Finite Group Databasebeta

← #27143382

Subgroups — #27143382

Order 12416.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#15690727C211yesmaximal, characteristic, complemented
2C2×#15690725C211yesmaximal, characteristic, complemented
3#20260525C212yesmaximal, complemented
4#20260522C212yesmaximal, complemented
5#534×C97C211yesmaximal, characteristic
6C2×#13095495—21no—
7C2×C22⋊Dic97—21no—
8#15690727—21no—
9#15690727C2212yes—
10#15690725—21no—
11#15690725C2212yes—
12C22×D4×C97C411yescharacteristic, complemented
13C22×D4×C97C2211yescharacteristic
14#123×C97—21no—
15#123×C97C411yescharacteristic, complemented
16#123×C97C2211yescharacteristic
17#121×C97—21no—
18#121×C97—21no—
19#120×C97—21no—
20#120×C97C411yescharacteristic, complemented
21#120×C97C2211yescharacteristic
22C2×C22⋊C4×C97C411yescharacteristic, complemented
23C22×Dic97—41no—
24#13095495—22no—
25#13095495—41no—
26C2×#11715895—41no—
27C22⋊Dic97—22no—
28C22⋊Dic97—41no—
29C24×C97C2×C411yescharacteristic
30C24×C97C2×C411yescharacteristic, complemented
31C24×C97D411yescharacteristic
32C2×D4×C97—21no—
33C2×D4×C97—21no—
34C2×D4×C97—21no—
35C2×D4×C97—21no—
36C2×D4×C97—41no—
37C2×D4×C97—41no—
38C2×D4×C97—41no—
39C2×D4×C97—41no—
40C2×D4×C97C2×C411yescharacteristic
41C2×D4×C97D411yescharacteristic
42C2×D4×C97D411yescharacteristic
43C2×D4×C97C2311yescharacteristic
44C22×C4×C97—21no—
45C22×C4×C97C2×C411yescharacteristic, complemented
46C22×C4×C97D411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C22⋊C4×C97C2×C411yescharacteristic, complemented
55C22⋊C4×C97C2×C411yescharacteristic, complemented
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97—21no—
61C23×C97—41no—
62C23×C97—81no—
63C23×C97—81no—
64C23×C97—81no—
65C23×C97C22⋊C411yescharacteristic
66C23×C97C22⋊C411yescharacteristic
67C23×C97C22⋊C411yescharacteristic
68C23×C97C22×C411yescharacteristic
69C23×C97C2×D411yescharacteristic
70D4×C97—41no—
71D4×C97—41no—
72D4×C97—41no—
73D4×C97—41no—
74D4×C97—42no—
75D4×C97—41no—
76D4×C97—41no—
77D4×C97—81no—
78D4×C97—81no—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—21no—
82C2×C4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—41no—
85C2×C4×C97—81no—
86C2×C4×C97C22⋊C411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×Dic97—42no—
89C2×Dic97—82no—
90#11715895—42no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97—42no—
101C22×C97—41no—
102C22×C97—41no—
103C22×C97—41no—
104C22×C97—81no—
105C22×C97—81no—
106C22×C97—81no—
107C22×C97—82no—
108C22×C97#9911yescharacteristic
109C22×C97#9911yescharacteristic
110C22×C97C2×C22⋊C411yescharacteristic
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—41no—
114C4×C97—41no—
115C4×C97—41no—
116Dic97—82no—
117C2×C97—21no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—41no—
121C2×C97—41no—
122C2×C97—41no—
123C2×C97—41no—
124C2×C97—81no—
125C2×C97C2×#9911yescharacteristic
126#2121—971nomaximal, Sylow
127C97#212111yescharacteristic, complemented, Sylow
128#534D9711yescharacteristic
129C2×#100—971no—
130C2×#99—971no—
131#352—972no—
132AΣL(1,𝔽16)—972no—
133C22×D4Dic9711yescharacteristic, complemented
134C22×D4C2×D9711yescharacteristic
135C2×M16—1941no—
136#123—21no—
137#123Dic9711yescharacteristic, complemented
138#123C2×D9711yescharacteristic
139#121—21no—
140#121—21no—
141#120—21no—
142#120Dic9711yescharacteristic, complemented
143#120C2×D9711yescharacteristic
144C2×C22⋊C4—1941no—
145C2×C22⋊C4Dic9711yescharacteristic, complemented
146#100—972no—
147#100—1941no—
148#99—972no—
149#99—1941no—
150C24C2×Dic9711yescharacteristic
151C24C2×Dic9711yescharacteristic, complemented
152C24C22⋊D9711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×Dic9711yescharacteristic
162C2×D4C22⋊D9711yescharacteristic
163C2×D4C22⋊D9711yescharacteristic
164C2×D4C22×D9711yescharacteristic
165C22×C4—21no—
166C22×C4—3881no—
167C22×C4C2×Dic9711yescharacteristic, complemented
168C22×C4C22⋊D9711yescharacteristic
169M16—3881no—
170M16—1942no—
171C2×C8—3881no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—3881no—
180C22⋊C4—1942no—
181C22⋊C4C2×Dic9711yescharacteristic, complemented
182C22⋊C4C2×Dic9711yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23C22⋊Dic9711yescharacteristic
193C23C22⋊Dic9711yescharacteristic
194C23C22⋊Dic9711yescharacteristic
195C23C22×Dic9711yescharacteristic
196C23C2×C22⋊D9711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—7762no—
214C2×C4—3882no—
215C2×C4C22⋊Dic9711yescharacteristic
216C2×C4C2×C22⋊D9711yescharacteristic
217C8—3882no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—82no—
235C22#1569072511yescharacteristic
236C22#1569072511yescharacteristic
237C22C2×C22⋊Dic9711yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—7762no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2C2×#1569072511yescharacteristic
subgroupG/H
centerC2C2×#15690725
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting, prime radical#534×C97C2
socleC2×C97C2×#99
cosocleC2×D4×C97C23
hypercenter#534D97
hypocenterC97#2121
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#15690727C2semidirect
2. C2×#15690725C2semidirect
3. #20260525C2semidirect
4. #20260522C2semidirect
12. C22×D4×C97C4semidirect
15. #123×C97C4semidirect
20. #120×C97C4semidirect
22. C2×C22⋊C4×C97C4semidirect
30. C24×C97C2×C4semidirect
45. C22×C4×C97C2×C4semidirect
54. C22⋊C4×C97C2×C4semidirect
55. C22⋊C4×C97C2×C4semidirect
127. C97#2121semidirect
133. C22×D4Dic97semidirect
137. #123Dic97semidirect
142. #120Dic97semidirect
145. C2×C22⋊C4Dic97semidirect
151. C24C2×Dic97semidirect
167. C22×C4C2×Dic97semidirect
181. C22⋊C4C2×Dic97semidirect
182. C22⋊C4C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907272
C2×#156907252
#202605252
#202605222
#534×C972
#212197