Finite Group Databasebeta

← #27157508

Subgroups — #27157508

Order 12416.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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×#15690873C212yesmaximal
2#20261694C211yesmaximal, characteristic
3C4×C4⋊#87604C211yesmaximal, characteristic
4#20261608C212yesmaximal
5C4×Q8×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8C2×C4×#87604—21no—
9#15690873C418yescomplemented
10#15690865—21no—
11#15690865—21no—
12(C2×#87604)⋊C4—21no—
13C2×Q8×D97C2211yescharacteristic
14Q8×Dic97C411yescharacteristic, complemented
15C4×Dic194—21no—
16C4×Dic194C411yescharacteristic, complemented
17#15690775—21no—
18#15690775C411yescharacteristic, complemented
19C4⋊C4×D97—21no—
20C4⋊C4×D97C2211yescharacteristic
21C42×D97—21no—
22C42×D97C2211yescharacteristic
23C4×Q8×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095519—22no—
26#13095519C2×C414yes—
27C2×#11715897—22no—
28C4⋊#87604—22no—
29C4⋊#87604C2×C414yes—
30C4×#87604—42no—
31C2×Dic194—21no—
32C2×Dic194C2×C411yescharacteristic
33Q8×D97—22no—
34Q8×D97C2×C414yes—
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97C2311yescharacteristic
40C4⋊Dic97—21no—
41C4⋊Dic97C2×C411yescharacteristic
42#13095490—21no—
43#13095490—21no—
44#13095490—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97C2×C411yescharacteristic
49C4×Dic97D411yescharacteristic
50C4×Dic97D411yescharacteristic
51C2×Q8×C97C2×C411yescharacteristic
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C42×C97—21no—
55C42×C97C2×C411yescharacteristic
56C22×D97C2×D411yescharacteristic
57C2×#87604—24no—
58C2×#87604—42no—
59C2×#87604—42no—
60Q8×C97—21no—
61Q8×C97C4212yescomplemented
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72C4×D97—24no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—22no—
76C4×D97—41no—
77C4×D97—41no—
78C4×D97C22×C412yes—
79C4×D97#4112yes—
80Dic194—21no—
81Dic194—24no—
82Dic194C4212yescomplemented
83#11715897—24no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97C4×D412yes—
86C2×D97AGL(1,ℤ/8)11yescharacteristic
87C2×D97#13712yes—
88C2×D97#13711yescharacteristic
89#87604—44no—
90C4×C97—22no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—41no—
94C4×C97C2×C4211yescharacteristic
95C4×C97#11711yescharacteristic
96Dic97—22no—
97Dic97—21no—
98Dic97—21no—
99Dic97—41no—
100Dic97C4×D412yes—
101C2×C97C4×C22⋊C411yescharacteristic
102C2×C97#41711yescharacteristic
103C2×C97#41911yescharacteristic
104D97#44012yes—
105D97#44112yes—
106#1763—971nomaximal, Sylow
107C97#176311yescharacteristic, complemented, Sylow
108C2×C4×Q8—971no—
109C2×#103—972no—
110C2×#97—971no—
111C4×C4⋊C4—971no—
112#340—972no—
113C22×Q8—971no—
114C22×C8—972no—
115C4×Q8—971no—
116C4×Q8—971no—
117C4×Q8—971no—
118C4×Q8—1941no—
119C4×Q8—1941no—
120C4×Q8#8760411yescharacteristic, complemented
121C2×C4⋊C4—972no—
122C2×C4⋊C4—971no—
123C2×C4⋊C4—1941no—
124C2×C42—971no—
125C2×C42—1941no—
126C2×C42—1941no—
127#103—978no—
128#97—1941no—
129#97—1941no—
130#95—1941no—
131C2×Q8—974no—
132C2×Q8—971no—
133C2×Q8—1941no—
134C2×Q8—1942no—
135C2×Q8C2×#8760411yescharacteristic
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—1941no—
140C22×C4—1941no—
141C22×C4—1942no—
142C2×C8—974no—
143C2×C8—1942no—
144C2×C8—1942no—
145C4⋊C4—21no—
146C4⋊C4—974no—
147C4⋊C4—971no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1942no—
152C4⋊C4—1941no—
153C4⋊C4C2×#8760411yescharacteristic
154C42—21no—
155C42—971no—
156C42—971no—
157C42—971no—
158C42—3882no—
159C42—1941no—
160C42—1941no—
161C42—1941no—
162C42C2×#8760411yescharacteristic
163C23—971no—
164Q8—21no—
165Q8—972no—
166Q8—1944no—
167Q8—1941no—
168Q8C4×#8760412yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—971no—
172C2×C4—972no—
173C2×C4—972no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4—1942no—
183C2×C4—1944no—
184C2×C4—1944no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22⋊#8760411yescharacteristic
189C2×C4C22×#8760411yescharacteristic
190C8—1944no—
191C22—972no—
192C22—971no—
193C22—971no—
194C22—972no—
195C22C2×C22⋊#8760411yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—972no—
201C4—3884no—
202C4—3881no—
203C4—1942no—
204C4—1941no—
205C4—1941no—
206C4C2×C4×#8760411yescharacteristic
207C4#1569088911yescharacteristic
208C2—972no—
209C2—972no—
210C2C4×C22⋊#8760411yescharacteristic
211C2#2026172611yescharacteristic
212C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1763
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
9. #15690873C4semidirect
14. Q8×Dic97C4semidirect
16. C4×Dic194C4semidirect
18. #15690775C4semidirect
23. C4×Q8×C97C4semidirect
61. Q8×C97C42semidirect
82. Dic194C42semidirect
107. C97#1763semidirect
120. C4×Q8#87604semidirect
168. Q8C4×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908732
#202616942
C4×C4⋊#876042
#202616082
C4×Q8×D972
#176397