Finite Group Databasebeta

← #27141926

Subgroups — #27141926

Order 12416.

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

normal subgroups only

This group has 360 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
1C22×C4⋊Dic97C211yesmaximal, characteristic
2C2×#15690744C212yesmaximal, complemented
3#20260595C211yesmaximal, characteristic
4#20260498C212yesmaximal, complemented
5C2×C22⋊C8×C97C211yesmaximal, characteristic
6C23×Dic97—21no—
7C2×C4⋊Dic97—24no—
8C2×C4⋊Dic97—21no—
9C2×C4⋊Dic97C2211yescharacteristic
10C2×C4⋊Dic97C2211yescharacteristic
11C2×C4⋊Dic97C2212yes—
12C2×C22⋊Dic97—21no—
13#15690744—24no—
14#15690717—21no—
15C23×C4×C97C2211yescharacteristic
16C22×C8×C97C2212yes—
17C22⋊C8×C97C414yescomplemented
18C22×Dic97—24no—
19C22×Dic97—22no—
20C22×Dic97—21no—
21C22×Dic97—21no—
22C22×Dic97—42no—
23C22×Dic97—42no—
24C4⋊Dic97—24no—
25C4⋊Dic97—22no—
26C4⋊Dic97—22no—
27C4⋊Dic97—22no—
28C4⋊Dic97D414yes—
29C22⋊Dic97—42no—
30C24×C97D411yescharacteristic
31C22×C4×C97—22no—
32C22×C4×C97—22no—
33C22×C4×C97C2×C412yes—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97Q812yes—
36C22×C4×C97C2311yescharacteristic
37C2×C8×C97—22no—
38C2×C8×C97—22no—
39C2×C8×C97C2×C414yescomplemented
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97C4⋊C412yes—
45C23×C97D812yes—
46C23×C97Q1612yes—
47C23×C97C2×D411yescharacteristic
48C2×C4×C97—21no—
49C2×C4×C97—21no—
50C2×C4×C97—24no—
51C2×C4×C97—24no—
52C2×C4×C97C4⋊C412yes—
53C2×C4×C97C22×C411yescharacteristic
54C2×C4×C97C2×D412yes—
55C2×C4×C97C2×Q811yescharacteristic
56C2×C4×C97#4112yes—
57C2×Dic97—24no—
58C2×Dic97—48no—
59C2×Dic97—42no—
60C2×Dic97—42no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97—41no—
65C2×Dic97—41no—
66C2×Dic97—41no—
67C8×C97—24no—
68C22×C97—22no—
69C22×C97—22no—
70C22×C97—22no—
71C22×C97—22no—
72C22×C97—22no—
73C22×C97—22no—
74C22×C97#10714yescomplemented
75C22×C97C2×C4⋊C411yescharacteristic
76C22×C97#12011yescharacteristic
77C22×C97#12311yescharacteristic
78C22×C97C2×D811yescharacteristic
79C22×C97C2×Q1611yescharacteristic
80C22×C97AGL(1,ℤ/8)11yescharacteristic
81C22×C97#13711yescharacteristic
82C4×C97—22no—
83C4×C97C4×D412yes—
84C4×C97C22⋊Q812yes—
85Dic97—44no—
86Dic97—81no—
87Dic97—81no—
88C2×C97—24no—
89C2×C97#38511yescharacteristic
90C2×C97C2×#10711yescharacteristic
91C2×C97#42811yescharacteristic
92C2×C97#44711yescharacteristic
93C2×C97#45111yescharacteristic
94C2×C97#48011yescharacteristic
95C2×C97#48411yescharacteristic
96#1852—971nomaximal, Sylow
97C97#185211yescharacteristic, complemented, Sylow
98C22×C4⋊C4—971no—
99C2×#107—972no—
100C2×C22⋊C8D9711yescharacteristic
101#380—971no—
102#340—972no—
103C23×C4—1941no—
104C23×C4C2×D9711yescharacteristic
105C22×C8C2×D9712yes—
106C2×C4⋊C4—972no—
107C2×C4⋊C4—971no—
108C2×C4⋊C4—971no—
109C2×C4⋊C4—1944no—
110C2×C4⋊C4—1941no—
111C2×C22⋊C4—1941no—
112#107—1944no—
113C22⋊C8Dic9714yescomplemented
114#95—1941no—
115C24D38811yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—3882no—
119C22×C4—3882no—
120C22×C4—1941no—
121C22×C4—1941no—
122C22×C4—1944no—
123C22×C4—1942no—
124C22×C4Dic19412yes—
125C22×C4D38811yescharacteristic
126C22×C4C2×Dic9712yes—
127C22×C4C22×D9711yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic9714yescomplemented
131C4⋊C4—974no—
132C4⋊C4—1944no—
133C4⋊C4—1942no—
134C4⋊C4—1942no—
135C4⋊C4—1942no—
136C22⋊C4—3882no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic9712yes—
142C23D77612yes—
143C23Dic38812yes—
144C23C2×D38811yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—3881no—
150C2×C4—3881no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—3888no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—1944no—
159C2×C4C4⋊Dic9712yes—
160C2×C4C22×Dic9711yescharacteristic
161C2×C4D4×D9712yes—
162C2×C4C2×Dic19411yescharacteristic
163C2×C4#1309551012yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1569074414yescomplemented
172C22C2×C4⋊Dic9711yescharacteristic
173C22#1569078011yescharacteristic
174C22#1569079411yescharacteristic
175C22C2×D77611yescharacteristic
176C22C2×Dic38811yescharacteristic
177C22#1569083811yescharacteristic
178C22#1569084311yescharacteristic
179C4—22no—
180C4—7761no—
181C4—7761no—
182C4—3884no—
183C4D4×Dic9712yes—
184C4#1569078712yes—
185C2—24no—
186C2#2026061011yescharacteristic
187C2C2×#1569074411yescharacteristic
188C2#2026077711yescharacteristic
189C2#2026087111yescharacteristic
190C2#2026089311yescharacteristic
191C2#2026104911yescharacteristic
192C2#2026107111yescharacteristic
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1852
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15690744C2semidirect
4. #20260498C2semidirect
17. C22⋊C8×C97C4semidirect
39. C2×C8×C97C2×C4semidirect
74. C22×C97#107semidirect
97. C97#1852semidirect
113. C22⋊C8Dic97semidirect
130. C2×C8C2×Dic97semidirect
171. C22#15690744semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic972
C2×#156907442
#202605952
#202604982
C2×C22⋊C8×C972
#185297