Finite Group Databasebeta

← #27152114

Subgroups — #27152114

Order 12416.

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

normal subgroups only

This group has 380 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
1#20261288C211yesmaximal, characteristic
2C2×#15690808C211yesmaximal, characteristic
3#20261102C214yesmaximal, complemented
4#20261084C214yesmaximal, complemented
5#20260745C211yesmaximal, characteristic
6#20260744C211yesmaximal, characteristic
7#20260743C211yesmaximal, characteristic
8M16×Dic97C211yesmaximal, characteristic
9#419×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×#13095495C2211yescharacteristic
12#15690809C2212yes—
13#15690808—22no—
14#15690808C2212yes—
15#15690808C2212yes—
16#15690801C2212yes—
17#15690787—22no—
18C4×Dic194—22no—
19#15690769C2211yescharacteristic
20C2×#13095490—22no—
21C2×C4×Dic97C2211yescharacteristic
22#15690735C2214yes—
23#15690734C2214yes—
24#15690733C2214yes—
25#15690720C2212yes—
26C8×Dic97C2212yes—
27C22×Q8×C97C2211yescharacteristic
28C2×M16×C97C2211yescharacteristic
29#117×C97C2211yescharacteristic
30#103×C97C2214yes—
31C2×Dic194—22no—
32C2×Dic194—22no—
33C2×Dic194—21no—
34C2×Dic194—22no—
35C2×Dic194C2312yes—
36C22×Dic97—21no—
37C22×Dic97D412yes—
38#13095495—22no—
39C2×#11715895C2312yes—
40C4⋊Dic97—22no—
41C4⋊Dic97C2312yes—
42#13095490—24no—
43#13095490—24no—
44#13095490—22no—
45#13095490—22no—
46C22⋊Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97D412yes—
49C4×Dic97C2311yescharacteristic
50C4×Dic97C2311yescharacteristic
51C2×Q8×C97—21no—
52C2×Q8×C97—22no—
53C2×Q8×C97—22no—
54C2×Q8×C97C2312yes—
55C22×C4×C97—21no—
56C22×C4×C97C2311yescharacteristic
57M16×C97—22no—
58C2×C8×C97C2312yes—
59C4⋊C4×C97C2312yes—
60C22⋊C4×C97—21no—
61C42×C97—21no—
62C23×C97C2×D411yescharacteristic
63Q8×C97—22no—
64Q8×C97—22no—
65Q8×C97—42no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—42no—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4112yes—
71C2×C4×C97#4112yes—
72C2×C4×C97C2411yescharacteristic
73C2×Dic97—24no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—42no—
77C2×Dic97C2×D412yes—
78C2×Dic97C2×D412yes—
79Dic194—22no—
80Dic194—22no—
81Dic194—42no—
82Dic194—44no—
83C8×C97—22no—
84#11715895—22no—
85C22×C97—22no—
86C22×C97#12412yes—
87C22×C97C22×D411yescharacteristic
88C4×C97—42no—
89C4×C97—42no—
90C4×C97#12412yes—
91C4×C97C2×#4111yescharacteristic
92C4×C97C2×#4111yescharacteristic
93Dic97—22no—
94Dic97—42no—
95Dic97—42no—
96Dic97#13714yes—
97C2×C97—22no—
98C2×C97C2×#12411yescharacteristic
99C2×C97C2×#13712yes—
100#3139—971nomaximal, Sylow
101C97#313911yescharacteristic, complemented, Sylow
102#533—971no—
103C2×C4⋊Q8—971no—
104#491—974no—
105#487—974no—
106#419—971no—
107#419—971no—
108#419—971no—
109#419D9711yescharacteristic
110C4×M16—971no—
111C22×Q8—971no—
112C22×Q8C2×D9711yescharacteristic
113C2×M16—971no—
114C2×M16C2×D9711yescharacteristic
115C4⋊Q8—972no—
116C4⋊Q8—972no—
117C4⋊Q8—972no—
118C4⋊Q8—1942no—
119#125—972no—
120C22⋊Q8—1942no—
121C4×Q8—1942no—
122#117—971no—
123#117C2×D9711yescharacteristic
124C2×C4⋊C4—1942no—
125C2×C42—971no—
126#103—974no—
127#103—974no—
128#103—974no—
129#103C2×D9714yes—
130#97—972no—
131C4×C8—972no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—972no—
136C2×Q8—1942no—
137C2×Q8—1942no—
138C2×Q8—1942no—
139C2×Q8—1941no—
140C2×Q8C22×D9712yes—
141C22×C4—21no—
142C22×C4—972no—
143C22×C4—1941no—
144C22×C4C22×D9711yescharacteristic
145M16—22no—
146M16—1942no—
147C2×C8—972no—
148C2×C8C22×D9712yes—
149C4⋊C4—972no—
150C4⋊C4—1944no—
151C4⋊C4—1944no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4C22×D9712yes—
156C22⋊C4—21no—
157C22⋊C4—1941no—
158C42—21no—
159C42—972no—
160C42—971no—
161C42—971no—
162C42—1941no—
163C23D4×D9711yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—3884no—
168Q8—3882no—
169Q8—1942no—
170Q8—1942no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—972no—
175C2×C4—972no—
176C2×C4—3882no—
177C2×C4—1944no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4D4×D9711yescharacteristic
181C2×C4#1309551012yes—
182C2×C4#1309551112yes—
183C2×C4C23×D9711yescharacteristic
184C8—22no—
185C8—1942no—
186C22—22no—
187C22#1569079612yes—
188C22C2×D4×D9711yescharacteristic
189C4—42no—
190C4—42no—
191C4—974no—
192C4—3882no—
193C4—3882no—
194C4—1942no—
195C4#1569079612yes—
196C4C2×#1309551011yescharacteristic
197C4C2×#1309551111yescharacteristic
198C2—22no—
199C2C2×#1569079611yescharacteristic
200C2#137×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenter#419D97
hypocenterC97#3139
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
3. #20261102C2semidirect
4. #20261084C2semidirect
101. C97#3139semidirect

Maximal subgroups

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

subgroupindex
#202612882
C2×#156908082
#202611022
#202610842
#202607452
#202607442
#202607432
M16×Dic972
#419×C972
#313997