Finite Group Databasebeta

← #16848923

Subgroups — #16848923

Order 3968.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×#92497C211yesmaximal, characteristic, complemented
2C2×D4×Dic31C212yesmaximal, complemented
3C2×(C22⋊D31)⋊C4C211yesmaximal, characteristic, complemented
4#14520970C211yesmaximal, characteristic, complemented
5#14520965C212yesmaximal, complemented
6#14520963C211yesmaximal, characteristic, complemented
7#14520961C211yesmaximal, characteristic, complemented
8#14520957C212yesmaximal, complemented
9C22⋊C4×Dic31C212yesmaximal, complemented
10#14520896C211yesmaximal, characteristic, complemented
11C31×#392C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C22×D4×C31C2212yes—
14C22×C22⋊D31C2212yes—
15C22×C22⋊D31C2211yescharacteristic
16C23×Dic31C2211yescharacteristic
17C23×Dic31C2212yes—
18C2×#92497—28no—
19C2×#92497—24no—
20C22×C4×D31C2211yescharacteristic
21C22×Dic62C2211yescharacteristic
22C2×C22⋊C4×C31C2212yes—
23C2×C22⋊C4×C31C2211yescharacteristic
24C2×C22⋊Dic31C2212yes—
25C2×C22⋊Dic31C2212yes—
26C2×C22⋊Dic31C2212yes—
27D4×Dic31—28no—
28C2×(C2×D31)⋊C4C2212yes—
29C2×(C2×D31)⋊C4C2211yescharacteristic
30C2×C4⋊Dic31C2212yes—
31C2×Dic31⋊C4C2211yescharacteristic
32C2×C4×Dic31C2212yes—
33C2×C4×Dic31C2211yescharacteristic
34(C22⋊D31)⋊C4—24no—
35C31×#95C2211yescharacteristic
36C31⋊#95C2211yescharacteristic
37C31⋊#95C2212yes—
38C31⋊#95C2212yes—
39C24×C31C2312yes—
40C24×C31C2311yescharacteristic
41C23×D31C2311yescharacteristic
42C2×D4×C31—24no—
43C2×D4×C31—24no—
44C2×D4×C31—24no—
45C2×D4×C31—22no—
46C2×D4×C31—24no—
47C22×C4×C31C2311yescharacteristic
48C22×C4×C31C2311yescharacteristic
49C22×C4×C31C2312yes—
50C2×C22⋊D31—24no—
51C2×C22⋊D31—22no—
52C2×C22⋊D31—22no—
53C2×C22⋊D31—24no—
54C2×C22⋊D31—24no—
55C2×C22⋊D31—22no—
56C22×Dic31—24no—
57C22×Dic31—24no—
58C22×Dic31—22no—
59C22×Dic31—24no—
60C22×Dic31D418yescomplemented
61C22×Dic31C2312yes—
62C22×Dic31C2311yescharacteristic
63C22×Dic31C2312yes—
64C22×Dic31C2311yescharacteristic
65C22×Dic31C2311yescharacteristic
66#92497—232no—
67C2×C4×D31—24no—
68C2×C4×D31D414yescomplemented
69C2×Dic62—22no—
70C2×Dic62—24no—
71C22⋊C4×C31—22no—
72C22⋊C4×C31—24no—
73C22⋊Dic31—24no—
74C22⋊Dic31—24no—
75C22⋊Dic31—24no—
76(C2×D31)⋊C4—22no—
77(C2×D31)⋊C4—24no—
78C4⋊Dic31—24no—
79Dic31⋊C4—22no—
80C4×Dic31—22no—
81C4×Dic31—24no—
82C23×C31—22no—
83C23×C31—21no—
84C23×C31—22no—
85C23×C31—22no—
86C23×C31—22no—
87C23×C31—22no—
88C23×C31—21no—
89C23×C31—22no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31C2×D414yes—
93C23×C31#4112yescomplemented
94C23×C31C2411yescharacteristic
95C22×D31—22no—
96C22×D31—22no—
97C22×D31—21no—
98C22×D31—21no—
99C22×D31C2×D412yes—
100D4×C31—28no—
101D4×C31—216no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31—21no—
106C2×C4×C31—22no—
107C2×C4×C31C2×D412yes—
108C2×C4×C31#4114yescomplemented
109C22⋊D31—216no—
110C22⋊D31—28no—
111C2×Dic31—21no—
112C2×Dic31—22no—
113C2×Dic31—22no—
114C2×Dic31—21no—
115C2×Dic31—216no—
116C2×Dic31—22no—
117C2×Dic31—21no—
118C2×Dic31—22no—
119C2×Dic31—22no—
120C2×Dic31—22no—
121C2×Dic31—44no—
122C2×Dic31C2×D414yes—
123C2×Dic31C2×D414yes—
124C2×Dic31C2×D412yes—
125C2×Dic31#4112yescomplemented
126C4×D31—28no—
127Dic62—28no—
128#2388—311nomaximal, Sylow
129C22×C31—24no—
130C22×C31—22no—
131C22×C31—24no—
132C22×C31—24no—
133C22×C31—24no—
134C22×C31—41no—
135C22×C31—42no—
136C22×C31—42no—
137C22×C31—42no—
138C22×C31—42no—
139C22×C31—41no—
140C22×C31—42no—
141C22×C31C22×D411yescharacteristic
142C22×C31C22×D412yes—
143C22×C31C2×#4111yescharacteristic
144C22×C31C2×#4112yes—
145C22×C31C2×#4111yescharacteristic
146C2×D31—22no—
147C2×D31—22no—
148C2×D31—22no—
149C2×D31—42no—
150C2×D31—42no—
151C4×C31—22no—
152C4×C31—24no—
153C4×C31—41no—
154Dic31—24no—
155Dic31—22no—
156Dic31—41no—
157Dic31—42no—
158Dic31#12014yes—
159C22×#41—311no—
160C2×C4×D4—312no—
161C2×C4×D4—311no—
162#395—311no—
163#394—311no—
164#394—312no—
165#393—311no—
166#392—312no—
167#392D3111yescharacteristic, complemented
168C4×C22⋊C4—312no—
169C4×C22⋊C4—311no—
170C2×C31—44no—
171C2×C31—42no—
172C2×C31C2×#12011yescharacteristic
173C2×C31#52512yes—
174C2×C31#52511yescharacteristic
175C2×C31#53212yes—
176C2×C31#53211yescharacteristic
177D31—42no—
178C2×#41—628no—
179C2×#41—624no—
180C22×Q8—311no—
181C22×D4—311no—
182C22×D4—312no—
183C22×D4C2×D3112yes—
184C22×D4C2×D3111yescharacteristic
185C23×C4—312no—
186C23×C4—311no—
187C23×C4—311no—
188C4×D4—624no—
189C4×D4—628no—
190C2×C4⋊C4—311no—
191C2×C4⋊C4—312no—
192C2×C22⋊C4—312no—
193C2×C22⋊C4—312no—
194C2×C22⋊C4—312no—
195C2×C22⋊C4—311no—
196C2×C22⋊C4—312no—
197C2×C22⋊C4C2×D3111yescharacteristic
198C2×C22⋊C4C2×D3112yes—
199C2×C42—311no—
200C2×C42—312no—
201#95—312no—
202#95—312no—
203#95—311no—
204#95C2×D3111yescharacteristic
205C31#238811yescharacteristic, complemented, Sylow
206C24—311no—
207C24C22×D3112yes—
208C24C22×D3111yescharacteristic
209#41—6232no—
210C2×Q8—624no—
211C2×Q8—622no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—622no—
218C2×D4—624no—
219C2×D4—622no—
220C2×D4—624no—
221C2×D4—624no—
222C2×D4—622no—
223C22×C4—311no—
224C22×C4—318no—
225C22×C4—311no—
226C22×C4—312no—
227C22×C4—314no—
228C22×C4—312no—
229C22×C4—311no—
230C22×C4—624no—
231C22×C4—624no—
232C22×C4—624no—
233C22×C4—622no—
234C22×C4—624no—
235C22×C4C22×D3111yescharacteristic
236C22×C4C22×D3111yescharacteristic
237C22×C4C22×D3112yes—
238C4⋊C4—624no—
239C4⋊C4—622no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—622no—
243C22⋊C4—624no—
244C22⋊C4—624no—
245C22⋊C4—624no—
246C22⋊C4—624no—
247C42—624no—
248C42—622no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—312no—
260C23—622no—
261C23—622no—
262C23—621no—
263C23—621no—
264C23D4×D3114yes—
265C23#9249712yescomplemented
266C23C23×D3111yescharacteristic
267Q8—628no—
268D4—216no—
269D4—28no—
270D4—628no—
271D4—6216no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—314no—
278C2×C4—312no—
279C2×C4—314no—
280C2×C4—312no—
281C2×C4—628no—
282C2×C4—6216no—
283C2×C4—621no—
284C2×C4—622no—
285C2×C4—622no—
286C2×C4—621no—
287C2×C4—622no—
288C2×C4—622no—
289C2×C4—621no—
290C2×C4—622no—
291C2×C4—622no—
292C2×C4—1244no—
293C2×C4D4×D3112yes—
294C2×C4#9249714yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—622no—
308C22—622no—
309C22—622no—
310C22—1242no—
311C22—1242no—
312C22C2×D4×D3112yes—
313C22C2×D4×D3111yescharacteristic
314C22C2×#9249711yescharacteristic
315C22C2×#9249712yes—
316C22#41×D3111yescharacteristic
317C4—22no—
318C4—24no—
319C4—41no—
320C4—314no—
321C4—624no—
322C4—622no—
323C4—1241no—
324C4—1242no—
325C2—44no—
326C2—42no—
327C2—1242no—
328C2#120×D3111yescharacteristic
329C2#1452155512yes—
330C2#1452155911yescharacteristic
331C2#1452159412yes—
332C2#1452159711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#392C2
hypercenter#392D31
hypocenterC31#2388
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#92497C2semidirect
2. C2×D4×Dic31C2semidirect
3. C2×(C22⋊D31)⋊C4C2semidirect
4. #14520970C2semidirect
5. #14520965C2semidirect
6. #14520963C2semidirect
7. #14520961C2semidirect
8. #14520957C2semidirect
9. C22⋊C4×Dic31C2semidirect
10. #14520896C2semidirect
11. C31×#392C2semidirect
60. C22×Dic31D4semidirect
68. C2×C4×D31D4semidirect
93. C23×C31#41semidirect
108. C2×C4×C31#41semidirect
125. C2×Dic31#41semidirect
167. #392D31semidirect
205. C31#2388semidirect
265. C23#92497semidirect
294. C2×C4#92497semidirect

Maximal subgroups

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

subgroupindex
C22×#924972
C2×D4×Dic312
C2×(C22⋊D31)⋊C42
#145209702
#145209652
#145209632
#145209612
#145209572
C22⋊C4×Dic312
#145208962
C31×#3922
#238831