Finite Group Databasebeta

← #16855291

Subgroups — #16855291

Order 3968.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#11758527C211yesmaximal, characteristic, complemented
2C22×C8×D31C211yesmaximal, characteristic, complemented
3#14521561C211yesmaximal, characteristic, complemented
4#14521556C211yesmaximal, characteristic, complemented
5#14521289C212yesmaximal, complemented
6#14521284C212yesmaximal, complemented
7#14521281C212yesmaximal, complemented
8#14521277C212yesmaximal, complemented
9#14521085C211yesmaximal, characteristic, complemented
10#14521046C211yesmaximal, characteristic, complemented
11C31×#416C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C22×C4×D31C2211yescharacteristic
15C22×C8×C31C2211yescharacteristic
16C31×#117C2211yescharacteristic
17#11758527—24no—
18#11758523C2212yes—
19C2×#92482C2211yescharacteristic, complemented
20C2×#92481C2211yescharacteristic, complemented
21#11758515C2212yes—
22#11758514—21no—
23#11758512—21no—
24C2×#92480C2211yescharacteristic, complemented
25C2×#92479C2211yescharacteristic, complemented
26C4×C22⋊D31—22no—
27#11758503C2211yescharacteristic
28C22×C31⋊C8C2211yescharacteristic
29C2×C8×D31—24no—
30C2×C8×D31C2212yes—
31C2×C8×D31C2212yes—
32#11758474C2212yes—
33#11758472C2212yes—
34#11758463—21no—
35#11758459—21no—
36C4×D124—22no—
37C31×#103C2212yescomplemented
38C31×#102C2212yescomplemented
39#11758409C2212yescomplemented
40#11758404C2212yescomplemented
41#11758396C2212yescomplemented
42#11758395C2212yescomplemented
43C23×D31D411yescharacteristic
44#41×C31—22no—
45#41×C31—22no—
46C2×Q8×C31C2311yescharacteristic
47C2×D4×C31—21no—
48C2×D4×C31C2311yescharacteristic
49C22×C4×C31—21no—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—21no—
52C2×C22⋊D31—22no—
53C22×Dic31D411yescharacteristic
54#92495—22no—
55#92495—22no—
56C2×D124—22no—
57C2×D124C2311yescharacteristic
58C2×C4×D31—22no—
59C2×C4×D31—22no—
60C2×C4×D31—21no—
61C2×C4×D31D412yes—
62C2×C4×D31C2312yes—
63C2×Dic62C2311yescharacteristic
64C2×C8×C31—22no—
65C2×C8×C31C2312yes—
66C4⋊C4×C31C2312yes—
67C22⋊C4×C31—21no—
68C42×C31—21no—
69C22⋊Dic31—21no—
70C22⋊Dic31—22no—
71#92482—22no—
72#92481—22no—
73#92480—22no—
74#92479—22no—
75(C2×D31)⋊C4—22no—
76(C2×D31)⋊C4—22no—
77(C2×D31)⋊C4—22no—
78C4⋊Dic31—22no—
79C4⋊Dic31C2312yes—
80Dic31⋊C4—22no—
81C4×Dic31—21no—
82C2×C31⋊C8D414yescomplemented
83C2×C31⋊C8C2311yescharacteristic
84C2×C31⋊C8C2311yescharacteristic
85C8×D31—24no—
86C8×D31—24no—
87C23×C31—21no—
88C23×C31C2×D411yescharacteristic
89C22×D31—22no—
90C22×D31—24no—
91C22×D31—21no—
92C22×D31C2×D412yes—
93Q8×C31—21no—
94Q8×C31—21no—
95D4×C31—21no—
96D4×C31—21no—
97D4×C31—42no—
98C2×C4×C31—21no—
99C2×C4×C31—22no—
100C2×C4×C31—42no—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31#4112yes—
104C2×C4×C31C2411yescharacteristic
105C22⋊D31—42no—
106C22⋊D31—44no—
107C2×Dic31—22no—
108C2×Dic31—22no—
109C2×Dic31—21no—
110C2×Dic31C2×D412yes—
111D124—21no—
112D124—21no—
113D124—44no—
114C4×D31—24no—
115C4×D31—22no—
116C4×D31—22no—
117C4×D31—42no—
118Dic62—21no—
119Dic62—21no—
120C8×C31—22no—
121C31⋊C8C2×D412yes—
122C31⋊C8C2×D412yes—
123#3050—311nomaximal, Sylow
124C22×C31—22no—
125C22×C31—41no—
126C22×C31—42no—
127C22×C31#12112yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—28no—
130C2×D31—22no—
131C2×D31—22no—
132C2×D31—41no—
133C2×D31—42no—
134C2×D31#13514yescomplemented
135C4×C31—41no—
136C4×C31—42no—
137C4×C31#12112yes—
138C4×C31C22×D411yescharacteristic
139C4×C31C2×#4111yescharacteristic
140Dic31—22no—
141Dic31—41no—
142Dic31—42no—
143C2×#135—311no—
144C23×C8—311no—
145#525—311no—
146#525—311no—
147#467—312no—
148#466—312no—
149#465—312no—
150#465—312no—
151#427—311no—
152#416—311no—
153#416D3111yescharacteristic, complemented
154C2×C31—22no—
155C2×C31—81no—
156C2×C31C2×#12111yescharacteristic
157C2×C31C2×#13512yes—
158D31—24no—
159D31—81no—
160C2×#41—311no—
161C2×#41C2×D3111yescharacteristic
162C23×C4—311no—
163#135—624no—
164C2×Q16—311no—
165C2×SD16—311no—
166C2×SD16—311no—
167C2×D8—311no—
168C22×C8—311no—
169C22×C8—312no—
170C22×C8—312no—
171C22×C8—624no—
172C22×C8C2×D3111yescharacteristic
173#123—621no—
174#123—621no—
175C22⋊Q8—312no—
176C22⋊Q8—312no—
177#121—312no—
178#121—312no—
179#120—621no—
180#120—621no—
181C4×D4—622no—
182C4×D4—622no—
183#117—311no—
184#117C2×D3111yescharacteristic
185#107—312no—
186#106—312no—
187#103—312no—
188#103C2×D3112yescomplemented
189#102—312no—
190#102C2×D3112yescomplemented
191C31#305011yescharacteristic, complemented, Sylow
192C24—311no—
193#41—22no—
194#41—22no—
195#41—622no—
196#41—622no—
197C2×Q8—311no—
198C2×Q8C22×D3111yescharacteristic
199C2×D4—21no—
200C2×D4—311no—
201C2×D4—622no—
202C2×D4—622no—
203C2×D4—621no—
204C2×D4C22×D3111yescharacteristic
205C22×C4—21no—
206C22×C4—311no—
207C22×C4—312no—
208C22×C4—312no—
209C22×C4—622no—
210C22×C4—622no—
211C22×C4—621no—
212C22×C4C22×D3111yescharacteristic
213Q16—622no—
214SD16—622no—
215SD16—622no—
216D8—622no—
217C2×C8—22no—
218C2×C8—314no—
219C2×C8—311no—
220C2×C8—311no—
221C2×C8—624no—
222C2×C8—624no—
223C2×C8C22×D3112yes—
224C4⋊C4—312no—
225C4⋊C4—622no—
226C4⋊C4—622no—
227C4⋊C4C22×D3112yes—
228C22⋊C4—21no—
229C22⋊C4—622no—
230C22⋊C4—622no—
231C22⋊C4—622no—
232C22⋊C4—622no—
233C22⋊C4—621no—
234C42—21no—
235C42—621no—
236C23—21no—
237C23—312no—
238C23—624no—
239C23—622no—
240C23—621no—
241C23D4×D3111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—621no—
245Q8—621no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—621no—
250D4—621no—
251D4—1244no—
252D4—1244no—
253D4—1242no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—312no—
258C2×C4—622no—
259C2×C4—624no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4—622no—
263C2×C4—621no—
264C2×C4—1242no—
265C2×C4#9249512yes—
266C2×C4D4×D3112yes—
267C2×C4D4×D3111yescharacteristic
268C2×C4C23×D3111yescharacteristic
269C8—22no—
270C8—312no—
271C8—312no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—314no—
276C22—628no—
277C22—622no—
278C22—622no—
279C22—1241no—
280C22—1242no—
281C22#1175846112yes—
282C22C2×D4×D3111yescharacteristic
283C4—41no—
284C4—42no—
285C4—622no—
286C4—1241no—
287C4—1242no—
288C4#1175846112yes—
289C4C2×#9249511yescharacteristic
290C4C2×D4×D3111yescharacteristic
291C2—22no—
292C2—81no—
293C2—624no—
294C2—2481no—
295C2C2×#1175846111yescharacteristic
296C2#135×D3112yes—
subgroupG/H
centerC2×C4D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#416C2
socleC22×C31C22×D4
hypercenter#416D31
hypocenterC31#3050
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. C22×C8×D31C2semidirect
3. #14521561C2semidirect
4. #14521556C2semidirect
5. #14521289C2semidirect
6. #14521284C2semidirect
7. #14521281C2semidirect
8. #14521277C2semidirect
9. #14521085C2semidirect
10. #14521046C2semidirect
11. C31×#416C2semidirect
19. C2×#92482C22semidirect
20. C2×#92481C22semidirect
24. C2×#92480C22semidirect
25. C2×#92479C22semidirect
37. C31×#103C22semidirect
38. C31×#102C22semidirect
39. #11758409C22semidirect
40. #11758404C22semidirect
41. #11758396C22semidirect
42. #11758395C22semidirect
82. C2×C31⋊C8D4semidirect
134. C2×D31#135semidirect
153. #416D31semidirect
188. #103C2×D31semidirect
190. #102C2×D31semidirect
191. C31#3050semidirect

Maximal subgroups

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

subgroupindex
C2×#117585272
C22×C8×D312
#145215612
#145215562
#145212892
#145212842
#145212812
#145212772
#145210852
#145210462
C31×#4162
#305031