Finite Group Databasebeta

← #27157067

Subgroups — #27157067

Order 12416.

246 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261756C211yesmaximal, characteristic, complemented
2#20261752C211yesmaximal, characteristic, complemented
3Q8⋊#11715897C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5#41×Dic97C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897C2212yes—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10#15690865—21no—
11C4×#11715897C2212yes—
12Q8×Dic97C411yescharacteristic
13Q8×Dic97C2211yescharacteristic
14D4×Dic97—21no—
15D4×Dic97—21no—
16D4×Dic97C411yescharacteristic
17D4×Dic97C2211yescharacteristic
18#15690769—21no—
19#15690769C411yescharacteristic
20C2×C4×Dic97—21no—
21C2×C4×Dic97C2211yescharacteristic
22C2×#41×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C22×Dic97—21no—
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32C4⋊Dic97C2×C411yescharacteristic
33C22⋊Dic97—21no—
34C22⋊Dic97—21no—
35C22⋊Dic97—21no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C4×Dic97—21no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97C2311yescharacteristic
43#41×C97—22no—
44#41×C97—22no—
45C2×Q8×C97C2×C411yescharacteristic
46C2×D4×C97—21no—
47C2×D4×C97C2×C411yescharacteristic
48C22×C4×C97—21no—
49C22×C4×C97C2×C411yescharacteristic
50C23×C97—21no—
51C23×C97C22⋊C411yescharacteristic
52Q8×C97—21no—
53Q8×C97M1612yes—
54D4×C97—24no—
55D4×C97—21no—
56D4×C97M1612yes—
57C2×C4×C97—24no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22×C411yescharacteristic
63C2×Dic97—22no—
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—41no—
67C2×Dic97—42no—
68C2×Dic97—42no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73#11715897—42no—
74#11715897—42no—
75C22×C97—21no—
76C22×C97—22no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C4×C97—21no—
83C4×C97—22no—
84C4×C97C2×M1611yescharacteristic
85C4×C97C2×M1611yescharacteristic
86Dic97—21no—
87Dic97—21no—
88Dic97—21no—
89Dic97—41no—
90Dic97—41no—
91Dic97#13512yes—
92C2×C97—41no—
93C2×C97—42no—
94C2×C97#40711yescharacteristic
95C2×C97#41611yescharacteristic
96C2×C97#42111yescharacteristic
97#1488—971nomaximal, Sylow
98C97#148811yescharacteristic, complemented, Sylow
99C4×#41—971no—
100#432—971no—
101#431—971no—
102Q8⋊C8—972no—
103D4⋊C8—972no—
104C2×#41#8760411yescharacteristic
105C4×Q8—971no—
106C4×Q8—971no—
107C4×D4—971no—
108C4×D4—971no—
109C4×D4—1941no—
110C4×D4—1941no—
111#117—971no—
112#117—1941no—
113C2×C42—971no—
114C2×C42—1941no—
115C4⋊C8—972no—
116C4⋊C8—1941no—
117C22⋊C8—1941no—
118C22⋊C8—1941no—
119#97—1941no—
120C4×C8—972no—
121#41—22no—
122#41—22no—
123C2×Q8C2×#8760411yescharacteristic
124C2×D4—21no—
125C2×D4C2×#8760411yescharacteristic
126C22×C4—21no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4C2×#8760411yescharacteristic
132C2×C8—1942no—
133C2×C8—1942no—
134C4⋊C4—971no—
135C4⋊C4—971no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C22⋊C4—1941no—
139C22⋊C4—1941no—
140C22⋊C4—1941no—
141C42—971no—
142C42—971no—
143C42—971no—
144C42—971no—
145C42—1941no—
146C42—1941no—
147C42—1941no—
148C23—21no—
149C23C22⋊#8760411yescharacteristic
150Q8—21no—
151Q8#1309552012yes—
152D4—24no—
153D4—21no—
154D4#1309552012yes—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3881no—
166C2×C4—1942no—
167C2×C4—1941no—
168C2×C4—1941no—
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—3882no—
172C8—3882no—
173C22—21no—
174C22—22no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊#8760411yescharacteristic
180C4—21no—
181C4—22no—
182C4—972no—
183C4—3881no—
184C4—3881no—
185C4—1941no—
186C4—1941no—
187C4—1941no—
188C4C2×#1309552011yescharacteristic
189C4C2×#1309552011yescharacteristic
190C2—41no—
191C2—42no—
192C2#2026170511yescharacteristic
193C2#2026172511yescharacteristic
194C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting, prime radicalC2×#41×C97C4
socleC22×C97C2×C22⋊C4
cosocleC4×Dic97C23
hypercenterC2×#41#87604
hypocenterC97#1488
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. #20261752C2semidirect
3. Q8⋊#11715897C2semidirect
4. D4⋊#11715897C2semidirect
98. C97#1488semidirect

Maximal subgroups

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

subgroupindex
#202617562
#202617522
Q8⋊#117158972
D4⋊#117158972
#41×Dic972
#148897