Finite Group Databasebeta

← #16846720

Subgroups — #16846720

Order 3968.

392 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 91 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×#92482C211yesmaximal, characteristic, complemented
2C2×#11758474C211yesmaximal, characteristic
3C2×#11758523C211yesmaximal, characteristic
4C2×#11758404C211yesmaximal, characteristic, complemented
5C2×(C2×D31)⋊C8C211yesmaximal, characteristic
6#14520811C211yesmaximal, characteristic, complemented
7C2×C31×#103C211yesmaximal, characteristic, complemented
8C22×Q8×C31C2211yescharacteristic
9C22×C4×D31C2211yescharacteristic
10C22×Dic62C2211yescharacteristic
11C22×C8×C31C2211yescharacteristic
12C2×C4⋊C4×C31C2211yescharacteristic
13#11758523—22no—
14#11758523—22no—
15C2×#92482—22no—
16C2×#92482—22no—
17C2×#92482—22no—
18C2×(C2×D31)⋊C4—21no—
19C2×(C2×D31)⋊C4—21no—
20C2×C4⋊Dic31—21no—
21C2×C4⋊Dic31C2211yescharacteristic
22C2×Dic31⋊C4—21no—
23C22×C31⋊C8C2211yescharacteristic
24#11758474—22no—
25#11758474—22no—
26C31×#103—22no—
27(C2×D31)⋊C8—22no—
28#11758404—22no—
29C23×D31D411yescharacteristic
30C2×Q8×C31—21no—
31C2×Q8×C31—21no—
32C2×Q8×C31—21no—
33C2×Q8×C31—21no—
34C2×Q8×C31—21no—
35C2×Q8×C31D412yes—
36C22×C4×C31—21no—
37C22×C4×C31—21no—
38C22×C4×C31C2311yescharacteristic
39C22×Dic31—21no—
40C22×Dic31—21no—
41C22×Dic31D411yescharacteristic
42C2×C4×D31—21no—
43C2×C4×D31—21no—
44C2×C4×D31—21no—
45C2×C4×D31—21no—
46C2×C4×D31—21no—
47C2×C4×D31—21no—
48C2×Dic62—21no—
49C2×Dic62—21no—
50C2×Dic62—21no—
51C2×Dic62—21no—
52C2×Dic62—21no—
53C2×Dic62D412yes—
54C2×C8×C31—21no—
55C2×C8×C31—21no—
56C2×C8×C31—21no—
57C4⋊C4×C31—21no—
58C4⋊C4×C31D412yes—
59#92482—28no—
60(C2×D31)⋊C4—24no—
61(C2×D31)⋊C4—24no—
62C4⋊Dic31—24no—
63C4⋊Dic31—21no—
64C4⋊Dic31D412yes—
65Dic31⋊C4—24no—
66C2×C31⋊C8—21no—
67C2×C31⋊C8—21no—
68C2×C31⋊C8D412yes—
69C23×C31C2×D411yescharacteristic
70C22×D31—21no—
71C22×D31—21no—
72C22×D31—21no—
73C22×D31—21no—
74C22×D31—21no—
75C22×D31—21no—
76C22×D31Q1612yescomplemented
77Q8×C31—22no—
78Q8×C31—22no—
79Q8×C31—41no—
80Q8×C31—41no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—41no—
84C2×C4×C31—41no—
85C2×C4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31#4111yescharacteristic
93C2×Dic31—21no—
94C2×Dic31—22no—
95C2×Dic31—21no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98C2×Dic31—41no—
99C2×Dic31—41no—
100C2×Dic31—41no—
101C2×Dic31—41no—
102C4×D31—41no—
103C4×D31—41no—
104C4×D31—41no—
105C4×D31—41no—
106Dic62—22no—
107Dic62—22no—
108Dic62—41no—
109Dic62—41no—
110C8×C31—41no—
111C31⋊C8—22no—
112#2001—311nomaximal, Sylow
113C22×C31#12011yescharacteristic
114C22×C31#12011yescharacteristic
115C22×C31#12111yescharacteristic
116C22×C31C2×Q1611yescharacteristic
117C22×C31#13511yescharacteristic
118C22×C31#13711yescharacteristic
119C22×C31#13711yescharacteristic
120C2×D31—22no—
121C2×D31—22no—
122C2×D31—22no—
123C2×D31—41no—
124C2×D31—41no—
125C2×D31—41no—
126C2×D31—41no—
127C4×C31—42no—
128C4×C31—42no—
129C4×C31#12011yescharacteristic
130C4×C31#12111yescharacteristic
131C4×C31#12111yescharacteristic
132C4×C31#12711yescharacteristic
133Dic31—42no—
134Dic31—41no—
135Dic31—42no—
136C22×Q16—311no—
137C2×C22⋊Q8—311no—
138C2×C22⋊Q8—311no—
139C2×#103—311no—
140C2×#103D3111yescharacteristic, complemented
141C2×C22⋊C8—311no—
142#340—311no—
143C2×C31#39211yescharacteristic
144C2×C31#45111yescharacteristic
145C2×C31#45111yescharacteristic
146C2×C31#45211yescharacteristic
147C2×C31#45211yescharacteristic
148C2×C31#46711yescharacteristic
149C2×C31#47011yescharacteristic
150D31—42no—
151C22×Q8—311no—
152C22×Q8C2×D3111yescharacteristic
153C23×C4—311no—
154C2×Q16—622no—
155C2×Q16—622no—
156C2×Q16—622no—
157C22×C8—311no—
158C22×C8C2×D3111yescharacteristic
159C22⋊Q8—622no—
160C22⋊Q8—622no—
161C22⋊Q8—622no—
162C22⋊Q8—622no—
163C2×C4⋊C4—311no—
164C2×C4⋊C4—621no—
165C2×C4⋊C4—621no—
166C2×C4⋊C4C2×D3111yescharacteristic
167C2×C22⋊C4—621no—
168C2×C22⋊C4—621no—
169#103—22no—
170#103—622no—
171C22⋊C8—622no—
172C31#200111yescharacteristic, complemented, Sylow
173C24—311no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—312no—
180C2×Q8—621no—
181C2×Q8—621no—
182C2×Q8—621no—
183C2×Q8—621no—
184C2×Q8—621no—
185C2×Q8D12412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—311no—
189C22×C4—621no—
190C22×C4—621no—
191C22×C4—621no—
192C22×C4—621no—
193C22×C4—621no—
194C22×C4—621no—
195C22×C4—621no—
196C22×C4—621no—
197C22×C4C22×D3111yescharacteristic
198Q16—628no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—312no—
203C2×C8—621no—
204C2×C8—621no—
205C4⋊C4—21no—
206C4⋊C4—312no—
207C4⋊C4—624no—
208C4⋊C4—624no—
209C4⋊C4—621no—
210C4⋊C4C22⋊D3112yes—
211C22⋊C4—624no—
212C22⋊C4—624no—
213C23—312no—
214C23—621no—
215C23—621no—
216C23—621no—
217C23—621no—
218C23—621no—
219C23—621no—
220C23D4×D3111yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—622no—
226Q8—622no—
227Q8—1241no—
228Q8—1241no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—622no—
236C2×C4—621no—
237C2×C4—621no—
238C2×C4—622no—
239C2×C4—621no—
240C2×C4—1241no—
241C2×C4—1241no—
242C2×C4—1241no—
243C2×C4—1241no—
244C2×C4—1241no—
245C2×C4—1241no—
246C2×C4—1241no—
247C2×C4—1241no—
248C2×C4C2×D12411yescharacteristic
249C2×C4#9249511yescharacteristic
250C2×C4D4×D3111yescharacteristic
251C2×C4D4×D3111yescharacteristic
252C2×C4D4×D3111yescharacteristic
253C2×C4C2×C22⋊D3111yescharacteristic
254C8—41no—
255C8—622no—
256C22—622no—
257C22—622no—
258C22—622no—
259C22—1241no—
260C22—1241no—
261C22—1241no—
262C22—1241no—
263C22#1175845911yescharacteristic
264C22#1175846111yescharacteristic
265C22#1175849211yescharacteristic
266C22#1175849311yescharacteristic
267C22Q16×D3111yescharacteristic
268C22#1175849511yescharacteristic
269C22#1175851411yescharacteristic
270C4—42no—
271C4—42no—
272C4—1241no—
273C4—1242no—
274C4—1242no—
275C4#1175845911yescharacteristic
276C4#1175846111yescharacteristic
277C4#1175850711yescharacteristic
278C4#1175851711yescharacteristic
279C2—1242no—
280C2#1452095811yescharacteristic
281C2#1452120311yescharacteristic
282C2#1452120611yescharacteristic
283C2#1452120811yescharacteristic
284C2#1452121411yescharacteristic
285C2#1452128911yescharacteristic
286C2#1452130611yescharacteristic
subgroupG/H
centerC23D4×D31
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC2×C31×#103C2
socleC23×C31C2×D4
prime radicalC23×D31D4
hypercenterC2×#103D31
hypocenterC31#2001
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#92482C2semidirect
4. C2×#11758404C2semidirect
6. #14520811C2semidirect
7. C2×C31×#103C2semidirect
76. C22×D31Q16semidirect
140. C2×#103D31semidirect
172. C31#2001semidirect

Maximal subgroups

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

subgroupindex
C22×#924822
C2×#117584742
C2×#117585232
C2×#117584042
C2×(C2×D31)⋊C82
#145208112
C2×C31×#1032
#200131