
269C22C2×Q8×Dic7114yes—
270C22C2×Q8×Dic7112yes—
271C22C22×#1176584912yes—
272C22#1798676612yes—
273C22C24×Dic7111yescharacteristic
274C4—26no—
275C4—26no—
276C4—26no—
277C4C2×Q8×Dic7114yes—
278C4C2×Q8×Dic7112yes—
279C4C22×#11765849112yes—
280C4C22×Q8×D7111yescharacteristic
281C4C22×#1176586611yescharacteristic
282C2—24no—
283C2Q8×#1176584914yesdirect product factor
284C2C22×Q8×Dic7111yescharacteristic
285C2C23×#1176584911yescharacteristic
286C2C2×#1798676611yescharacteristic
subgroupG/H
centerC22×C4C23×D71
commutatorC22×C71C24×C4
FrattiniC2×C4C24×D71
FittingC22×C4×Q8×C71C2
socleC23×C71C23×C4
prime radicalC24×C71C22×C4
cosocleC2×C4×C71C25
hypercenterC22×C4×Q8D71
hypocenterC71C2×Q8×M16
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#11765849C2direct (N × H)
2. C2×#17986130C2semidirect
3. C2×Q8×C71⋊C8C2semidirect
9. #17986130C22semidirect
10. Q8×C71⋊C8C22semidirect
27. C2×#11765849Q8direct (N × H)
60. #11765849C2×Q8direct (N × H)
62. C2×C71⋊C8C2×Q8semidirect
68. C2×Q8×C71M16semidirect
100. Q8×C71C2×M16semidirect
121. C71⋊C8C22×Q8semidirect
140. C2×C71Q8×M16semidirect
150. C71C2×Q8×M16semidirect
206. C2×Q8#11765849direct (N × H)
244. Q8C2×#11765849direct (N × H)
283. C2Q8×#11765849direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#117658492
C2×#179861302
C2×Q8×C71⋊C82
C2×#179860232
C2×C4×#117658492
C22×C4×Q8×C712
C2×Q8×M1671