Finite Group Databasebeta

← C2×#27132823 #37581727

Subgroups — C2×#27132823 #37581727

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×#15690287C211yesmaximal, characteristic
2C22×#15690277C211yesmaximal, characteristic, complemented
3C22×#15690274C211yesmaximal, characteristic, complemented
4#27132823C218yesmaximal, direct product factor
5C2×#20259245C211yesmaximal, characteristic, complemented
6C2×#20259242C211yesmaximal, characteristic, complemented
7C2×C22⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C191C211yesmaximal, characteristic, complemented
9C2×#15690287—24no—
10C2×#15690287—24no—
11C2×#15690287—22no—
12C2×#15690287C2214yes—
13C22×Dic382C2211yescharacteristic
14C22×D764C2211yescharacteristic
15C22×C22⋊D191—21no—
16C22×C4×D191—21no—
17C2×#15690277—24no—
18C2×#15690277C2214yescomplemented
19C2×#15690274—24no—
20C2×#15690274C2214yescomplemented
21C22×C191⋊C8C2211yescharacteristic
22C22×C191⋊C8C2211yescharacteristic
23#20259245C2214yescomplemented
24#20259242C2214yescomplemented
25C22⋊(C191⋊C8)C2214yescomplemented
26C22×Q8×C191C2211yescharacteristic
27C23×C4×C191C2211yescharacteristic
28C22⋊Q8×C191—22no—
29C22⋊Q8×C191C2214yescomplemented
30C2×C4⋊C4×C191—21no—
31C2×C4⋊C4×C191C2211yescharacteristic
32C2×C22⋊C4×C191—21no—
33C23×D191—21no—
34#15690287—216no—
35#15690287—28no—
36#15690287—28no—
37C2×Dic382—22no—
38C2×Dic382—21no—
39C2×Dic382D414yes—
40C2×Dic382C2312yes—
41C2×D764—22no—
42C2×D764—21no—
43C2×D764D414yes—
44C2×D764C2312yes—
45C2×C22⋊D191—28no—
46C2×C22⋊D191—24no—
47C22×Dic191—21no—
48C2×C4×D191—28no—
49C2×C4×D191—24no—
50#15690277—28no—
51#15690274—28no—
52C2×C191⋊C8—22no—
53C2×C191⋊C8—22no—
54C2×C191⋊C8C2312yes—
55C2×C191⋊C8C2312yes—
56C24×C191D411yescharacteristic
57C2×Q8×C191—21no—
58C2×Q8×C191—22no—
59C2×Q8×C191—22no—
60C2×Q8×C191C2312yes—
61C22×C4×C191—22no—
62C22×C4×C191—22no—
63C22×C4×C191—21no—
64C22×C4×C191—21no—
65C22×C4×C191D411yescharacteristic
66C22×C4×C191D412yes—
67C22×C4×C191C2312yes—
68C22×C4×C191C2311yescharacteristic
69C4⋊C4×C191—24no—
70C4⋊C4×C191—21no—
71C4⋊C4×C191C2312yes—
72C22⋊C4×C191—24no—
73C22×D191—24no—
74C22×D191—22no—
75C22×D191—42no—
76C22×D191—42no—
77Dic382—24no—
78Dic382—22no—
79Dic382C2×D414yes—
80D764—24no—
81D764—22no—
82D764C2×D414yes—
83C22⋊D191—216no—
84C2×Dic191—24no—
85C2×Dic191—22no—
86C4×D191—216no—
87C191⋊C8—22no—
88C191⋊C8—22no—
89C23×C191—22no—
90C23×C191—22no—
91C23×C191—22no—
92C23×C191C2×D412yes—
93C23×C191C2×D411yescharacteristic
94Q8×C191—22no—
95Q8×C191—22no—
96Q8×C191—42no—
97C2×C4×C191—22no—
98C2×C4×C191—24no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—42no—
104C2×C4×C191—42no—
105C2×C4×C191C2×D412yes—
106C2×C4×C191C2×D412yes—
107C2×C4×C191C2×D412yes—
108C2×C4×C191C2×D411yescharacteristic
109C2×C4×C191C2411yescharacteristic
110C22×C191—22no—
111C22×C191—22no—
112C22×C191—22no—
113C22×C191—44no—
114C22×C191#12012yes—
115C22×C191#13512yes—
116C22×C191#13712yes—
117C22×C191C22×D411yescharacteristic
118C2×D191—24no—
119C2×D191—42no—
120C2×D191—42no—
121C2×D191—48no—
122C4×C191—22no—
123C4×C191—42no—
124C4×C191—42no—
125C4×C191#12012yes—
126C4×C191C22×D411yescharacteristic
127C4×C191C22×D411yescharacteristic
128Dic191—24no—
129C2×C191—42no—
130C2×C191#45214yescomplemented
131C2×C191C2×#12011yescharacteristic
132C2×C191C2×#13511yescharacteristic
133C2×C191C2×#13711yescharacteristic
134D191—44no—
135C191C2×#45211yescharacteristic, complemented, Sylow
136C2×#452—1911nomaximal, Sylow
137C22×#41—1911no—
138C22×Q16—1911no—
139C22×SD16—1911no—
140C2×C22⋊Q8D19111yescharacteristic, complemented
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×D4—1911no—
153C23×C4—3821no—
154C23×C4C2×D19111yescharacteristic
155C2×Q16—3824no—
156C2×Q16—1914no—
157C2×SD16—3824no—
158C2×SD16—1914no—
159C22×C8—1911no—
160C22×C8—1911no—
161C22⋊Q8—22no—
162C22⋊Q8C2×D19114yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×D19111yescharacteristic
165C2×C22⋊C4—21no—
166#103—1914no—
167#102—1914no—
168C22⋊C8—1914no—
169C24—3821no—
170C24C22⋊D19111yescharacteristic
171#41—38216no—
172#41—3828no—
173#41—3828no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—3822no—
178C2×Q8—3821no—
179C2×Q8—1914no—
180C2×Q8—1912no—
181C2×Q8C22×D19112yes—
182C2×D4—3828no—
183C2×D4—3824no—
184C2×D4—3822no—
185C2×D4—3821no—
186C2×D4—1914no—
187C2×D4—1912no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—3828no—
193C22×C4—3821no—
194C22×C4—3824no—
195C22×C4C22⋊D19112yes—
196C22×C4C22⋊D19111yescharacteristic
197C22×C4C22×D19112yes—
198C22×C4C22×D19111yescharacteristic
199Q16—3828no—
200SD16—3828no—
201C2×C8—3822no—
202C2×C8—3822no—
203C2×C8—1912no—
204C2×C8—1912no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D19112yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—3822no—
213C23—3824no—
214C23—7642no—
215C23—7642no—
216C23C2×C22⋊D19112yes—
217C23C2×C22⋊D19111yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—3822no—
222Q8—3824no—
223Q8—1914no—
224D4—38216no—
225D4—3822no—
226D4—3824no—
227D4—1914no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—38216no—
237C2×C4—3822no—
238C2×C4—3824no—
239C2×C4D4×D19112yes—
240C2×C4D4×D19112yes—
241C2×C4C2×C22⋊D19112yes—
242C2×C4C2×C22⋊D19111yescharacteristic
243C2×C4C23×D19111yescharacteristic
244C8—3822no—
245C8—3822no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—3824no—
251C22—7642no—
252C22—7642no—
253C22—7648no—
254C22#2025929012yes—
255C22#2025934212yes—
256C22#2025935312yes—
257C22C22×C22⋊D19111yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—3824no—
262C4#2025929012yes—
263C4C2×D4×D19111yescharacteristic
264C4C2×D4×D19111yescharacteristic
265C2—42no—
266C2—7644no—
267C2#2713282314yesdirect product factor
268C2C2×#2025929011yescharacteristic
269C2C2×#2025934211yescharacteristic
270C2C2×#2025935311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×C22⋊Q8×C191C2
socleC23×C191C2×D4
prime radicalC22×#15690287C2
hypercenterC2×C22⋊Q8D191
hypocenterC191C2×#452
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#15690277C2semidirect
3. C22×#15690274C2semidirect
4. #27132823C2direct (N × H)
5. C2×#20259245C2semidirect
6. C2×#20259242C2semidirect
7. C2×C22⋊(C191⋊C8)C2semidirect
8. C2×C22⋊Q8×C191C2semidirect
18. C2×#15690277C22semidirect
20. C2×#15690274C22semidirect
23. #20259245C22semidirect
24. #20259242C22semidirect
25. C22⋊(C191⋊C8)C22semidirect
29. C22⋊Q8×C191C22semidirect
130. C2×C191#452semidirect
135. C191C2×#452semidirect
140. C2×C22⋊Q8D191semidirect
162. C22⋊Q8C2×D191semidirect
267. C2#27132823direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156902872
C22×#156902772
C22×#156902742
#271328232
C2×#202592452
C2×#202592422
C2×C22⋊(C191⋊C8)2
C2×C22⋊Q8×C1912
C2×#452191