Finite Group Databasebeta

← #27151854

Subgroups — #27151854

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
1C2×C4×Dic194C211yesmaximal, characteristic, complemented
2#20261046C212yesmaximal, complemented
3#20261027C212yesmaximal, complemented
4#20260932C212yesmaximal, complemented
5#20260873C212yesmaximal, complemented
6#20260835C214yesmaximal, complemented
7#20260792C211yesmaximal, characteristic, complemented
8#532×C97C211yesmaximal, characteristic
9C22×Dic194C2211yescharacteristic
10C2×#13095499C2214yes—
11C4×Dic194—21no—
12C4×Dic194—21no—
13C4×Dic194C2212yes—
14C4×Dic194C2212yes—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C2×C4×Dic97—21no—
18#15690743C2214yes—
19C4⋊#11715895C2212yes—
20#15690733C2214yes—
21#15690731C2214yes—
22C22⋊#11715895C2212yes—
23#15690721C2212yes—
24C2×#41×C97—21no—
25C4⋊Q8×C97C2211yescharacteristic
26#127×C97C2211yescharacteristic
27#124×C97—21no—
28#121×C97—21no—
29#121×C97C2212yes—
30C4×D4×C97—21no—
31C4×D4×C97C2212yes—
32C2×C42×C97C2211yescharacteristic
33C2×Dic194—21no—
34C2×Dic194—24no—
35C2×Dic194C2312yes—
36C22×Dic97—21no—
37C22×Dic97—21no—
38#13095499—28no—
39C2×#11715895C2314yes—
40C4⋊Dic97—21no—
41C4⋊Dic97C2312yes—
42#13095490—22no—
43#13095490—22no—
44C4×Dic97—22no—
45C4×Dic97—22no—
46#41×C97—44no—
47C2×Q8×C97—21no—
48C2×D4×C97—22no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97C2312yes—
52C22×C4×C97—22no—
53C22×C4×C97D411yescharacteristic
54C22×C4×C97D411yescharacteristic
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2312yes—
58C22⋊C4×C97—22no—
59C22⋊C4×C97—22no—
60C42×C97D411yescharacteristic
61C42×C97D411yescharacteristic
62C42×C97C2311yescharacteristic
63C42×C97C2311yescharacteristic
64C23×C97—22no—
65C23×C97C2×D411yescharacteristic
66Q8×C97—42no—
67D4×C97—24no—
68D4×C97—42no—
69D4×C97—44no—
70D4×C97—42no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—44no—
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2411yescharacteristic
82C2×Dic97—24no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—42no—
86Dic194—24no—
87Dic194—22no—
88Dic194#4114yescomplemented
89#11715895—24no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97AGL(1,ℤ/8)12yes—
96C22×C97C22×D411yescharacteristic
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—42no—
102C4×C97#13712yes—
103C4×C97C2×#4112yes—
104Dic97—24no—
105Dic97—42no—
106C2×C97—22no—
107C2×C97—82no—
108C2×C97#52511yescharacteristic
109C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C97C2×#13711yescharacteristic
111#3112—971nomaximal, Sylow
112C97#311211yescharacteristic, complemented, Sylow
113#532D9711yescharacteristic
114C2×C4×Q8—971no—
115#480—972no—
116#477—972no—
117#460—972no—
118#448—972no—
119#440—974no—
120#432—971no—
121C2×#41—21no—
122C22×Q8—971no—
123C2×SD16—974no—
124C4⋊Q8C2×D9711yescharacteristic
125#127C2×D9711yescharacteristic
126#124—21no—
127#121—21no—
128#121C2×D9712yes—
129C4×Q8—972no—
130C4×Q8—972no—
131C4×Q8—1941no—
132C4×Q8—1941no—
133C4×D4—21no—
134C4×D4C2×D9712yes—
135C2×C4⋊C4—971no—
136C2×C4⋊C4—1941no—
137C2×C42—1941no—
138C2×C42C2×D9711yescharacteristic
139#107—974no—
140C4⋊C8—972no—
141#103—974no—
142#102—974no—
143C22⋊C8—972no—
144#97—972no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—972no—
148C2×Q8—1944no—
149C2×Q8—1941no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D9712yes—
154C22×C4—22no—
155C22×C4—1941no—
156C22×C4—1941no—
157C22×C4C22⋊D9711yescharacteristic
158C22×C4C22⋊D9711yescharacteristic
159C22×C4C22×D9711yescharacteristic
160SD16—1948no—
161C2×C8—974no—
162C4⋊C4—21no—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C4⋊C4—1942no—
166C4⋊C4—1941no—
167C4⋊C4C22×D9712yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—1942no—
171C42—1942no—
172C42C22⋊D9711yescharacteristic
173C42C22⋊D9711yescharacteristic
174C42C22×D9711yescharacteristic
175C42C22×D9711yescharacteristic
176C23—22no—
177C23C2×C22⋊D9711yescharacteristic
178Q8—42no—
179Q8—974no—
180Q8—1944no—
181Q8—1942no—
182D4—24no—
183D4—42no—
184D4—44no—
185D4—42no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—3882no—
192C2×C4—1942no—
193C2×C4—1944no—
194C2×C4—1942no—
195C2×C4C2×C22⋊D9711yescharacteristic
196C2×C4C2×C22⋊D9711yescharacteristic
197C2×C4C2×C22⋊D9711yescharacteristic
198C2×C4C2×C22⋊D9711yescharacteristic
199C2×C4C2×C22⋊D9711yescharacteristic
200C2×C4C23×D9711yescharacteristic
201C8—1944no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22#1569083612yes—
208C22C22×C22⋊D9711yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—3882no—
215C4—1944no—
216C4#1569084012yes—
217C4#41×D9712yes—
218C2—22no—
219C2—82no—
220C2#2026124211yescharacteristic
221C2C2×#1569083611yescharacteristic
222C2C2×#1569084011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#532×C97C2
socleC22×C97C22×D4
hypercenter#532D97
hypocenterC97#3112
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic194C2semidirect
2. #20261046C2semidirect
3. #20261027C2semidirect
4. #20260932C2semidirect
5. #20260873C2semidirect
6. #20260835C2semidirect
7. #20260792C2semidirect
88. Dic194#41semidirect
112. C97#3112semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic1942
#202610462
#202610272
#202609322
#202608732
#202608352
#202607922
#532×C972
#311297