Finite Group Databasebeta

← #27141769

Subgroups — #27141769

Order 12416.

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

normal subgroups only

This group has 316 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
1C2×#15690744C211yesmaximal, characteristic, complemented
2C2×#15690742C211yesmaximal, characteristic, complemented
3#20260595C212yesmaximal
4#20260498C212yesmaximal, complemented
5C22×M16×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2212yes—
7C2×C4⋊Dic97C2212yes—
8C2×C22⋊Dic97—22no—
9#15690744—22no—
10#15690742—22no—
11#15690717—22no—
12C23×C4×C97C2211yescharacteristic
13C2×M16×C97—22no—
14C2×M16×C97—22no—
15C2×M16×C97C414yescomplemented
16C22×C8×C97C2211yescharacteristic
17C22×C8×C97C2211yescharacteristic
18C22×Dic97—22no—
19C22×Dic97—22no—
20C4⋊Dic97—22no—
21C4⋊Dic97—22no—
22C4⋊Dic97—22no—
23C4⋊Dic97—22no—
24C22⋊Dic97—44no—
25C24×C97D411yescharacteristic
26C22×C4×C97—21no—
27C22×C4×C97—21no—
28C22×C4×C97—21no—
29C22×C4×C97—21no—
30C22×C4×C97C2×C412yes—
31C22×C4×C97D411yescharacteristic
32C22×C4×C97Q812yes—
33C22×C4×C97C2311yescharacteristic
34M16×C97—28no—
35C2×C8×C97—21no—
36C2×C8×C97—21no—
37C2×C8×C97C2×C412yes—
38C2×C8×C97C2×C412yes—
39C2×C8×C97D412yes—
40C2×C8×C97D412yes—
41C23×C97—21no—
42C23×C97—21no—
43C23×C97—22no—
44C23×C97—22no—
45C23×C97C4⋊C412yes—
46C23×C97C2×D411yescharacteristic
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97C4⋊C412yes—
54C2×C4×C97C22×C411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×Q811yescharacteristic
58C2×C4×C97#4111yescharacteristic
59C2×C4×C97#4111yescharacteristic
60C2×Dic97—42no—
61C2×Dic97—42no—
62C2×Dic97—42no—
63C2×Dic97—42no—
64C2×Dic97—42no—
65C2×Dic97—42no—
66C8×C97—22no—
67C8×C97C22⋊C414yescomplemented
68C22×C97—22no—
69C22×C97—24no—
70C22×C97—42no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97C2×C4⋊C411yescharacteristic
74C22×C97#12111yescharacteristic
75C22×C97#12111yescharacteristic
76C22×C97AGL(1,ℤ/8)12yes—
77C22×C97#13712yes—
78C4×C97—22no—
79C4×C97C2×C22⋊C411yescharacteristic
80C4×C97#11711yescharacteristic
81C4×C97C22⋊Q811yescharacteristic
82C4×C97C22⋊Q811yescharacteristic
83Dic97—82no—
84Dic97—82no—
85C2×C97—42no—
86C2×C97#38011yescharacteristic
87C2×C97#42812yes—
88C2×C97#46812yes—
89C2×C97#46911yescharacteristic
90C2×C97#47011yescharacteristic
91#1809—971nomaximal, Sylow
92C97#180911yescharacteristic, complemented, Sylow
93C22×M16D9711yescharacteristic
94C2×#107—971no—
95C2×#106—971no—
96#380—972no—
97#340—972no—
98C23×C4C2×D9711yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16Dic9714yescomplemented
102C22×C8C2×D9711yescharacteristic
103C22×C8C2×D9711yescharacteristic
104C2×C4⋊C4—972no—
105C2×C4⋊C4—972no—
106C2×C22⋊C4—1942no—
107#107—1942no—
108#106—1942no—
109#95—1942no—
110C24D38811yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—1942no—
116C22×C4—1942no—
117C22×C4Dic19412yes—
118C22×C4D38811yescharacteristic
119C22×C4C2×Dic9712yes—
120C22×C4C22×D9711yescharacteristic
121M16—28no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C2×Dic9712yes—
125C2×C8C2×Dic9712yes—
126C2×C8C22⋊D9712yes—
127C2×C8C22⋊D9712yes—
128C4⋊C4—1942no—
129C4⋊C4—1942no—
130C4⋊C4—1942no—
131C4⋊C4—1942no—
132C22⋊C4—3884no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—22no—
137C23C4⋊Dic9712yes—
138C23C2×D38811yescharacteristic
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3882no—
146C2×C4—3882no—
147C2×C4—3882no—
148C2×C4—3882no—
149C2×C4—3882no—
150C2×C4—3882no—
151C2×C4C4⋊Dic9712yes—
152C2×C4C22×Dic9711yescharacteristic
153C2×C4C2×C22⋊D9711yescharacteristic
154C2×C4C2×C22⋊D9711yescharacteristic
155C2×C4C2×Dic19411yescharacteristic
156C2×C4#1309551111yescharacteristic
157C2×C4#1309551111yescharacteristic
158C8—22no—
159C8C22⋊Dic9714yescomplemented
160C22—24no—
161C22—22no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22C2×C4⋊Dic9711yescharacteristic
166C22#1569078411yescharacteristic
167C22#1569078411yescharacteristic
168C22#1569083812yes—
169C22#1569084312yes—
170C4—22no—
171C4—7762no—
172C4—7762no—
173C4C2×C22⋊Dic9711yescharacteristic
174C4#1569076911yescharacteristic
175C4#1569078911yescharacteristic
176C4#1569078911yescharacteristic
177C2—42no—
178C2#2026059511yescharacteristic
179C2#2026077712yes—
180C2#2026098112yes—
181C2#2026098811yescharacteristic
182C2#2026099311yescharacteristic
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×M16×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC22×M16D97
hypocenterC97#1809
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C2×#15690744C2semidirect
2. C2×#15690742C2semidirect
4. #20260498C2semidirect
15. C2×M16×C97C4semidirect
67. C8×C97C22⋊C4semidirect
92. C97#1809semidirect
101. C2×M16Dic97semidirect
159. C8C22⋊Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907442
C2×#156907422
#202605952
#202604982
C22×M16×C972
#180997