Finite Group Databasebeta

← #27140915

Subgroups — #27140915

Order 12416.

308 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261081C211yesmaximal, characteristic, complemented
2#20261015C211yesmaximal, characteristic, complemented
3#20260940C211yesmaximal, characteristic, complemented
4#20260479C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7D42×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic, complemented
10#15690743—21no—
11C4⋊#11715895C2211yescharacteristic, complemented
12C4⋊#11715895C2211yescharacteristic, complemented
13#15690733—21no—
14#15690731—21no—
15#15690731—22no—
16C4×#11715895C2211yescharacteristic, complemented
17C22×D4×C97—21no—
18C22×D4×C97—21no—
19#127×C97C2211yescharacteristic
20#121×C97—21no—
21#121×C97—21no—
22#120×C97—22no—
23C4×D4×C97C2211yescharacteristic
24C4×D4×C97C2211yescharacteristic
25C2×Dic194—21no—
26#13095499—42no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C24×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97—22no—
35C2×D4×C97—21no—
36C2×D4×C97—21no—
37C2×D4×C97—42no—
38C2×D4×C97—42no—
39C2×D4×C97—42no—
40C2×D4×C97—42no—
41C2×D4×C97D411yescharacteristic
42C2×D4×C97D411yescharacteristic
43C2×D4×C97D412yes—
44C22×C4×C97—21no—
45C22×C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—22no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C42×C97C2311yescharacteristic
52C23×C97—21no—
53C23×C97—21no—
54C23×C97—22no—
55C23×C97—44no—
56C23×C97—44no—
57C23×C97—82no—
58D4×C97—21no—
59D4×C97—21no—
60D4×C97—42no—
61D4×C97—44no—
62D4×C97—44no—
63D4×C97—41no—
64D4×C97—41no—
65D4×C97—41no—
66D4×C97—41no—
67D4×C97D812yes—
68D4×C97SD1612yescomplemented
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×Dic97—41no—
79Dic194—41no—
80Dic194—41no—
81#11715895—41no—
82#11715895—41no—
83#11715895—41no—
84C22×C97—22no—
85C22×C97—22no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97—42no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—88no—
94C22×C97#12011yescharacteristic
95C4×C97—21no—
96C4×C97—41no—
97C4×C97—41no—
98C4×C97C2×D811yescharacteristic
99C4×C97C2×SD1611yescharacteristic
100C4×C97AGL(1,ℤ/8)11yescharacteristic
101C4×C97AGL(1,ℤ/8)11yescharacteristic
102Dic97—81no—
103C2×C97—42no—
104C2×C97—42no—
105C2×C97—82no—
106C2×C97#44711yescharacteristic
107C2×C97#45011yescharacteristic
108C2×C97#45511yescharacteristic
109#1639—971nomaximal, Sylow
110C97#163911yescharacteristic, complemented, Sylow
111D42D9711yescharacteristic
112#486—971no—
113#474—971no—
114#461—971no—
115#331—971no—
116D4⋊C8—971no—
117D4⋊C8—971no—
118C22×D4—21no—
119C22×D4—21no—
120C2×SD16—1941no—
121C4⋊Q8—971no—
122#127C2×D9711yescharacteristic
123#121—21no—
124#121—21no—
125#120—22no—
126C4×D4C2×D9711yescharacteristic
127C4×D4C2×D9711yescharacteristic
128#107—1941no—
129C4⋊C8—971no—
130C4⋊C8—971no—
131#103—1941no—
132#102—1941no—
133#102—1942no—
134C4×C8—971no—
135C24—22no—
136C2×Q8—1941no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C22⋊D9711yescharacteristic
146C2×D4C22⋊D9711yescharacteristic
147C2×D4C22⋊D9712yes—
148C22×C4—21no—
149C22×C4—21no—
150SD16—3882no—
151C2×C8—1941no—
152C2×C8—1941no—
153C2×C8—1941no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4C22⋊D9711yescharacteristic
157C4⋊C4C22⋊D9711yescharacteristic
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C42C22×D9711yescharacteristic
162C23—21no—
163C23—21no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—82no—
168Q8—3881no—
169Q8—3881no—
170D4—21no—
171D4—21no—
172D4—42no—
173D4—44no—
174D4—44no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4#1309549612yes—
180D4#1309549912yescomplemented
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—3881no—
188C2×C4C2×C22⋊D9711yescharacteristic
189C2×C4C2×C22⋊D9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C8—3881no—
192C8—3881no—
193C8—3881no—
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—88no—
204C22#1569077911yescharacteristic
205C4—21no—
206C4—41no—
207C4—41no—
208C4—7761no—
209C4C2×#1309549611yescharacteristic
210C4C2×#1309549911yescharacteristic
211C4#1569083611yescharacteristic
212C4#1569083611yescharacteristic
213C2—42no—
214C2—42no—
215C2—82no—
216C2#2026086911yescharacteristic
217C2#2026088611yescharacteristic
218C2#2026091311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radicalD42×C97C2
socleC22×C97#120
hypercenterD42D97
hypocenterC97#1639
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20261081C2semidirect
2. #20261015C2semidirect
3. #20260940C2semidirect
4. #20260479C2semidirect
5. D4⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
9. #15690810C22semidirect
11. C4⋊#11715895C22semidirect
12. C4⋊#11715895C22semidirect
16. C4×#11715895C22semidirect
68. D4×C97SD16semidirect
110. C97#1639semidirect
180. D4#13095499semidirect

Maximal subgroups

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

subgroupindex
#202610812
#202610152
#202609402
#202604792
D4⋊#117158952
D42×C972
#163997