Finite Group Databasebeta

← #16855294

Subgroups — #16855294

Order 3968.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11758496C211yesmaximal, characteristic, complemented
2C22×C8×D31C211yesmaximal, characteristic, complemented
3#14521562C212yesmaximal, complemented
4#14521287C211yesmaximal, characteristic, complemented
5#14521286C211yesmaximal, characteristic, complemented
6#14521285C211yesmaximal, characteristic, complemented
7#14521281C212yesmaximal, complemented
8#14521280C212yesmaximal, complemented
9#14521086C211yesmaximal, characteristic, complemented
10#14521044C212yesmaximal, complemented
11C31×#467C211yesmaximal, characteristic, complemented
12C2×#92499C2212yes—
13C22×C4×D31C2211yescharacteristic
14C2×Q16×C31C2211yescharacteristic, complemented
15C22×C8×C31C2211yescharacteristic
16C31×C22⋊Q8C2212yes—
17#11758523C2212yes—
18C2×#92481C2212yescomplemented
19#11758507C2212yes—
20C22×C31⋊C8C2211yescharacteristic
21#11758496—24no—
22C2×D248C2211yescharacteristic, complemented
23C2×C8×D31—24no—
24C2×C8×D31C2211yescharacteristic
25C2×C8×D31C2211yescharacteristic
26C2×C8×D31C2211yescharacteristic
27C2×C8×D31C2211yescharacteristic
28#11758472C2212yes—
29#11758471—22no—
30#11758470—22no—
31#11758469C2212yes—
32#11758459—22no—
33(C22⋊D31)⋊C4—22no—
34C31×#107C2211yescharacteristic, complemented
35C31×#103C2212yescomplemented
36#11758423C2212yescomplemented
37#11758409C2212yescomplemented
38#11758406C2211yescharacteristic, complemented
39#11758397C2212yescomplemented
40#11758396C2212yescomplemented
41C23×D31D411yescharacteristic
42C2×Q8×C31C2312yes—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—22no—
45C22×Dic31D411yescharacteristic
46#92499—24no—
47#92499—24no—
48C2×D124—22no—
49C2×D124—22no—
50C2×D124C2312yes—
51C2×C4×D31—22no—
52C2×C4×D31—22no—
53C2×C4×D31—22no—
54C2×C4×D31D411yescharacteristic
55C2×C4×D31D411yescharacteristic
56C2×C4×D31C2311yescharacteristic
57C2×C4×D31C2311yescharacteristic
58Q16×C31—22no—
59C2×C8×C31—22no—
60C2×C8×C31C2311yescharacteristic
61C2×C8×C31C2311yescharacteristic
62C4⋊C4×C31—22no—
63C4⋊C4×C31C2312yes—
64C22⋊C4×C31—22no—
65#92481—24no—
66(C2×D31)⋊C4—22no—
67(C2×D31)⋊C4—22no—
68(C2×D31)⋊C4—22no—
69(C2×D31)⋊C4—22no—
70C4⋊Dic31C2312yes—
71Dic31⋊C4—22no—
72C4×Dic31—22no—
73C2×C31⋊C8—22no—
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2311yescharacteristic
76D248—22no—
77C8×D31—22no—
78C8×D31—22no—
79C8×D31—22no—
80C8×D31D414yescomplemented
81C23×C31C2×D411yescharacteristic
82C22×D31—22no—
83C22×D31—22no—
84C22×D31—22no—
85C22×D31—22no—
86C22×D31C2×D411yescharacteristic
87C22×D31C2×D411yescharacteristic
88Q8×C31—22no—
89Q8×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—44no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31C2×D411yescharacteristic
100C2×Dic31C2×D411yescharacteristic
101D124—22no—
102D124—22no—
103D124—44no—
104D124—44no—
105C4×D31—22no—
106C4×D31—21no—
107C4×D31—21no—
108C4×D31—22no—
109C4×D31—44no—
110C4×D31C2×D412yes—
111C4×D31#4112yes—
112C8×C31—21no—
113C8×C31C2×D412yes—
114C31⋊C8—21no—
115C31⋊C8C2×D412yes—
116#3050—311nomaximal, Sylow
117C22×C31—21no—
118C22×C31—21no—
119C22×C31#13512yescomplemented
120C22×C31C22×D411yescharacteristic
121C2×D31—28no—
122C2×D31—21no—
123C2×D31—21no—
124C2×D31—21no—
125C2×D31—21no—
126C2×D31—42no—
127C2×D31—42no—
128C2×D31—42no—
129C2×D31#12112yes—
130C2×D31#13512yescomplemented
131C4×C31—21no—
132C4×C31—42no—
133C4×C31—42no—
134C4×C31C22×D411yescharacteristic
135C4×C31C2×#4111yescharacteristic
136Dic31—21no—
137Dic31—42no—
138Dic31#12112yes—
139C2×#135—311no—
140C23×C8—311no—
141#525—312no—
142#467—311no—
143#467D3111yescharacteristic, complemented
144#466—311no—
145#466—311no—
146#465—312no—
147#465—312no—
148#427—311no—
149#416—312no—
150C2×C31—22no—
151C2×C31C2×#12111yescharacteristic
152C2×C31C2×#13511yescharacteristic
153C2×C31C2×#13511yescharacteristic
154D31—22no—
155D31—22no—
156D31—82no—
157C2×#41—312no—
158C23×C4—311no—
159#135—624no—
160C2×Q16C2×D3111yescharacteristic, complemented
161C2×SD16—312no—
162C2×D8—311no—
163C22×C8—311no—
164C22×C8—311no—
165C22×C8—311no—
166C22×C8—311no—
167C22×C8—311no—
168C22×C8—624no—
169C22×C8C2×D3111yescharacteristic
170#123—622no—
171C22⋊Q8—312no—
172C22⋊Q8C2×D3112yes—
173#121—312no—
174#121—312no—
175#120—622no—
176C4×D4—622no—
177C4×D4—622no—
178#117—312no—
179#107—311no—
180#107C2×D3111yescharacteristic, complemented
181#106—312no—
182#103—312no—
183#103C2×D3112yescomplemented
184#102—312no—
185#102—312no—
186C31#305011yescharacteristic, complemented, Sylow
187C24—311no—
188#41—624no—
189#41—624no—
190C2×Q8C22×D3112yes—
191C2×D4—312no—
192C2×D4—622no—
193C2×D4—622no—
194C2×D4—622no—
195C22×C4—311no—
196C22×C4—311no—
197C22×C4—311no—
198C22×C4—311no—
199C22×C4—311no—
200C22×C4—622no—
201C22×C4—622no—
202C22×C4—622no—
203C22×C4C22×D3111yescharacteristic
204Q16—22no—
205SD16—624no—
206D8—622no—
207C2×C8—22no—
208C2×C8—311no—
209C2×C8—314no—
210C2×C8—311no—
211C2×C8—622no—
212C2×C8—622no—
213C2×C8—622no—
214C2×C8—622no—
215C2×C8C22×D3111yescharacteristic
216C2×C8C22×D3111yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—312no—
219C4⋊C4—622no—
220C4⋊C4C22×D3112yes—
221C22⋊C4—22no—
222C22⋊C4—622no—
223C22⋊C4—622no—
224C22⋊C4—622no—
225C22⋊C4—622no—
226C42—622no—
227C23—311no—
228C23—311no—
229C23—622no—
230C23—622no—
231C23—622no—
232C23—622no—
233C23D4×D3111yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—622no—
237D4—622no—
238D4—1244no—
239D4—1244no—
240D4—1244no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—311no—
246C2×C4—312no—
247C2×C4—312no—
248C2×C4—311no—
249C2×C4—622no—
250C2×C4—622no—
251C2×C4—622no—
252C2×C4—621no—
253C2×C4—621no—
254C2×C4—622no—
255C2×C4—1244no—
256C2×C4D4×D3111yescharacteristic
257C2×C4C23×D3111yescharacteristic
258C8—21no—
259C8—312no—
260C8—621no—
261C8D4×D3112yes—
262C22—21no—
263C22—21no—
264C22—312no—
265C22—312no—
266C22—628no—
267C22—621no—
268C22—621no—
269C22—621no—
270C22—621no—
271C22—1242no—
272C22—1242no—
273C22—1242no—
274C22#1175849612yescomplemented
275C22C2×D4×D3111yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—312no—
280C4—621no—
281C4—1242no—
282C4C2×D4×D3111yescharacteristic
283C4#41×D3111yescharacteristic
284C2—22no—
285C2—622no—
286C2—622no—
287C2—2482no—
288C2#121×D3111yescharacteristic
289C2#135×D3111yescharacteristic
290C2C2×#1175849611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#467C2
socleC22×C31C22×D4
hypercenter#467D31
hypocenterC31#3050
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#11758496C2semidirect
2. C22×C8×D31C2semidirect
3. #14521562C2semidirect
4. #14521287C2semidirect
5. #14521286C2semidirect
6. #14521285C2semidirect
7. #14521281C2semidirect
8. #14521280C2semidirect
9. #14521086C2semidirect
10. #14521044C2semidirect
11. C31×#467C2semidirect
14. C2×Q16×C31C22semidirect
18. C2×#92481C22semidirect
22. C2×D248C22semidirect
34. C31×#107C22semidirect
35. C31×#103C22semidirect
36. #11758423C22semidirect
37. #11758409C22semidirect
38. #11758406C22semidirect
39. #11758397C22semidirect
40. #11758396C22semidirect
80. C8×D31D4semidirect
119. C22×C31#135semidirect
130. C2×D31#135semidirect
143. #467D31semidirect
160. C2×Q16C2×D31semidirect
180. #107C2×D31semidirect
183. #103C2×D31semidirect
186. C31#3050semidirect
274. C22#11758496semidirect

Maximal subgroups

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

subgroupindex
C2×#117584962
C22×C8×D312
#145215622
#145212872
#145212862
#145212852
#145212812
#145212802
#145210862
#145210442
C31×#4672
#305031