Finite Group Databasebeta

← #16844281

Subgroups — #16844281

Order 3968.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D31C211yesmaximal, characteristic
2C22⋊C8×D31C211yesmaximal, characteristic
3#14520908C211yesmaximal, characteristic
4#14520787C211yesmaximal, characteristic, complemented
5#14520786C211yesmaximal, characteristic, complemented
6#14520785C211yesmaximal, characteristic, complemented
7C31×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D31C411yescharacteristic, complemented
9C22×C4×D31C2211yescharacteristic
10C31×C22⋊Q8C2211yescharacteristic
11#11758523C411yescharacteristic, complemented
12C2×(C2×D31)⋊C4—21no—
13#11758501C2211yescharacteristic
14C2×Dic31⋊C4—21no—
15C2×C8×D31—21no—
16#11758474C411yescharacteristic, complemented
17#11758473—21no—
18C4⋊C4×D31—21no—
19C4⋊C4×D31C411yescharacteristic, complemented
20C22⋊C4×D31—21no—
21#11758455—21no—
22C31×C22⋊C8C2211yescharacteristic
23C31×#95C2211yescharacteristic
24C31⋊#95C2211yescharacteristic
25C22⋊(C31⋊C8)C2211yescharacteristic
26(C2×D31)⋊C8—21no—
27C23×D31D411yescharacteristic
28C2×Q8×C31C2×C411yescharacteristic
29C22×C4×C31—21no—
30C22×C4×C31C2311yescharacteristic
31C22×Dic31—21no—
32C22×Dic31D411yescharacteristic
33Q8×D31—42no—
34C2×C4×D31—21no—
35C2×C4×D31—21no—
36C2×C4×D31—42no—
37C2×C4×D31C2×C411yescharacteristic
38C2×C4×D31C2×C411yescharacteristic
39C2×C4×D31D411yescharacteristic
40C2×C4×D31D411yescharacteristic
41C2×Dic62—21no—
42C2×Dic62C2×C411yescharacteristic
43C2×C8×C31—21no—
44C4⋊C4×C31—21no—
45C4⋊C4×C31C2×C411yescharacteristic
46C22⋊C4×C31—21no—
47C22⋊Dic31—21no—
48(C2×D31)⋊C4—21no—
49(C2×D31)⋊C4—21no—
50(C2×D31)⋊C4—42no—
51C4⋊Dic31—21no—
52C4⋊Dic31C2×C411yescharacteristic
53Dic31⋊C4—21no—
54Dic31⋊C4—21no—
55Dic31⋊C4—21no—
56Dic31⋊C4—42no—
57C2×C31⋊C8—21no—
58C8×D31—42no—
59C23×C31C2×D411yescharacteristic
60C22×D31—41no—
61C22×D31—41no—
62C22×D31C22⋊C411yescharacteristic
63C22×D31C22⋊C411yescharacteristic
64C22×D31SD1612yes—
65C22×D31Q1612yes—
66Q8×C31—41no—
67C2×C4×C31—21no—
68C2×C4×C31—21no—
69C2×C4×C31—41no—
70C2×C4×C31—41no—
71C2×C4×C31—41no—
72C2×C4×C31—41no—
73C2×C4×C31C22×C411yescharacteristic
74C2×C4×C31C2×D411yescharacteristic
75C2×Dic31—22no—
76C2×Dic31—21no—
77C2×Dic31—21no—
78C2×Dic31—41no—
79C2×Dic31—41no—
80C2×Dic31—41no—
81C2×Dic31C22⋊C411yescharacteristic
82C2×Dic31C22⋊C411yescharacteristic
83C4×D31—22no—
84C4×D31—22no—
85C4×D31—41no—
86C4×D31—41no—
87C4×D31—42no—
88C4×D31—42no—
89Dic62—42no—
90Dic62—41no—
91C8×C31—41no—
92C31⋊C8—41no—
93#1520—311nomaximal, Sylow
94C22×C31—21no—
95C22×C31—21no—
96C22×C31C2×C22⋊C411yescharacteristic
97C22×C31C2×SD1611yescharacteristic
98C22×C31C2×Q1611yescharacteristic
99C2×D31—21no—
100C2×D31—21no—
101C2×D31—21no—
102C2×D31—21no—
103C2×D31—42no—
104C2×D31—42no—
105C2×D31#10312yes—
106C2×D31#10312yes—
107C4×C31—21no—
108C4×C31—21no—
109C4×C31—41no—
110C4×C31—41no—
111C4×C31—81no—
112Dic31—21no—
113Dic31—41no—
114Dic31—41no—
115Dic31—81no—
116Dic31#9912yes—
117C2×C22⋊Q8—311no—
118C2×C22⋊C8—311no—
119#380—311no—
120#328—311no—
121#328—311no—
122#328—311no—
123#328D3111yescharacteristic, complemented
124C2×C31—21no—
125C2×C31—21no—
126C2×C31C2×#9911yescharacteristic
127C2×C31C2×#10311yescharacteristic
128C2×C31#42111yescharacteristic
129D31—22no—
130D31—22no—
131C22×Q8—311no—
132C23×C4—311no—
133C22×C8—621no—
134C22⋊Q8—311no—
135C22⋊Q8—311no—
136C22⋊Q8—311no—
137C22⋊Q8—621no—
138C22⋊Q8—621no—
139C22⋊Q8C2×D3111yescharacteristic
140C2×C4⋊C4—311no—
141C2×C4⋊C4—621no—
142C2×C4⋊C4—621no—
143C2×C22⋊C4—621no—
144C2×C22⋊C4—621no—
145C22⋊C8—311no—
146C22⋊C8—621no—
147C22⋊C8C2×D3111yescharacteristic
148#95—311no—
149#95C2×D3111yescharacteristic
150C31#152011yescharacteristic, complemented, Sylow
151C24—311no—
152C2×Q8—311no—
153C2×Q8—621no—
154C2×Q8—1242no—
155C2×Q8C4×D3111yescharacteristic
156C22×C4—21no—
157C22×C4—311no—
158C22×C4—311no—
159C22×C4—311no—
160C22×C4—311no—
161C22×C4—311no—
162C22×C4—621no—
163C22×C4—621no—
164C22×C4—621no—
165C22×C4—1242no—
166C22×C4C22×D3111yescharacteristic
167C2×C8—21no—
168C2×C8—621no—
169C2×C8—1242no—
170C4⋊C4—21no—
171C4⋊C4—311no—
172C4⋊C4—621no—
173C4⋊C4—621no—
174C4⋊C4—621no—
175C4⋊C4—621no—
176C4⋊C4—1242no—
177C4⋊C4C4×D3111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—621no—
180C22⋊C4—621no—
181C22⋊C4—621no—
182C22⋊C4—1242no—
183C23—311no—
184C23—311no—
185C23—312no—
186C23—312no—
187C23—1241no—
188C23—1241no—
189C23D4×D3111yescharacteristic
190Q8—41no—
191Q8—1241no—
192Q8—1242no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—311no—
200C2×C4—311no—
201C2×C4—622no—
202C2×C4—622no—
203C2×C4—622no—
204C2×C4—621no—
205C2×C4—621no—
206C2×C4—1241no—
207C2×C4—1241no—
208C2×C4—1242no—
209C2×C4—1242no—
210C2×C4—1241no—
211C2×C4—1241no—
212C2×C4—1241no—
213C2×C4C2×C4×D3111yescharacteristic
214C2×C4D4×D3111yescharacteristic
215C8—41no—
216C8—1241no—
217C22—21no—
218C22—21no—
219C22—312no—
220C22—312no—
221C22—621no—
222C22—621no—
223C22—621no—
224C22—621no—
225C22—1242no—
226C22—1242no—
227C22C22⋊C4×D3111yescharacteristic
228C22SD16×D3111yescharacteristic
229C22Q16×D3111yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—312no—
236C4—621no—
237C4—1241no—
238C4—1241no—
239C4—2481no—
240C2—21no—
241C2—21no—
242C2—622no—
243C2—622no—
244C2#99×D3111yescharacteristic
245C2#103×D3111yescharacteristic
246C2#1452106411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#328C2
socleC22×C31C2×C22⋊C4
prime radicalC23×D31D4
cosocleC22×C4×C31C23
hypercenter#328D31
hypocenterC31#1520
LCSG ▷ C2×C4×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #14520787C2semidirect
5. #14520786C2semidirect
6. #14520785C2semidirect
7. C31×#328C2semidirect
8. C2×Q8×D31C4semidirect
11. #11758523C4semidirect
16. #11758474C4semidirect
19. C4⋊C4×D31C4semidirect
123. #328D31semidirect
150. C31#1520semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D312
C22⋊C8×D312
#145209082
#145207872
#145207862
#145207852
C31×#3282
#152031