Finite Group Databasebeta

← #16858846

Subgroups — #16858846

Order 3968.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#135×D31C211yesmaximal, characteristic, complemented
3#14521491C212yesmaximal, complemented
4#14521490C211yesmaximal, characteristic, complemented
5#14521487C211yesmaximal, characteristic, complemented
6#14521486C212yesmaximal, complemented
7#14521484C211yesmaximal, characteristic, complemented
8#14521482C212yesmaximal, complemented
9#14521480C212yesmaximal, complemented
10M32×D31C211yesmaximal, characteristic, complemented
11C31×#510C211yesmaximal, characteristic, complemented
12#41×D31—21no—
13C2×#92499—21no—
14C31×#135C2211yescharacteristic
15C2×Q16×C31C2211yescharacteristic, complemented
16#11758527—21no—
17C2×#92481—21no—
18#11758496—21no—
19#11758496—21no—
20#11758496C2212yes—
21#11758496C2212yescomplemented
22#11758496C2211yescharacteristic, complemented
23Q16×D31C2211yescharacteristic, complemented
24#11758493—21no—
25SD16×D31—21no—
26#11758489C2211yescharacteristic, complemented
27D8×D31C2211yescharacteristic, complemented
28#11758481C2211yescharacteristic
29C2×D248C2211yescharacteristic, complemented
30C2×C8×D31C2211yescharacteristic
31C31×Q32C2212yescomplemented
32C31×SD32C2212yescomplemented
33C31×M32C2211yescharacteristic, complemented
34#11758417C2212yescomplemented
35#11758416C2212yescomplemented
36#11758416C2212yescomplemented
37#11758414C2212yescomplemented
38#11758400C2211yescharacteristic, complemented
39#11758388C2212yescomplemented
40D496C2212yescomplemented
41#11758386C2212yescomplemented
42C16×D31C2212yescomplemented
43#41×C31—21no—
44C2×Q8×C31—21no—
45#92499—21no—
46#92499—22no—
47#92499—22no—
48#92499—41no—
49#92499—41no—
50Q8×D31—21no—
51#92497—21no—
52#92497—41no—
53D4×D31—21no—
54D4×D31—41no—
55#92495—21no—
56#92495—41no—
57C2×D124—21no—
58C2×D124—41no—
59C2×C4×D31—41no—
60C2×C4×D31—41no—
61C2×C4×D31D411yescharacteristic
62Q16×C31—21no—
63Q16×C31C2311yescharacteristic
64Q16×C31C2312yes—
65SD16×C31—21no—
66D8×C31C2311yescharacteristic
67C2×C8×C31C2311yescharacteristic
68#92482—21no—
69#92481—21no—
70#92481—22no—
71#92481—22no—
72#92480—21no—
73#92479—21no—
74C2×C31⋊C8D411yescharacteristic
75Dic124C2311yescharacteristic
76D248—21no—
77D248C2311yescharacteristic
78D248C2312yes—
79#92470—21no—
80C8×D31D411yescharacteristic
81C8×D31D411yescharacteristic
82C8×D31C2311yescharacteristic
83C8×D31C2311yescharacteristic
84C16×C31C2312yes—
85C31⋊C16C2312yes—
86C22×D31—41no—
87C22×D31—41no—
88C22×D31D811yescharacteristic
89Q8×C31—22no—
90Q8×C31—21no—
91Q8×C31—41no—
92D4×C31—21no—
93D4×C31—41no—
94C2×C4×C31—41no—
95C2×C4×C31—41no—
96C2×C4×C31C2×D411yescharacteristic
97C22⋊D31—41no—
98C22⋊D31—41no—
99C22⋊D31—41no—
100C2×Dic31—41no—
101C2×Dic31D811yescharacteristic
102D124—21no—
103D124—22no—
104D124—42no—
105D124—42no—
106D124—41no—
107D124—41no—
108C4×D31—41no—
109C4×D31—41no—
110C4×D31—41no—
111C4×D31—42no—
112C4×D31—42no—
113C4×D31D811yescharacteristic
114C4×D31D811yescharacteristic
115C4×D31C2×D411yescharacteristic
116C4×D31C2×D411yescharacteristic
117Dic62—21no—
118Dic62—41no—
119C8×C31C2×D411yescharacteristic
120C8×C31C2411yescharacteristic
121C31⋊C8C2×D411yescharacteristic
122C31⋊C8C2×D411yescharacteristic
123#3414—311nomaximal, Sylow
124C22×C31—41no—
125C22×C31C2×D811yescharacteristic
126C2×D31—41no—
127C2×D31—42no—
128C2×D31—41no—
129C2×D31—82no—
130C2×D31—82no—
131C2×D31C2×D811yescharacteristic
132C2×D31C2×D811yescharacteristic
133C4×C31—41no—
134C4×C31—42no—
135C4×C31C2×D811yescharacteristic
136C4×C31C22×D411yescharacteristic
137Dic31—41no—
138Dic31C2×D811yescharacteristic
139Dic31C2×D811yescharacteristic
140C2×#135—311no—
141C2×#135—311no—
142#510—311no—
143#510—312no—
144#510D3111yescharacteristic, complemented
145#509—311no—
146#509—312no—
147#509—311no—
148#508—312no—
149#508—312no—
150C2×M32—311no—
151C2×C31—21no—
152C2×C31—81no—
153C2×C31C22×D811yescharacteristic
154D31—21no—
155D31—21no—
156D31—82no—
157D31—81no—
158C2×#41—621no—
159C2×#41—621no—
160#135—312no—
161#135—311no—
162#135—312no—
163#135—311no—
164#135—311no—
165#135—621no—
166#135—621no—
167#135—621no—
168#135—621no—
169#135C2×D3111yescharacteristic
170C2×Q16—311no—
171C2×Q16C2×D3111yescharacteristic, complemented
172C2×SD16—621no—
173C2×SD16—621no—
174C2×D8—311no—
175C2×D8—311no—
176C22×C8—311no—
177Q32—312no—
178Q32C2×D3112yescomplemented
179SD32—312no—
180SD32—312no—
181SD32—312no—
182SD32C2×D3112yescomplemented
183D16—312no—
184D16—312no—
185M32—311no—
186M32—312no—
187M32C2×D3111yescharacteristic, complemented
188C2×C16—312no—
189C31#341411yescharacteristic, complemented, Sylow
190#41—21no—
191#41—621no—
192#41—622no—
193#41—621no—
194#41—622no—
195#41—621no—
196#41—1241no—
197#41—1241no—
198#41—1241no—
199#41—1241no—
200C2×Q8—21no—
201C2×Q8—621no—
202C2×D4—621no—
203C2×D4—621no—
204C2×D4—1241no—
205C2×D4—1241no—
206C22×C4—311no—
207C22×C4—1241no—
208C22×C4—1241no—
209Q16—21no—
210Q16—311no—
211Q16—621no—
212Q16C22×D3111yescharacteristic
213Q16C22×D3112yes—
214SD16—21no—
215SD16—621no—
216SD16—622no—
217SD16—621no—
218SD16—622no—
219SD16—621no—
220D8—311no—
221D8—312no—
222D8—621no—
223D8—621no—
224D8C22×D3111yescharacteristic
225C2×C8—311no—
226C2×C8—311no—
227C2×C8—311no—
228C2×C8—311no—
229C2×C8—311no—
230C2×C8C22×D3111yescharacteristic
231C16—312no—
232C16C22×D3112yes—
233C23—311no—
234C23—1241no—
235C23—1241no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—621no—
240Q8—1241no—
241D4—21no—
242D4—41no—
243D4—621no—
244D4—622no—
245D4—1241no—
246D4—1242no—
247D4—1242no—
248D4—1241no—
249D4—1241no—
250D4—1241no—
251D4—1241no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—311no—
255C2×C4—311no—
256C2×C4—311no—
257C2×C4—311no—
258C2×C4—311no—
259C2×C4—1241no—
260C2×C4—1242no—
261C2×C4—1242no—
262C2×C4—1241no—
263C2×C4—1241no—
264C2×C4—1241no—
265C2×C4D4×D3111yescharacteristic
266C8—311no—
267C8—311no—
268C8D4×D3111yescharacteristic
269C8C23×D3111yescharacteristic
270C22—41no—
271C22—311no—
272C22—311no—
273C22—1241no—
274C22—1242no—
275C22—1241no—
276C22—2482no—
277C22—2482no—
278C22D8×D3111yescharacteristic
279C4—41no—
280C4—42no—
281C4—311no—
282C4—311no—
283C4—1241no—
284C4D8×D3111yescharacteristic
285C4C2×D4×D3111yescharacteristic
286C2—21no—
287C2—81no—
288C2—621no—
289C2—621no—
290C2—2482no—
291C2—2481no—
292C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC2C2×D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#510C2
socleC2×C31C22×D8
hypercenter#510D31
hypocenterC31#3414
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#11758496C2semidirect
2. #135×D31C2semidirect
3. #14521491C2semidirect
4. #14521490C2semidirect
5. #14521487C2semidirect
6. #14521486C2semidirect
7. #14521484C2semidirect
8. #14521482C2semidirect
9. #14521480C2semidirect
10. M32×D31C2semidirect
11. C31×#510C2semidirect
15. C2×Q16×C31C22semidirect
21. #11758496C22semidirect
22. #11758496C22semidirect
23. Q16×D31C22semidirect
26. #11758489C22semidirect
27. D8×D31C22semidirect
29. C2×D248C22semidirect
31. C31×Q32C22semidirect
32. C31×SD32C22semidirect
33. C31×M32C22semidirect
34. #11758417C22semidirect
35. #11758416C22semidirect
36. #11758416C22semidirect
37. #11758414C22semidirect
38. #11758400C22semidirect
39. #11758388C22semidirect
40. D496C22semidirect
41. #11758386C22semidirect
42. C16×D31C22semidirect
144. #510D31semidirect
171. C2×Q16C2×D31semidirect
178. Q32C2×D31semidirect
182. SD32C2×D31semidirect
187. M32C2×D31semidirect
189. C31#3414semidirect

Maximal subgroups

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

subgroupindex
C2×#117584962
#135×D312
#145214912
#145214902
#145214872
#145214862
#145214842
#145214822
#145214802
M32×D312
C31×#5102
#341431