Finite Group Databasebeta

← #27140228

Subgroups — #27140228

Order 12416.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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#20261246C211yesmaximal, characteristic
2C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
3#20260473C212yesmaximal, complemented
4#20260470C212yesmaximal, complemented
5C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7#15690793—21no—
8#15690789C411yescharacteristic, complemented
9#15690789C2211yescharacteristic
10#15690784C411yescharacteristic, complemented
11#15690784C2211yescharacteristic
12#15690779—21no—
13C4×C22⋊D97—21no—
14C4×C22⋊D97—21no—
15#15690769C411yescharacteristic, complemented
16C2×C22⋊Dic97—21no—
17C2×C4×Dic97—21no—
18#15690717C2212yes—
19C23×C4×C97C2211yescharacteristic
20C22×C8×C97—21no—
21C22⋊C8×C97—21no—
22C22⋊C8×C97C2212yes—
23#13095511—42no—
24C2×Dic194C2×C411yescharacteristic
25C2×D388C2×C411yescharacteristic
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C22×Dic97—22no—
30C2×C4×D97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35C22⋊Dic97—24no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97—42no—
43C24×C97D411yescharacteristic
44C22×C4×C97—42no—
45C22×C4×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97D411yescharacteristic
50C22×C4×C97C2311yescharacteristic
51C2×C8×C97—22no—
52C2×C8×C97—42no—
53C23×C97—22no—
54C23×C97—22no—
55C23×C97—41no—
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22⋊C411yescharacteristic
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C22⋊D97—42no—
71C22⋊D97—42no—
72C22⋊D97—42no—
73C2×Dic97—24no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—42no—
78C2×Dic97—42no—
79D388—41no—
80C4×D97—42no—
81Dic194—41no—
82C8×C97—42no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—44no—
91C22×C97#10412yes—
92C22×C97C2×C22⋊C411yescharacteristic
93C22×C97#13512yes—
94C2×D97—41no—
95C2×D97—41no—
96C2×D97—41no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97#9912yes—
101Dic97—44no—
102Dic97—41no—
103Dic97—41no—
104Dic97—41no—
105C2×C97—22no—
106C2×C97—22no—
107C2×C97—41no—
108C2×C97C2×#9911yescharacteristic
109C2×C97#41611yescharacteristic
110C2×C97C2×#10411yescharacteristic
111D97—81no—
112#1501—971nomaximal, Sylow
113C97#150111yescharacteristic, complemented, Sylow
114#525—971no—
115C2×C22⋊C8D9711yescharacteristic, complemented
116C4×C22⋊C4—971no—
117#328—972no—
118#327—972no—
119C2×#41—971no—
120C23×C4C2×D9711yescharacteristic
121C22×C8—21no—
122#123—1941no—
123C22⋊Q8—971no—
124C22⋊Q8—971no—
125#121—971no—
126#121—971no—
127#120—1941no—
128C4×D4—1941no—
129C4×D4—1941no—
130#117—971no—
131C2×C22⋊C4—1941no—
132C2×C42—1941no—
133C22⋊C8—21no—
134C22⋊C8C2×D9712yes—
135#95—972no—
136C24D38811yescharacteristic
137#41—3882no—
138C2×Q8—971no—
139C2×D4—971no—
140C2×D4—1941no—
141C2×D4—1941no—
142C2×D4—1941no—
143C22×C4—42no—
144C22×C4—1942no—
145C22×C4—1941no—
146C22×C4C4×D9711yescharacteristic
147C22×C4C4×D9711yescharacteristic
148C22×C4D38811yescharacteristic
149C22×C4C22⋊D9711yescharacteristic
150C22×C4C22⋊D9711yescharacteristic
151C22×C4C22×D9711yescharacteristic
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—971no—
155C4⋊C4—971no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C22⋊C4—1944no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C42—3882no—
165C42—1941no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—1941no—
170C23#1309548811yescharacteristic
171C23#1309548811yescharacteristic
172C23C2×D38811yescharacteristic
173Q8—3881no—
174D4—3881no—
175D4—3882no—
176D4—3882no—
177D4—3882no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—3882no—
185C2×C4—3882no—
186C2×C4—3882no—
187C2×C4—1944no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4—1941no—
191C2×C4#1309548811yescharacteristic
192C2×C4#1309548811yescharacteristic
193C2×C4C2×C4×D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—3881no—
205C22—3881no—
206C22—3881no—
207C22#1569073612yes—
208C22C2×#1309548811yescharacteristic
209C22#1569083212yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3884no—
214C4—3881no—
215C4—3881no—
216C4—3881no—
217C4#1569072412yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—7761no—
222C2C2×#1569072411yescharacteristic
223C2#2026073211yescharacteristic
224C2C2×#1569073611yescharacteristic
subgroupG/H
centerC2×C4#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1501
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic97C2semidirect
3. #20260473C2semidirect
4. #20260470C2semidirect
5. C2×C22⋊C8×C97C2semidirect
6. C2×#13095511C4semidirect
8. #15690789C4semidirect
10. #15690784C4semidirect
15. #15690769C4semidirect
113. C97#1501semidirect
115. C2×C22⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
C4×C22⋊Dic972
#202604732
#202604702
C2×C22⋊C8×C972
#150197