Finite Group Databasebeta

← #27145015

Subgroups — #27145015

Order 12416.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 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×C4×D388C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D97C214yesmaximal, complemented
3#20260671C212yesmaximal, complemented
4#20260670C212yesmaximal, complemented
5#20260644C212yesmaximal, complemented
6#20260643C212yesmaximal, complemented
7#20260581C211yesmaximal, characteristic, complemented
8C2×C4×D4×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97C2214yes—
11C22×C4×D97C2212yes—
12C4×D388—24no—
13C4×C22⋊D97—216no—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2214yes—
16C2×C22⋊Dic97C2214yes—
17C2×#13095488C2212yes—
18C2×#13095488C2214yes—
19C2×C4×Dic97C2212yes—
20#15690717C2212yes—
21#15690717C2212yes—
22C22×D4×C97C2211yescharacteristic
23C23×C4×C97C2212yes—
24C4×D4×C97—24no—
25C2×C4⋊C4×C97C2211yescharacteristic
26C2×C22⋊C4×C97C2212yes—
27C2×C42×C97C2211yescharacteristic
28C23×D97C2312yes—
29C2×D388—24no—
30C2×D388—22no—
31C2×C22⋊D97—28no—
32C2×C22⋊D97—28no—
33C2×C22⋊D97—28no—
34C22×Dic97C2312yes—
35C22×Dic97C2314yes—
36C2×C4×D97—28no—
37C2×C4×D97—24no—
38C4⋊Dic97—22no—
39#13095490—28no—
40C22⋊Dic97—28no—
41#13095488—24no—
42#13095488—28no—
43C4×Dic97D418yescomplemented
44C24×C97C2312yes—
45C2×D4×C97—24no—
46C2×D4×C97—22no—
47C22×C4×C97—28no—
48C22×C4×C97—24no—
49C22×C4×C97C2312yes—
50C22×C4×C97C2311yescharacteristic
51C22×C4×C97C2311yescharacteristic
52C22×C4×C97C2311yescharacteristic
53C4⋊C4×C97—22no—
54C22⋊C4×C97—24no—
55C42×C97D414yescomplemented
56C23×C97—22no—
57C23×C97—24no—
58C23×C97—22no—
59C23×C97—24no—
60C23×C97#4114yes—
61C23×C97C2411yescharacteristic
62C22×D97—24no—
63C22×D97—22no—
64C22×D97—24no—
65C22×D97—22no—
66C22×D97#4114yes—
67D4×C97—28no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—24no—
72C2×C4×C97—48no—
73C2×C4×C97C2×D414yes—
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C22⋊D97—232no—
78C2×Dic97—24no—
79C2×Dic97—24no—
80C2×Dic97—22no—
81C2×Dic97—24no—
82C2×Dic97C2×D418yes—
83D388—28no—
84C4×D97—48no—
85C22×C97—28no—
86C22×C97—24no—
87C22×C97—42no—
88C22×C97—44no—
89C22×C97—42no—
90C22×C97C22×D412yes—
91C22×C97C22×D411yescharacteristic
92C22×C97C2×#4112yes—
93C22×C97C2×#4112yes—
94C2×D97—24no—
95C2×D97—28no—
96C2×D97—44no—
97C2×D97—42no—
98C2×D97—42no—
99C4×C97—22no—
100C4×C97—22no—
101C4×C97—42no—
102C4×C97#12714yes—
103Dic97—28no—
104Dic97—42no—
105C2×C97—44no—
106C2×C97#52511yescharacteristic
107C2×C97#52514yes—
108C2×C97#52511yescharacteristic
109C2×C97C2×#12711yescharacteristic
110D97—44no—
111#2397—971nomaximal, Sylow
112C97#239711yescharacteristic, complemented, Sylow
113C2×C4×D4—974no—
114C2×C4×D4—971no—
115C2×C4×D4D9711yescharacteristic, complemented
116#399—972no—
117#399—972no—
118#392—972no—
119#392—972no—
120#376—971no—
121C22×D4—974no—
122C22×D4—971no—
123C22×D4C2×D9711yescharacteristic
124C23×C4—972no—
125C23×C4C2×D9712yes—
126C4×D4—24no—
127C4×D4—19416no—
128C4×D4—1944no—
129C2×C4⋊C4—974no—
130C2×C4⋊C4—971no—
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C22⋊C4—974no—
133C2×C22⋊C4—974no—
134C2×C22⋊C4—972no—
135C2×C22⋊C4C2×D9712yes—
136C2×C42—972no—
137C2×C42C2×D9711yescharacteristic
138#95—972no—
139#95—972no—
140C24—972no—
141C24C22×D9712yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—1944no—
145C2×D4—1948no—
146C2×D4—1948no—
147C2×D4—1948no—
148C2×D4—1942no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—974no—
152C22×C4—972no—
153C22×C4—1948no—
154C22×C4—1944no—
155C22×C4C22×D9712yes—
156C22×C4C22×D9711yescharacteristic
157C22×C4C22×D9711yescharacteristic
158C22×C4C22×D9711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—1948no—
161C4⋊C4—1942no—
162C22⋊C4—24no—
163C22⋊C4—1948no—
164C22⋊C4—1948no—
165C22⋊C4—1944no—
166C42—978no—
167C42C22⋊D9714yescomplemented
168C23—24no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—974no—
173C23—1944no—
174C23—1942no—
175C23—1942no—
176C23—1944no—
177C23#1309551114yes—
178C23C23×D9711yescharacteristic
179D4—28no—
180D4—1948no—
181D4—19432no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—978no—
188C2×C4—3888no—
189C2×C4—1944no—
190C2×C4—1942no—
191C2×C4—1944no—
192C2×C4—1944no—
193C2×C4D4×D9714yes—
194C2×C4C2×C22⋊D9712yes—
195C2×C4C2×C22⋊D9712yes—
196C2×C4C2×C22⋊D9712yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—3884no—
203C22—3882no—
204C22—3882no—
205C22—1944no—
206C22—1948no—
207C22C2×D4×D9712yes—
208C22C22×C22⋊D9711yescharacteristic
209C22#41×D9712yes—
210C22C2×#1309551112yes—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—3882no—
215C4—1948no—
216C4#1569080614yes—
217C2—44no—
218C2—3884no—
219C2#2026124211yescharacteristic
220C2#2026124414yes—
221C2#2026124611yescharacteristic
222C2C2×#1569080611yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC2×C4×D4×C97C2
hypercenterC2×C4×D4D97
hypocenterC97#2397
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D388C2semidirect
2. C2×C4×C22⋊D97C2semidirect
3. #20260671C2semidirect
4. #20260670C2semidirect
5. #20260644C2semidirect
6. #20260643C2semidirect
7. #20260581C2semidirect
8. C2×C4×D4×C97C2semidirect
43. C4×Dic97D4semidirect
55. C42×C97D4semidirect
112. C97#2397semidirect
115. C2×C4×D4D97semidirect
167. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3882
C2×C4×C22⋊D972
#202606712
#202606702
#202606442
#202606432
#202605812
C2×C4×D4×C972
#239797