Finite Group Databasebeta

← #27149998

Subgroups — #27149998

Order 12416.

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

normal subgroups only

This group has 396 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, complemented
2#20260668C212yesmaximal, complemented
3#20260665C212yesmaximal, complemented
4#20260658C212yesmaximal
5#20260655C212yesmaximal, complemented
6#20260647C211yesmaximal, characteristic
7#20260596C211yesmaximal, characteristic
8#20260593C212yesmaximal
9#399×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C23×Dic97C2211yescharacteristic
12C2×C4⋊Dic97C2212yes—
13C2×#13095490C2212yes—
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×C22⋊Dic97C2212yes—
18C2×C22⋊Dic97C2211yescharacteristic
19#15690717C2211yescharacteristic
20#15690717C2212yes—
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26C2×C4⋊C4×C97C2212yes—
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C22⋊C4×C97C2211yescharacteristic
29C2×C22⋊C4×C97C2212yes—
30#95×C97C2211yescharacteristic
31C2×Dic194—42no—
32C2×Dic194—41no—
33C22×Dic97—41no—
34C22×Dic97—41no—
35C22×Dic97Q814yescomplemented
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C22×Dic97C2312yes—
40C4⋊Dic97—42no—
41#13095490—42no—
42#13095490—42no—
43#13095490—42no—
44#13095490—42no—
45C22⋊Dic97—41no—
46C22⋊Dic97—42no—
47C24×C97C2311yescharacteristic
48C22×C4×C97C2312yes—
49C22×C4×C97C2312yes—
50C22×C4×C97C2311yescharacteristic
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—41no—
53C4⋊C4×C97—42no—
54C22⋊C4×C97—42no—
55C22⋊C4×C97—41no—
56C23×C97—22no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—22no—
60C23×C97C2×Q812yes—
61C23×C97C2411yescharacteristic
62C2×C4×C97—22no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×Dic97—22no—
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—44no—
84C2×Dic97C2×Q814yes—
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)12yes—
94C22×C97H(5,𝔽2)11yescharacteristic
95C22×C972−1+412yes—
96C4×C97—41no—
97C4×C97—42no—
98C4×C97—42no—
99C4×C97—41no—
100Dic97—24no—
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#399D9711yescharacteristic
114#398—972no—
115#398—972no—
116#396—972no—
117#395—972no—
118#393—971no—
119#381—971no—
120#380—972no—
121C22×Q8—971no—
122C23×C4—971no—
123C2×C4⋊C4—972no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—972no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—972no—
128C2×C4⋊C4C2×D9712yes—
129C2×C4⋊C4C2×D9711yescharacteristic
130C2×C22⋊C4—972no—
131C2×C22⋊C4—971no—
132C2×C22⋊C4C2×D9712yes—
133C2×C22⋊C4C2×D9711yescharacteristic
134#95—972no—
135#95—972no—
136#95—972no—
137#95—972no—
138#95—971no—
139#95—972no—
140#95—972no—
141#95C2×D9711yescharacteristic
142C24C22×D9711yescharacteristic
143C2×Q8—3881no—
144C2×Q8—3882no—
145C22×C4—974no—
146C22×C4—972no—
147C22×C4—972no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—3881no—
151C22×C4—3881no—
152C22×C4C22×D9711yescharacteristic
153C22×C4C22×D9711yescharacteristic
154C22×C4C22×D9712yes—
155C22×C4C22×D9712yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—3882no—
159C4⋊C4—3882no—
160C4⋊C4—3882no—
161C4⋊C4—3882no—
162C4⋊C4—3882no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—3882no—
166C22⋊C4—3881no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D9712yes—
172C23C23×D9711yescharacteristic
173Q8—3884no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—974no—
185C2×C4—3884no—
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×C4—1942no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D9711yescharacteristic
203C22#1569085512yes—
204C22#1569085511yescharacteristic
205C22#1569085611yescharacteristic
206C22#1569085712yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—3882no—
212C4—3882no—
213C4—3882no—
214C4—1944no—
215C2—42no—
216C2#2026136011yescharacteristic
217C2#2026136412yes—
218C2#2026147211yescharacteristic
219C2#2026148212yes—
220C2#2026148811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
prime radicalC24×C97C23
hypercenter#399D97
hypocenterC97#2841
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #20260676C2semidirect
2. #20260668C2semidirect
3. #20260665C2semidirect
5. #20260655C2semidirect
35. C22×Dic97Q8semidirect
111. C97#2841semidirect

Maximal subgroups

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

subgroupindex
#202606762
#202606682
#202606652
#202606582
#202606552
#202606472
#202605962
#202605932
#399×C972
#284197