Finite Group Databasebeta

← #27150083

Subgroups — #27150083

Order 12416.

644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690786C212yesmaximal
2C2×C42×D97C211yesmaximal, characteristic
3#20260642C212yesmaximal, complemented
4#20260631C214yesmaximal, complemented
5#20260605C211yesmaximal, characteristic, complemented
6#20260592C214yesmaximal
7C2×C4⋊Q8×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C22×C4×D97C2212yes—
10#15690786—28no—
11C2×C4⋊Dic97C2214yes—
12C2×C4⋊Dic97C2212yes—
13C2×#13095490C2214yes—
14C2×#13095488C2214yes—
15C2×C4×Dic97C2212yes—
16C2×C4×Dic97C2211yescharacteristic
17C42×D97—24no—
18#15690717C2218yes—
19C22×Q8×C97C2212yes—
20C4⋊Q8×C97—24no—
21C2×C4⋊C4×C97C2214yes—
22C2×C42×C97C2211yescharacteristic
23C23×D97C2311yescharacteristic
24C22×Dic97C2314yes—
25C22×Dic97C2311yescharacteristic
26C22×Dic97C2312yes—
27C2×C4×D97—22no—
28C2×C4×D97—24no—
29C2×C4×D97—22no—
30C2×C4×D97—24no—
31C2×C4×D97D414yes—
32C2×C4×D97Q818yescomplemented
33C4⋊Dic97—24no—
34C4⋊Dic97—28no—
35#13095490—28no—
36#13095488—28no—
37C4×Dic97—22no—
38C4×Dic97—24no—
39C2×Q8×C97—24no—
40C2×Q8×C97—24no—
41C2×Q8×C97—24no—
42C22×C4×C97C2314yes—
43C22×C4×C97C2311yescharacteristic
44C22×C4×C97C2312yes—
45C4⋊C4×C97—28no—
46C42×C97D414yes—
47C23×C97C2411yescharacteristic
48C22×D97—21no—
49C22×D97—21no—
50C22×D97—22no—
51C22×D97C2×D412yes—
52C22×D97C2×Q814yes—
53Q8×C97—48no—
54C2×C4×C97—24no—
55C2×C4×C97—24no—
56C2×C4×C97—24no—
57C2×C4×C97C2×D412yes—
58C2×C4×C97C2×D412yes—
59C2×C4×C97C2×D414yes—
60C2×C4×C97C2×Q814yes—
61C2×C4×C97#4112yes—
62C2×C4×C97#4114yes—
63C2×Dic97—21no—
64C2×Dic97—22no—
65C2×Dic97—21no—
66C2×Dic97—22no—
67C2×Dic97—24no—
68C2×Dic97—28no—
69C2×Dic97C2×D412yes—
70C2×Dic97C2×Q814yes—
71C4×D97—28no—
72C4×D97—216no—
73C22×C97C22×D411yescharacteristic
74C22×C97C22×D411yescharacteristic
75C22×C97C22×Q812yes—
76C22×C97C2×#4111yescharacteristic
77C22×C97C2×#4112yes—
78C2×D97—22no—
79C2×D97—24no—
80C2×D97—22no—
81C2×D97—24no—
82C2×D97C4⋊Q814yescomplemented
83C4×C97—44no—
84C4×C97#12114yes—
85C4×C97C22⋊Q818yescomplemented
86Dic97—22no—
87Dic97—24no—
88Dic97—44no—
89C2×C97C2×#12111yescharacteristic
90C2×C97C2×C22⋊Q812yes—
91C2×C97C2×C4⋊Q811yescharacteristic
92C2×C97#53211yescharacteristic
93C2×C97#53312yes—
94D97—24no—
95#2857—971nomaximal, Sylow
96C97#285711yescharacteristic, complemented, Sylow
97C2×C4⋊Q8D9711yescharacteristic, complemented
98C2×C22⋊Q8—972no—
99C22×C42—971no—
100#391—972no—
101#389—974no—
102#384—971no—
103#380—974no—
104C22×Q8C2×D9712yes—
105C23×C4—971no—
106C23×C4—972no—
107C4⋊Q8—24no—
108C22⋊Q8—1948no—
109C2×C4⋊C4—974no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—974no—
112C2×C4⋊C4C2×D9714yes—
113C2×C22⋊C4—974no—
114C2×C42—972no—
115C2×C42—971no—
116C2×C42—1944no—
117C2×C42C2×D9711yescharacteristic
118#95—978no—
119C24—971no—
120C2×Q8—24no—
121C2×Q8—24no—
122C2×Q8—24no—
123C22×C4—972no—
124C22×C4—971no—
125C22×C4—974no—
126C22×C4—978no—
127C22×C4—974no—
128C22×C4—1942no—
129C22×C4—1944no—
130C22×C4—1942no—
131C22×C4—1944no—
132C22×C4C22×D9714yes—
133C22×C4C22×D9711yescharacteristic
134C22×C4C22×D9712yes—
135C4⋊C4—28no—
136C4⋊C4—1948no—
137C4⋊C4—1944no—
138C4⋊C4—1948no—
139C22⋊C4—1948no—
140C42—1942no—
141C42—1944no—
142C42C22⋊D9714yes—
143C23—972no—
144C23—974no—
145C23—1941no—
146C23—1941no—
147C23—1942no—
148C23C23×D9711yescharacteristic
149Q8—48no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—972no—
154C2×C4—974no—
155C2×C4—1944no—
156C2×C4—1948no—
157C2×C4—1941no—
158C2×C4—1942no—
159C2×C4—1941no—
160C2×C4—1942no—
161C2×C4—1948no—
162C2×C4—19416no—
163C2×C4D4×D9712yes—
164C2×C4C2×C22⋊D9712yes—
165C2×C4C2×C22⋊D9714yes—
166C2×C4Q8×D9714yes—
167C2×C4#1309551012yes—
168C2×C4#1309551214yes—
169C22—974no—
170C22—1942no—
171C22—1944no—
172C22—1942no—
173C22—1944no—
174C22C2×D4×D9711yescharacteristic
175C22C22×C22⋊D9711yescharacteristic
176C22C2×Q8×D9712yes—
177C22C2×#1309551011yescharacteristic
178C22C2×#1309551212yes—
179C4—44no—
180C4—3884no—
181C4—1942no—
182C4—1944no—
183C4#1569078114yes—
184C4#1569078618yescomplemented
185C2—1944no—
186C2C2×#1569078111yescharacteristic
187C2C2×#1569078612yes—
188C2C4⋊Q8×D9711yescharacteristic
189C2#2026128611yescharacteristic
190C2#2026129012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×C4⋊Q8×C97C2
prime radicalC23×D97C23
hypercenterC2×C4⋊Q8D97
hypocenterC97#2857
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20260642C2semidirect
4. #20260631C2semidirect
5. #20260605C2semidirect
7. C2×C4⋊Q8×C97C2semidirect
32. C2×C4×D97Q8semidirect
82. C2×D97C4⋊Q8semidirect
85. C4×C97C22⋊Q8semidirect
96. C97#2857semidirect
97. C2×C4⋊Q8D97semidirect
184. C4#15690786semidirect

Maximal subgroups

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

subgroupindex
C2×#156907862
C2×C42×D972
#202606422
#202606312
#202606052
#202605922
C2×C4⋊Q8×C972
#285797