Finite Group Databasebeta

← #27157065

Subgroups — #27157065

Order 12416.

248 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 76 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#20261754C211yesmaximal, characteristic
2#20261739C211yesmaximal, characteristic
3#20261585C212yesmaximal, complemented
4Q8⋊#11715897C212yesmaximal, complemented
5C4×Q8×D97C211yesmaximal, characteristic
6C2×#13095521—21no—
7C4⋊#11715897—21no—
8C4⋊#11715897—21no—
9C4⋊#11715897C2212yes—
10#15690867—21no—
11C4×#11715897C2212yes—
12C2×Q8×D97C411yescharacteristic
13Q8×Dic97C2211yescharacteristic
14C4×Dic194—21no—
15C4×Dic194C411yescharacteristic
16#15690775—21no—
17#15690775C2211yescharacteristic
18C4⋊C4×D97—21no—
19C4⋊C4×D97C411yescharacteristic
20C42×D97—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095521—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28Q8×D97—22no—
29Q8×D97—22no—
30C2×C4×D97—21no—
31C2×C4×D97—21no—
32C2×C4×D97C2×C411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40C4×Dic97—21no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97C2311yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C42×C97—21no—
50C42×C97C2×C411yescharacteristic
51C22×D97C22⋊C411yescharacteristic
52Q8×C97—21no—
53Q8×C97M1612yescomplemented
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—24no—
65C4×D97—21no—
66C4×D97—21no—
67C4×D97—41no—
68C4×D97—42no—
69C4×D97SD1612yes—
70C4×D97Q1612yes—
71Dic194—24no—
72Dic194—21no—
73Dic194M1612yescomplemented
74#11715897—42no—
75#11715897—42no—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—21no—
78C2×D97—21no—
79C2×D97#10312yes—
80C4×C97—22no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97#10312yes—
84C4×C97C2×M1611yescharacteristic
85C4×C97C2×M1611yescharacteristic
86Dic97—21no—
87Dic97—22no—
88Dic97—21no—
89Dic97—41no—
90Dic97C2×SD1611yescharacteristic
91Dic97C2×Q1611yescharacteristic
92C2×C97#40711yescharacteristic
93C2×C97C2×#10311yescharacteristic
94C2×C97#42111yescharacteristic
95D97—22no—
96#1487—971nomaximal, Sylow
97C97#148711yescharacteristic, complemented, Sylow
98C2×C4×Q8—971no—
99#431—971no—
100#423—971no—
101Q8⋊C8—972no—
102Q8⋊C8—972no—
103C22×Q8—971no—
104C2×M16—1941no—
105C4×Q8—971no—
106C4×Q8—971no—
107C4×Q8—971no—
108C4×Q8—1941no—
109C4×Q8—1941no—
110C4×Q8#8760411yescharacteristic
111C2×C4⋊C4—971no—
112C2×C4⋊C4—1941no—
113C2×C42—971no—
114C2×C42—1941no—
115C4⋊C8—972no—
116C4⋊C8—1941no—
117C4⋊C8—1941no—
118C22⋊C8—1941no—
119C4×C8—972no—
120C2×Q8—971no—
121C2×Q8—1941no—
122C2×Q8—1942no—
123C2×Q8—1942no—
124C2×Q8C2×#8760411yescharacteristic
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—1941no—
129C22×C4—1941no—
130M16—3882no—
131C2×C8—1942no—
132C2×C8—1942no—
133C4⋊C4—21no—
134C4⋊C4—971no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C4⋊C4C2×#8760411yescharacteristic
140C42—21no—
141C42—971no—
142C42—971no—
143C42—971no—
144C42—1941no—
145C42—1941no—
146C42—1941no—
147C42C2×#8760411yescharacteristic
148C23—971no—
149Q8—21no—
150Q8—972no—
151Q8—1944no—
152Q8—1941no—
153Q8#1309552112yescomplemented
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—971no—
157C2×C4—972no—
158C2×C4—972no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—3881no—
162C2×C4—3882no—
163C2×C4—1944no—
164C2×C4—1941no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4—1941no—
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—3882no—
172C8—3882no—
173C22—972no—
174C22—1941no—
175C22—1941no—
176C22C2×C22⋊#8760411yescharacteristic
177C4—22no—
178C4—21no—
179C4—41no—
180C4—971no—
181C4—971no—
182C4—3881no—
183C4—1941no—
184C4—1942no—
185C4—1941no—
186C4#1569087312yes—
187C4D97⋊M1611yescharacteristic
188C4C2×#1309552111yescharacteristic
189C2—1942no—
190C2#2026170611yescharacteristic
191C2C2×#1569087311yescharacteristic
192C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1487
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20261585C2semidirect
4. Q8⋊#11715897C2semidirect
53. Q8×C97M16semidirect
73. Dic194M16semidirect
97. C97#1487semidirect
153. Q8#13095521semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202617392
#202615852
Q8⋊#117158972
C4×Q8×D972
#148797