Finite Group Databasebeta

← #27140281

Subgroups — #27140281

Order 12416.

252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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#20261280C211yesmaximal, characteristic
2#20260789C211yesmaximal, characteristic, complemented
3#20260602C211yesmaximal, characteristic, complemented
4#20260470C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260468C211yesmaximal, characteristic, complemented
7#327×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9#15690809C411yescharacteristic, complemented
10#15690806C411yescharacteristic, complemented
11#15690796—21no—
12#15690784C2211yescharacteristic
13#15690782—21no—
14D4×Dic97C411yescharacteristic, complemented
15#15690772C411yescharacteristic, complemented
16#15690770—21no—
17C2×C4×Dic97C2211yescharacteristic
18#15690739—21no—
19C22⋊#11715895C2211yescharacteristic
20#15690720—21no—
21#15690717—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24#121×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27#13095510—42no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C22×Dic97—21no—
35C22×Dic97—21no—
36C22×Dic97D411yescharacteristic
37C22×Dic97D411yescharacteristic
38C2×C4×D97—21no—
39C2×#11715895—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×D4×C97—21no—
50C2×D4×C97C2×C411yescharacteristic
51C22×C4×C97—21no—
52C22×C4×C97C2311yescharacteristic
53C2×C8×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C23×C97—21no—
57C23×C97C2×D411yescharacteristic
58C22×D97—21no—
59D4×C97—41no—
60D4×C97—41no—
61D4×C97—41no—
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C22×C411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C22⋊D97—42no—
70C22⋊D97—42no—
71C22⋊D97—41no—
72C22⋊D97—41no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—41no—
77C2×Dic97—42no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97C22⋊C411yescharacteristic
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C22⋊C411yescharacteristic
84C2×Dic97C22⋊C411yescharacteristic
85D388—41no—
86C4×D97—42no—
87Dic194—42no—
88C8×C97—41no—
89#11715895—41no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97AGL(1,ℤ/8)11yescharacteristic
97C22×C97AGL(1,ℤ/8)11yescharacteristic
98C2×D97—41no—
99C2×D97—41no—
100C2×D97—41no—
101C4×C97—21no—
102C4×C97—21no—
103C4×C97—41no—
104C4×C97—81no—
105Dic97—21no—
106Dic97—21no—
107Dic97—21no—
108Dic97—21no—
109Dic97—41no—
110Dic97—81no—
111C2×C97—21no—
112C2×C97—21no—
113C2×C97—81no—
114C2×C97#41011yescharacteristic
115C2×C97#41811yescharacteristic
116C2×C97#42111yescharacteristic
117D97—81no—
118#1514—971nomaximal, Sylow
119C97#151411yescharacteristic, complemented, Sylow
120#532—971no—
121#432—971no—
122#383—971no—
123#327—971no—
124#327—971no—
125#327—971no—
126#327D9711yescharacteristic, complemented
127C2×#41—1941no—
128C4⋊Q8—971no—
129#127—971no—
130#124—1941no—
131#121—971no—
132#121—1941no—
133#121C2×D9711yescharacteristic
134C4×D4—971no—
135C4×D4—971no—
136C4×D4—1941no—
137C2×C42—971no—
138C4⋊C8—1941no—
139C22⋊C8—971no—
140C22⋊C8C2×D9711yescharacteristic
141#97—1941no—
142#95—971no—
143#95—1941no—
144#95—1941no—
145#95C2×D9711yescharacteristic
146#41—3882no—
147#41—3882no—
148C2×Q8—1941no—
149C2×D4—21no—
150C2×D4—971no—
151C2×D4—1941no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4C4×D9711yescharacteristic
155C22×C4—21no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—1941no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4C22×D9711yescharacteristic
162C2×C8—21no—
163C2×C8—1941no—
164C4⋊C4—971no—
165C4⋊C4—1941no—
166C4⋊C4C4×D9711yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C42—971no—
172C42—971no—
173C42—971no—
174C42—971no—
175C23—21no—
176C23—1941no—
177C23D4×D9711yescharacteristic
178Q8—3882no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—3881no—
183D4—3881no—
184D4—3881no—
185D4—3882no—
186D4—3882no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—971no—
193C2×C4—971no—
194C2×C4—971no—
195C2×C4—971no—
196C2×C4—3882no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—3881no—
200C2×C4—3882no—
201C2×C4—3881no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4C2×C4×D9711yescharacteristic
206C2×C4D4×D9711yescharacteristic
207C8—41no—
208C8—3881no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—3881no—
215C22—3881no—
216C22—3881no—
217C22C22⋊C4×D9711yescharacteristic
218C22#1569083411yescharacteristic
219C22#1569083711yescharacteristic
220C4—21no—
221C4—21no—
222C4—41no—
223C4—81no—
224C4—7761no—
225C4—3881no—
226C4—1941no—
227C4—1941no—
228C4—1941no—
229C4—1941no—
230C2—21no—
231C2—21no—
232C2—81no—
233C2—7761no—
234C2#2026070711yescharacteristic
235C2#2026073811yescharacteristic
236C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#327×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261280C2
cosocleC22×C4×C97C23
hypercenter#327D97
hypocenterC97#1514
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260602C2semidirect
4. #20260470C2semidirect
5. #20260469C2semidirect
6. #20260468C2semidirect
7. #327×C97C2semidirect
9. #15690809C4semidirect
10. #15690806C4semidirect
14. D4×Dic97C4semidirect
15. #15690772C4semidirect
119. C97#1514semidirect
126. #327D97semidirect

Maximal subgroups

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

subgroupindex
#202612802
#202607892
#202606022
#202604702
#202604692
#202604682
#327×C972
#151497