Finite Group Databasebeta

← #37576159

Subgroups — #37576159

Order 24448.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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×Q8×D191C211yesmaximal, characteristic
2C2×#20259321C212yesmaximal, complemented
3C2×#20259300C211yesmaximal, characteristic
4#27132584C211yesmaximal, characteristic, complemented
5#27132583C211yesmaximal, characteristic, complemented
6#27132582C211yesmaximal, characteristic, complemented
7#27132567C212yesmaximal, complemented
8#27132537C212yesmaximal
9#27132534C212yesmaximal
10#27132510C211yesmaximal, characteristic
11#395×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×Dic382C2211yescharacteristic
14C22×Dic382C2212yes—
15C2×Q8×D191—28no—
16C2×Q8×D191—24no—
17C22×C4×D191C2212yes—
18C22×C4×D191C2211yescharacteristic
19#20259321—28no—
20#20259300—24no—
21C2×C4⋊Dic191C2212yes—
22C2×Dic191⋊C4C2212yes—
23C2×Dic191⋊C4C2212yes—
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×(C2×D191)⋊C4C2211yescharacteristic
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2212yes—
29C2×C4×Dic191C2212yes—
30C191⋊#95C2211yescharacteristic
31C191⋊#95C2212yes—
32C191⋊#95C2212yes—
33C191⋊#95C2212yes—
34C22×Q8×C191C2211yescharacteristic
35C2×C4⋊C4×C191C2212yes—
36C2×C4⋊C4×C191C2211yescharacteristic
37#95×C191C2211yescharacteristic
38#95×C191C2212yes—
39C23×D191C2311yescharacteristic
40C2×Dic382—24no—
41C2×Dic382—24no—
42C2×Dic382—24no—
43C2×Dic382—22no—
44C2×Dic382—24no—
45C2×Dic382—22no—
46C2×Dic382D418yes—
47Q8×D191—232no—
48C22×Dic191C2311yescharacteristic
49C22×Dic191C2311yescharacteristic
50C22×Dic191C2312yes—
51C22×Dic191C2312yes—
52C22×Dic191C2311yescharacteristic
53C2×C4×D191—24no—
54C2×C4×D191—24no—
55C2×C4×D191—24no—
56C2×C4×D191—24no—
57C2×C4×D191D414yes—
58C4⋊Dic191—24no—
59Dic191⋊C4—22no—
60Dic191⋊C4—24no—
61Dic191⋊C4—22no—
62Dic191⋊C4—24no—
63(C2×D191)⋊C4—22no—
64(C2×D191)⋊C4—22no—
65(C2×D191)⋊C4Q818yescomplemented
66C4×Dic191—24no—
67C2×Q8×C191—24no—
68C2×Q8×C191—22no—
69C22×C4×C191C2311yescharacteristic
70C22×C4×C191C2311yescharacteristic
71C22×C4×C191C2312yes—
72C22×C4×C191C2311yescharacteristic
73C22×C4×C191C2312yes—
74C4⋊C4×C191—22no—
75C4⋊C4×C191—24no—
76C22×D191—22no—
77C22×D191—21no—
78C22×D191—21no—
79C22×D191C2×D412yes—
80C22×D191C2×Q814yes—
81Dic382—28no—
82Dic382—216no—
83Dic382—44no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191—22no—
88C2×Dic191—21no—
89C2×Dic191—21no—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—21no—
93C2×Dic191C2×D414yes—
94C2×Dic191C2×D414yes—
95C2×Dic191C2×D412yes—
96C2×Dic191C2×Q814yescomplemented
97C4×D191—216no—
98C4×D191—28no—
99C23×C191C2411yescharacteristic
100Q8×C191—28no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—22no—
104C2×C4×C191—22no—
105C2×C4×C191—22no—
106C2×C4×C191—21no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—21no—
110C2×C4×C191—21no—
111C2×C4×C191C2×D414yes—
112C2×C4×C191C2×D412yes—
113C2×C4×C191C2×Q814yescomplemented
114C22×C191C22×D411yescharacteristic
115C22×C191C22×D412yes—
116C22×C191C22×Q812yes—
117C22×C1912−1+411yescharacteristic
118C22×C1912−1+411yescharacteristic
119C2×D191—22no—
120C2×D191—22no—
121C2×D191—24no—
122C2×D191—24no—
123C2×D191C4⋊Q814yescomplemented
124C4×C191—24no—
125C4×C191—22no—
126C4×C191—24no—
127C4×C191—41no—
128C4×C191—41no—
129Dic191—24no—
130Dic191—24no—
131Dic191—41no—
132Dic191—41no—
133Dic191#12014yes—
134C2×C191C2×#12011yescharacteristic
135C2×C191C2×C4⋊Q811yescharacteristic
136C2×C191#53611yescharacteristic
137C2×C191D4×Q812yes—
138C2×C191D4×Q812yes—
139D191—24no—
140C191#243411yescharacteristic, complemented, Sylow
141#2434—1911nomaximal, Sylow
142C23×Q8—1911no—
143C2×C4⋊Q8—1912no—
144C2×C22⋊Q8—1911no—
145#395—1911no—
146#395—1911no—
147#395—1911no—
148#395D19111yescharacteristic, complemented
149#391—1912no—
150#385—1912no—
151#385—1912no—
152C4×C22⋊C4—1911no—
153C22×Q8—3828no—
154C22×Q8—3824no—
155C22×Q8—1911no—
156C22×Q8—1911no—
157C22×Q8—1912no—
158C22×Q8C2×D19111yescharacteristic
159C23×C4—1912no—
160C23×C4—1911no—
161C4⋊Q8—3828no—
162C22⋊Q8—3824no—
163C2×C4⋊C4—1911no—
164C2×C4⋊C4—1911no—
165C2×C4⋊C4—1912no—
166C2×C4⋊C4—1912no—
167C2×C4⋊C4—1912no—
168C2×C4⋊C4C2×D19111yescharacteristic
169C2×C4⋊C4C2×D19112yes—
170C2×C22⋊C4—1911no—
171C2×C22⋊C4—1911no—
172C2×C22⋊C4—1912no—
173C2×C42—1912no—
174#95—1911no—
175#95—1912no—
176#95—1912no—
177#95—1912no—
178#95C2×D19111yescharacteristic
179#95C2×D19112yes—
180C24—1911no—
181C2×Q8—24no—
182C2×Q8—22no—
183C2×Q8—3824no—
184C2×Q8—3824no—
185C2×Q8—3824no—
186C2×Q8—3822no—
187C2×Q8—3824no—
188C2×Q8—38232no—
189C2×Q8—3822no—
190C2×Q8—1918no—
191C22×C4—3824no—
192C22×C4—3824no—
193C22×C4—3824no—
194C22×C4—3824no—
195C22×C4—1911no—
196C22×C4—1911no—
197C22×C4—1912no—
198C22×C4—1912no—
199C22×C4—1914no—
200C22×C4—1911no—
201C22×C4C22×D19111yescharacteristic
202C22×C4C22×D19111yescharacteristic
203C22×C4C22×D19112yes—
204C22×C4C22×D19111yescharacteristic
205C22×C4C22×D19112yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—3822no—
209C4⋊C4—3824no—
210C4⋊C4—3824no—
211C4⋊C4—3822no—
212C4⋊C4—3824no—
213C22⋊C4—3822no—
214C22⋊C4—3822no—
215C22⋊C4—1918no—
216C42—3824no—
217C23—3822no—
218C23—3821no—
219C23—3821no—
220C23—1912no—
221C23—1914no—
222C23C23×D19111yescharacteristic
223Q8—28no—
224Q8—3828no—
225Q8—38216no—
226Q8—7644no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—38216no—
238C2×C4—3828no—
239C2×C4—3822no—
240C2×C4—3822no—
241C2×C4—3822no—
242C2×C4—3822no—
243C2×C4—3821no—
244C2×C4—3821no—
245C2×C4—3822no—
246C2×C4—3821no—
247C2×C4—3821no—
248C2×C4—1914no—
249C2×C4—1914no—
250C2×C4—1914no—
251C2×C4—1912no—
252C2×C4D4×D19114yes—
253C2×C4D4×D19112yes—
254C2×C4Q8×D19114yescomplemented
255C22—3822no—
256C22—3824no—
257C22—3824no—
258C22—3822no—
259C22—1914no—
260C22C2×D4×D19111yescharacteristic
261C22C2×D4×D19112yes—
262C22C2×Q8×D19112yes—
263C22#2025936911yescharacteristic
264C22#2025937011yescharacteristic
265C4—24no—
266C4—24no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—3824no—
271C4—3824no—
272C4—1914no—
273C4—7641no—
274C4—7641no—
275C2—3824no—
276C2#120×D19111yescharacteristic
277C2C4⋊Q8×D19111yescharacteristic
278C2#2713323611yescharacteristic
279C2#2713332612yes—
280C2#2713332812yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#395×C191C2
prime radicalC23×D191C23
hypercenter#395D191
hypocenterC191#2434
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#20259321C2semidirect
4. #27132584C2semidirect
5. #27132583C2semidirect
6. #27132582C2semidirect
7. #27132567C2semidirect
11. #395×C191C2semidirect
65. (C2×D191)⋊C4Q8semidirect
96. C2×Dic191C2×Q8semidirect
113. C2×C4×C191C2×Q8semidirect
123. C2×D191C4⋊Q8semidirect
140. C191#2434semidirect
148. #395D191semidirect
254. C2×C4Q8×D191semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C2×#202593212
C2×#202593002
#271325842
#271325832
#271325822
#271325672
#271325372
#271325342
#271325102
#395×C1912
#2434191