Finite Group Databasebeta

← #37582917

Subgroups — #37582917

Order 24448.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#27133212C211yesmaximal, characteristic
2#125×D191C211yesmaximal, characteristic, complemented
3#27133030C212yesmaximal, complemented
4#27133026C212yesmaximal, complemented
5#27133020C211yesmaximal, characteristic, complemented
6#27133018C211yesmaximal, characteristic, complemented
7#27133017C211yesmaximal, characteristic, complemented
8#27132663C212yesmaximal, complemented
9#27132661C212yesmaximal, complemented
10C4×#15690270C211yesmaximal, characteristic, complemented
11#488×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic
13C2×#15690286C2211yescharacteristic
14C2×#15690270C2212yes—
15#20259320C2211yescharacteristic
16#20259318C2211yescharacteristic
17#20259314C2211yescharacteristic
18#20259313C2212yes—
19#20259312—22no—
20#20259310C2211yescharacteristic
21#20259295—22no—
22(C22⋊D191)⋊C4—22no—
23C4⋊C4×D191—22no—
24C4⋊C4×D191C2212yes—
25C42×D191C2211yescharacteristic
26#20259247C2212yes—
27#20259246C2212yes—
28#20259245C2212yes—
29#20259244C2212yes—
30#20259243C2212yes—
31#20259242C2212yes—
32#20259232C2212yes—
33C4×C191⋊C8C2211yescharacteristic
34#125×C191C2211yescharacteristic
35#124×C191C2211yescharacteristic
36#103×C191C2212yes—
37#102×C191C2212yes—
38C4×C8×C191C2211yescharacteristic
39#15690288—24no—
40#15690286—24no—
41C2×Dic382C2311yescharacteristic
42C2×D764—21no—
43C2×D764C2311yescharacteristic
44C2×C22⋊D191—21no—
45C2×C22⋊D191—22no—
46C22×Dic191—21no—
47C2×C4×D191—22no—
48C2×C4×D191—21no—
49C2×C4×D191D412yes—
50C2×C4×D191C2311yescharacteristic
51#15690270—24no—
52C2×C191⋊C8C2312yes—
53C4⋊Dic191—22no—
54C4⋊Dic191C2312yes—
55Dic191⋊C4—22no—
56Dic191⋊C4—22no—
57Dic191⋊C4—22no—
58Dic191⋊C4—22no—
59(C2×D191)⋊C4—22no—
60C4×Dic191D412yes—
61C4×Dic191C2311yescharacteristic
62C2×Q8×C191C2311yescharacteristic
63C2×D4×C191C2311yescharacteristic
64C2×C8×C191C2312yes—
65C4⋊C4×C191—22no—
66C4⋊C4×C191C2312yes—
67C22⋊C4×C191—22no—
68C42×C191C2311yescharacteristic
69C22×D191—21no—
70C22×D191C2×D411yescharacteristic
71Dic382—22no—
72D764—22no—
73D764—42no—
74C22⋊D191—44no—
75C22⋊D191—42no—
76C2×Dic191—21no—
77C2×Dic191—22no—
78C2×Dic191—42no—
79C2×Dic191C2×D412yes—
80C2×Dic191C2×D411yescharacteristic
81C4×D191—42no—
82C4×D191—44no—
83C4×D191—42no—
84C4×D191#4114yes—
85C191⋊C8—22no—
86C23×C191—21no—
87Q8×C191—22no—
88D4×C191—22no—
89C2×C4×C191—21no—
90C2×C4×C191—22no—
91C2×C4×C191C2×D412yes—
92C2×C4×C191C2411yescharacteristic
93C8×C191—22no—
94C22×C191—41no—
95C22×C191—42no—
96C22×C191C22×D411yescharacteristic
97C2×D191—22no—
98C2×D191—41no—
99C2×D191—42no—
100C2×D191#12412yes—
101C4×C191—22no—
102C4×C191—41no—
103C4×C191—42no—
104C4×C191C2×#4112yes—
105Dic191—22no—
106Dic191—41no—
107Dic191—42no—
108Dic191#12412yes—
109C2×C191—81no—
110C2×C191C2×#12411yescharacteristic
111C2×C191#57512yes—
112D191—22no—
113D191—81no—
114C191#314211yescharacteristic, complemented, Sylow
115#3142—1911nomaximal, Sylow
116#532—1911no—
117C2×#125—1911no—
118#491—1912no—
119#490—1912no—
120#488—1911no—
121#488—1911no—
122#488—1911no—
123#488D19111yescharacteristic, complemented
124#417—1912no—
125#417—1912no—
126C4×M16—1911no—
127C2×#41—1911no—
128C2×#41—1911no—
129C2×M16—1912no—
130C4⋊Q8—1911no—
131#127—1911no—
132#125—3822no—
133#125—1912no—
134#125—1911no—
135#125C2×D19111yescharacteristic
136#124—1911no—
137#124C2×D19111yescharacteristic
138#121—3822no—
139C4×D4—3822no—
140C2×C4⋊C4—3822no—
141C2×C4⋊C4—1912no—
142C2×C42—1911no—
143#103—1912no—
144#103—1912no—
145#103—1912no—
146#103C2×D19112yes—
147#102—1912no—
148#102—1912no—
149#102—1912no—
150#102C2×D19112yes—
151#97—1912no—
152C4×C8—1911no—
153C4×C8C2×D19111yescharacteristic
154#41—3824no—
155#41—3824no—
156C2×Q8—1911no—
157C2×Q8C22×D19111yescharacteristic
158C2×D4—3821no—
159C2×D4—3821no—
160C2×D4—3822no—
161C2×D4—1911no—
162C2×D4C22×D19111yescharacteristic
163C22×C4—3822no—
164C22×C4—3821no—
165C22×C4—3821no—
166C22×C4—1911no—
167C22×C4—1912no—
168M16—3824no—
169C2×C8—1912no—
170C2×C8C22×D19112yes—
171C4⋊C4—22no—
172C4⋊C4—3822no—
173C4⋊C4—3822no—
174C4⋊C4—3822no—
175C4⋊C4—3822no—
176C4⋊C4—3822no—
177C4⋊C4—1912no—
178C4⋊C4C22×D19112yes—
179C22⋊C4—22no—
180C22⋊C4—3822no—
181C42—1912no—
182C42—1911no—
183C42C22×D19111yescharacteristic
184C23—21no—
185C23—3821no—
186C23—1911no—
187Q8—22no—
188Q8—3822no—
189D4—22no—
190D4—3822no—
191D4—7642no—
192D4—7644no—
193D4—7642no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—3821no—
197C2×C4—3822no—
198C2×C4—1912no—
199C2×C4—1911no—
200C2×C4—1914no—
201C2×C4—7642no—
202C2×C4—7642no—
203C2×C4—7644no—
204C2×C4—7642no—
205C2×C4D4×D19112yes—
206C2×C4C23×D19111yescharacteristic
207C8—22no—
208C8—3822no—
209C22—42no—
210C22—41no—
211C22—3822no—
212C22—1912no—
213C22—7642no—
214C22—7641no—
215C22C2×D4×D19111yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—3822no—
220C4—1912no—
221C4—7642no—
222C4—7641no—
223C4#41×D19112yes—
224C2—81no—
225C2—3822no—
226C2—15281no—
227C2#124×D19111yescharacteristic
228C2#2713346012yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#488×C191C2
socleC22×C191C22×D4
prime radical#27133212C2
hypercenter#488D191
hypocenterC191#3142
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D191C2semidirect
3. #27133030C2semidirect
4. #27133026C2semidirect
5. #27133020C2semidirect
6. #27133018C2semidirect
7. #27133017C2semidirect
8. #27132663C2semidirect
9. #27132661C2semidirect
10. C4×#15690270C2semidirect
11. #488×C191C2semidirect
114. C191#3142semidirect
123. #488D191semidirect

Maximal subgroups

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

subgroupindex
#271332122
#125×D1912
#271330302
#271330262
#271330202
#271330182
#271330172
#271326632
#271326612
C4×#156902702
#488×C1912
#3142191