Finite Group Databasebeta

← #27144899

Subgroups — #27144899

Order 12416.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690789C211yesmaximal, characteristic
2#20260639C212yesmaximal, complemented
3#20260630C212yesmaximal, complemented
4#20260629C212yesmaximal
5#20260623C212yesmaximal
6#20260618C212yesmaximal, complemented
7#20260616C212yesmaximal, complemented
8#20260598C211yesmaximal, characteristic
9#380×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690789C418yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C2212yes—
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×C22⋊Dic97C2212yes—
17C2×C22⋊Dic97C2212yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20#15690717C2211yescharacteristic
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2211yescharacteristic
27C23×C4×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2212yes—
29C2×C22⋊C4×C97C2211yescharacteristic
30C2×C22⋊C4×C97C2211yescharacteristic
31#95×C97C2212yes—
32C2×Dic194—42no—
33C2×Dic194C2×C414yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C4⋊Dic97C2×C414yes—
39#13095490—42no—
40#13095490—42no—
41#13095490C2×C418yes—
42C22⋊Dic97—42no—
43C22⋊Dic97C2×C418yes—
44C4×Dic97—42no—
45C4×Dic97—42no—
46C24×C97C2311yescharacteristic
47C22×C4×C97—42no—
48C22×C4×C97C2×C414yes—
49C22×C4×C97C2312yes—
50C22×C4×C97C2312yes—
51C22×C4×C97C2311yescharacteristic
52C22×C4×C97C2311yescharacteristic
53C4⋊C4×C97—42no—
54C22⋊C4×C97—41no—
55C22⋊C4×C97—41no—
56C23×C97—21no—
57C23×C97—22no—
58C23×C97—22no—
59C23×C97—21no—
60C23×C97C22×C412yes—
61C23×C97C2411yescharacteristic
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—44no—
71C2×C4×C97C22×C412yes—
72C2×C4×C97C22×C412yes—
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×Dic97C22×C414yes—
84C2×Dic97C22×C414yes—
85Dic194—44no—
86C22×C97—22no—
87C22×C97—22no—
88C22×C97—22no—
89C22×C97—42no—
90C22×C97—42no—
91C22×C97C23×C411yescharacteristic
92C22×C97H(5,𝔽2)12yes—
93C22×C97H(5,𝔽2)11yescharacteristic
94C22×C972−1+412yes—
95C22×C972−1+411yescharacteristic
96C4×C97—22no—
97C4×C97—22no—
98C4×C97—42no—
99C4×C97—42no—
100Dic97—24no—
101Dic97—24no—
102Dic97—42no—
103Dic97—42no—
104C2×C97—42no—
105C2×C97#51811yescharacteristic
106C2×C97#51911yescharacteristic
107C2×C97#52011yescharacteristic
108C2×C97#56212yes—
109C2×C97#56312yes—
110#2378—971nomaximal, Sylow
111C97#237811yescharacteristic, complemented, Sylow
112C2×C22⋊Q8—971no—
113#391—972no—
114#389—972no—
115#388—972no—
116#388—972no—
117#387—972no—
118#387—972no—
119#381—971no—
120#380D9711yescharacteristic
121C22×Q8—971no—
122C23×C4C2×D9711yescharacteristic
123C22⋊Q8—978no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—971no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—972no—
128C2×C4⋊C4C2×D9712yes—
129C2×C22⋊C4—972no—
130C2×C22⋊C4—972no—
131C2×C22⋊C4C2×D9711yescharacteristic
132C2×C22⋊C4C2×D9711yescharacteristic
133C2×C42—972no—
134C2×C42—972no—
135#95—971no—
136#95—972no—
137#95—971no—
138#95—972no—
139#95—972no—
140#95—972no—
141#95—972no—
142#95C2×D9712yes—
143C24C22×D9711yescharacteristic
144C2×Q8—974no—
145C2×Q8—3882no—
146C22×C4—42no—
147C22×C4—972no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—972no—
151C22×C4C4×D9714yes—
152C22×C4C22×D9712yes—
153C22×C4C22×D9712yes—
154C22×C4C22×D9711yescharacteristic
155C22×C4C22×D9711yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—978no—
158C4⋊C4—974no—
159C4⋊C4—3882no—
160C4⋊C4—3882no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C22⋊C4—978no—
164C22⋊C4—3882no—
165C42—3882no—
166C42—3882no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23C2×C4×D9712yes—
172C23C23×D9711yescharacteristic
173Q8—3884no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—974no—
184C2×C4—974no—
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×C4C2×C4×D9712yes—
196C2×C4C2×C4×D9712yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D9711yescharacteristic
203C22#1569085512yes—
204C22#1569085511yescharacteristic
205C22#1569085712yes—
206C22#1569085811yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—3882no—
212C4—3882no—
213C4—1944no—
214C4—1944no—
215C2—42no—
216C2#2026120811yescharacteristic
217C2#2026121211yescharacteristic
218C2#2026121611yescharacteristic
219C2#2026148012yes—
220C2#2026148612yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#380×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenter#380D97
hypocenterC97#2378
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #20260639C2semidirect
3. #20260630C2semidirect
6. #20260618C2semidirect
7. #20260616C2semidirect
11. #15690789C4semidirect
111. C97#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#156907892
#202606392
#202606302
#202606292
#202606232
#202606182
#202606162
#202605982
#380×C972
#237897