Finite Group Databasebeta

← #37582029

Subgroups — #37582029

Order 24448.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259348C211yesmaximal, characteristic
2#27133237C211yesmaximal, characteristic, complemented
3#27133138C211yesmaximal, characteristic, complemented
4#27132880C214yesmaximal, complemented
5#27132866C214yesmaximal, complemented
6#27132689C211yesmaximal, characteristic, complemented
7#27132672C211yesmaximal, characteristic, complemented
8#27132660C211yesmaximal, characteristic, complemented
9C22×SD16×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11#20259348—24no—
12C2×#15690275C2212yes—
13C2×#15690272C2212yes—
14C2×#15690271C2211yescharacteristic
15#20259320C2212yescomplemented
16#20259311C2212yescomplemented
17#20259305—22no—
18#20259298—22no—
19D4×Dic191—22no—
20D4×Dic191C2214yes—
21#20259281C2211yescharacteristic
22#20259281C2211yescharacteristic
23C2×C22⋊Dic191—22no—
24Dic191⋊C8C2214yescomplemented
25#20259247C2212yescomplemented
26#20259246C2214yescomplemented
27#20259244C2212yescomplemented
28C22×Q8×C191C2211yescharacteristic
29C22×D4×C191C2211yescharacteristic
30C2×SD16×C191—24no—
31C2×SD16×C191C2214yes—
32C22×C8×C191C2211yescharacteristic
33#15690287—22no—
34#15690287—22no—
35C2×Dic382C2311yescharacteristic
36C2×D764C2311yescharacteristic
37C2×C22⋊D191—21no—
38C22×Dic191—24no—
39C2×C4×D191—21no—
40#15690275—24no—
41#15690272—24no—
42#15690271—22no—
43C2×C191⋊C8C2312yes—
44C4⋊Dic191D412yes—
45C4⋊Dic191C2312yes—
46Dic191⋊C4—24no—
47C22⋊Dic191—24no—
48C22⋊Dic191—21no—
49C22⋊Dic191D412yes—
50(C2×D191)⋊C4—24no—
51C4×Dic191—21no—
52C4×Dic191C2312yes—
53C24×C191—21no—
54C2×Q8×C191—21no—
55C2×Q8×C191—22no—
56C2×Q8×C191—22no—
57C2×Q8×C191C2312yes—
58C2×D4×C191—21no—
59C2×D4×C191—22no—
60C2×D4×C191D414yes—
61C2×D4×C191C2312yes—
62C22×C4×C191—21no—
63C22×C4×C191C2311yescharacteristic
64SD16×C191—28no—
65C2×C8×C191—22no—
66C2×C8×C191C2312yes—
67C22×D191—21no—
68Dic382—21no—
69Dic382—21no—
70D764—21no—
71D764—21no—
72C22⋊D191—42no—
73C2×Dic191—22no—
74C2×Dic191—21no—
75C2×Dic191—84no—
76C2×Dic191C2×D414yes—
77C4×D191—42no—
78C191⋊C8—22no—
79C23×C191—22no—
80C23×C191—24no—
81C23×C191—82no—
82C23×C191C2×D411yescharacteristic
83Q8×C191—22no—
84Q8×C191—22no—
85Q8×C191—42no—
86D4×C191—24no—
87D4×C191—22no—
88D4×C191C2×D414yes—
89C2×C4×C191—22no—
90C2×C4×C191—42no—
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2×D412yes—
93C2×C4×C191#4112yes—
94C2×C4×C191C2411yescharacteristic
95C8×C191—22no—
96C22×C191—22no—
97C22×C191—24no—
98C22×C191—44no—
99C22×C191—84no—
100C22×C191#12112yes—
101C22×C191C22×D411yescharacteristic
102C2×D191—41no—
103C2×D191—42no—
104C4×C191—42no—
105C4×C191#12112yes—
106C4×C191C22×D411yescharacteristic
107C4×C191C2×#4111yescharacteristic
108Dic191—24no—
109Dic191—41no—
110Dic191—42no—
111C2×C191—22no—
112C2×C191—44no—
113C2×C191C2×#12111yescharacteristic
114C2×C191#57712yes—
115D191—81no—
116C191#304211yescharacteristic, complemented, Sylow
117#3042—1911nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1911no—
119C22×SD16D19111yescharacteristic, complemented
120#536—1911no—
121#518—1911no—
122#463—1914no—
123#461—1914no—
124#424—1911no—
125#419—1911no—
126#416—1911no—
127C2×#41—1911no—
128C22×Q8C2×D19111yescharacteristic
129C22×D4C2×D19111yescharacteristic
130AGL(1,ℤ/8)—3824no—
131C2×SD16—24no—
132C2×SD16—1912no—
133C2×SD16C2×D19114yes—
134C2×D8—1912no—
135C2×M16—1911no—
136C22×C8C2×D19111yescharacteristic
137C4⋊Q8—1912no—
138#124—1912no—
139#123—3822no—
140C22⋊Q8—3822no—
141C4×D4—3822no—
142C4×D4—1914no—
143#117—1911no—
144#117—1911no—
145C2×C22⋊C4—3822no—
146C4⋊C8—1914no—
147#103—1914no—
148#103—1912no—
149#102—1912no—
150C24—21no—
151#41—3822no—
152#41—3822no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1911no—
157C2×Q8C22×D19112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—3821no—
161C2×D4—1911no—
162C2×D4C22⋊D19114yes—
163C2×D4C22×D19112yes—
164C22×C4—21no—
165C22×C4—3824no—
166C22×C4—3821no—
167C22×C4C22×D19111yescharacteristic
168SD16—28no—
169SD16—3824no—
170D8—3824no—
171M16—3822no—
172C2×C8—22no—
173C2×C8—1912no—
174C2×C8C22×D19112yes—
175C4⋊C4—3824no—
176C4⋊C4—1912no—
177C4⋊C4—1912no—
178C22⋊C4—3824no—
179C22⋊C4—3821no—
180C22⋊C4—3824no—
181C22⋊C4—1912no—
182C42—3821no—
183C42—1912no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—3821no—
188C23D4×D19111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3821no—
193Q8—3821no—
194D4—22no—
195D4—24no—
196D4—3821no—
197D4—3821no—
198D4—7642no—
199D4C2×C22⋊D19114yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—3822no—
203C2×C4—3821no—
204C2×C4—1914no—
205C2×C4—15284no—
206C2×C4—7642no—
207C2×C4D4×D19111yescharacteristic
208C2×C4C2×C22⋊D19112yes—
209C2×C4#1569028612yes—
210C2×C4C23×D19111yescharacteristic
211C8—22no—
212C8—3822no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—7642no—
218C22—7641no—
219C22#2025929412yes—
220C22C2×D4×D19111yescharacteristic
221C4—42no—
222C4—3824no—
223C4—7642no—
224C4—7641no—
225C4#2025929412yes—
226C4C22×C22⋊D19111yescharacteristic
227C4C2×#1569028611yescharacteristic
228C2—22no—
229C2—44no—
230C2—15281no—
231C2C2×#2025929411yescharacteristic
232C2#2713346912yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×SD16×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259348C2
hypercenterC22×SD16D191
hypocenterC191#3042
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #27133237C2semidirect
3. #27133138C2semidirect
4. #27132880C2semidirect
5. #27132866C2semidirect
6. #27132689C2semidirect
7. #27132672C2semidirect
8. #27132660C2semidirect
9. C22×SD16×C191C2semidirect
15. #20259320C22semidirect
16. #20259311C22semidirect
24. Dic191⋊C8C22semidirect
25. #20259247C22semidirect
26. #20259246C22semidirect
27. #20259244C22semidirect
116. C191#3042semidirect
119. C22×SD16D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
#271332372
#271331382
#271328802
#271328662
#271326892
#271326722
#271326602
C22×SD16×C1912
#3042191