Finite Group Databasebeta

← #27157500

Subgroups — #27157500

Order 12416.

320 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 92 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×#15690875C212yesmaximal
2C4×#13095521C211yesmaximal, characteristic, complemented
3#20261671C211yesmaximal, characteristic, complemented
4#20261605C212yesmaximal, complemented
5C4×#13095512C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×C4×#87604C2212yes—
8#15690875C418yescomplemented
9#15690865—21no—
10C4×#11715897—21no—
11(C2×#87604)⋊C4—21no—
12(C2×#87604)⋊C4—21no—
13C2×#13095512C2211yescharacteristic
14Q8×Dic97C411yescharacteristic, complemented
15C4×D388—21no—
16C4×D388C411yescharacteristic, complemented
17#15690772—21no—
18#15690772C411yescharacteristic, complemented
19#15690768—21no—
20#15690768C2211yescharacteristic
21C42×D97—21no—
22C42×D97C2211yescharacteristic
23C4×Q8×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095521—22no—
26#13095521C2×C414yescomplemented
27C2×#11715897—22no—
28C4×#87604—22no—
29C4×#87604C2×C414yescomplemented
30#13095512—22no—
31#13095512C2×C414yes—
32C2×D388—21no—
33C2×D388C2×C411yescharacteristic
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97C2311yescharacteristic
40C4⋊Dic97—21no—
41C4⋊Dic97C2×C411yescharacteristic
42#13095488—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D411yescharacteristic
49C4×Dic97D411yescharacteristic
50C2×Q8×C97C2×C411yescharacteristic
51C4⋊C4×C97—21no—
52C4⋊C4×C97C2×C411yescharacteristic
53C42×C97—21no—
54C42×C97C2×C411yescharacteristic
55C22×D97—21no—
56C22×D97C2×D411yescharacteristic
57C2×#87604—24no—
58C2×#87604—42no—
59C2×#87604—42no—
60Q8×C97—21no—
61Q8×C97C4212yes—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—21no—
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71D388—24no—
72D388—21no—
73D388C4212yes—
74C4×D97—24no—
75C4×D97—41no—
76C4×D97—41no—
77C4×D97—42no—
78C4×D97—42no—
79C4×D97C22×C412yes—
80C4×D97#4112yes—
81#11715897—24no—
82C22×C97C2×C22⋊C411yescharacteristic
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—22no—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97C4×D412yes—
89#87604—44no—
90C4×C97—22no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—41no—
94C4×C97C2×C4211yescharacteristic
95C4×C97#11711yescharacteristic
96Dic97—21no—
97Dic97—21no—
98Dic97—41no—
99Dic97C4×D412yes—
100C2×C97C4×C22⋊C411yescharacteristic
101C2×C97#42111yescharacteristic
102C2×C97#42111yescharacteristic
103D97—22no—
104D97—42no—
105#1759—971nomaximal, Sylow
106C97#175911yescharacteristic, complemented, Sylow
107C4×#41—971no—
108C2×#104—972no—
109C4×M16—971no—
110#376—971no—
111#339—972no—
112C2×#41—971no—
113C2×M16—972no—
114C4×Q8—971no—
115C4×Q8#8760411yescharacteristic, complemented
116C4×D4—971no—
117C4×D4—971no—
118C4×D4—1941no—
119C4×D4—1941no—
120#117—971no—
121#117—1941no—
122C2×C42—971no—
123C2×C42—972no—
124C2×C42—1941no—
125#104—978no—
126#97—1941no—
127C4×C8—1941no—
128#95—1941no—
129#95—1941no—
130#41—974no—
131#41—1942no—
132C2×Q8C2×#8760411yescharacteristic
133C2×D4—971no—
134C2×D4—1941no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139C22×C4—1941no—
140C22×C4—1942no—
141C22×C4—1941no—
142M16—974no—
143M16—1942no—
144C2×C8—1942no—
145C4⋊C4—21no—
146C4⋊C4—971no—
147C4⋊C4—1941no—
148C4⋊C4C2×#8760411yescharacteristic
149C22⋊C4—1941no—
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C42—21no—
153C42—971no—
154C42—971no—
155C42—971no—
156C42—974no—
157C42—1941no—
158C42—1941no—
159C42—1942no—
160C42C2×#8760411yescharacteristic
161C23—971no—
162C23—1941no—
163Q8—21no—
164Q8C4×#8760412yes—
165D4—972no—
166D4—1944no—
167D4—1941no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—971no—
171C2×C4—972no—
172C2×C4—972no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—3882no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—1944no—
182C2×C4—1944no—
183C2×C4—1941no—
184C2×C4C22⋊#8760411yescharacteristic
185C2×C4C22⋊#8760411yescharacteristic
186C2×C4C22×#8760411yescharacteristic
187C8—1944no—
188C22—972no—
189C22—3881no—
190C22—3881no—
191C22—1942no—
192C22—1941no—
193C22—1941no—
194C22C2×C22⋊#8760411yescharacteristic
195C4—22no—
196C4—21no—
197C4—21no—
198C4—41no—
199C4—972no—
200C4—3884no—
201C4—3881no—
202C4—1941no—
203C4—1941no—
204C4C2×C4×#8760411yescharacteristic
205C4#1569088911yescharacteristic
206C2—3882no—
207C2—1942no—
208C2C4×C22⋊#8760411yescharacteristic
209C2#2026173211yescharacteristic
210C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095512C22
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1759
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. C4×#13095521C2semidirect
3. #20261671C2semidirect
4. #20261605C2semidirect
8. #15690875C4semidirect
14. Q8×Dic97C4semidirect
16. C4×D388C4semidirect
18. #15690772C4semidirect
23. C4×Q8×C97C4semidirect
26. #13095521C2×C4semidirect
29. C4×#87604C2×C4semidirect
106. C97#1759semidirect
115. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908752
C4×#130955212
#202616712
#202616052
C4×#130955122
#175997