Finite Group Databasebeta

← #16848978

Subgroups — #16848978

Order 3968.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×Dic31C211yesmaximal, characteristic, complemented
2C2×#11758454C211yesmaximal, characteristic
3#14520989C211yesmaximal, characteristic, complemented
4#14520985C211yesmaximal, characteristic
5#14520983C212yesmaximal, complemented
6#14520972C212yesmaximal
7#14520968C211yesmaximal, characteristic, complemented
8#14520900C211yesmaximal, characteristic, complemented
9#14520900C212yesmaximal, complemented
10C22⋊C4×Dic31C212yesmaximal
11C31×#393C211yesmaximal, characteristic
12C22×Q8×C31C2211yescharacteristic
13C23×Dic31C2211yescharacteristic
14C2×C22⋊C4×C31C2212yes—
15C2×C22⋊C4×C31C2211yescharacteristic
16C2×C22⋊Dic31C2212yes—
17C2×C22⋊Dic31C2211yescharacteristic
18Q8×Dic31—24no—
19C2×C4⋊Dic31C2212yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×Dic31⋊C4C2212yes—
22C2×Dic31⋊C4C2212yes—
23C2×Dic31⋊C4C2212yes—
24C2×Dic31⋊C4C2211yescharacteristic
25C2×Dic31⋊C4C2211yescharacteristic
26C2×C4×Dic31C2212yes—
27C2×C4×Dic31C2212yes—
28C2×C4×Dic31C2211yescharacteristic
29C2×C4×Dic31C2211yescharacteristic
30#11758454—24no—
31C31×#95C2212yes—
32C31×#95C2211yescharacteristic
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2211yescharacteristic
36C31⋊#95C2212yes—
37C24×C31C2311yescharacteristic
38C2×Q8×C31—22no—
39C2×Q8×C31—24no—
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2312yes—
44C22×Dic31—22no—
45C22×Dic31—22no—
46C22×Dic31Q814yescomplemented
47C22×Dic31C2311yescharacteristic
48C22×Dic31C2312yes—
49C22×Dic31C2311yescharacteristic
50C22×Dic31C2311yescharacteristic
51C22×Dic31C2312yes—
52C22×Dic31C2311yescharacteristic
53C22⋊C4×C31—22no—
54C22⋊C4×C31—24no—
55C22⋊Dic31—22no—
56C22⋊Dic31—24no—
57C4⋊Dic31—22no—
58C4⋊Dic31—24no—
59Dic31⋊C4—22no—
60Dic31⋊C4—24no—
61Dic31⋊C4—22no—
62Dic31⋊C4—24no—
63Dic31⋊C4—24no—
64C4×Dic31—22no—
65C4×Dic31—24no—
66C4×Dic31—24no—
67C4×Dic31D414yes—
68C23×C31—21no—
69C23×C31—22no—
70C23×C31—22no—
71C23×C31—21no—
72C23×C31C2×Q812yes—
73C23×C31C2411yescharacteristic
74Q8×C31—28no—
75C2×C4×C31—22no—
76C2×C4×C31—22no—
77C2×C4×C31—22no—
78C2×C4×C31—21no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31C2×D412yes—
84C2×C4×C31#4114yes—
85C2×Dic31—21no—
86C2×Dic31—22no—
87C2×Dic31—22no—
88C2×Dic31—22no—
89C2×Dic31—21no—
90C2×Dic31—28no—
91C2×Dic31—21no—
92C2×Dic31—22no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—22no—
96C2×Dic31C2×D412yes—
97C2×Dic31C2×D412yes—
98C2×Dic31C2×Q812yes—
99C2×Dic31C2×Q812yes—
100C2×Dic31#4114yes—
101C2×Dic31#4112yes—
102#2396—311nomaximal, Sylow
103C22×C31—22no—
104C22×C31—24no—
105C22×C31—41no—
106C22×C31—42no—
107C22×C31—41no—
108C22×C31C22×D411yescharacteristic
109C22×C31C22×Q811yescharacteristic
110C22×C31C2×#4112yes—
111C22×C31C2×#4111yescharacteristic
112C22×C31C2×#4112yes—
113C4×C31—22no—
114C4×C31—24no—
115C4×C31—42no—
116C4×C31—41no—
117Dic31—22no—
118Dic31—24no—
119Dic31—22no—
120Dic31—41no—
121Dic31—42no—
122Dic31C22⋊Q814yes—
123C2×C4×Q8—311no—
124C2×#117—311no—
125#400—311no—
126#399—311no—
127#398—312no—
128#396—312no—
129#395—311no—
130#393D3111yescharacteristic
131C4×C4⋊C4—311no—
132C4×C4⋊C4—312no—
133C4×C22⋊C4—312no—
134C2×C31—42no—
135C2×C31C2×C22⋊Q811yescharacteristic
136C2×C31#52511yescharacteristic
137C2×C31#52912yes—
138C2×C31#53211yescharacteristic
139C2×C31#53312yes—
140C22×Q8C2×D3111yescharacteristic
141C23×C4—311no—
142C4×Q8—624no—
143#117—624no—
144C2×C4⋊C4—311no—
145C2×C4⋊C4—311no—
146C2×C4⋊C4—312no—
147C2×C4⋊C4—311no—
148C2×C4⋊C4—312no—
149C2×C4⋊C4—312no—
150C2×C4⋊C4—312no—
151C2×C22⋊C4—311no—
152C2×C22⋊C4—312no—
153C2×C22⋊C4C2×D3111yescharacteristic
154C2×C22⋊C4C2×D3112yes—
155C2×C42—311no—
156C2×C42—311no—
157C2×C42—312no—
158C2×C42—312no—
159#95—312no—
160#95—312no—
161#95—311no—
162#95—312no—
163#95C2×D3112yes—
164#95C2×D3111yescharacteristic
165C31#239611yescharacteristic, complemented, Sylow
166C24C22×D3111yescharacteristic
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—311no—
170C22×C4—314no—
171C22×C4—311no—
172C22×C4—312no—
173C22×C4—311no—
174C22×C4—311no—
175C22×C4—312no—
176C22×C4—622no—
177C22×C4—622no—
178C22×C4C22×D3112yes—
179C22×C4C22×D3111yescharacteristic
180C22×C4C22×D3111yescharacteristic
181C22×C4C22×D3112yes—
182C4⋊C4—624no—
183C4⋊C4—624no—
184C4⋊C4—622no—
185C4⋊C4—624no—
186C4⋊C4—622no—
187C4⋊C4—622no—
188C4⋊C4—624no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—624no—
192C22⋊C4—622no—
193C42—314no—
194C42—624no—
195C42—622no—
196C42—624no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23Q8×D3112yes—
202C23C23×D3111yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—312no—
213C2×C4—312no—
214C2×C4—312no—
215C2×C4—312no—
216C2×C4—312no—
217C2×C4—314no—
218C2×C4—628no—
219C2×C4—621no—
220C2×C4—622no—
221C2×C4—622no—
222C2×C4—622no—
223C2×C4—621no—
224C2×C4—621no—
225C2×C4—622no—
226C2×C4—622no—
227C2×C4—622no—
228C2×C4—622no—
229C2×C4D4×D3112yes—
230C2×C4#9249714yes—
231C22—24no—
232C22—22no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D3111yescharacteristic
237C22C2×#9249712yes—
238C22C2×Q8×D3111yescharacteristic
239C22#41×D3112yes—
240C22#41×D3111yescharacteristic
241C4—22no—
242C4—24no—
243C4—42no—
244C4—41no—
245C4—314no—
246C4—622no—
247C4—624no—
248C4—622no—
249C4—1241no—
250C4—1242no—
251C2—42no—
252C2C22⋊Q8×D3111yescharacteristic
253C2#1452155511yescharacteristic
254C2#1452157812yes—
255C2#1452159811yescharacteristic
256C2#1452160112yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#393C2
prime radicalC24×C31C23
hypercenter#393D31
hypocenterC31#2396
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic31C2semidirect
3. #14520989C2semidirect
5. #14520983C2semidirect
7. #14520968C2semidirect
8. #14520900C2semidirect
9. #14520900C2semidirect
46. C22×Dic31Q8semidirect
165. C31#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic312
C2×#117584542
#145209892
#145209852
#145209832
#145209722
#145209682
#145209002
C22⋊C4×Dic312
C31×#3932
#239631