Finite Group Databasebeta

← #27143403

Subgroups — #27143403

Order 12416.

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

normal subgroups only

This group has 286 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#20261302C211yesmaximal, characteristic
2C2×#15690725C211yesmaximal, characteristic, complemented
3#20260529C212yesmaximal, complemented
4#20260527C212yesmaximal, complemented
5C2×#99×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7#15690806C411yescharacteristic, complemented
8#15690806C2211yescharacteristic
9#15690797C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690781—21no—
12#15690781—21no—
13#15690779—21no—
14#15690779—21no—
15#15690769C411yescharacteristic, complemented
16C2×C22⋊Dic97—21no—
17#15690725—21no—
18#15690725C2212yes—
19C22×D4×C97C2211yescharacteristic
20C2×C22⋊C4×C97—21no—
21#99×C97—21no—
22#99×C97C2212yes—
23#13095511—41no—
24#13095511—41no—
25C2×Dic194C2×C411yescharacteristic, complemented
26C2×D388C2×C411yescharacteristic, complemented
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C22×Dic97—41no—
33C2×C4×D97—21no—
34C4⋊Dic97—21no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97—22no—
41C22⋊Dic97—41no—
42C4×Dic97C2×C411yescharacteristic, complemented
43C4×Dic97C2×C411yescharacteristic, complemented
44C24×C97D411yescharacteristic
45C24×C97D411yescharacteristic
46C2×D4×C97—41no—
47C2×D4×C97—41no—
48C2×D4×C97C2×C411yescharacteristic
49C2×D4×C97D411yescharacteristic
50C2×D4×C97D411yescharacteristic
51C2×D4×C97C2311yescharacteristic
52C22×C4×C97—41no—
53C22×C4×C97C2×C411yescharacteristic
54C22⋊C4×C97—22no—
55C22⋊C4×C97—41no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97—81no—
61C23×C97—81no—
62C23×C97C22⋊C411yescharacteristic
63C23×C97C22⋊C411yescharacteristic
64C23×C97C22⋊C411yescharacteristic
65C23×C97C2×D411yescharacteristic
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68D4×C97—41no—
69D4×C97—41no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—42no—
75C2×C4×C97—82no—
76C2×C4×C97C22⋊C411yescharacteristic
77C2×C4×C97C22×C411yescharacteristic
78C22⋊D97—41no—
79C22⋊D97—41no—
80C22⋊D97—81no—
81C22⋊D97—81no—
82C22⋊D97—81no—
83C22⋊D97—81no—
84C2×Dic97—21no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—42no—
88C2×Dic97—82no—
89D388—41no—
90C4×D97—42no—
91Dic194—41no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—21no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97—41no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97—81no—
104C22×C97—81no—
105C22×C97#9911yescharacteristic
106C22×C97#9911yescharacteristic
107C22×C97C2×C22⋊C411yescharacteristic
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—41no—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—82no—
114Dic97—41no—
115Dic97—41no—
116Dic97—41no—
117Dic97—82no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—41no—
122C2×C97—41no—
123C2×C97—41no—
124C2×C97—41no—
125C2×C97C2×#9911yescharacteristic
126D97—81no—
127#2127—971nomaximal, Sylow
128C97#212711yescharacteristic, complemented, Sylow
129#535—971no—
130C2×#99—971no—
131C2×#99D9711yescharacteristic, complemented
132#354—972no—
133#353—972no—
134C2×#41—971no—
135C22×D4C2×D9711yescharacteristic
136#127—971no—
137#127—971no—
138#124—971no—
139#124—971no—
140#121—1941no—
141#121—1941no—
142#120—1941no—
143#120—1941no—
144#117—971no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—1941no—
147#99—21no—
148#99—972no—
149#99—1941no—
150#99C2×D9712yes—
151C24D38811yescharacteristic
152C24C22⋊D9711yescharacteristic
153#41—3881no—
154#41—3881no—
155C2×Q8—971no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—971no—
159C2×D4—1941no—
160C2×D4—1941no—
161C2×D4—1941no—
162C2×D4—1941no—
163C2×D4—1941no—
164C2×D4C4×D9711yescharacteristic
165C2×D4D38811yescharacteristic
166C2×D4C22⋊D9711yescharacteristic
167C2×D4C22×D9711yescharacteristic
168C22×C4—41no—
169C22×C4—3881no—
170C22×C4—1941no—
171C22×C4C4×D9711yescharacteristic
172C4⋊C4—1941no—
173C22⋊C4—22no—
174C22⋊C4—41no—
175C22⋊C4—3881no—
176C22⋊C4—1942no—
177C22⋊C4—1941no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4—1941no—
181C22⋊C4—1941no—
182C42—971no—
183C42—971no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—81no—
189C23—81no—
190C23—1941no—
191C23#1309548811yescharacteristic
192C23#1309548811yescharacteristic
193C23#1309548811yescharacteristic
194C23C2×C22⋊D9711yescharacteristic
195C23C2×D38811yescharacteristic
196Q8—3881no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—7761no—
202D4—7761no—
203D4—7761no—
204D4—7761no—
205D4—3881no—
206D4—3881no—
207D4—3881no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—7762no—
213C2×C4—3882no—
214C2×C4—3882no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1941no—
218C2×C4#1309548811yescharacteristic
219C2×C4C2×C4×D9711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—3881no—
234C22—3881no—
235C22—3881no—
236C22#1569072411yescharacteristic
237C22#1569072411yescharacteristic
238C22C2×#1309548811yescharacteristic
239C4—21no—
240C4—21no—
241C4—82no—
242C4—7762no—
243C4—3881no—
244C4—3881no—
245C4—3881no—
246C2—21no—
247C2—21no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—41no—
252C2—41no—
253C2—7761no—
254C2C2×#1569072411yescharacteristic
subgroupG/H
centerC2C2×#15690724
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
FittingC2×#99×C97C2
socleC2×C97C2×#99
prime radical#20261302C2
cosocleC2×D4×C97C23
hypercenterC2×#99D97
hypocenterC97#2127
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#15690725C2semidirect
3. #20260529C2semidirect
4. #20260527C2semidirect
5. C2×#99×C97C2semidirect
6. C2×#13095511C4semidirect
7. #15690806C4semidirect
9. #15690797C4semidirect
15. #15690769C4semidirect
25. C2×Dic194C2×C4semidirect
26. C2×D388C2×C4semidirect
42. C4×Dic97C2×C4semidirect
43. C4×Dic97C2×C4semidirect
128. C97#2127semidirect
131. C2×#99D97semidirect

Maximal subgroups

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

subgroupindex
#202613022
C2×#156907252
#202605292
#202605272
C2×#99×C972
#212797