Finite Group Databasebeta

← #16854032

Subgroups — #16854032

Order 3968.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#11758512C211yesmaximal, characteristic
2C22×C4×Dic31C211yesmaximal, characteristic, complemented
3#14520943C212yesmaximal, complemented
4#14520942C212yesmaximal, complemented
5#14520932C212yesmaximal, complemented
6#14520931C212yesmaximal, complemented
7#14520929C211yesmaximal, characteristic
8#14520924C211yesmaximal, characteristic, complemented
9#14520922C211yesmaximal, characteristic, complemented
10#14520910C211yesmaximal, characteristic, complemented
11C31×#386C211yesmaximal, characteristic
12C22×D4×C31C2211yescharacteristic
13C23×C4×C31C2211yescharacteristic
14C23×Dic31C2211yescharacteristic
15C23×Dic31C2211yescharacteristic
16C2×C22⋊C4×C31C2211yescharacteristic
17C2×C22⋊C4×C31C2212yes—
18C2×C22⋊Dic31C2211yescharacteristic
19C2×C22⋊Dic31C2211yescharacteristic
20C2×C22⋊Dic31C2212yes—
21C2×C22⋊Dic31C2211yescharacteristic
22C2×C22⋊Dic31C2211yescharacteristic
23#11758512—24no—
24C2×C4⋊Dic31C2212yes—
25C2×Dic31⋊C4C2212yes—
26C2×Dic31⋊C4C2211yescharacteristic
27C2×Dic31⋊C4C2211yescharacteristic
28C2×C4×Dic31—24no—
29C2×C4×Dic31C2212yes—
30C2×C4×Dic31C2212yes—
31C31×#95C2212yes—
32C31⋊#95C2212yes—
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C24×C31C2311yescharacteristic
38C24×C31C2311yescharacteristic
39C2×D4×C31—22no—
40C2×D4×C31—22no—
41C2×D4×C31—22no—
42C22×C4×C31—22no—
43C22×C4×C31—24no—
44C22×C4×C31C2312yes—
45C22×C4×C31C2311yescharacteristic
46C22×C4×C31C2312yes—
47C22×Dic31—22no—
48C22×Dic31—22no—
49C22×Dic31—22no—
50C22×Dic31—24no—
51C22×Dic31D414yes—
52C22×Dic31C2312yes—
53C22×Dic31C2311yescharacteristic
54C22×Dic31C2311yescharacteristic
55C22×Dic31C2311yescharacteristic
56C22×Dic31C2311yescharacteristic
57C22×Dic31C2312yes—
58C22⋊C4×C31—22no—
59C22⋊C4×C31—24no—
60C22⋊Dic31—22no—
61C22⋊Dic31—22no—
62C22⋊Dic31—22no—
63C22⋊Dic31—22no—
64C22⋊Dic31—24no—
65C4⋊Dic31—24no—
66Dic31⋊C4—24no—
67Dic31⋊C4—22no—
68Dic31⋊C4—22no—
69C4×Dic31—24no—
70C4×Dic31—24no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—21no—
74C23×C31—22no—
75C23×C31—22no—
76C23×C31—21no—
77C23×C31—21no—
78C23×C31C2×D412yes—
79C23×C31#4112yes—
80C23×C31#4112yes—
81C23×C31C2411yescharacteristic
82D4×C31—44no—
83C2×C4×C31—28no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31#4114yes—
93C2×Dic31—28no—
94C2×Dic31—28no—
95C2×Dic31—22no—
96C2×Dic31—21no—
97C2×Dic31—21no—
98C2×Dic31—21no—
99C2×Dic31—21no—
100C2×Dic31—22no—
101C2×Dic31—21no—
102C2×Dic31—21no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—22no—
108C2×Dic31C2×D412yes—
109C2×Dic31C2×D412yes—
110C2×Dic31#4114yes—
111#2853—311nomaximal, Sylow
112C22×C31—22no—
113C22×C31—22no—
114C22×C31—24no—
115C22×C31—24no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31—42no—
120C22×C31—42no—
121C22×C31#12314yes—
122C22×C31C22×D411yescharacteristic
123C22×C31C2×#4111yescharacteristic
124C22×C31C2×#4111yescharacteristic
125C22×C31C2×#4112yes—
126C22×C31C2×#4112yes—
127C4×C31—24no—
128C4×C31—41no—
129C4×C31—42no—
130Dic31—22no—
131Dic31—22no—
132Dic31—24no—
133Dic31—41no—
134Dic31—41no—
135Dic31—42no—
136C2×#123—311no—
137C22×C42—311no—
138#388—312no—
139#388—312no—
140#387—312no—
141#387—312no—
142#386—311no—
143#386D3111yescharacteristic
144#385—311no—
145#385—311no—
146#381—311no—
147C2×C31—24no—
148C2×C31—81no—
149C2×C31C2×#12311yescharacteristic
150C2×C31#52511yescharacteristic
151C2×C31#52511yescharacteristic
152C2×C31#52912yes—
153C2×C31#52912yes—
154C22×D4C2×D3111yescharacteristic
155C23×C4—311no—
156C23×C4—311no—
157C23×C4C2×D3111yescharacteristic
158#123—624no—
159C2×C4⋊C4—311no—
160C2×C4⋊C4—311no—
161C2×C4⋊C4—312no—
162C2×C4⋊C4—312no—
163C2×C22⋊C4—311no—
164C2×C22⋊C4—311no—
165C2×C22⋊C4—311no—
166C2×C22⋊C4—311no—
167C2×C22⋊C4—312no—
168C2×C22⋊C4C2×D3111yescharacteristic
169C2×C22⋊C4C2×D3112yes—
170C2×C42—312no—
171C2×C42—312no—
172C2×C42—624no—
173#95—312no—
174#95—312no—
175#95—312no—
176#95—312no—
177#95—312no—
178#95C2×D3112yes—
179C31#285311yescharacteristic, complemented, Sylow
180C24C22×D3111yescharacteristic
181C24C22×D3111yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—314no—
188C22×C4—311no—
189C22×C4—311no—
190C22×C4—312no—
191C22×C4—312no—
192C22×C4—311no—
193C22×C4—311no—
194C22×C4—624no—
195C22×C4—622no—
196C22×C4—622no—
197C22×C4—622no—
198C22×C4C22×D3112yes—
199C22×C4C22×D3111yescharacteristic
200C22×C4C22×D3112yes—
201C4⋊C4—622no—
202C4⋊C4—622no—
203C4⋊C4—624no—
204C4⋊C4—624no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—624no—
208C22⋊C4—622no—
209C22⋊C4—622no—
210C22⋊C4—622no—
211C22⋊C4—622no—
212C42—624no—
213C42—624no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—21no—
220C23—22no—
221C23#9249512yes—
222C23D4×D3112yes—
223C23#9249712yes—
224C23C23×D3111yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—312no—
236C2×C4—312no—
237C2×C4—314no—
238C2×C4—628no—
239C2×C4—628no—
240C2×C4—621no—
241C2×C4—621no—
242C2×C4—622no—
243C2×C4—621no—
244C2×C4—621no—
245C2×C4—622no—
246C2×C4—622no—
247C2×C4—621no—
248C2×C4—621no—
249C2×C4—622no—
250C2×C4—622no—
251C2×C4—622no—
252C2×C4—622no—
253C2×C4#9249714yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1175846014yes—
264C22C2×#9249511yescharacteristic
265C22C2×D4×D3111yescharacteristic
266C22C2×#9249711yescharacteristic
267C22C2×#9249712yes—
268C22#41×D3112yes—
269C4—24no—
270C4—41no—
271C4—42no—
272C4—622no—
273C4—622no—
274C4—624no—
275C4—1241no—
276C4—1241no—
277C4—1242no—
278C2—24no—
279C2—81no—
280C2C2×#1175846011yescharacteristic
281C2#1452155511yescharacteristic
282C2#1452155711yescharacteristic
283C2#1452157912yes—
284C2#1452158012yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#386C2
prime radicalC22×D4×C31C22
hypercenter#386D31
hypocenterC31#2853
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic31C2semidirect
3. #14520943C2semidirect
4. #14520942C2semidirect
5. #14520932C2semidirect
6. #14520931C2semidirect
8. #14520924C2semidirect
9. #14520922C2semidirect
10. #14520910C2semidirect
179. C31#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#117585122
C22×C4×Dic312
#145209432
#145209422
#145209322
#145209312
#145209292
#145209242
#145209222
#145209102
C31×#3862
#285331