Finite Group Databasebeta

← C2×#27132822 #37581726

Subgroups — C2×#27132822 #37581726

Order 24448.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690286C211yesmaximal, characteristic
2C22×#15690277C211yesmaximal, characteristic, complemented
3C2×#20259298C211yesmaximal, characteristic, complemented
4#27132822C218yesmaximal, direct product factor
5C2×#20259247C211yesmaximal, characteristic, complemented
6C2×#20259243C211yesmaximal, characteristic, complemented
7C2×(C2×D191)⋊C8C211yesmaximal, characteristic, complemented
8C22×SD16×C191C211yesmaximal, characteristic, complemented
9C2×#15690286—24no—
10C2×#15690286—22no—
11C2×#15690286—24no—
12C2×#15690286C2214yes—
13C22×Dic382C2211yescharacteristic
14C22×C22⋊D191—21no—
15C23×Dic191—21no—
16C22×C4×D191C2211yescharacteristic
17C2×#15690277—24no—
18C2×#15690277C2214yescomplemented
19C22×C191⋊C8C2211yescharacteristic
20#20259298—22no—
21#20259298C2214yescomplemented
22C2×C4⋊Dic191C2211yescharacteristic
23C2×Dic191⋊C4—21no—
24C2×(C2×D191)⋊C4—21no—
25#20259247C2214yescomplemented
26#20259243C2214yescomplemented
27(C2×D191)⋊C8C2214yescomplemented
28C22×Q8×C191C2211yescharacteristic
29C22×D4×C191C2211yescharacteristic
30C2×SD16×C191—24no—
31C2×SD16×C191C2214yescomplemented
32C22×C8×C191C2211yescharacteristic
33C23×D191D411yescharacteristic
34#15690286—216no—
35#15690286—28no—
36#15690286—28no—
37C2×Dic382—21no—
38C2×Dic382—22no—
39C2×Dic382D414yes—
40C2×Dic382C2312yes—
41C2×C22⋊D191—28no—
42C2×C22⋊D191—24no—
43C22×Dic191—28no—
44C22×Dic191—24no—
45C22×Dic191—21no—
46C22×Dic191—21no—
47C22×Dic191D411yescharacteristic
48C2×C4×D191—22no—
49C2×C4×D191—22no—
50C2×C4×D191D412yes—
51C2×C4×D191C2312yes—
52#15690277—28no—
53C2×C191⋊C8—22no—
54C2×C191⋊C8C2312yes—
55C4⋊Dic191—21no—
56C4⋊Dic191C2312yes—
57Dic191⋊C4—24no—
58(C2×D191)⋊C4—24no—
59C24×C191—21no—
60C2×Q8×C191—21no—
61C2×Q8×C191—22no—
62C2×Q8×C191—22no—
63C2×Q8×C191C2312yes—
64C2×D4×C191—21no—
65C2×D4×C191—22no—
66C2×D4×C191D414yes—
67C2×D4×C191C2312yes—
68C22×C4×C191—21no—
69C22×C4×C191C2311yescharacteristic
70SD16×C191—28no—
71C2×C8×C191—22no—
72C2×C8×C191C2312yes—
73C22×D191—22no—
74C22×D191—22no—
75C22×D191—22no—
76C22×D191C2×D412yes—
77Dic382—24no—
78Dic382—22no—
79Dic382C2×D414yes—
80C22⋊D191—216no—
81C2×Dic191—216no—
82C2×Dic191—22no—
83C2×Dic191—24no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—42no—
87C2×Dic191C2×D412yes—
88C4×D191—24no—
89C4×D191—22no—
90C4×D191—22no—
91C191⋊C8—22no—
92C23×C191—24no—
93C23×C191—22no—
94C23×C191—42no—
95C23×C191—42no—
96C23×C191C2×D411yescharacteristic
97Q8×C191—22no—
98Q8×C191—22no—
99Q8×C191—42no—
100D4×C191—24no—
101D4×C191—22no—
102D4×C191C2×D414yes—
103C2×C4×C191—22no—
104C2×C4×C191—42no—
105C2×C4×C191C2×D411yescharacteristic
106C2×C4×C191C2×D412yes—
107C2×C4×C191C2×D412yes—
108C2×C4×C191C2411yescharacteristic
109C8×C191—22no—
110C22×C191—24no—
111C22×C191—42no—
112C22×C191—42no—
113C22×C191—48no—
114C22×C191#12012yes—
115C22×C191#13512yes—
116C22×C191#13712yes—
117C22×C191C22×D411yescharacteristic
118C2×D191—22no—
119C2×D191—22no—
120C2×D191—22no—
121C2×D191—44no—
122C4×C191—42no—
123C4×C191#12012yes—
124C4×C191C22×D411yescharacteristic
125C4×C191C22×D411yescharacteristic
126Dic191—22no—
127Dic191—24no—
128Dic191—42no—
129C2×C191—44no—
130C2×C191#45214yescomplemented
131C2×C191C2×#12011yescharacteristic
132C2×C191C2×#13511yescharacteristic
133C2×C191C2×#13711yescharacteristic
134D191—42no—
135C191C2×#45211yescharacteristic, complemented, Sylow
136C2×#452—1911nomaximal, Sylow
137C22×#41—1911no—
138C22×Q16—1911no—
139C22×SD16D19111yescharacteristic, complemented
140C2×C22⋊Q8—1911no—
141#452—1918no—
142C2×#103—1911no—
143C2×#102—1911no—
144C2×C22⋊C8—1911no—
145C2×#41—3824no—
146C2×#41—3824no—
147C2×#41—3822no—
148C2×#41—1914no—
149C22×Q8—1911no—
150C22×Q8C2×D19111yescharacteristic
151C22×D4—3821no—
152C22×D4C2×D19111yescharacteristic
153C23×C4—3821no—
154C23×C4—1911no—
155C2×Q16—3824no—
156C2×Q16—1914no—
157C2×SD16—24no—
158C2×SD16C2×D19114yescomplemented
159C22×C8—1911no—
160C22×C8C2×D19111yescharacteristic
161C22⋊Q8—3822no—
162C22⋊Q8—1914no—
163C2×C4⋊C4—3821no—
164C2×C4⋊C4—1911no—
165C2×C22⋊C4—3821no—
166#103—1914no—
167#102—1914no—
168C22⋊C8—1914no—
169C24—21no—
170C24—1911no—
171#41—38216no—
172#41—3828no—
173#41—3828no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—3821no—
178C2×Q8—3822no—
179C2×Q8—1914no—
180C2×Q8—1912no—
181C2×Q8C22×D19112yes—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—3828no—
185C2×D4—3824no—
186C2×D4C22⋊D19114yes—
187C2×D4C22×D19112yes—
188C22×C4—21no—
189C22×C4—3828no—
190C22×C4—3821no—
191C22×C4—3824no—
192C22×C4—3822no—
193C22×C4—3822no—
194C22×C4—3821no—
195C22×C4—1911no—
196C22×C4—1912no—
197C22×C4—1912no—
198C22×C4C22×D19111yescharacteristic
199Q16—3828no—
200SD16—28no—
201C2×C8—22no—
202C2×C8—3822no—
203C2×C8—1912no—
204C2×C8C22×D19112yes—
205C4⋊C4—3824no—
206C4⋊C4—3821no—
207C4⋊C4—1912no—
208C22⋊C4—3824no—
209C23—22no—
210C23—24no—
211C23—42no—
212C23—42no—
213C23—3822no—
214C23—3822no—
215C23—3822no—
216C23—1912no—
217C23D4×D19111yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—3822no—
222Q8—3824no—
223Q8—1914no—
224D4—22no—
225D4—24no—
226D4—38216no—
227D4C2×C22⋊D19114yes—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—38216no—
231C2×C4—3822no—
232C2×C4—3822no—
233C2×C4—3824no—
234C2×C4—3822no—
235C2×C4—3822no—
236C2×C4—3824no—
237C2×C4—3822no—
238C2×C4—1912no—
239C2×C4—7642no—
240C2×C4D4×D19112yes—
241C2×C4D4×D19111yescharacteristic
242C2×C4C2×C22⋊D19112yes—
243C2×C4C23×D19111yescharacteristic
244C8—22no—
245C8—3822no—
246C22—24no—
247C22—42no—
248C22—42no—
249C22—48no—
250C22—3822no—
251C22—3822no—
252C22—3822no—
253C22—7644no—
254C22#2025929012yes—
255C22#2025934312yes—
256C22#2025935112yes—
257C22C2×D4×D19111yescharacteristic
258C4—42no—
259C4—3822no—
260C4—3824no—
261C4—7642no—
262C4#2025929012yes—
263C4C2×D4×D19111yescharacteristic
264C4C22×C22⋊D19111yescharacteristic
265C2—44no—
266C2—7642no—
267C2#2713282214yesdirect product factor
268C2C2×#2025929011yescharacteristic
269C2C2×#2025934311yescharacteristic
270C2C2×#2025935111yescharacteristic
subgroupG/H
centerC23D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×SD16×C191C2
socleC23×C191C2×D4
prime radicalC22×#15690286C2
hypercenterC22×SD16D191
hypocenterC191C2×#452
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C22×#15690277C2semidirect
3. C2×#20259298C2semidirect
4. #27132822C2direct (N × H)
5. C2×#20259247C2semidirect
6. C2×#20259243C2semidirect
7. C2×(C2×D191)⋊C8C2semidirect
8. C22×SD16×C191C2semidirect
18. C2×#15690277C22semidirect
21. #20259298C22semidirect
25. #20259247C22semidirect
26. #20259243C22semidirect
27. (C2×D191)⋊C8C22semidirect
31. C2×SD16×C191C22semidirect
130. C2×C191#452semidirect
135. C191C2×#452semidirect
139. C22×SD16D191semidirect
158. C2×SD16C2×D191semidirect
267. C2#27132822direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156902862
C22×#156902772
C2×#202592982
#271328222
C2×#202592472
C2×#202592432
C2×(C2×D191)⋊C82
C22×SD16×C1912
C2×#452191