Finite Group Databasebeta

← #27151216

Subgroups — #27151216

Order 12416.

584 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 157 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×#15690835C212yesmaximal, complemented
2C2×#15690807C211yesmaximal, characteristic, complemented
3#20260935C214yesmaximal, complemented
4#20260930C214yesmaximal, complemented
5#20260759C211yesmaximal, characteristic, complemented
6#20260739C212yesmaximal, complemented
7C4×#41×C97C211yesmaximal, characteristic, complemented
8C22×D388—22no—
9C22×D388C2212yes—
10#15690835—28no—
11C2×#13095498C2214yescomplemented
12C2×#13095496C2214yescomplemented
13C2×#13095495C2212yes—
14#15690807—21no—
15#15690807—21no—
16#15690807C2214yes—
17C4⋊#11715895C2214yescomplemented
18#15690730C2218yescomplemented
19C2×#41×C97C2211yescharacteristic
20C4×Q8×C97C2212yescomplemented
21C4×D4×C97—22no—
22C4×D4×C97C2212yescomplemented
23#117×C97—21no—
24#117×C97C2211yescharacteristic
25C2×C42×C97—21no—
26C2×C42×C97C2211yescharacteristic
27C23×D97—22no—
28C2×D388—24no—
29C2×D388—24no—
30C2×D388—22no—
31C2×D388—24no—
32C2×D388—24no—
33C2×D388—48no—
34C2×D388C2314yes—
35#13095498—28no—
36#13095496—28no—
37#13095495—24no—
38C2×#11715895C2314yes—
39#41×C97—22no—
40#41×C97D414yescomplemented
41C2×Q8×C97C2311yescharacteristic
42C2×D4×C97—21no—
43C2×D4×C97C2311yescharacteristic
44C22×C4×C97—21no—
45C22×C4×C97—22no—
46C22×C4×C97D412yes—
47C22×C4×C97C2311yescharacteristic
48C4⋊C4×C97—22no—
49C4⋊C4×C97C2312yes—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—22no—
52C42×C97—21no—
53C42×C97—22no—
54C42×C97D411yescharacteristic
55C42×C97D411yescharacteristic
56C42×C97C2312yes—
57C23×C97—21no—
58C23×C97C2×D411yescharacteristic
59C22×D97—24no—
60C22×D97—48no—
61C22×D97—44no—
62Q8×C97—21no—
63Q8×C97C2×D412yescomplemented
64D4×C97—24no—
65D4×C97—21no—
66D4×C97C2×D412yescomplemented
67C2×C4×C97—21no—
68C2×C4×C97—24no—
69C2×C4×C97—24no—
70C2×C4×C97—22no—
71C2×C4×C97—44no—
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D412yes—
76C2×C4×C97#4112yes—
77C2×C4×C97C2411yescharacteristic
78D388—24no—
79D388—24no—
80D388—48no—
81D388—44no—
82D388—48no—
83#11715895—24no—
84C22×C97—22no—
85C22×C97—22no—
86C22×C97—42no—
87C22×C97#12112yes—
88C22×C97C22×D411yescharacteristic
89C2×D97—44no—
90C2×D97—48no—
91C2×D97—88no—
92C4×C97—22no—
93C4×C97—22no—
94C4×C97—42no—
95C4×C97#12112yes—
96C4×C97AGL(1,ℤ/8)14yescomplemented
97C4×C97C22×D411yescharacteristic
98C4×C97C2×#4111yescharacteristic
99C2×C97—22no—
100C2×C97—42no—
101C2×C97C2×#12111yescharacteristic
102C2×C97C2×AGL(1,ℤ/8)12yes—
103D97—84no—
104#3038—971nomaximal, Sylow
105C97#303811yescharacteristic, complemented, Sylow
106C2×AGL(1,ℤ/8)—972no—
107C2×#127—971no—
108C4×#41D9711yescharacteristic, complemented
109#460—974no—
110#459—974no—
111#423—971no—
112#418—972no—
113C2×#41C2×D9711yescharacteristic
114C22×D4—972no—
115C22×D4—1942no—
116AGL(1,ℤ/8)—1948no—
117C2×SD16—974no—
118C2×D8—974no—
119C2×M16—972no—
120#127—974no—
121#127—1941no—
122#127—1941no—
123C4×Q8C2×D9712yescomplemented
124C4×D4—22no—
125C4×D4C2×D9712yescomplemented
126#117—21no—
127#117C2×D9711yescharacteristic
128C2×C42—21no—
129C2×C42C2×D9711yescharacteristic
130C4⋊C8—974no—
131#102—978no—
132C24—1942no—
133#41—22no—
134#41D38814yescomplemented
135C2×Q8C22×D9711yescharacteristic
136C2×D4—21no—
137C2×D4—974no—
138C2×D4—3888no—
139C2×D4—1944no—
140C2×D4—1944no—
141C2×D4—1942no—
142C2×D4—1944no—
143C2×D4—1944no—
144C2×D4C22×D9711yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4C22⋊D9712yes—
148C22×C4C22×D9711yescharacteristic
149SD16—1948no—
150D8—1948no—
151M16—1944no—
152C2×C8—974no—
153C4⋊C4—22no—
154C4⋊C4C22×D9712yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42C22⋊D9711yescharacteristic
160C42C22⋊D9711yescharacteristic
161C42C22×D9712yes—
162C23—21no—
163C23—3888no—
164C23—3884no—
165C23—1944no—
166C23C2×C22⋊D9711yescharacteristic
167Q8—21no—
168Q8C2×D38812yescomplemented
169D4—24no—
170D4—21no—
171D4—3888no—
172D4—3884no—
173D4—3888no—
174D4—1944no—
175D4—1944no—
176D4C2×D38812yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×C22⋊D9712yes—
183C2×C4C2×C22⋊D9711yescharacteristic
184C2×C4C2×C22⋊D9712yes—
185C2×C4C2×D38812yes—
186C2×C4#1309551112yes—
187C2×C4C23×D9711yescharacteristic
188C8—1944no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—7768no—
193C22—3884no—
194C22—3888no—
195C22#1569078412yes—
196C22C22×C22⋊D9711yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4#1569078412yes—
201C4#1569083514yescomplemented
202C4C22×D38811yescharacteristic
203C4C2×#1309551111yescharacteristic
204C2—22no—
205C2—42no—
206C2—7764no—
207C2C2×#1569078411yescharacteristic
208C2C2×#1569083512yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×#41×C97C2
socleC22×C97C22×D4
hypercenterC4×#41D97
hypocenterC97#3038
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690807C2semidirect
3. #20260935C2semidirect
4. #20260930C2semidirect
5. #20260759C2semidirect
6. #20260739C2semidirect
7. C4×#41×C97C2semidirect
11. C2×#13095498C22semidirect
12. C2×#13095496C22semidirect
17. C4⋊#11715895C22semidirect
18. #15690730C22semidirect
20. C4×Q8×C97C22semidirect
22. C4×D4×C97C22semidirect
40. #41×C97D4semidirect
63. Q8×C97C2×D4semidirect
66. D4×C97C2×D4semidirect
96. C4×C97AGL(1,ℤ/8)semidirect
105. C97#3038semidirect
108. C4×#41D97semidirect
123. C4×Q8C2×D97semidirect
125. C4×D4C2×D97semidirect
134. #41D388semidirect
168. Q8C2×D388semidirect
176. D4C2×D388semidirect
201. C4#15690835semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156908072
#202609352
#202609302
#202607592
#202607392
C4×#41×C972
#303897