Finite Group Databasebeta

← C2×Q8×C22⋊C4×C23 #19181726

Subgroups — C2×Q8×C22⋊C4×C23 #19181726

Order 5888.

5414 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×Q8×C23C211yesmaximal, characteristic
2C22×C4×Q8×C23C212yesmaximal, complemented
3Q8×C22⋊C4×C23C2116yesmaximal, direct product factor
4C2×C23×#391C216yesmaximal, complemented
5C2×C23×#380C213yesmaximal
6C2×C4×C22⋊C4×C23C213yesmaximal
7C23×Q8×C23—216no—
8C23×Q8×C23C4116yescomplemented
9C23×Q8×C23C2213yes—
10C23×Q8×C23C2218yes—
11C23×Q8×C23C2211yescharacteristic
12C24×C4×C23C2213yes—
13C2×C4×Q8×C23—216no—
14C2×C4×Q8×C23C22116yescomplemented
15C22×C4⋊C4×C23C2216yes—
16C22×C22⋊C4×C23C2213yes—
17C22×C22⋊C4×C23C2211yescharacteristic
18C22×C42×C23C2216yes—
19C23×#391C22148yescomplemented
20C23×#380C22124yes—
21C4×C22⋊C4×C23C22124yes—
22C2×C23×#95C2216yes—
23C2×C23×#95C2216yes—
24C25×C23C2311yescharacteristic
25C22×Q8×C23—2128no—
26C22×Q8×C23—264no—
27C22×Q8×C23—224no—
28C22×Q8×C23C2×C4124yes—
29C22×Q8×C23C2×C418yes—
30C22×Q8×C23C2×C4164yescomplemented
31C22×Q8×C23D4116yes—
32C22×Q8×C23C23112yes—
33C22×Q8×C23C2314yes—
34C23×C4×C23—224no—
35C23×C4×C23C2×C4124yes—
36C23×C4×C23C2316yes—
37C23×C4×C23C23112yes—
38C23×C4×C23C2313yes—
39C23×C4×C23C2312yes—
40C23×C4×C23C2313yes—
41C4×Q8×C23—264no—
42C2×C4⋊C4×C23—224no—
43C2×C4⋊C4×C23C23124yes—
44C2×C22⋊C4×C23—212no—
45C2×C22⋊C4×C23Q818yesdirect product factor
46C2×C22⋊C4×C23C23112yes—
47C2×C22⋊C4×C23C2314yes—
48C2×C42×C23—224no—
49C2×C42×C23C23124yes—
50C23×#95C23124yes—
51C23×#95C23124yes—
52C24×C23—24no—
53C24×C23—24no—
54C24×C23C22×C414yes—
55C24×C23C2×Q814yes—
56C24×C23#4114yes—
57C24×C23C2412yes—
58C24×C23C2411yescharacteristic
59C2×Q8×C23—2256no—
60C2×Q8×C23—296no—
61C2×Q8×C23—248no—
62C2×Q8×C23C22⋊C4164yesdirect product factor
63C2×Q8×C23C22×C4148yes—
64C2×Q8×C23C22×C4116yes—
65C2×Q8×C23C2×D4132yes—
66C22×C4×C23—24no—
67C22×C4×C23—212no—
68C22×C4×C23—24no—
69C22×C4×C23—296no—
70C22×C4×C23—248no—
71C22×C4×C23—212no—
72C22×C4×C23—212no—
73C22×C4×C23—212no—
74C22×C4×C23C22×C4112yes—
75C22×C4×C23C22×C4148yes—
76C22×C4×C23C22×C4112yes—
77C22×C4×C23C2×D4124yes—
78C22×C4×C23C2×Q818yescomplemented
79C22×C4×C23C24112yes—
80C22×C4×C23C2416yes—
81C22×C4×C23C2414yes—
82C22×C4×C23C2416yes—
83C4⋊C4×C23—248no—
84C22⋊C4×C23—224no—
85C22⋊C4×C23C2×Q8116yesdirect product factor
86C42×C23—248no—
87C2×Q8×C22⋊C4C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—28no—
89C23×C23—28no—
90C23×C23—216no—
91C23×C23—216no—
92C23×C23—24no—
93C23×C23—24no—
94C23×C23C4×Q8116yescomplemented
95C23×C23C23×C414yes—
96C23×C23C23×C412yes—
97C23×C23C22×D414yes—
98C23×C23C22×Q814yes—
99C23×C23C22×Q812yes—
100C23×C23C2×#4114yes—
101C23×C23C2×#4112yes—
102C23×C232−1+414yes—
103C23×C23C2511yescharacteristic
104Q8×C23—296no—
105Q8×C23C2×C22⋊C4164yesdirect product factor
106C2×C4×C23—224no—
107C2×C4×C23—216no—
108C2×C4×C23—24no—
109C2×C4×C23—212no—
110C2×C4×C23—24no—
111C2×C4×C23—296no—
112C2×C4×C23—212no—
113C2×C4×C23—212no—
114C2×C4×C23—212no—
115C2×C4×C23—424no—
116C2×C4×C23C2×C22⋊C4148yes—
117C2×C4×C23C23×C4112yes—
118C2×C4×C23C23×C4112yes—
119C2×C4×C23C22×D4124yes—
120C2×C4×C23C22×Q818yescomplemented
121C24×Q8C2×C2311yescharacteristic
122C22×C4×Q8C2×C2312yescomplemented
123Q8×C22⋊C4C2×C23116yesdirect product factor
124C2×#391C2×C2316yescomplemented
125C2×#380C2×C2313yes—
126C2×C4×C22⋊C4C2×C2313yes—
127C22×C23—232no—
128C22×C23—24no—
129C22×C23—24no—
130C22×C23—28no—
131C22×C23—28no—
132C22×C23C22×C22⋊C414yes—
133C22×C23C2×C4×Q814yes—
134C22×C23C2×C4×Q818yescomplemented
135C22×C23#51914yes—
136C22×C23#54818yes—
137C22×C23D4×Q818yes—
138C22×C23C24×C411yescharacteristic
139C22×C23C23×D412yes—
140C22×C23C23×Q811yescharacteristic
141C22×C23C22×#4111yescharacteristic
142C22×C23C2×2−1+412yes—
143C4×C23—212no—
144C4×C23—28no—
145C4×C23—412no—
146C4×C23C22×C22⋊C4124yes—
147C23×Q8—216no—
148C23×Q8C4×C23116yescomplemented
149C23×Q8C22×C2313yes—
150C23×Q8C22×C2318yes—
151C23×Q8C22×C2311yescharacteristic
152C24×C4C22×C2313yes—
153C2×C4×Q8—216no—
154C2×C4×Q8C22×C23116yescomplemented
155C22×C4⋊C4C22×C2316yes—
156C22×C22⋊C4C22×C2313yes—
157C22×C22⋊C4C22×C2311yescharacteristic
158C22×C42C22×C2316yes—
159#391C22×C23148yescomplemented
160#380C22×C23124yes—
161C4×C22⋊C4C22×C23124yes—
162C2×#95C22×C2316yes—
163C2×#95C22×C2316yes—
164C2×C23—28no—
165C2×C23Q8×C22⋊C418yesdirect product factor
166C2×C23C23×C22⋊C411yescharacteristic
167C2×C23C22×C4×Q811yescharacteristic
168C2×C23C2×#51911yescharacteristic
169C2×C23C2×#54812yes—
170C2×C23C2×D4×Q812yes—
171C25C23×C2311yescharacteristic
172C22×Q8—2128no—
173C22×Q8—264no—
174C22×Q8—224no—
175C22×Q8C2×C4×C23124yes—
176C22×Q8C2×C4×C2318yes—
177C22×Q8C2×C4×C23164yescomplemented
178C22×Q8D4×C23116yes—
179C22×Q8C23×C23112yes—
180C22×Q8C23×C2314yes—
181C23×C4—224no—
182C23×C4C2×C4×C23124yes—
183C23×C4C23×C2316yes—
184C23×C4C23×C2313yes—
185C23×C4C23×C23112yes—
186C23×C4C23×C2312yes—
187C23×C4C23×C2313yes—
188C4×Q8—264no—
189C2×C4⋊C4—224no—
190C2×C4⋊C4C23×C23124yes—
191C2×C22⋊C4—212no—
192C2×C22⋊C4Q8×C2318yesdirect product factor
193C2×C22⋊C4C23×C23112yes—
194C2×C22⋊C4C23×C2314yes—
195C2×C42—224no—
196C2×C42C23×C23124yes—
197#95C23×C23124yes—
198#95C23×C23124yes—
199C23C2×Q8×C22⋊C411yescharacteristic, direct product factor, Sylow
200C24—24no—
201C24—24no—
202C24C22×C4×C2314yes—
203C24C2×Q8×C2314yes—
204C24#41×C2314yes—
205C24C24×C2312yes—
206C24C24×C2311yescharacteristic
207C2×Q8—2256no—
208C2×Q8—296no—
209C2×Q8—248no—
210C2×Q8C22⋊C4×C23164yesdirect product factor
211C2×Q8C22×C4×C23148yes—
212C2×Q8C22×C4×C23116yes—
213C2×Q8C2×D4×C23132yes—
214C22×C4—24no—
215C22×C4—212no—
216C22×C4—24no—
217C22×C4—296no—
218C22×C4—248no—
219C22×C4—212no—
220C22×C4—212no—
221C22×C4—212no—
222C22×C4C22×C4×C23112yes—
223C22×C4C22×C4×C23148yes—
224C22×C4C22×C4×C23112yes—
225C22×C4C2×D4×C23124yes—
226C22×C4C2×Q8×C2318yescomplemented
227C22×C4C24×C23112yes—
228C22×C4C24×C2316yes—
229C22×C4C24×C2314yes—
230C22×C4C24×C2316yes—
231C4⋊C4—248no—
232C22⋊C4—224no—
233C22⋊C4C2×Q8×C23116yesdirect product factor
234C42—248no—
235C23—216no—
236C23—216no—
237C23—24no—
238C23—24no—
239C23—28no—
240C23—28no—
241C23C4×Q8×C23116yescomplemented
242C23C23×C4×C2314yes—
243C23C23×C4×C2312yes—
244C23C22×D4×C2314yes—
245C23C22×Q8×C2314yes—
246C23C22×Q8×C2312yes—
247C23C2×#41×C2314yes—
248C23C2×#41×C2312yes—
249C23C23×2−1+414yes—
250C23C25×C2311yescharacteristic
251Q8—296no—
252Q8C2×C22⋊C4×C23164yesdirect product factor
253C2×C4—216no—
254C2×C4—24no—
255C2×C4—212no—
256C2×C4—24no—
257C2×C4—224no—
258C2×C4—296no—
259C2×C4—212no—
260C2×C4—212no—
261C2×C4—212no—
262C2×C4—424no—
263C2×C4C2×C22⋊C4×C23148yes—
264C2×C4C23×C4×C23112yes—
265C2×C4C23×C4×C23112yes—
266C2×C4C22×D4×C23124yes—
267C2×C4C22×Q8×C2318yescomplemented
268C22—24no—
269C22—28no—
270C22—28no—
271C22—232no—
272C22—24no—
273C22C22×C22⋊C4×C2314yes—
274C22C2×C4×Q8×C2314yes—
275C22C2×C4×Q8×C2318yescomplemented
276C22C23×#51914yes—
277C22C23×#54818yes—
278C22D4×Q8×C2318yes—
279C22C24×C4×C2311yescharacteristic
280C22C23×D4×C2312yes—
281C22C23×Q8×C2311yescharacteristic
282C22C22×#41×C2311yescharacteristic
283C22C2×C23×2−1+412yes—
284C4—28no—
285C4—212no—
286C4—412no—
287C4C22×C22⋊C4×C23124yes—
288C2—28no—
289C2Q8×C22⋊C4×C2318yesdirect product factor
290C2C23×C22⋊C4×C2311yescharacteristic
291C2C22×C4×Q8×C2311yescharacteristic
292C2C2×C23×#51911yescharacteristic
293C2C2×C23×#54812yes—
294C2C2×D4×Q8×C2312yes—
subgroupG/H
center, socleC24×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
prime radicalC25×C23C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q8×C23C2semidirect
3. Q8×C22⋊C4×C23C2direct (N × H)
4. C2×C23×#391C2semidirect
8. C23×Q8×C23C4semidirect
14. C2×C4×Q8×C23C22semidirect
19. C23×#391C22semidirect
30. C22×Q8×C23C2×C4semidirect
45. C2×C22⋊C4×C23Q8direct (N × H)
62. C2×Q8×C23C22⋊C4direct (N × H)
78. C22×C4×C23C2×Q8semidirect
85. C22⋊C4×C23C2×Q8direct (N × H)
87. C2×Q8×C22⋊C4C23direct (N × H)
94. C23×C23C4×Q8semidirect
105. Q8×C23C2×C22⋊C4direct (N × H)
120. C2×C4×C23C22×Q8semidirect
122. C22×C4×Q8C2×C23semidirect
123. Q8×C22⋊C4C2×C23direct (N × H)
124. C2×#391C2×C23semidirect
134. C22×C23C2×C4×Q8semidirect
148. C23×Q8C4×C23semidirect
154. C2×C4×Q8C22×C23semidirect
159. #391C22×C23semidirect
165. C2×C23Q8×C22⋊C4direct (N × H)
177. C22×Q8C2×C4×C23semidirect
192. C2×C22⋊C4Q8×C23direct (N × H)
199. C23C2×Q8×C22⋊C4direct (N × H)
210. C2×Q8C22⋊C4×C23direct (N × H)
226. C22×C4C2×Q8×C23semidirect
233. C22⋊C4C2×Q8×C23direct (N × H)
241. C23C4×Q8×C23semidirect
252. Q8C2×C22⋊C4×C23direct (N × H)
267. C2×C4C22×Q8×C23semidirect
275. C22C2×C4×Q8×C23semidirect
289. C2Q8×C22⋊C4×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C24×Q8×C232
C22×C4×Q8×C232
Q8×C22⋊C4×C232
C2×C23×#3912
C2×C23×#3802
C2×C4×C22⋊C4×C232
C2×Q8×C22⋊C423