Finite Group Databasebeta

← #16846507

Subgroups — #16846507

Order 3968.

384 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 105 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×C4×C22⋊D31C211yesmaximal, characteristic
2C2×Dic31⋊C8C211yesmaximal, characteristic, complemented
3C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
4C2×(C2×D31)⋊C8C211yesmaximal, characteristic, complemented
5C2×(C2×D31)⋊C8C211yesmaximal, characteristic, complemented
6C2×#11758403C211yesmaximal, characteristic, complemented
7C2×C31×C22⋊C8C211yesmaximal, characteristic, complemented
8C23×C4×C31C2211yescharacteristic
9C22×C22⋊D31C411yescharacteristic
10C22×C4×D31C2211yescharacteristic
11C22×C8×C31C2211yescharacteristic
12C22×C8×C31C2211yescharacteristic
13C2×C22⋊Dic31C411yescharacteristic
14C4×C22⋊D31—22no—
15C4×C22⋊D31—22no—
16C2×(C2×D31)⋊C4C411yescharacteristic
17C2×Dic31⋊C4C411yescharacteristic
18C2×C4×Dic31C2211yescharacteristic
19C22×C31⋊C8C2211yescharacteristic
20C22×C31⋊C8C2211yescharacteristic
21C31×C22⋊C8—22no—
22C22⋊(C31⋊C8)—22no—
23(C2×D31)⋊C8—22no—
24(C2×D31)⋊C8—22no—
25#11758403—22no—
26Dic31⋊C8—22no—
27C24×C31C2×C411yescharacteristic
28C23×D31C2×C411yescharacteristic
29C22×C4×C31—21no—
30C22×C4×C31—21no—
31C22×C4×C31—21no—
32C22×C4×C31—21no—
33C22×C4×C31—21no—
34C22×C4×C31—21no—
35C22×C4×C31C2×C411yescharacteristic
36C22×C4×C31C2311yescharacteristic
37C2×C22⋊D31—22no—
38C2×C22⋊D31—22no—
39C2×C22⋊D31—22no—
40C22×Dic31C2×C411yescharacteristic
41C22×Dic31C2×C411yescharacteristic
42C22×Dic31C2×C411yescharacteristic
43C2×C4×D31—21no—
44C2×C4×D31—21no—
45C2×C4×D31—21no—
46C2×C4×D31—21no—
47C2×C4×D31—21no—
48C2×C4×D31—21no—
49C2×C8×C31—21no—
50C2×C8×C31—21no—
51C2×C8×C31—21no—
52C2×C8×C31—21no—
53C2×C8×C31—21no—
54C2×C8×C31D412yescomplemented
55C22⋊Dic31—22no—
56(C2×D31)⋊C4—22no—
57Dic31⋊C4—22no—
58C4×Dic31D412yes—
59C4×Dic31D412yes—
60C2×C31⋊C8—21no—
61C2×C31⋊C8—21no—
62C2×C31⋊C8—21no—
63C2×C31⋊C8—21no—
64C2×C31⋊C8—21no—
65C2×C31⋊C8D412yescomplemented
66C23×C31—21no—
67C23×C31—21no—
68C23×C31—21no—
69C23×C31—21no—
70C23×C31—21no—
71C23×C31—21no—
72C23×C31M1612yes—
73C23×C31C22×C411yescharacteristic
74C22×D31—21no—
75C22×D31—21no—
76C22×D31—21no—
77C22×D31—21no—
78C22×D31—21no—
79C22×D31—21no—
80C22×D31M1612yes—
81C2×C4×C31—21no—
82C2×C4×C31—21no—
83C2×C4×C31—21no—
84C2×C4×C31—41no—
85C2×C4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31—41no—
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31#4111yescharacteristic
93C2×C4×C31#4111yescharacteristic
94C22⋊D31—28no—
95C2×Dic31—21no—
96C2×Dic31—21no—
97C2×Dic31—21no—
98C2×Dic31—21no—
99C2×Dic31—21no—
100C2×Dic31—21no—
101C2×Dic31—21no—
102C2×Dic31C22⋊C412yes—
103C2×Dic31C22⋊C412yes—
104C4×D31—41no—
105C4×D31—41no—
106C4×D31—41no—
107C4×D31—41no—
108C8×C31—22no—
109C8×C31—41no—
110C31⋊C8—22no—
111C31⋊C8—41no—
112#1955—311nomaximal, Sylow
113C22×C31—22no—
114C22×C31—22no—
115C22×C31—22no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31—41no—
120C22×C31C2×C22⋊C411yescharacteristic
121C22×C31C4×D411yescharacteristic
122C22×C31C4×D411yescharacteristic
123C22×C31C2×M1611yescharacteristic
124C22×C31C2×M1611yescharacteristic
125C22×C31#13111yescharacteristic
126C22×C31#13111yescharacteristic
127C2×D31—22no—
128C2×D31—22no—
129C2×D31—22no—
130C2×D31—41no—
131C2×D31—41no—
132C2×D31—41no—
133C2×D31—41no—
134C4×C31—41no—
135C4×C31#12111yescharacteristic
136C4×C31#12111yescharacteristic
137C4×C31#12411yescharacteristic
138C4×C31#12711yescharacteristic
139Dic31—22no—
140Dic31—22no—
141Dic31—41no—
142C2×C4×D4—311no—
143C2×C4⋊C8—311no—
144C2×C22⋊C8—311no—
145C2×C22⋊C8—311no—
146C2×C22⋊C8—311no—
147C2×C22⋊C8D3111yescharacteristic, complemented
148C2×#97—311no—
149C2×C31—42no—
150C2×C31#39011yescharacteristic
151C2×C31#40711yescharacteristic
152C2×C31#40811yescharacteristic
153C2×C31#43511yescharacteristic
154C2×C31#43511yescharacteristic
155C2×C31#43511yescharacteristic
156C2×C31#43511yescharacteristic
157D31—42no—
158C22×D4—311no—
159C23×C4—311no—
160C23×C4C2×D3111yescharacteristic
161C22×C8—311no—
162C22×C8—311no—
163C22×C8C2×D3111yescharacteristic
164C22×C8C2×D3111yescharacteristic
165C4×D4—622no—
166C4×D4—622no—
167C2×C4⋊C4—311no—
168C2×C22⋊C4—311no—
169C2×C22⋊C4—311no—
170C2×C42—311no—
171C4⋊C8—622no—
172C22⋊C8—22no—
173C22⋊C8—622no—
174C22⋊C8—622no—
175C22⋊C8—622no—
176#97—622no—
177C31#195511yescharacteristic, complemented, Sylow
178C24—311no—
179C24C4×D3111yescharacteristic
180C2×D4—622no—
181C2×D4—622no—
182C2×D4—622no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—311no—
190C22×C4—311no—
191C22×C4—311no—
192C22×C4—621no—
193C22×C4—621no—
194C22×C4—621no—
195C22×C4—621no—
196C22×C4—621no—
197C22×C4—621no—
198C22×C4C4×D3111yescharacteristic
199C22×C4C22×D3111yescharacteristic
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—312no—
206C2×C8—621no—
207C2×C8—621no—
208C2×C8—621no—
209C2×C8—621no—
210C2×C8—621no—
211C2×C8C22⋊D3112yescomplemented
212C4⋊C4—622no—
213C22⋊C4—622no—
214C22⋊C4—622no—
215C42—312no—
216C42—312no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—312no—
224C23—621no—
225C23—621no—
226C23—621no—
227C23—621no—
228C23—621no—
229C23—621no—
230C23#9246912yes—
231C23C2×C4×D3111yescharacteristic
232D4—628no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—312no—
241C2×C4—312no—
242C2×C4—621no—
243C2×C4—621no—
244C2×C4—621no—
245C2×C4—621no—
246C2×C4—621no—
247C2×C4—621no—
248C2×C4—621no—
249C2×C4—1241no—
250C2×C4—1241no—
251C2×C4—1241no—
252C2×C4—1241no—
253C2×C4#9249511yescharacteristic
254C2×C4D4×D3111yescharacteristic
255C2×C4D4×D3111yescharacteristic
256C2×C4D4×D3111yescharacteristic
257C2×C4#9249711yescharacteristic
258C2×C4C2×C22⋊D3111yescharacteristic
259C8—22no—
260C8—41no—
261C8—622no—
262C8—1241no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—622no—
271C22—622no—
272C22—622no—
273C22—1241no—
274C22—1241no—
275C22—1241no—
276C22—1241no—
277C22C22⋊C4×D3111yescharacteristic
278C22(C22⋊D31)⋊C411yescharacteristic
279C22C2×#9246911yescharacteristic
280C22#1175847811yescharacteristic
281C22M16×D3111yescharacteristic
282C22#1175848411yescharacteristic
283C22C4×C22⋊D3111yescharacteristic
284C4—41no—
285C4—622no—
286C4—622no—
287C4—1241no—
288C4#1175846111yescharacteristic
289C4#1175846211yescharacteristic
290C4#1175851611yescharacteristic
291C4#1175851711yescharacteristic
292C2—42no—
293C2—1242no—
294C2#1452094911yescharacteristic
295C2#1452101111yescharacteristic
296C2#1452101411yescharacteristic
297C2#1452111711yescharacteristic
298C2#1452111811yescharacteristic
299C2#1452112011yescharacteristic
300C2#1452112311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×D31
commutator, socleC23×C31C22×C4
FittingC2×C31×C22⋊C8C2
prime radicalC22×C22⋊D31C4
cosocleC22×C4×C31C23
hypercenterC2×C22⋊C8D31
hypocenterC31#1955
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×Dic31⋊C8C2semidirect
3. C2×C22⋊(C31⋊C8)C2semidirect
4. C2×(C2×D31)⋊C8C2semidirect
5. C2×(C2×D31)⋊C8C2semidirect
6. C2×#11758403C2semidirect
7. C2×C31×C22⋊C8C2semidirect
54. C2×C8×C31D4semidirect
65. C2×C31⋊C8D4semidirect
147. C2×C22⋊C8D31semidirect
177. C31#1955semidirect
211. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D312
C2×Dic31⋊C82
C2×C22⋊(C31⋊C8)2
C2×(C2×D31)⋊C82
C2×#117584032
C2×C31×C22⋊C82
#195531