Finite Group Databasebeta

← #27142126

Subgroups — #27142126

Order 12416.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690808C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic
3#20260744C211yesmaximal, characteristic
4C2×#15690728C211yesmaximal, characteristic
5#20260496C211yesmaximal, characteristic
6#20260494C211yesmaximal, characteristic
7#419×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690808—22no—
11#15690808C414yescomplemented
12C4×Dic194—22no—
13#15690775—22no—
14#15690769C2211yescharacteristic
15#15690766—22no—
16C2×#13095490—22no—
17C2×C4×Dic97C2211yescharacteristic
18#15690734—22no—
19#15690728—22no—
20C22×Q8×C97C2211yescharacteristic
21C2×M16×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23#103×C97—22no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194C2×C412yes—
29C2×Dic194D412yes—
30C22×Dic97—21no—
31C22×Dic97D412yes—
32#13095495—21no—
33#13095495—21no—
34C2×#11715895—21no—
35C4⋊Dic97D412yes—
36#13095490—24no—
37#13095490—28no—
38C22⋊Dic97—21no—
39C4×Dic97—24no—
40C4×Dic97—21no—
41C4×Dic97—21no—
42C4×Dic97C2×C412yes—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97C2×C412yes—
49C22×C4×C97—21no—
50C22×C4×C97C2311yescharacteristic
51M16×C97—21no—
52M16×C97—21no—
53C2×C8×C97—21no—
54C4⋊C4×C97D412yes—
55C22⋊C4×C97—21no—
56C42×C97—21no—
57C23×C97C2×D411yescharacteristic
58Q8×C97—22no—
59Q8×C97—22no—
60Q8×C97—41no—
61Q8×C97—41no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4111yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×Dic97—24no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97C22⋊C414yes—
78Dic194—22no—
79Dic194—22no—
80Dic194—22no—
81Dic194—22no—
82C8×C97—41no—
83#11715895—41no—
84C22×C97—22no—
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97#12011yescharacteristic
87C22×C97#12311yescharacteristic
88C4×C97—42no—
89C4×C97—42no—
90C4×C97C4×D411yescharacteristic
91C4×C97C4×D411yescharacteristic
92C4×C97#12111yescharacteristic
93C4×C97#12111yescharacteristic
94Dic97—24no—
95Dic97—24no—
96Dic97—42no—
97C2×C97—21no—
98C2×C97—21no—
99C2×C97#38611yescharacteristic
100C2×C97#45612yes—
101#1886—971nomaximal, Sylow
102C97#188611yescharacteristic, complemented, Sylow
103C2×C4⋊Q8—971no—
104#519—971no—
105#419—971no—
106#419D9711yescharacteristic
107C2×#101—971no—
108#339—971no—
109#339—971no—
110C22×Q8—971no—
111C22×Q8C2×D9711yescharacteristic
112C2×M16—971no—
113C2×M16C2×D9711yescharacteristic
114C4⋊Q8—974no—
115C4⋊Q8—1942no—
116C4×Q8—1942no—
117C4×Q8—1942no—
118#117—971no—
119#117—1942no—
120#117C2×D9711yescharacteristic
121C2×C4⋊C4—1942no—
122C2×C42—971no—
123#103—22no—
124#103—1942no—
125#101—1942no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—972no—
132C2×Q8—972no—
133C2×Q8—1941no—
134C2×Q8—1941no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×Q8C4×D9712yes—
138C22×C4—21no—
139C22×C4—972no—
140C22×C4—1941no—
141C22×C4C22×D9711yescharacteristic
142M16—21no—
143M16—21no—
144M16—1941no—
145M16—1941no—
146C2×C8—21no—
147C2×C8—1941no—
148C4⋊C4—972no—
149C4⋊C4—1948no—
150C4⋊C4—1944no—
151C4⋊C4C22⋊D9712yes—
152C22⋊C4—21no—
153C22⋊C4—1941no—
154C42—21no—
155C42—972no—
156C42—1944no—
157C42—1941no—
158C42—1941no—
159C23D4×D9711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—1942no—
165Q8—1942no—
166Q8—1942no—
167Q8—1942no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—974no—
173C2×C4—1942no—
174C2×C4—1944no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4C2×C4×D9711yescharacteristic
179C2×C4D4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4C2×C22⋊D9711yescharacteristic
182C2×C4#1309551011yescharacteristic
183C2×C4#1309551111yescharacteristic
184C8—41no—
185C8—3881no—
186C22—22no—
187C22C22⋊C4×D9711yescharacteristic
188C22#1569077811yescharacteristic
189C22#1569079211yescharacteristic
190C4—42no—
191C4—42no—
192C4—3882no—
193C4—1944no—
194C4—1944no—
195C4#1569077011yescharacteristic
196C4C4×C22⋊D9711yescharacteristic
197C4#1569078111yescharacteristic
198C4#1569078311yescharacteristic
199C2—21no—
200C2—21no—
201C2#2026061111yescharacteristic
202C2#2026091612yes—
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#419×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#419D97
hypocenterC97#1886
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #15690808C4semidirect
102. C97#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
#202612122
#202607442
C2×#156907282
#202604962
#202604942
#419×C972
#188697