Finite Group Databasebeta

← #16846714

Subgroups — #16846714

Order 3968.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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×#92480C211yesmaximal, characteristic, complemented
2C2×#11758501C211yesmaximal, characteristic, complemented
3C2×#11758398C211yesmaximal, characteristic, complemented
4C2×#11758420C211yesmaximal, characteristic, complemented
5C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
6#14520811C211yesmaximal, characteristic, complemented
7C2×C31×#121C211yesmaximal, characteristic
8C22×D4×C31—21no—
9C22×D4×C31C2211yescharacteristic
10C23×C4×C31C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C31×#121—22no—
13C31×#121—22no—
14C2×C4⋊C4×C31C2211yescharacteristic
15C2×C22⋊C4×C31—21no—
16C2×C22⋊Dic31—21no—
17C2×#92480—22no—
18C2×#92480—22no—
19C2×#92480—22no—
20C2×C4⋊Dic31C2211yescharacteristic
21#11758501—22no—
22#11758501—22no—
23C2×Dic31⋊C4—21no—
24C22×C31⋊C8C2211yescharacteristic
25C22×C31⋊C8C2211yescharacteristic
26#11758420—22no—
27C22⋊(C31⋊C8)—22no—
28#11758398—22no—
29C24×C31—21no—
30C24×C31D411yescharacteristic
31C2×D4×C31—24no—
32C2×D4×C31—21no—
33C2×D4×C31—21no—
34C2×D4×C31—21no—
35C2×D4×C31—21no—
36C2×D4×C31—21no—
37C2×D4×C31—42no—
38C2×D4×C31—42no—
39C2×D4×C31D412yes—
40C22×C4×C31—21no—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31—21no—
44C22×C4×C31—21no—
45C22×C4×C31—21no—
46C22×C4×C31—21no—
47C22×C4×C31D411yescharacteristic
48C22×C4×C31C2311yescharacteristic
49C22×Dic31—21no—
50C22×Dic31—21no—
51C2×Dic62—21no—
52C2×Dic62—21no—
53C2×Dic62—21no—
54C2×Dic62—21no—
55C2×Dic62—21no—
56C2×Dic62D412yescomplemented
57C4⋊C4×C31—21no—
58C4⋊C4×C31D412yes—
59C22⋊C4×C31—24no—
60C22⋊Dic31—24no—
61#92480—28no—
62C4⋊Dic31—21no—
63C4⋊Dic31D412yescomplemented
64Dic31⋊C4—24no—
65C2×C31⋊C8—21no—
66C2×C31⋊C8—21no—
67C2×C31⋊C8—21no—
68C2×C31⋊C8—21no—
69C2×C31⋊C8—21no—
70C2×C31⋊C8D412yescomplemented
71C23×C31—21no—
72C23×C31—21no—
73C23×C31—21no—
74C23×C31—21no—
75C23×C31—21no—
76C23×C31—21no—
77C23×C31—22no—
78C23×C31—41no—
79C23×C31—41no—
80C23×C31—41no—
81C23×C31—41no—
82C23×C31—41no—
83C23×C31—41no—
84C23×C31SD1612yescomplemented
85C23×C31C2×D411yescharacteristic
86D4×C31—22no—
87D4×C31—22no—
88D4×C31—48no—
89D4×C31—41no—
90D4×C31—41no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—22no—
94C2×C4×C31—21no—
95C2×C4×C31—41no—
96C2×C4×C31—41no—
97C2×C4×C31—41no—
98C2×C4×C31—41no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2×D411yescharacteristic
104C2×C4×C31C2×D411yescharacteristic
105C2×C4×C31C2×D411yescharacteristic
106C2×C4×C31#4111yescharacteristic
107C2×Dic31—22no—
108C2×Dic31—22no—
109C2×Dic31—41no—
110C2×Dic31—41no—
111C2×Dic31—41no—
112C2×Dic31—41no—
113Dic62—22no—
114Dic62—22no—
115Dic62—41no—
116Dic62—41no—
117C31⋊C8—22no—
118C31⋊C8—41no—
119#2000—311nomaximal, Sylow
120C22×C31—22no—
121C22×C31—22no—
122C22×C31—22no—
123C22×C31—41no—
124C22×C31—41no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—42no—
128C22×C31—42no—
129C22×C31—42no—
130C22×C31—81no—
131C22×C31—81no—
132C22×C31—81no—
133C22×C31—81no—
134C22×C31#12011yescharacteristic
135C22×C31#12011yescharacteristic
136C22×C31#12111yescharacteristic
137C22×C31C2×SD1611yescharacteristic
138C22×C31#13511yescharacteristic
139C22×C31AGL(1,ℤ/8)11yescharacteristic
140C22×C31#13711yescharacteristic
141C4×C31—41no—
142C4×C31—42no—
143C4×C31#12011yescharacteristic
144C4×C31#12111yescharacteristic
145C4×C31#12111yescharacteristic
146C4×C31#12711yescharacteristic
147Dic31—42no—
148Dic31—42no—
149C22×SD16—311no—
150C2×C22⋊Q8—311no—
151C2×#121D3111yescharacteristic
152C2×#103—311no—
153C2×#102—311no—
154C2×C22⋊C8—311no—
155#340—311no—
156C2×C31—42no—
157C2×C31—82no—
158C2×C31#39211yescharacteristic
159C2×C31#44811yescharacteristic
160C2×C31#44911yescharacteristic
161C2×C31#45011yescharacteristic
162C2×C31#45211yescharacteristic
163C2×C31#46511yescharacteristic
164C2×C31#46811yescharacteristic
165C22×Q8—311no—
166C22×D4—21no—
167C22×D4C2×D3111yescharacteristic
168C23×C4C2×D3111yescharacteristic
169C2×SD16—622no—
170C2×SD16—622no—
171C2×SD16—622no—
172C22×C8—311no—
173C22×C8—311no—
174C22⋊Q8—622no—
175C22⋊Q8—622no—
176#121—22no—
177#121—22no—
178C2×C4⋊C4—311no—
179C2×C4⋊C4—621no—
180C2×C4⋊C4C2×D3111yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—621no—
183#103—622no—
184#102—622no—
185C22⋊C8—622no—
186C31#200011yescharacteristic, complemented, Sylow
187C24—21no—
188C24C22⋊D3111yescharacteristic
189C2×Q8—312no—
190C2×Q8—621no—
191C2×Q8—621no—
192C2×Q8—621no—
193C2×Q8—621no—
194C2×Q8—621no—
195C2×D4—24no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4C22⋊D3112yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—621no—
212C22×C4—621no—
213C22×C4C22⋊D3111yescharacteristic
214C22×C4C22×D3111yescharacteristic
215SD16—628no—
216C2×C8—312no—
217C2×C8—621no—
218C2×C8—621no—
219C2×C8—621no—
220C2×C8—621no—
221C2×C8—621no—
222C4⋊C4—21no—
223C4⋊C4—312no—
224C4⋊C4—624no—
225C4⋊C4—621no—
226C4⋊C4C22⋊D3112yes—
227C22⋊C4—24no—
228C22⋊C4—624no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#9248012yescomplemented
243C23C2×C22⋊D3111yescharacteristic
244Q8—622no—
245Q8—622no—
246Q8—1241no—
247Q8—1241no—
248D4—22no—
249D4—22no—
250D4—48no—
251D4—41no—
252D4—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—622no—
264C2×C4—622no—
265C2×C4—1241no—
266C2×C4—1241no—
267C2×C4—1241no—
268C2×C4—1241no—
269C2×C4D4×D3111yescharacteristic
270C2×C4D4×D3111yescharacteristic
271C2×C4D4×D3111yescharacteristic
272C2×C4#9249711yescharacteristic
273C2×C4C2×C22⋊D3111yescharacteristic
274C2×C4C2×C22⋊D3111yescharacteristic
275C8—622no—
276C8—1241no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22#1175850911yescharacteristic
292C22C2×#9248011yescharacteristic
293C22#1175851411yescharacteristic
294C22#1175851611yescharacteristic
295C22#1175852711yescharacteristic
296C22#1175852811yescharacteristic
297C22#1175853011yescharacteristic
298C4—42no—
299C4—41no—
300C4—1242no—
301C4—1242no—
302C4#1175851411yescharacteristic
303C4#1175851511yescharacteristic
304C4#1175851611yescharacteristic
305C4#1175851711yescharacteristic
306C2—42no—
307C2—82no—
308C2#1452095711yescharacteristic
309C2#1452118711yescharacteristic
310C2#1452119211yescharacteristic
311C2#1452120011yescharacteristic
312C2#1452121111yescharacteristic
313C2#1452127611yescharacteristic
314C2#1452129311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
Fitting, prime radicalC2×C31×#121C2
socleC23×C31C2×D4
hypercenterC2×#121D31
hypocenterC31#2000
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#92480C2semidirect
2. C2×#11758501C2semidirect
3. C2×#11758398C2semidirect
4. C2×#11758420C2semidirect
5. C2×C22⋊(C31⋊C8)C2semidirect
6. #14520811C2semidirect
56. C2×Dic62D4semidirect
63. C4⋊Dic31D4semidirect
70. C2×C31⋊C8D4semidirect
84. C23×C31SD16semidirect
186. C31#2000semidirect
242. C23#92480semidirect

Maximal subgroups

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

subgroupindex
C22×#924802
C2×#117585012
C2×#117583982
C2×#117584202
C2×C22⋊(C31⋊C8)2
#145208112
C2×C31×#1212
#200031