Finite Group Databasebeta

← C2×Q8×M16×C61 #29563623

Subgroups — C2×Q8×M16×C61 #29563623

Order 15616.

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
1C22×C4×Q8×C61C211yesmaximal, characteristic
2Q8×M16×C61C2116yesmaximal, direct product factor
3C2×C61×#446C216yesmaximal, complemented
4C2×C8×Q8×C61C212yesmaximal, complemented
5C2×C61×#423C213yesmaximal
6C2×C4×M16×C61C213yesmaximal
7C23×Q8×C61C411yescharacteristic
8C22×M16×C61C2213yes—
9C22×M16×C61C2211yescharacteristic
10C2×C4×Q8×C61C413yes—
11C2×C4×Q8×C61C418yes—
12C2×C4×Q8×C61C411yescharacteristic
13C2×C4×Q8×C61C2211yescharacteristic
14C2×C4×Q8×C61C2213yes—
15C2×C4×Q8×C61C2218yes—
16C22×C4⋊C4×C61C413yes—
17C22×C42×C61C2213yes—
18C61×#446C22148yescomplemented
19C8×Q8×C61C22116yescomplemented
20C61×#423C22124yes—
21C2×C4⋊C8×C61C2216yes—
22C2×C4⋊C8×C61C2216yes—
23C4×M16×C61C22124yes—
24C2×#97×C61C2216yes—
25C2×C4×C8×C61C2216yes—
26C22×Q8×C61—28no—
27C22×Q8×C61C2×C418yes—
28C22×Q8×C61C2×C413yes—
29C22×Q8×C61C2×C411yescharacteristic
30C23×C4×C61C2×C413yes—
31C23×C4×C61C2×C413yes—
32C23×C4×C61C2311yescharacteristic
33C2×M16×C61—212no—
34C2×M16×C61Q818yesdirect product factor
35C2×M16×C61C23112yes—
36C2×M16×C61C2314yes—
37C22×C8×C61C2316yes—
38C22×C8×C61C2312yes—
39C4×Q8×C61C2×C4112yes—
40C4×Q8×C61C2×C4124yes—
41C4×Q8×C61C2×C414yes—
42C4×Q8×C61C2×C418yes—
43C4×Q8×C61C2314yes—
44C4×Q8×C61C23112yes—
45C2×C4⋊C4×C61C2×C416yes—
46C2×C4⋊C4×C61C2×C413yes—
47C2×C4⋊C4×C61C2×C413yes—
48C2×C4⋊C4×C61C2×C4124yes—
49C2×C42×C61C2×C413yes—
50C2×C42×C61C2×C413yes—
51C2×C42×C61C2×C4112yes—
52C2×C42×C61C23112yes—
53C2×C42×C61C2313yes—
54C2×C42×C61C2313yes—
55C4⋊C8×C61C23124yes—
56C4⋊C8×C61C23124yes—
57#97×C61C23124yes—
58C4×C8×C61C23124yes—
59C24×C61C22×C411yescharacteristic
60C2×Q8×C61—232no—
61C2×Q8×C61—212no—
62C2×Q8×C61M1618yesdirect product factor
63C2×Q8×C61C22×C4112yes—
64C2×Q8×C61C22×C414yes—
65C22×C4×C61—212no—
66C22×C4×C61—46no—
67C22×C4×C61C22×C413yes—
68C22×C4×C61C22×C413yes—
69C22×C4×C61C22×C413yes—
70C22×C4×C61C22×C4112yes—
71C22×C4×C61C22×C4112yes—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C22×C412yes—
74C22×C4×C61C22×C413yes—
75C22×C4×C61C2×Q814yes—
76C22×C4×C61#4114yes—
77C22×C4×C61C2412yes—
78C22×C4×C61C2411yescharacteristic
79M16×C61—224no—
80M16×C61C2×Q8116yesdirect product factor
81C2×C8×C61—212no—
82C2×C8×C61C2×Q818yescomplemented
83C2×C8×C61C24112yes—
84C2×C8×C61C2414yes—
85C4⋊C4×C61C22×C4124yes—
86C4⋊C4×C61C22×C4112yes—
87C4⋊C4×C61C22×C4112yes—
88C42×C61C22×C4112yes—
89C42×C61C22×C416yes—
90C42×C61C22×C4112yes—
91C42×C61C22×C416yes—
92C42×C61C2416yes—
93C42×C61C2416yes—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61C4×Q814yes—
97C23×C61C23×C412yes—
98C23×C61C23×C411yescharacteristic
99Q8×C61—224no—
100Q8×C61C2×M16116yesdirect product factor
101C2×C4×C61—26no—
102C2×C4×C61—26no—
103C2×C4×C61—26no—
104C2×C4×C61—224no—
105C2×C4×C61—412no—
106C2×C4×C61C4×Q818yes—
107C2×C4×C61C4×Q814yes—
108C2×C4×C61C2×M16112yes—
109C2×C4×C61C23×C416yes—
110C2×C4×C61C23×C416yes—
111C2×C4×C61C23×C416yes—
112C2×C4×C61C23×C412yes—
113C2×C4×C61C23×C411yescharacteristic
114C2×C4×C61C23×C416yes—
115C2×C4×C61C22×Q812yes—
116C2×C4×C61C22×Q814yes—
117C2×C4×C61C2×#4112yes—
118C2×C4×C61C2×#4114yes—
119C2×C4×C61C2511yescharacteristic
120C8×C61—212no—
121C8×C61C22×Q818yescomplemented
122C2×Q8×M16C6111yesmaximal, characteristic, direct product factor, Sylow
123C22×C61—28no—
124C22×C61—22no—
125C22×C61—22no—
126C22×C61C2×C4×Q814yes—
127C22×C61C2×C4×Q812yes—
128C22×C61C22×M1612yes—
129C22×C61#56812yes—
130C22×C61C24×C411yescharacteristic
131C4×C61—26no—
132C4×C61—26no—
133C4×C61—26no—
134C4×C61C2×C4×Q814yes—
135C4×C61C2×C4×Q812yes—
136C4×C61C22×M16112yes—
137C4×C61C23×Q811yescharacteristic
138C4×C61C22×#4111yescharacteristic
139C22×C4×Q8C2×C6111yescharacteristic
140Q8×M16C2×C61116yesdirect product factor
141C2×#446C2×C6116yescomplemented
142C2×C8×Q8C2×C6112yescomplemented
143C2×#423C2×C6113yes—
144C2×C4×M16C2×C6113yes—
145C2×C61—24no—
146C2×C61Q8×M1614yesdirect product factor
147C2×C61C22×C4×Q811yescharacteristic
148C2×C61C23×M1611yescharacteristic
149C2×C61C2×#56811yescharacteristic
150C23×Q8C4×C6111yescharacteristic
151C22×M16C22×C6113yes—
152C22×M16C22×C6111yescharacteristic
153C2×C4×Q8C4×C6113yes—
154C2×C4×Q8C4×C6118yes—
155C2×C4×Q8C4×C6111yescharacteristic
156C2×C4×Q8C22×C6111yescharacteristic
157C2×C4×Q8C22×C6113yes—
158C2×C4×Q8C22×C6118yes—
159C22×C4⋊C4C4×C6113yes—
160C22×C42C22×C6113yes—
161#446C22×C61148yescomplemented
162C8×Q8C22×C61116yescomplemented
163#423C22×C61124yes—
164C2×C4⋊C8C22×C6116yes—
165C2×C4⋊C8C22×C6116yes—
166C4×M16C22×C61124yes—
167C2×#97C22×C6116yes—
168C2×C4×C8C22×C6116yes—
169C61C2×Q8×M1611yescharacteristic, direct product factor, Sylow
170C22×Q8—28no—
171C22×Q8C2×C4×C6118yes—
172C22×Q8C2×C4×C6113yes—
173C22×Q8C2×C4×C6111yescharacteristic
174C23×C4C2×C4×C6113yes—
175C23×C4C2×C4×C6113yes—
176C23×C4C23×C6111yescharacteristic
177C2×M16—212no—
178C2×M16Q8×C6118yesdirect product factor
179C2×M16C23×C61112yes—
180C2×M16C23×C6114yes—
181C22×C8C23×C6116yes—
182C22×C8C23×C6112yes—
183C4×Q8C2×C4×C61112yes—
184C4×Q8C2×C4×C61124yes—
185C4×Q8C2×C4×C6114yes—
186C4×Q8C2×C4×C6118yes—
187C4×Q8C23×C6114yes—
188C4×Q8C23×C61112yes—
189C2×C4⋊C4C2×C4×C6116yes—
190C2×C4⋊C4C2×C4×C6113yes—
191C2×C4⋊C4C2×C4×C6113yes—
192C2×C4⋊C4C2×C4×C61124yes—
193C2×C42C2×C4×C6113yes—
194C2×C42C2×C4×C6113yes—
195C2×C42C2×C4×C61112yes—
196C2×C42C23×C6113yes—
197C2×C42C23×C6113yes—
198C2×C42C23×C61112yes—
199C4⋊C8C23×C61124yes—
200C4⋊C8C23×C61124yes—
201#97C23×C61124yes—
202C4×C8C23×C61124yes—
203C24C22×C4×C6111yescharacteristic
204C2×Q8—232no—
205C2×Q8—212no—
206C2×Q8M16×C6118yesdirect product factor
207C2×Q8C22×C4×C61112yes—
208C2×Q8C22×C4×C6114yes—
209C22×C4—212no—
210C22×C4—46no—
211C22×C4C22×C4×C6113yes—
212C22×C4C22×C4×C6113yes—
213C22×C4C22×C4×C6113yes—
214C22×C4C22×C4×C61112yes—
215C22×C4C22×C4×C61112yes—
216C22×C4C22×C4×C6112yes—
217C22×C4C22×C4×C6111yescharacteristic
218C22×C4C22×C4×C6113yes—
219C22×C4C2×Q8×C6114yes—
220C22×C4#41×C6114yes—
221C22×C4C24×C6111yescharacteristic
222C22×C4C24×C6112yes—
223M16—224no—
224M16C2×Q8×C61116yesdirect product factor
225C2×C8—212no—
226C2×C8C2×Q8×C6118yescomplemented
227C2×C8C24×C61112yes—
228C2×C8C24×C6114yes—
229C4⋊C4C22×C4×C61124yes—
230C4⋊C4C22×C4×C61112yes—
231C4⋊C4C22×C4×C61112yes—
232C42C22×C4×C6116yes—
233C42C22×C4×C61112yes—
234C42C22×C4×C6116yes—
235C42C22×C4×C61112yes—
236C42C24×C6116yes—
237C42C24×C6116yes—
238C23—22no—
239C23—22no—
240C23C4×Q8×C6114yes—
241C23C23×C4×C6112yes—
242C23C23×C4×C6111yescharacteristic
243Q8—224no—
244Q8C2×M16×C61116yesdirect product factor
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—224no—
249C2×C4—412no—
250C2×C4C4×Q8×C6118yes—
251C2×C4C4×Q8×C6114yes—
252C2×C4C2×M16×C61112yes—
253C2×C4C23×C4×C6116yes—
254C2×C4C23×C4×C6116yes—
255C2×C4C23×C4×C6116yes—
256C2×C4C23×C4×C6116yes—
257C2×C4C23×C4×C6112yes—
258C2×C4C23×C4×C6111yescharacteristic
259C2×C4C22×Q8×C6112yes—
260C2×C4C22×Q8×C6114yes—
261C2×C4C2×#41×C6112yes—
262C2×C4C2×#41×C6114yes—
263C2×C4C25×C6111yescharacteristic
264C8—212no—
265C8C22×Q8×C6118yescomplemented
266C22—22no—
267C22—22no—
268C22—28no—
269C22C2×C4×Q8×C6114yes—
270C22C2×C4×Q8×C6112yes—
271C22C22×M16×C6112yes—
272C22C61×#56812yes—
273C22C24×C4×C6111yescharacteristic
274C4—26no—
275C4—26no—
276C4—26no—
277C4C2×C4×Q8×C6112yes—
278C4C2×C4×Q8×C6114yes—
279C4C22×M16×C61112yes—
280C4C23×Q8×C6111yescharacteristic
281C4C22×#41×C6111yescharacteristic
282C2—24no—
283C2Q8×M16×C6114yesdirect product factor
284C2C22×C4×Q8×C6111yescharacteristic
285C2C23×M16×C6111yescharacteristic
286C2C2×C61×#56811yescharacteristic
subgroupG/H
centerC22×C4×C61C24
commutatorC22C24×C4×C61
Frattini, cosocleC2×C4C25×C61
socleC23×C61C23×C4
prime radicalC24×C61C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×M16×C61C2direct (N × H)
3. C2×C61×#446C2semidirect
4. C2×C8×Q8×C61C2semidirect
18. C61×#446C22semidirect
19. C8×Q8×C61C22semidirect
34. C2×M16×C61Q8direct (N × H)
62. C2×Q8×C61M16direct (N × H)
80. M16×C61C2×Q8direct (N × H)
82. C2×C8×C61C2×Q8semidirect
100. Q8×C61C2×M16direct (N × H)
121. C8×C61C22×Q8semidirect
122. C2×Q8×M16C61direct (N × H)
140. Q8×M16C2×C61direct (N × H)
141. C2×#446C2×C61semidirect
142. C2×C8×Q8C2×C61semidirect
146. C2×C61Q8×M16direct (N × H)
161. #446C22×C61semidirect
162. C8×Q8C22×C61semidirect
169. C61C2×Q8×M16direct (N × H)
178. C2×M16Q8×C61direct (N × H)
206. C2×Q8M16×C61direct (N × H)
224. M16C2×Q8×C61direct (N × H)
226. C2×C8C2×Q8×C61semidirect
244. Q8C2×M16×C61direct (N × H)
265. C8C22×Q8×C61semidirect
283. C2Q8×M16×C61direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C612
Q8×M16×C612
C2×C61×#4462
C2×C8×Q8×C612
C2×C61×#4232
C2×C4×M16×C612
C2×Q8×M1661