Finite Group Databasebeta

← #27149993

Subgroups — #27149993

Order 12416.

396 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 133 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#20260677C212yesmaximal, complemented
2#20260671C211yesmaximal, characteristic
3#20260669C212yesmaximal, complemented
4#20260667C212yesmaximal, complemented
5#20260658C212yesmaximal
6#20260656C212yesmaximal, complemented
7#20260647C211yesmaximal, characteristic
8#20260595C212yesmaximal
9#381×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×C4⋊Dic97C2212yes—
12C2×C4⋊Dic97C2212yes—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C4⋊Dic97C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×C22⋊Dic97C2212yes—
19C2×C22⋊Dic97C2212yes—
20#15690717C2212yes—
21#15690717C2211yescharacteristic
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2211yescharacteristic
26C23×C4×C97C2211yescharacteristic
27C2×C22⋊C4×C97C2211yescharacteristic
28C2×C22⋊C4×C97C2211yescharacteristic
29#95×C97C2212yes—
30#95×C97C2212yes—
31C2×Dic194—42no—
32C2×Dic194—41no—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C4⋊Dic97—42no—
38C4⋊Dic97—42no—
39C4⋊Dic97—41no—
40C4⋊Dic97—42no—
41#13095490—42no—
42#13095490—42no—
43#13095490—42no—
44C22⋊Dic97—42no—
45C22⋊Dic97—42no—
46C24×C97C2311yescharacteristic
47C22×C4×C97—41no—
48C22×C4×C97—41no—
49C22×C4×C97Q814yescomplemented
50C22×C4×C97C2312yes—
51C22×C4×C97C2311yescharacteristic
52C22×C4×C97C2311yescharacteristic
53C22×C4×C97C2312yes—
54C22⋊C4×C97—41no—
55C22⋊C4×C97—41no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—22no—
59C23×C97—22no—
60C23×C97C2×Q812yes—
61C23×C97C2411yescharacteristic
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—44no—
72C2×C4×C97C2×Q814yes—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85Dic194—44no—
86C22×C97—22no—
87C22×C97—24no—
88C22×C97—41no—
89C22×C97—42no—
90C22×C97—41no—
91C22×C97C22×Q811yescharacteristic
92C22×C97H(5,𝔽2)11yescharacteristic
93C22×C97H(5,𝔽2)11yescharacteristic
94C22×C97H(5,𝔽2)12yes—
95C22×C972−1+412yes—
96C4×C97—24no—
97C4×C97—41no—
98C4×C97—42no—
99C4×C97—41no—
100Dic97—42no—
101Dic97—42no—
102Dic97—42no—
103Dic97—42no—
104C2×C97—42no—
105C2×C97#54311yescharacteristic
106C2×C97#54412yes—
107C2×C97#56011yescharacteristic
108C2×C97#56212yes—
109C2×C97#56311yescharacteristic
110#2841—971nomaximal, Sylow
111C97#284111yescharacteristic, complemented, Sylow
112#400—972no—
113#399—971no—
114#398—972no—
115#398—972no—
116#396—972no—
117#395—972no—
118#393—971no—
119#381D9711yescharacteristic
120#380—972no—
121C22×Q8—971no—
122C23×C4C2×D9711yescharacteristic
123C2×C4⋊C4—972no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—972no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—972no—
128C2×C4⋊C4—972no—
129C2×C4⋊C4—971no—
130C2×C22⋊C4—972no—
131C2×C22⋊C4—972no—
132C2×C22⋊C4C2×D9711yescharacteristic
133C2×C22⋊C4C2×D9711yescharacteristic
134#95—972no—
135#95—972no—
136#95—971no—
137#95—972no—
138#95—972no—
139#95—971no—
140#95C2×D9712yes—
141#95C2×D9712yes—
142C24C22×D9711yescharacteristic
143C2×Q8—3881no—
144C2×Q8—3882no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—972no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—972no—
151C22×C4Dic19414yescomplemented
152C22×C4C22×D9711yescharacteristic
153C22×C4C22×D9712yes—
154C22×C4C22×D9711yescharacteristic
155C22×C4C22×D9712yes—
156C4⋊C4—3882no—
157C4⋊C4—3882no—
158C4⋊C4—3882no—
159C4⋊C4—3882no—
160C4⋊C4—3882no—
161C4⋊C4—3881no—
162C4⋊C4—3882no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C22⋊C4—3882no—
166C22⋊C4—3882no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23C2×Dic19412yes—
172C23C23×D9711yescharacteristic
173Q8—3884no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4—1942no—
196C2×C4C2×Dic19414yes—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22×Dic19411yescharacteristic
203C22#1569085511yescharacteristic
204C22#1569085511yescharacteristic
205C22#1569085612yes—
206C22#1569085712yes—
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—3882no—
212C4—3882no—
213C4—3882no—
214C4—3882no—
215C2—42no—
216C2#2026136111yescharacteristic
217C2#2026136312yes—
218C2#2026147511yescharacteristic
219C2#2026148212yes—
220C2#2026148811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#381×C97C2
prime radicalC24×C97C23
hypercenter#381D97
hypocenterC97#2841
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #20260677C2semidirect
3. #20260669C2semidirect
4. #20260667C2semidirect
6. #20260656C2semidirect
49. C22×C4×C97Q8semidirect
111. C97#2841semidirect
151. C22×C4Dic194semidirect

Maximal subgroups

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

subgroupindex
#202606772
#202606712
#202606692
#202606672
#202606582
#202606562
#202606472
#202605952
#381×C972
#284197