Finite Group Databasebeta

← #16854048

Subgroups — #16854048

Order 3968.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#11758455C211yesmaximal, characteristic
2C22×C4×Dic31C211yesmaximal, characteristic
3#14520953C212yesmaximal, complemented
4#14520946C212yesmaximal, complemented
5#14520936C212yesmaximal, complemented
6#14520932C212yesmaximal, complemented
7#14520924C211yesmaximal, characteristic
8#14520921C211yesmaximal, characteristic
9#14520912C211yesmaximal, characteristic
10#14520907C211yesmaximal, characteristic
11C31×#381C211yesmaximal, characteristic
12C23×C4×C31C2211yescharacteristic
13C23×Dic31C2211yescharacteristic
14C23×Dic31C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C2×C22⋊C4×C31C2211yescharacteristic
17C2×C22⋊C4×C31C2211yescharacteristic
18C2×C22⋊Dic31C2211yescharacteristic
19C2×C22⋊Dic31C2211yescharacteristic
20C2×C4⋊Dic31C2212yes—
21C2×C4⋊Dic31C2211yescharacteristic
22C2×Dic31⋊C4C2212yes—
23C2×Dic31⋊C4C2212yes—
24C2×Dic31⋊C4C2211yescharacteristic
25C2×Dic31⋊C4C2211yescharacteristic
26C2×C4×Dic31—24no—
27C2×C4×Dic31C2212yes—
28C2×C4×Dic31C2212yes—
29#11758455—24no—
30C31×#95C2212yes—
31C31×#95C2212yes—
32C31⋊#95C2212yes—
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C24×C31C2311yescharacteristic
38C22×C4×C31—22no—
39C22×C4×C31—22no—
40C22×C4×C31Q814yescomplemented
41C22×C4×C31C2312yes—
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2312yes—
45C22×Dic31—22no—
46C22×Dic31—24no—
47C22×Dic31—24no—
48C22×Dic31D414yes—
49C22×Dic31C2312yes—
50C22×Dic31C2311yescharacteristic
51C22×Dic31C2311yescharacteristic
52C22×Dic31C2311yescharacteristic
53C22×Dic31C2312yes—
54C22×Dic31C2311yescharacteristic
55C2×Dic62—22no—
56C2×Dic62—24no—
57C22⋊C4×C31—22no—
58C22⋊C4×C31—22no—
59C22⋊Dic31—22no—
60C22⋊Dic31—22no—
61C4⋊Dic31—22no—
62C4⋊Dic31—24no—
63Dic31⋊C4—22no—
64Dic31⋊C4—22no—
65Dic31⋊C4—24no—
66Dic31⋊C4—24no—
67C4×Dic31—24no—
68C4×Dic31—24no—
69C23×C31—22no—
70C23×C31—22no—
71C23×C31C2×D412yes—
72C23×C31C2×Q812yes—
73C23×C31#4112yes—
74C23×C31C2411yescharacteristic
75C2×C4×C31—28no—
76C2×C4×C31—22no—
77C2×C4×C31—22no—
78C2×C4×C31—21no—
79C2×C4×C31—21no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31C2×Q814yes—
86C2×Dic31—28no—
87C2×Dic31—22no—
88C2×Dic31—22no—
89C2×Dic31—22no—
90C2×Dic31—28no—
91C2×Dic31—21no—
92C2×Dic31—21no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98C2×Dic31C2×D412yes—
99C2×Dic31C2×D412yes—
100C2×Dic31#4114yes—
101C2×Dic31#4114yes—
102Dic62—44no—
103#2855—311nomaximal, Sylow
104C22×C31—22no—
105C22×C31—22no—
106C22×C31—24no—
107C22×C31—24no—
108C22×C31C22⋊Q814yes—
109C22×C31C22×D411yescharacteristic
110C22×C31C22×Q811yescharacteristic
111C22×C31C2×#4111yescharacteristic
112C22×C31C2×#4112yes—
113C22×C31C2×#4112yes—
114C4×C31—24no—
115C4×C31—41no—
116C4×C31—41no—
117C4×C31—42no—
118Dic31—22no—
119Dic31—24no—
120Dic31—41no—
121Dic31—41no—
122Dic31—42no—
123Dic31#12314yes—
124C2×C22⋊Q8—311no—
125C22×C42—311no—
126#391—312no—
127#389—312no—
128#387—312no—
129#387—312no—
130#385—311no—
131#385—311no—
132#381—311no—
133#381D3111yescharacteristic
134#380—311no—
135C2×C31—24no—
136C2×C31C2×C22⋊Q811yescharacteristic
137C2×C31C2×#12311yescharacteristic
138C2×C31#52511yescharacteristic
139C2×C31#52912yes—
140C2×C31#53312yes—
141C22×Q8—311no—
142C23×C4—311no—
143C23×C4—311no—
144C23×C4C2×D3111yescharacteristic
145C22⋊Q8—624no—
146C2×C4⋊C4—311no—
147C2×C4⋊C4—311no—
148C2×C4⋊C4—311no—
149C2×C4⋊C4—312no—
150C2×C4⋊C4—312no—
151C2×C4⋊C4—312no—
152C2×C22⋊C4—311no—
153C2×C22⋊C4—311no—
154C2×C22⋊C4C2×D3111yescharacteristic
155C2×C22⋊C4C2×D3111yescharacteristic
156C2×C42—312no—
157C2×C42—312no—
158C2×C42—624no—
159#95—312no—
160#95—312no—
161#95—312no—
162#95—312no—
163#95—312no—
164#95C2×D3112yes—
165#95C2×D3112yes—
166C31#285511yescharacteristic, complemented, Sylow
167C24C22×D3111yescharacteristic
168C2×Q8—622no—
169C2×Q8—624no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—311no—
173C22×C4—314no—
174C22×C4—311no—
175C22×C4—312no—
176C22×C4—312no—
177C22×C4—311no—
178C22×C4—311no—
179C22×C4—624no—
180C22×C4—624no—
181C22×C4—622no—
182C22×C4Dic6214yescomplemented
183C22×C4C22×D3112yes—
184C22×C4C22×D3111yescharacteristic
185C22×C4C22×D3111yescharacteristic
186C22×C4C22×D3112yes—
187C4⋊C4—624no—
188C4⋊C4—624no—
189C4⋊C4—622no—
190C4⋊C4—622no—
191C4⋊C4—624no—
192C4⋊C4—622no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—622no—
196C22⋊C4—622no—
197C42—624no—
198C42—624no—
199C23—22no—
200C23—22no—
201C23C2×Dic6212yes—
202C23D4×D3112yes—
203C23#9249712yes—
204C23C23×D3111yescharacteristic
205Q8—1244no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—312no—
217C2×C4—314no—
218C2×C4—312no—
219C2×C4—314no—
220C2×C4—628no—
221C2×C4—628no—
222C2×C4—622no—
223C2×C4—622no—
224C2×C4—622no—
225C2×C4—621no—
226C2×C4—621no—
227C2×C4—622no—
228C2×C4—622no—
229C2×C4—622no—
230C2×C4—622no—
231C2×C4—622no—
232C2×C4C2×Dic6214yes—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22#1175845514yes—
238C22C22×Dic6211yescharacteristic
239C22C2×D4×D3111yescharacteristic
240C22C2×#9249711yescharacteristic
241C22#41×D3112yes—
242C22#41×D3112yes—
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—314no—
248C4—622no—
249C4—624no—
250C4—1241no—
251C4—1241no—
252C4—1242no—
253C2—24no—
254C2C2×#1175845511yescharacteristic
255C2#123×D3111yescharacteristic
256C2#1452155711yescharacteristic
257C2#1452157612yes—
258C2#1452160212yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#381C2
prime radicalC24×C31C23
hypercenter#381D31
hypocenterC31#2855
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #14520953C2semidirect
4. #14520946C2semidirect
5. #14520936C2semidirect
6. #14520932C2semidirect
40. C22×C4×C31Q8semidirect
166. C31#2855semidirect
182. C22×C4Dic62semidirect

Maximal subgroups

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

subgroupindex
C2×#117584552
C22×C4×Dic312
#145209532
#145209462
#145209362
#145209322
#145209242
#145209212
#145209122
#145209072
C31×#3812
#285531