Finite Group Databasebeta

← #37576148

Subgroups — #37576148

Order 24448.

836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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
2C2×#20259322C211yesmaximal, characteristic, complemented
3C2×#20259310C211yesmaximal, characteristic, complemented
4C2×#20259301C211yesmaximal, characteristic, complemented
5#27132579C214yesmaximal, complemented
6#27132569C211yesmaximal, characteristic, complemented
7#27132565C211yesmaximal, characteristic, complemented
8#27132537C214yesmaximal, complemented
9C4×C22⋊C4×C191C211yesmaximal, characteristic, complemented
10C2×#15690287—24no—
11C2×#15690287—24no—
12C2×#15690287—24no—
13C22×Dic382C2211yescharacteristic
14C22×Dic382C2211yescharacteristic
15C22×D764C2211yescharacteristic
16C22×C22⋊D191C2212yes—
17C22×C4×D191C2212yes—
18#20259322—24no—
19#20259310—24no—
20#20259301—24no—
21C2×C4⋊Dic191C2214yes—
22C2×C4⋊Dic191C2211yescharacteristic
23C2×Dic191⋊C4C2212yes—
24C2×C22⋊Dic191C2212yes—
25C2×(C2×D191)⋊C4C2214yes—
26C2×(C2×D191)⋊C4C2214yes—
27C191⋊#95C2214yes—
28C23×C4×C191C2211yescharacteristic
29C2×C22⋊C4×C191C2212yes—
30C2×C42×C191C2211yescharacteristic
31C2×C42×C191C2211yescharacteristic
32#95×C191C2212yes—
33C23×D191C2312yes—
34#15690287—232no—
35C2×Dic382—22no—
36C2×Dic382—22no—
37C2×Dic382—22no—
38C2×Dic382—22no—
39C2×Dic382—22no—
40C2×Dic382D414yes—
41C2×D764—22no—
42C2×D764—22no—
43C2×D764D414yes—
44C2×C22⋊D191—24no—
45C2×C22⋊D191—24no—
46C2×C22⋊D191—24no—
47C22×Dic191C2312yes—
48C22×Dic191C2312yes—
49C22×Dic191C2312yes—
50C2×C4×D191—24no—
51C2×C4×D191—24no—
52C2×C4×D191—24no—
53C4⋊Dic191—28no—
54C4⋊Dic191—22no—
55Dic191⋊C4—24no—
56C22⋊Dic191—24no—
57(C2×D191)⋊C4—28no—
58(C2×D191)⋊C4—28no—
59C24×C191C2311yescharacteristic
60C22×C4×C191—22no—
61C22×C4×C191—22no—
62C22×C4×C191D414yes—
63C22×C4×C191C2312yes—
64C22×C4×C191C2311yescharacteristic
65C22×C4×C191C2312yes—
66C22×C4×C191C2311yescharacteristic
67C22⋊C4×C191—24no—
68C42×C191—22no—
69C42×C191—22no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—24no—
73C22×D191—24no—
74C22×D191C2×D414yes—
75Dic382—28no—
76Dic382—44no—
77D764—28no—
78C22⋊D191—216no—
79C2×Dic191—22no—
80C2×Dic191—22no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—24no—
86C2×Dic191C2×D414yes—
87C4×D191—216no—
88C23×C191—21no—
89C23×C191—21no—
90C23×C191—22no—
91C23×C191—22no—
92C23×C191C2×D412yes—
93C23×C191C2411yescharacteristic
94C2×C4×C191—22no—
95C2×C4×C191—21no—
96C2×C4×C191—21no—
97C2×C4×C191—22no—
98C2×C4×C191—28no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191C2×D412yes—
103C2×C4×C191C2×D412yes—
104C2×C4×C191C2×D412yes—
105C2×C4×C191C2×D412yes—
106C2×C4×C191#4114yescomplemented
107C22×C191—22no—
108C22×C191—22no—
109C22×C191—22no—
110C22×C191—42no—
111C22×C191—42no—
112C22×C191C22×D411yescharacteristic
113C22×C191C22×D411yescharacteristic
114C22×C191C22×D411yescharacteristic
115C22×C191C2×#4112yes—
116C22×C1912−1+412yes—
117C2×D191—24no—
118C2×D191—28no—
119C2×D191—42no—
120C2×D191—42no—
121C2×D191—44no—
122C4×C191—22no—
123C4×C191—24no—
124C4×C191—42no—
125C4×C191#12014yes—
126Dic191—24no—
127Dic191—42no—
128Dic191—42no—
129C2×C191—42no—
130C2×C191C2×#12011yescharacteristic
131C2×C191#53211yescharacteristic
132C2×C191#53611yescharacteristic
133C2×C191#54712yes—
134C2×C191#54812yes—
135D191—44no—
136C191#243211yescharacteristic, complemented, Sylow
137#2432—1911nomaximal, Sylow
138C22×#41—1911no—
139C2×C4⋊Q8—1911no—
140C2×#124—1911no—
141C2×C22⋊Q8—1911no—
142#394—1914no—
143#391—1911no—
144#390—1911no—
145#385—1914no—
146C4×C22⋊C4D19111yescharacteristic, complemented
147C2×#41—3824no—
148C2×#41—3824no—
149C2×#41—3824no—
150C22×Q8—1911no—
151C22×Q8—1911no—
152C22×D4—1912no—
153C22×D4—1911no—
154C23×C4—1912no—
155C23×C4C2×D19111yescharacteristic
156C4⋊Q8—3824no—
157#124—3824no—
158C22⋊Q8—3824no—
159C2×C4⋊C4—1912no—
160C2×C4⋊C4—1911no—
161C2×C4⋊C4—1914no—
162C2×C22⋊C4—1912no—
163C2×C22⋊C4—1914no—
164C2×C22⋊C4—1914no—
165C2×C22⋊C4C2×D19112yes—
166C2×C42C2×D19111yescharacteristic
167C2×C42C2×D19111yescharacteristic
168#95—1914no—
169#95C2×D19112yes—
170C24—1912no—
171C24C22×D19111yescharacteristic
172#41—38232no—
173C2×Q8—3822no—
174C2×Q8—3822no—
175C2×Q8—3822no—
176C2×Q8—3822no—
177C2×Q8—3822no—
178C2×Q8—1914no—
179C2×D4—3824no—
180C2×D4—3824no—
181C2×D4—3824no—
182C2×D4—3822no—
183C2×D4—3822no—
184C2×D4—1914no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—3824no—
188C22×C4—3824no—
189C22×C4—3824no—
190C22×C4—1912no—
191C22×C4—1912no—
192C22×C4—1912no—
193C22×C4D76414yes—
194C22×C4C22×D19112yes—
195C22×C4C22×D19111yescharacteristic
196C22×C4C22×D19112yes—
197C22×C4C22×D19111yescharacteristic
198C4⋊C4—3828no—
199C4⋊C4—3824no—
200C4⋊C4—3822no—
201C22⋊C4—24no—
202C22⋊C4—3828no—
203C22⋊C4—3824no—
204C22⋊C4—3828no—
205C42—22no—
206C42—22no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—3822no—
212C23—3822no—
213C23—3824no—
214C23—3824no—
215C23—1914no—
216C23C2×D76412yes—
217C23C23×D19111yescharacteristic
218Q8—3828no—
219Q8—7644no—
220D4—38216no—
221D4—3828no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—38216no—
231C2×C4—3822no—
232C2×C4—3822no—
233C2×C4—3822no—
234C2×C4—3822no—
235C2×C4—3822no—
236C2×C4—3822no—
237C2×C4—3824no—
238C2×C4—1914no—
239C2×C4D4×D19112yes—
240C2×C4D4×D19112yes—
241C2×C4C2×D76412yes—
242C2×C4C2×D76412yes—
243C2×C4#1569028714yescomplemented
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—3828no—
250C22—3824no—
251C22—7642no—
252C22—7642no—
253C22—7644no—
254C22C2×D4×D19111yescharacteristic
255C22C2×D4×D19111yescharacteristic
256C22C22×D76411yescharacteristic
257C22C2×#1569028712yes—
258C22#2025936912yes—
259C4—24no—
260C4—22no—
261C4—42no—
262C4—3824no—
263C4—7642no—
264C4—7642no—
265C4#2025929214yes—
266C2—42no—
267C2—7644no—
268C2C2×#2025929211yescharacteristic
269C2#2713321311yescharacteristic
270C2#2713323911yescharacteristic
271C2#2713331812yes—
272C2#2713332212yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC4×C22⋊C4×C191C2
prime radicalC22×#15690287C2
hypercenterC4×C22⋊C4D191
hypocenterC191#2432
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#20259322C2semidirect
3. C2×#20259310C2semidirect
4. C2×#20259301C2semidirect
5. #27132579C2semidirect
6. #27132569C2semidirect
7. #27132565C2semidirect
8. #27132537C2semidirect
9. C4×C22⋊C4×C191C2semidirect
106. C2×C4×C191#41semidirect
136. C191#2432semidirect
146. C4×C22⋊C4D191semidirect
243. C2×C4#15690287semidirect

Maximal subgroups

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

subgroupindex
C22×#156902872
C2×#202593222
C2×#202593102
C2×#202593012
#271325792
#271325692
#271325652
#271325372
C4×C22⋊C4×C1912
#2432191