Finite Group Databasebeta

← #16848999

Subgroups — #16848999

Order 3968.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#11758464C211yesmaximal, characteristic, complemented
2C2×D4×Dic31C211yesmaximal, characteristic
3#14520989C211yesmaximal, characteristic, complemented
4#14520988C211yesmaximal, characteristic, complemented
5#14520977C212yesmaximal, complemented
6#14520963C211yesmaximal, characteristic
7#14520961C212yesmaximal, complemented
8C22⋊C4×Dic31C212yesmaximal, complemented
9C22⋊C4×Dic31C212yesmaximal, complemented
10#14520894C211yesmaximal, characteristic, complemented
11C31×#394C211yesmaximal, characteristic
12C22×D4×C31C2211yescharacteristic
13C23×Dic31C2212yes—
14C22×Dic62C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C22⋊C4×C31C2212yes—
17C2×C22⋊C4×C31C2212yes—
18C2×C22⋊Dic31C2212yes—
19C2×C22⋊Dic31C2212yes—
20C2×C22⋊Dic31C2212yes—
21D4×Dic31—24no—
22C2×C4⋊Dic31C2211yescharacteristic
23C2×Dic31⋊C4C2211yescharacteristic
24C2×Dic31⋊C4C2211yescharacteristic
25C2×C4×Dic31C2212yes—
26C2×C4×Dic31C2211yescharacteristic
27C2×C4×Dic31C2211yescharacteristic
28C2×C4×Dic31C2211yescharacteristic
29#11758464—24no—
30C31×#95C2211yescharacteristic
31C31⋊#95C2212yes—
32C31⋊#95C2211yescharacteristic
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C24×C31C2312yes—
38C2×D4×C31—24no—
39C2×D4×C31—22no—
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C22×Dic31—24no—
45C22×Dic31—24no—
46C22×Dic31—24no—
47C22×Dic31C2312yes—
48C22×Dic31C2312yes—
49C22×Dic31C2311yescharacteristic
50C22×Dic31C2311yescharacteristic
51C22×Dic31C2311yescharacteristic
52C22×Dic31C2311yescharacteristic
53C2×Dic62—22no—
54C2×Dic62—22no—
55C2×Dic62—22no—
56C4⋊C4×C31—22no—
57C22⋊C4×C31—24no—
58C22⋊C4×C31—24no—
59C22⋊Dic31—24no—
60C22⋊Dic31—24no—
61C22⋊Dic31—24no—
62C4⋊Dic31—22no—
63Dic31⋊C4—22no—
64Dic31⋊C4—22no—
65C4×Dic31—22no—
66C4×Dic31—22no—
67C4×Dic31—24no—
68C4×Dic31D414yes—
69C23×C31—22no—
70C23×C31—22no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31#4114yes—
76C23×C31C2411yescharacteristic
77D4×C31—28no—
78C2×C4×C31—21no—
79C2×C4×C31—21no—
80C2×C4×C31—22no—
81C2×C4×C31—21no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31C2×D412yes—
87C2×C4×C31#4112yes—
88C2×Dic31—21no—
89C2×Dic31—21no—
90C2×Dic31—21no—
91C2×Dic31—21no—
92C2×Dic31—21no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—48no—
101C2×Dic31C2×D412yes—
102C2×Dic31C2×D412yes—
103C2×Dic31#4112yes—
104C2×Dic31#4114yes—
105Dic62—28no—
106#2398—311nomaximal, Sylow
107C22×C31—24no—
108C22×C31—24no—
109C22×C31—24no—
110C22×C31—42no—
111C22×C31—42no—
112C22×C31—42no—
113C22×C31—42no—
114C22×C31C22×D411yescharacteristic
115C22×C31C2×#4112yes—
116C22×C31C2×#4111yescharacteristic
117C22×C31C2×#4111yescharacteristic
118C22×C31C2×#4112yes—
119C4×C31—22no—
120C4×C31—22no—
121C4×C31—41no—
122C4×C31—42no—
123Dic31—22no—
124Dic31—22no—
125Dic31—41no—
126Dic31—42no—
127Dic31—42no—
128Dic31#12414yes—
129C2×C4×Q8—311no—
130C2×C4×D4—311no—
131#400—311no—
132#400—311no—
133#397—312no—
134#394—311no—
135#394D3111yescharacteristic
136#393—312no—
137C4×C22⋊C4—312no—
138C4×C22⋊C4—312no—
139#376—311no—
140C2×C31—44no—
141C2×C31#52511yescharacteristic
142C2×C31#52511yescharacteristic
143C2×C31C2×#12411yescharacteristic
144C2×C31#52912yes—
145C2×C31#52912yes—
146C22×Q8—311no—
147C22×D4C2×D3111yescharacteristic
148C23×C4—312no—
149C4×Q8—624no—
150C4×D4—624no—
151C2×C4⋊C4—311no—
152C2×C4⋊C4—311no—
153C2×C4⋊C4—311no—
154C2×C4⋊C4C2×D3111yescharacteristic
155C2×C22⋊C4—312no—
156C2×C22⋊C4—312no—
157C2×C22⋊C4—312no—
158C2×C22⋊C4C2×D3112yes—
159C2×C22⋊C4C2×D3112yes—
160C2×C42—311no—
161C2×C42—311no—
162C2×C42—311no—
163C2×C42—312no—
164#95—312no—
165#95—311no—
166#95—312no—
167#95—312no—
168#95—312no—
169#95—312no—
170#95C2×D3111yescharacteristic
171C31#239811yescharacteristic, complemented, Sylow
172C24C22×D3112yes—
173C2×Q8—622no—
174C2×Q8—622no—
175C2×Q8—622no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—311no—
179C22×C4—311no—
180C22×C4—311no—
181C22×C4—312no—
182C22×C4—312no—
183C22×C4—311no—
184C22×C4—624no—
185C22×C4—624no—
186C22×C4—624no—
187C22×C4C22×D3112yes—
188C22×C4C22×D3111yescharacteristic
189C22×C4C22×D3111yescharacteristic
190C22×C4C22×D3111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—622no—
193C4⋊C4—622no—
194C4⋊C4—622no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—624no—
198C22⋊C4—624no—
199C22⋊C4—624no—
200C42—314no—
201C42—624no—
202C42—622no—
203C42—622no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#9249714yes—
211C23C23×D3111yescharacteristic
212Q8—628no—
213D4—28no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—312no—
223C2×C4—314no—
224C2×C4—312no—
225C2×C4—312no—
226C2×C4—621no—
227C2×C4—621no—
228C2×C4—621no—
229C2×C4—621no—
230C2×C4—621no—
231C2×C4—622no—
232C2×C4—622no—
233C2×C4—622no—
234C2×C4—622no—
235C2×C4—622no—
236C2×C4—622no—
237C2×C4—622no—
238C2×C4—1248no—
239C2×C4D4×D3112yes—
240C2×C4#9249712yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D3111yescharacteristic
249C22C2×#9249712yes—
250C22C2×#9249711yescharacteristic
251C22#41×D3111yescharacteristic
252C22#41×D3112yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—314no—
258C4—622no—
259C4—622no—
260C4—1241no—
261C4—1242no—
262C4—1242no—
263C2—44no—
264C2#1452155511yescharacteristic
265C2#1452155911yescharacteristic
266C2#124×D3111yescharacteristic
267C2#1452157612yes—
268C2#1452158012yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#394C2
prime radicalC22×D4×C31C22
hypercenter#394D31
hypocenterC31#2398
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#11758464C2semidirect
3. #14520989C2semidirect
4. #14520988C2semidirect
5. #14520977C2semidirect
7. #14520961C2semidirect
8. C22⋊C4×Dic31C2semidirect
9. C22⋊C4×Dic31C2semidirect
10. #14520894C2semidirect
171. C31#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#117584642
C2×D4×Dic312
#145209892
#145209882
#145209772
#145209632
#145209612
C22⋊C4×Dic312
#145208942
C31×#3942
#239831