Finite Group Databasebeta

← #27150009

Subgroups — #27150009

Order 12416.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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#20260676C212yesmaximal
2#20260675C212yesmaximal
3#20260674C211yesmaximal, characteristic
4#20260656C212yesmaximal
5#20260656C212yesmaximal
6#20260655C212yesmaximal
7#20260600C211yesmaximal, characteristic
8#20260599C212yesmaximal
9#400×C97C211yesmaximal, characteristic
10C22×Dic194C2212yes—
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C4⋊Dic97C2212yes—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×C4×Dic97C2211yescharacteristic
18#15690717C2212yes—
19#15690717C2212yes—
20#15690717C2212yes—
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28#95×C97C2212yes—
29#95×C97C2212yes—
30#95×C97C2211yescharacteristic
31C2×Dic194—42no—
32C2×Dic194—42no—
33C2×Dic194—42no—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C4⋊Dic97—41no—
39C4⋊Dic97—41no—
40C4⋊Dic97—42no—
41#13095490—42no—
42#13095490—42no—
43#13095490—42no—
44C4×Dic97Q814yescomplemented
45C22×C4×C97C2312yes—
46C22×C4×C97C2312yes—
47C22×C4×C97C2311yescharacteristic
48C22×C4×C97C2311yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—41no—
51C4⋊C4×C97—41no—
52C23×C97C2411yescharacteristic
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97C2×Q812yes—
66C2×Dic97—22no—
67C2×Dic97—22no—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97C2×Q814yes—
78Dic194—48no—
79C22×C97C22×Q811yescharacteristic
80C22×C97H(5,𝔽2)11yescharacteristic
81C22×C97H(5,𝔽2)12yes—
82C22×C972−1+411yescharacteristic
83C22×C972−1+412yes—
84C4×C97—22no—
85C4×C97—42no—
86C4×C97—42no—
87C4×C97—41no—
88C4×C97—41no—
89Dic97—24no—
90Dic97—42no—
91Dic97—42no—
92Dic97—42no—
93C2×C97#54411yescharacteristic
94C2×C97#54412yes—
95C2×C97H(3,𝔽4)11yescharacteristic
96C2×C97#56312yes—
97C2×C97#56311yescharacteristic
98#2844—971nomaximal, Sylow
99C97#284411yescharacteristic, complemented, Sylow
100#400—971no—
101#400—972no—
102#400—972no—
103#400D9711yescharacteristic
104#395—972no—
105#395—972no—
106#395—972no—
107#382—971no—
108#382—972no—
109C22×Q8—972no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—972no—
112C2×C4⋊C4—972no—
113C2×C4⋊C4—971no—
114C2×C4⋊C4—971no—
115C2×C4⋊C4—972no—
116C2×C4⋊C4C2×D9711yescharacteristic
117C2×C4⋊C4C2×D9711yescharacteristic
118C2×C42—971no—
119#95—972no—
120#95—972no—
121#95—972no—
122#95—972no—
123#95—972no—
124#95—972no—
125#95—972no—
126#95—971no—
127#95C2×D9712yes—
128#95C2×D9712yes—
129#95C2×D9711yescharacteristic
130C2×Q8—3882no—
131C2×Q8—3882no—
132C2×Q8—3882no—
133C22×C4—972no—
134C22×C4—972no—
135C22×C4—972no—
136C22×C4—972no—
137C22×C4C22×D9711yescharacteristic
138C22×C4C22×D9711yescharacteristic
139C22×C4C22×D9712yes—
140C22×C4C22×D9712yes—
141C22×C4C22×D9711yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—3882no—
145C4⋊C4—3882no—
146C4⋊C4—3882no—
147C4⋊C4—3881no—
148C4⋊C4—3881no—
149C4⋊C4—3882no—
150C42—974no—
151C23C23×D9711yescharacteristic
152Q8—3888no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—974no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4—1942no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4Q8×D9712yes—
178C22C2×Q8×D9711yescharacteristic
179C22#1569085511yescharacteristic
180C22#1569085612yes—
181C22#1569085711yescharacteristic
182C22#1569085712yes—
183C4—22no—
184C4—42no—
185C4—42no—
186C4—41no—
187C4—41no—
188C4—3882no—
189C4—3882no—
190C4—3882no—
191C4—1944no—
192C2#2026136411yescharacteristic
193C2#2026136512yes—
194C2#2026147711yescharacteristic
195C2#2026148712yes—
196C2#2026148811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, prime radical, cosocleC23×C97C24
Fitting#400×C97C2
hypercenter#400D97
hypocenterC97#2844
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
44. C4×Dic97Q8semidirect
99. C97#2844semidirect

Maximal subgroups

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

subgroupindex
#202606762
#202606752
#202606742
#202606562
#202606552
#202606002
#202605992
#400×C972
#284497