Finite Group Databasebeta

← #27142165

Subgroups — #27142165

Order 12416.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C23×#11715895C211yesmaximal, characteristic, complemented
2C2×#15690731C212yesmaximal, complemented
3#20260595C211yesmaximal, characteristic, complemented
4#20260497C212yesmaximal, complemented
5C2×#121×C97C211yesmaximal, characteristic
6C22×#11715895—24no—
7C22×#11715895—22no—
8C22×#11715895C2212yes—
9C2×C4⋊Dic97C2212yes—
10C2×C22⋊Dic97—21no—
11#15690731—24no—
12#15690717—21no—
13C22×D4×C97—21no—
14C22×D4×C97C2211yescharacteristic
15C23×C4×C97C2211yescharacteristic
16#121×C97—22no—
17#121×C97C414yescomplemented
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21C2×#11715895—28no—
22C2×#11715895—22no—
23C2×#11715895—22no—
24C2×#11715895D414yescomplemented
25C4⋊Dic97—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—42no—
28C24×C97—21no—
29C24×C97D411yescharacteristic
30C2×D4×C97—24no—
31C2×D4×C97—22no—
32C2×D4×C97—21no—
33C2×D4×C97—22no—
34C2×D4×C97—42no—
35C2×D4×C97—42no—
36C2×D4×C97C2×C412yes—
37C22×C4×C97—22no—
38C22×C4×C97—22no—
39C22×C4×C97—21no—
40C22×C4×C97C2×C412yes—
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97C2×C412yes—
46C22⋊C4×C97—24no—
47C23×C97—22no—
48C23×C97—22no—
49C23×C97—22no—
50C23×C97—42no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97—42no—
54C23×C97C22⋊C412yes—
55C23×C97D812yes—
56C23×C97SD1612yes—
57C23×C97C2×D411yescharacteristic
58D4×C97—24no—
59D4×C97—48no—
60D4×C97—41no—
61D4×C97—41no—
62C2×C4×C97—22no—
63C2×C4×C97—24no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C22⋊C412yes—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D412yes—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97#4112yes—
74C2×Dic97—42no—
75C2×Dic97—42no—
76C2×Dic97—42no—
77#11715895—24no—
78C22×C97—22no—
79C22×C97—22no—
80C22×C97—24no—
81C22×C97—24no—
82C22×C97—42no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97—82no—
86C22×C97—82no—
87C22×C97#10214yescomplemented
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97#12112yes—
90C22×C97C2×D811yescharacteristic
91C22×C97C2×SD1611yescharacteristic
92C22×C97#13512yes—
93C4×C97—22no—
94C4×C97—42no—
95C4×C97C4×D412yes—
96C4×C97#12011yescharacteristic
97C4×C97#12311yescharacteristic
98Dic97—82no—
99C2×C97—24no—
100C2×C97—82no—
101C2×C97#38611yescharacteristic
102C2×C97C2×#10211yescharacteristic
103C2×C97#41611yescharacteristic
104C2×C97#46512yes—
105C2×C97#46612yes—
106#1893—971nomaximal, Sylow
107C97#189311yescharacteristic, complemented, Sylow
108C23×C8—971no—
109C2×#121D9711yescharacteristic
110C2×#102—972no—
111#380—971no—
112#340—972no—
113C22×D4—21no—
114C22×D4C2×D9711yescharacteristic
115C23×C4C2×D9711yescharacteristic
116C22×C8—972no—
117C22×C8—1944no—
118C22×C8—1942no—
119#121—22no—
120#121Dic9714yescomplemented
121C2×C4⋊C4—972no—
122C2×C4⋊C4C2×D9711yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—1941no—
125#102—1944no—
126#95—1941no—
127C24—21no—
128C24C22⋊D9711yescharacteristic
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×Dic9712yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—1942no—
140C22×C4C2×Dic9712yes—
141C22×C4C22⋊D9711yescharacteristic
142C22×C4C22⋊D9712yes—
143C22×C4C22×D9711yescharacteristic
144C2×C8—974no—
145C2×C8—1948no—
146C2×C8—1942no—
147C2×C8—1942no—
148C4⋊C4—21no—
149C4⋊C4—1942no—
150C4⋊C4—1942no—
151C4⋊C4C2×Dic9712yes—
152C22⋊C4—24no—
153C22⋊C4—3882no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—42no—
158C23—41no—
159C23—41no—
160C23—42no—
161C23C22⋊Dic9712yes—
162C23#1309549612yes—
163C23#1309549912yes—
164C23C2×C22⋊D9711yescharacteristic
165D4—24no—
166D4—48no—
167D4—41no—
168D4—41no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—3882no—
177C2×C4—3882no—
178C2×C4—3882no—
179C2×C4C22⋊Dic9712yes—
180C2×C4C22×Dic9711yescharacteristic
181C2×C4D4×D9712yes—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4#1309551012yes—
184C8—1944no—
185C22—22no—
186C22—24no—
187C22—24no—
188C22—22no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—82no—
194C22#1569073114yescomplemented
195C22C2×C22⋊Dic9711yescharacteristic
196C22#1569078212yes—
197C22C2×#1309549611yescharacteristic
198C22C2×#1309549911yescharacteristic
199C22#1569083012yes—
200C4—22no—
201C4—42no—
202C4—7762no—
203C4D4×Dic9712yes—
204C4#1569077811yescharacteristic
205C4#1569079111yescharacteristic
206C2—24no—
207C2—82no—
208C2#2026061511yescharacteristic
209C2C2×#1569073111yescharacteristic
210C2#2026073111yescharacteristic
211C2#2026096212yes—
212C2#2026096912yes—
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×#121×C97C2
socleC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC2×#121D97
hypocenterC97#1893
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×#11715895C2semidirect
2. C2×#15690731C2semidirect
3. #20260595C2semidirect
4. #20260497C2semidirect
17. #121×C97C4semidirect
24. C2×#11715895D4semidirect
87. C22×C97#102semidirect
107. C97#1893semidirect
120. #121Dic97semidirect
194. C22#15690731semidirect

Maximal subgroups

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

subgroupindex
C23×#117158952
C2×#156907312
#202605952
#202604972
C2×#121×C972
#189397