Finite Group Databasebeta

← #27142059

Subgroups — #27142059

Order 12416.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15690769C211yesmaximal, characteristic, complemented
2C2×#15690731C212yesmaximal, complemented
3C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
4#20260497C212yesmaximal, complemented
5C2×#121×C97C211yesmaximal, characteristic
6C22×#11715895C2212yes—
7#15690769—24no—
8C2×C4⋊Dic97C2212yes—
9C2×C22⋊Dic97—21no—
10C2×C4×Dic97—21no—
11#15690731—24no—
12C22⋊#11715895—22no—
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—22no—
22C2×#11715895—22no—
23C2×#11715895—22no—
24C4⋊Dic97—22no—
25C4⋊Dic97D414yescomplemented
26C22⋊Dic97—24no—
27C4×Dic97—24no—
28C24×C97—21no—
29C24×C97D411yescharacteristic
30C2×D4×C97—24no—
31C2×D4×C97—22no—
32C2×D4×C97—21no—
33C2×D4×C97—22no—
34C2×D4×C97—44no—
35C2×D4×C97C2×C412yes—
36C22×C4×C97—22no—
37C22×C4×C97—22no—
38C22×C4×C97—21no—
39C22×C4×C97C2×C412yes—
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—21no—
44C4⋊C4×C97C2×C412yes—
45C22⋊C4×C97—24no—
46C23×C97—21no—
47C23×C97—21no—
48C23×C97—22no—
49C23×C97—22no—
50C23×C97—22no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97—42no—
54C23×C97—42no—
55C23×C97C22⋊C412yes—
56C23×C97C2×D411yescharacteristic
57D4×C97—24no—
58D4×C97—48no—
59D4×C97—42no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—24no—
63C2×C4×C97—24no—
64C2×C4×C97—22no—
65C2×C4×C97—42no—
66C2×C4×C97C22⋊C412yes—
67C2×C4×C97C22×C411yescharacteristic
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4112yes—
71C2×Dic97—24no—
72C2×Dic97—42no—
73C2×Dic97—42no—
74#11715895—42no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—82no—
84C22×C97—82no—
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97#12011yescharacteristic
87C22×C97#12311yescharacteristic
88C22×C97#13512yes—
89C22×C97AGL(1,ℤ/8)12yes—
90C4×C97—22no—
91C4×C97—42no—
92C4×C97C4×D412yes—
93C4×C97#12112yes—
94Dic97—44no—
95C2×C97—42no—
96C2×C97—82no—
97C2×C97#38611yescharacteristic
98C2×C97#41611yescharacteristic
99C2×C97#41811yescharacteristic
100C2×C97#44912yes—
101C2×C97#48212yes—
102#1875—971nomaximal, Sylow
103C97#187511yescharacteristic, complemented, Sylow
104C2×#121D9711yescharacteristic
105C2×#117—971no—
106C2×#102—972no—
107C2×C22⋊C8—971no—
108#340—972no—
109C22×D4—21no—
110C22×D4C2×D9711yescharacteristic
111C23×C4C2×D9711yescharacteristic
112C22×C8—972no—
113#121—22no—
114#121Dic9714yescomplemented
115#117—1944no—
116C2×C4⋊C4—972no—
117C2×C4⋊C4C2×D9711yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—1941no—
120C2×C42—1941no—
121#102—1944no—
122C22⋊C8—1942no—
123C24—21no—
124C24C22⋊D9711yescharacteristic
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4C2×Dic9712yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—1942no—
135C22×C4C2×Dic9712yes—
136C22×C4C22⋊D9711yescharacteristic
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139C2×C8—1942no—
140C2×C8—1942no—
141C2×C8—1942no—
142C4⋊C4—21no—
143C4⋊C4—974no—
144C4⋊C4—1942no—
145C4⋊C4C2×Dic9712yes—
146C22⋊C4—24no—
147C22⋊C4—1944no—
148C42—1944no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23C22⋊Dic9712yes—
159C23C2×C22⋊D9711yescharacteristic
160D4—24no—
161D4—48no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—1944no—
172C2×C4C22⋊Dic9712yes—
173C2×C4C22×Dic9711yescharacteristic
174C2×C4D4×D9712yes—
175C2×C4C2×C22⋊D9711yescharacteristic
176C2×C4#1309551012yes—
177C8—3882no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—82no—
187C22—82no—
188C22C2×C22⋊Dic9711yescharacteristic
189C22#1569077811yescharacteristic
190C22#1569079111yescharacteristic
191C22#1569083012yes—
192C22#1569083612yes—
193C4—22no—
194C4—42no—
195C4—3884no—
196C4D4×Dic9712yes—
197C4#1569078212yes—
198C2—42no—
199C2—82no—
200C2#2026061511yescharacteristic
201C2#2026073111yescharacteristic
202C2#2026074011yescharacteristic
203C2#2026087812yes—
204C2#2026105612yes—
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#1875
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. C2×#15690769C2semidirect
2. C2×#15690731C2semidirect
3. C2×C22⋊#11715895C2semidirect
4. #20260497C2semidirect
17. #121×C97C4semidirect
25. C4⋊Dic97D4semidirect
103. C97#1875semidirect
114. #121Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
C2×#156907312
C2×C22⋊#117158952
#202604972
C2×#121×C972
#187597