Finite Group Databasebeta

← #27151859

Subgroups — #27151859

Order 12416.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#20261285C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic, complemented
3#20261049C212yesmaximal, complemented
4#20261031C212yesmaximal, complemented
5#20260936C212yesmaximal, complemented
6#20260876C212yesmaximal, complemented
7#20260837C214yesmaximal, complemented
8#432×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—21no—
10C22×Dic194C2211yescharacteristic
11C2×#13095500C2214yes—
12#15690810C2211yescharacteristic
13#15690807C2211yescharacteristic
14#15690798—21no—
15#15690784—21no—
16#15690784C2212yes—
17C4×Dic194—21no—
18C4×Dic194—21no—
19C4×Dic194C2212yes—
20C4×Dic194C2212yes—
21C4×D388—21no—
22C4×D388C2212yes—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490—21no—
25C2×C4×Dic97—21no—
26#15690744C2214yes—
27#15690735C2214yes—
28#15690732C2214yes—
29C2×C42×C97C2211yescharacteristic
30C4⋊C8×C97C2212yes—
31C22⋊C8×C97C2212yes—
32#97×C97C2212yes—
33#13095511—44no—
34C2×Dic194—21no—
35C2×Dic194—24no—
36C2×Dic194—21no—
37C2×Dic194C2312yes—
38C2×D388—21no—
39C2×D388—21no—
40C2×D388C2312yes—
41C2×C22⋊D97—22no—
42C22×Dic97—21no—
43C22×Dic97—21no—
44C2×C4×D97—22no—
45#13095500—28no—
46C4⋊Dic97—21no—
47C4⋊Dic97—21no—
48C4⋊Dic97C2312yes—
49C4⋊Dic97C2312yes—
50#13095490—22no—
51#13095490—22no—
52#13095488—22no—
53#13095488—22no—
54C4×Dic97—22no—
55C4×Dic97—22no—
56C22×C4×C97D411yescharacteristic
57C22×C4×C97D411yescharacteristic
58C22×C4×C97C2311yescharacteristic
59C2×C8×C97C2314yes—
60C42×C97D411yescharacteristic
61C42×C97D411yescharacteristic
62C42×C97C2311yescharacteristic
63C42×C97C2311yescharacteristic
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2411yescharacteristic
75C22⋊D97—44no—
76C2×Dic97—24no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—42no—
81D388—24no—
82D388—42no—
83D388—42no—
84C4×D97—44no—
85Dic194—24no—
86Dic194—22no—
87Dic194—42no—
88Dic194#4114yescomplemented
89C8×C97—24no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97AGL(1,ℤ/8)12yes—
93C22×C97C22×D411yescharacteristic
94C2×D97—42no—
95C2×D97—42no—
96C2×D97—42no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—21no—
101C4×C97#13712yes—
102C4×C97C2×#4112yes—
103Dic97—24no—
104Dic97—42no—
105Dic97—42no—
106C2×C97—22no—
107C2×C97#52511yescharacteristic
108C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C97C2×#13711yescharacteristic
110D97—82no—
111#3112—971nomaximal, Sylow
112C97#311211yescharacteristic, complemented, Sylow
113#532—971no—
114C2×C4×Q8—971no—
115#480—972no—
116#477—972no—
117#460—972no—
118#448—972no—
119#440—974no—
120#432D9711yescharacteristic, complemented
121C2×#41—1941no—
122C22×Q8—971no—
123C2×SD16—974no—
124C4⋊Q8—971no—
125#127—971no—
126#124—1941no—
127#121—972no—
128#121—1941no—
129C4×Q8—972no—
130C4×Q8—972no—
131C4×Q8—1941no—
132C4×Q8—1941no—
133C4×D4—972no—
134C4×D4—1941no—
135C2×C4⋊C4—971no—
136C2×C4⋊C4—1941no—
137C2×C42—1941no—
138C2×C42C2×D9711yescharacteristic
139#107—974no—
140C4⋊C8C2×D9712yes—
141#103—974no—
142#102—974no—
143C22⋊C8C2×D9712yes—
144#97C2×D9712yes—
145#41—3884no—
146C2×Q8—972no—
147C2×Q8—1944no—
148C2×Q8—1941no—
149C2×Q8—1941no—
150C2×D4—972no—
151C2×D4—1942no—
152C2×D4—1941no—
153C2×D4—1941no—
154C22×C4—1941no—
155C22×C4—1942no—
156C22×C4—1941no—
157C22×C4D38811yescharacteristic
158C22×C4D38811yescharacteristic
159C22×C4C22×D9711yescharacteristic
160SD16—1948no—
161C2×C8C22×D9714yes—
162C4⋊C4—972no—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C4⋊C4—1942no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C22⋊C4—1942no—
169C22⋊C4—1942no—
170C42—1942no—
171C42—1942no—
172C42D38811yescharacteristic
173C42D38811yescharacteristic
174C42C22×D9711yescharacteristic
175C42C22×D9711yescharacteristic
176C23—1942no—
177C23C2×D38811yescharacteristic
178Q8—974no—
179Q8—3882no—
180Q8—1944no—
181Q8—1942no—
182D4—3882no—
183D4—3884no—
184D4—3882no—
185D4—1944no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—3884no—
190C2×C4—3882no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4—1944no—
194C2×C4—1942no—
195C2×C4C2×D38811yescharacteristic
196C2×C4C2×D38811yescharacteristic
197C2×C4C2×D38811yescharacteristic
198C2×C4C2×D38811yescharacteristic
199C2×C4C2×D38811yescharacteristic
200C2×C4C23×D9711yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—3882no—
205C22—3882no—
206C22—3882no—
207C22#1569083812yes—
208C22C22×D38811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3882no—
214C4—3882no—
215C4—1944no—
216C4#1569084312yes—
217C4#41×D9712yes—
218C2—22no—
219C2—7762no—
220C2#2026124711yescharacteristic
221C2C2×#1569083811yescharacteristic
222C2C2×#1569084311yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#432×C97C2
socleC22×C97C22×D4
prime radical#20261285C2
hypercenter#432D97
hypocenterC97#3112
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic194C2semidirect
3. #20261049C2semidirect
4. #20261031C2semidirect
5. #20260936C2semidirect
6. #20260876C2semidirect
7. #20260837C2semidirect
8. #432×C97C2semidirect
88. Dic194#41semidirect
112. C97#3112semidirect
120. #432D97semidirect

Maximal subgroups

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

subgroupindex
#202612852
C2×C4×Dic1942
#202610492
#202610312
#202609362
#202608762
#202608372
#432×C972
#311297