Finite Group Databasebeta

← #27140238

Subgroups — #27140238

Order 12416.

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

normal subgroups only

This group has 334 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
2#20260699C211yesmaximal, characteristic, complemented
3#20260471C212yesmaximal, complemented
4#20260468C212yesmaximal, complemented
5C2×#95×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7C2×#13095495—21no—
8#15690793—21no—
9#15690789C411yescharacteristic, complemented
10#15690789C2211yescharacteristic
11#15690784C411yescharacteristic, complemented
12#15690784C2211yescharacteristic
13#15690779—21no—
14C4×C22⋊D97—21no—
15C4×C22⋊D97—21no—
16#15690769C411yescharacteristic, complemented
17C22⋊#11715895—21no—
18C22⋊#11715895C2212yes—
19C23×C4×C97—21no—
20C23×C4×C97C2211yescharacteristic
21#95×C97—21no—
22#95×C97C2212yes—
23#13095511—42no—
24C2×Dic194C2×C411yescharacteristic, complemented
25C2×D388C2×C411yescharacteristic, complemented
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C2×C4×D97—21no—
30#13095495—42no—
31C2×#11715895—22no—
32C4⋊Dic97C2×C411yescharacteristic, complemented
33C4⋊Dic97C2×C411yescharacteristic, complemented
34#13095490—21no—
35#13095490—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C24×C97D411yescharacteristic
43C22×C4×C97—24no—
44C22×C4×C97—22no—
45C22×C4×C97—42no—
46C22×C4×C97—42no—
47C22×C4×C97—42no—
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C22×C4×C97D411yescharacteristic
51C22×C4×C97D411yescharacteristic
52C22×C4×C97D411yescharacteristic
53C22×C4×C97C2311yescharacteristic
54C23×C97—22no—
55C23×C97—22no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97C22⋊C411yescharacteristic
59C23×C97C22⋊C411yescharacteristic
60C23×C97C2×D411yescharacteristic
61C22×D97—21no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—24no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—48no—
70C2×C4×C97—42no—
71C2×C4×C97—42no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22×C411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C22⋊D97—42no—
77C22⋊D97—42no—
78C22⋊D97—41no—
79C22⋊D97—41no—
80C2×Dic97—21no—
81C2×Dic97—21no—
82C2×Dic97—21no—
83D388—41no—
84C4×D97—42no—
85Dic194—41no—
86#11715895—42no—
87C22×C97—24no—
88C22×C97—24no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97#9912yes—
96C22×C97#10412yes—
97C22×C97C2×C22⋊C411yescharacteristic
98C22×C97AGL(1,ℤ/8)11yescharacteristic
99C22×C97#13711yescharacteristic
100C2×D97—41no—
101C2×D97—41no—
102C2×D97—41no—
103C4×C97—21no—
104C4×C97—21no—
105C4×C97—21no—
106C4×C97—21no—
107C4×C97—44no—
108Dic97—41no—
109Dic97—41no—
110Dic97—41no—
111C2×C97—21no—
112C2×C97—22no—
113C2×C97—22no—
114C2×C97—21no—
115C2×C97C2×#9911yescharacteristic
116C2×C97#41711yescharacteristic
117C2×C97C2×#10411yescharacteristic
118D97—81no—
119#1504—971nomaximal, Sylow
120C97#150411yescharacteristic, complemented, Sylow
121#525—971no—
122#407—971no—
123C2×#95D9711yescharacteristic, complemented
124#328—972no—
125#327—972no—
126C2×#41—971no—
127C23×C4—21no—
128C23×C4C2×D9711yescharacteristic
129C2×M16—1941no—
130#123—1941no—
131C22⋊Q8—971no—
132C22⋊Q8—971no—
133#121—971no—
134#121—971no—
135#120—1941no—
136C4×D4—1941no—
137C4×D4—1941no—
138#117—971no—
139C22⋊C8—972no—
140C22⋊C8—1941no—
141#95—21no—
142#95C2×D9712yes—
143C24C22⋊D9711yescharacteristic
144#41—3882no—
145C2×Q8—971no—
146C2×D4—971no—
147C2×D4—1941no—
148C2×D4—1941no—
149C2×D4—1941no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—1941no—
156C22×C4C4×D9711yescharacteristic
157C22×C4C4×D9711yescharacteristic
158C22×C4D38811yescharacteristic
159C22×C4D38811yescharacteristic
160C22×C4C22⋊D9711yescharacteristic
161C22×C4C22×D9711yescharacteristic
162M16—3882no—
163C2×C8—1942no—
164C4⋊C4—971no—
165C4⋊C4—971no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C22⋊C4—1941no—
172C22⋊C4—1941no—
173C42—1941no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—1941no—
179C23#1309548811yescharacteristic
180C23#1309548811yescharacteristic
181C23C2×C22⋊D9711yescharacteristic
182Q8—3881no—
183D4—3881no—
184D4—3881no—
185D4—3881no—
186D4—3882no—
187D4—3882no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—3882no—
199C2×C4—1941no—
200C2×C4—1941no—
201C2×C4—1941no—
202C2×C4#1309548811yescharacteristic
203C2×C4#1309548811yescharacteristic
204C2×C4C2×C4×D9711yescharacteristic
205C2×C4C2×D38811yescharacteristic
206C8—3882no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—3881no—
216C22—3881no—
217C22—3881no—
218C22#1569072412yes—
219C22#1569073812yes—
220C22C2×#1309548811yescharacteristic
221C22#1569083611yescharacteristic
222C22#1569084111yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—3881no—
229C4—3881no—
230C4—3881no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—7761no—
236C2C2×#1569072411yescharacteristic
237C2#2026073411yescharacteristic
238C2C2×#1569073811yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#95×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenterC2×#95D97
hypocenterC97#1504
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #20260699C2semidirect
3. #20260471C2semidirect
4. #20260468C2semidirect
5. C2×#95×C97C2semidirect
6. C2×#13095511C4semidirect
9. #15690789C4semidirect
11. #15690784C4semidirect
16. #15690769C4semidirect
24. C2×Dic194C2×C4semidirect
25. C2×D388C2×C4semidirect
32. C4⋊Dic97C2×C4semidirect
33. C4⋊Dic97C2×C4semidirect
120. C97#1504semidirect
123. C2×#95D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202606992
#202604712
#202604682
C2×#95×C972
#150497