Finite Group Databasebeta

← #37581998

Subgroups — #37581998

Order 24448.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#20259347C211yesmaximal, characteristic, complemented
2C2×#20259318C211yesmaximal, characteristic, complemented
3#41×Dic191C211yesmaximal, characteristic, complemented
4#27132861C214yesmaximal, complemented
5#27132856C214yesmaximal, complemented
6#27132686C211yesmaximal, characteristic, complemented
7#27132669C211yesmaximal, characteristic, complemented
8#27132668C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
10C22×D764C2211yescharacteristic
11C22×C22⋊D191—22no—
12#20259347—24no—
13C2×#15690274C2212yescomplemented
14C2×#15690272C2212yescomplemented
15C2×#15690271C2211yescharacteristic
16#20259318—21no—
17#20259318—21no—
18#20259318C2214yes—
19Q8×Dic191C2212yescomplemented
20D4×Dic191—22no—
21D4×Dic191C2212yescomplemented
22#20259281—21no—
23#20259281C2211yescharacteristic
24C2×C4×Dic191—21no—
25C2×C4×Dic191C2211yescharacteristic
26Dic191⋊C8C2214yescomplemented
27#20259244C2214yescomplemented
28#20259243C2214yescomplemented
29C2×#41×C191C2211yescharacteristic
30C22×D4×C191C2211yescharacteristic
31AGL(1,ℤ/8)×C191—24no—
32C2×SD16×C191C2212yescomplemented
33C2×D8×C191C2212yescomplemented
34C2×M16×C191C2211yescharacteristic
35C23×D191—21no—
36C2×D764—22no—
37C2×D764—21no—
38C2×D764—22no—
39C2×D764C2312yes—
40C2×C22⋊D191—24no—
41C2×C22⋊D191—24no—
42C2×C22⋊D191—48no—
43C22×Dic191—22no—
44C22×Dic191D412yes—
45#15690274—24no—
46#15690272—24no—
47#15690271—22no—
48C2×C191⋊C8C2312yes—
49C4⋊Dic191—22no—
50C4⋊Dic191C2312yes—
51C22⋊Dic191—22no—
52C22⋊Dic191—21no—
53C4×Dic191—22no—
54C4×Dic191—21no—
55C4×Dic191D411yescharacteristic
56C4×Dic191D411yescharacteristic
57C4×Dic191C2312yes—
58C24×C191—21no—
59#41×C191—22no—
60#41×C191D414yescomplemented
61C2×Q8×C191C2311yescharacteristic
62C2×D4×C191—21no—
63C2×D4×C191—22no—
64C2×D4×C191—21no—
65C2×D4×C191—22no—
66C2×D4×C191C2312yes—
67C2×D4×C191C2311yescharacteristic
68C22×C4×C191—21no—
69C22×C4×C191C2311yescharacteristic
70SD16×C191—24no—
71D8×C191—24no—
72M16×C191—22no—
73C2×C8×C191C2312yes—
74C22×D191—22no—
75C22×D191—44no—
76C22×D191—42no—
77D764—22no—
78D764—22no—
79D764—42no—
80C22⋊D191—48no—
81C22⋊D191—48no—
82C2×Dic191—24no—
83C2×Dic191—22no—
84C2×Dic191—44no—
85C2×Dic191C2×D412yes—
86C2×Dic191C2×D412yes—
87C191⋊C8—22no—
88C23×C191—21no—
89C23×C191—22no—
90C23×C191—44no—
91C23×C191—42no—
92C23×C191C2×D411yescharacteristic
93Q8×C191—21no—
94Q8×C191C2×D412yescomplemented
95D4×C191—24no—
96D4×C191—22no—
97D4×C191—21no—
98D4×C191—22no—
99D4×C191—42no—
100D4×C191C2×D412yescomplemented
101C2×C4×C191—21no—
102C2×C4×C191—24no—
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191C2×D412yes—
105C2×C4×C191#4112yes—
106C2×C4×C191C2411yescharacteristic
107C8×C191—22no—
108C22×C191—22no—
109C22×C191—22no—
110C22×C191—42no—
111C22×C191—42no—
112C22×C191—44no—
113C22×C191—84no—
114C22×C191#12112yes—
115C22×C191C22×D411yescharacteristic
116C2×D191—42no—
117C2×D191—44no—
118C2×D191—84no—
119C4×C191—22no—
120C4×C191#12112yes—
121C4×C191C22×D411yescharacteristic
122C4×C191C2×#4111yescharacteristic
123Dic191—22no—
124Dic191—42no—
125Dic191AGL(1,ℤ/8)14yescomplemented
126C2×C191—22no—
127C2×C191—42no—
128C2×C191—82no—
129C2×C191C2×#12111yescharacteristic
130C2×C191C2×AGL(1,ℤ/8)12yes—
131D191—82no—
132C191#303811yescharacteristic, complemented, Sylow
133#3038—1911nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1911no—
135C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
136C2×#127—1911no—
137C4×#41—1911no—
138#460—1914no—
139#459—1914no—
140#423—1911no—
141#418—1911no—
142#418—1911no—
143C2×#41C2×D19111yescharacteristic
144C22×D4—3822no—
145C22×D4—1911no—
146C22×D4C2×D19111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—3824no—
149C2×SD16—1912no—
150C2×SD16C2×D19112yescomplemented
151C2×D8—1912no—
152C2×D8C2×D19112yescomplemented
153C2×M16—1911no—
154C2×M16C2×D19111yescharacteristic
155#127—3821no—
156#127—3821no—
157#127—1914no—
158C4×Q8—1912no—
159C4×D4—3822no—
160C4×D4—1912no—
161#117—3821no—
162#117—1911no—
163C2×C42—3821no—
164C2×C42—1911no—
165C4⋊C8—1914no—
166#102—1914no—
167#102—1914no—
168C24—21no—
169C24—3821no—
170#41—22no—
171#41C22⋊D19114yescomplemented
172C2×Q8C22×D19111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—3824no—
178C2×D4—3824no—
179C2×D4—3821no—
180C2×D4—3822no—
181C2×D4—3822no—
182C2×D4—1912no—
183C2×D4—7648no—
184C2×D4C22×D19112yes—
185C2×D4C22×D19111yescharacteristic
186C22×C4—21no—
187C22×C4—3822no—
188C22×C4—1912no—
189C22×C4C22×D19111yescharacteristic
190SD16—24no—
191SD16—3824no—
192D8—24no—
193D8—3824no—
194M16—22no—
195M16—3822no—
196C2×C8—1912no—
197C2×C8C22×D19112yes—
198C4⋊C4—3822no—
199C4⋊C4—1912no—
200C22⋊C4—3821no—
201C22⋊C4—3822no—
202C42—3822no—
203C42—3821no—
204C42—1911no—
205C42—1911no—
206C42—1912no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—3822no—
212C23—7644no—
213C23—7642no—
214C23D4×D19111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D19112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—3822no—
223D4—3822no—
224D4—7648no—
225D4—7648no—
226D4—7642no—
227D4C2×C22⋊D19112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—3824no—
231C2×C4—3822no—
232C2×C4—1912no—
233C2×C4—1912no—
234C2×C4—7644no—
235C2×C4D4×D19111yescharacteristic
236C2×C4C2×C22⋊D19112yes—
237C2×C4#1569028612yes—
238C2×C4C23×D19111yescharacteristic
239C8—22no—
240C8—3822no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—15284no—
248C22—7644no—
249C22—7642no—
250C22#2025929412yes—
251C22C2×D4×D19111yescharacteristic
252C4—22no—
253C4—3822no—
254C4—1914no—
255C4—7642no—
256C4#2025929412yes—
257C4C22×C22⋊D19111yescharacteristic
258C4C2×#1569028611yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—15282no—
263C2C2×#2025929411yescharacteristic
264C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3038
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. C2×#20259318C2semidirect
3. #41×Dic191C2semidirect
4. #27132861C2semidirect
5. #27132856C2semidirect
6. #27132686C2semidirect
7. #27132669C2semidirect
8. #27132668C2semidirect
9. C2×AGL(1,ℤ/8)×C191C2semidirect
13. C2×#15690274C22semidirect
14. C2×#15690272C22semidirect
19. Q8×Dic191C22semidirect
21. D4×Dic191C22semidirect
26. Dic191⋊C8C22semidirect
27. #20259244C22semidirect
28. #20259243C22semidirect
32. C2×SD16×C191C22semidirect
33. C2×D8×C191C22semidirect
60. #41×C191D4semidirect
94. Q8×C191C2×D4semidirect
100. D4×C191C2×D4semidirect
125. Dic191AGL(1,ℤ/8)semidirect
132. C191#3038semidirect
135. C2×AGL(1,ℤ/8)D191semidirect
150. C2×SD16C2×D191semidirect
152. C2×D8C2×D191semidirect
171. #41C22⋊D191semidirect
216. Q8C2×C22⋊D191semidirect
227. D4C2×C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593472
C2×#202593182
#41×Dic1912
#271328612
#271328562
#271326862
#271326692
#271326682
C2×AGL(1,ℤ/8)×C1912
#3038191