Finite Group Databasebeta

← #16855540

Subgroups — #16855540

Order 3968.

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

normal subgroups only

This group has 398 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
1#14521680C211yesmaximal, characteristic, complemented
2#14521523C211yesmaximal, characteristic
3#14521352C212yesmaximal, complemented
4#14521348C212yesmaximal, complemented
5#14521338C212yesmaximal, complemented
6#14521328C212yesmaximal, complemented
7#14521085C211yesmaximal, characteristic, complemented
8#14521077C211yesmaximal, characteristic, complemented
9#14521055C211yesmaximal, characteristic
10C2×#11758397C211yesmaximal, characteristic
11C31×#428C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C2×M16×C31C2211yescharacteristic
14C31×#117C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16#11758503C2211yescharacteristic
17#11758503C2211yescharacteristic
18#11758501—22no—
19C22×C31⋊C8C2211yescharacteristic
20#11758470C2212yes—
21#11758470C2212yes—
22#11758467—21no—
23#11758467C2212yescomplemented
24#11758465—21no—
25#11758465C2212yescomplemented
26C22⋊C4×D31—22no—
27C4×D124—21no—
28C4×D124—21no—
29C31×#107C2212yescomplemented
30C31×#106C2212yescomplemented
31#11758409C2212yes—
32#11758409C2212yes—
33Dic31⋊C8C2212yescomplemented
34Dic31⋊C8C2212yescomplemented
35#11758397C2212yes—
36#11758397C2212yes—
37#11758396C2211yescharacteristic, complemented
38#11758396C2211yescharacteristic, complemented
39#11758395C2211yescharacteristic, complemented
40#11758395C2211yescharacteristic, complemented
41C23×D31—21no—
42C22×C4×C31—21no—
43C22×C4×C31C2311yescharacteristic
44C2×D124—21no—
45C2×D124—22no—
46C2×D124Q814yescomplemented
47C2×D124C2312yes—
48C2×C4×D31—22no—
49C2×C4×D31—22no—
50C2×Dic62—22no—
51M16×C31—22no—
52C2×C8×C31C2312yes—
53C4⋊C4×C31—21no—
54C4⋊C4×C31C2311yescharacteristic
55C4⋊C4×C31C2311yescharacteristic
56C4⋊C4×C31C2312yes—
57C22⋊C4×C31—21no—
58C42×C31—21no—
59C22⋊Dic31—21no—
60C22⋊Dic31D412yes—
61(C2×D31)⋊C4—22no—
62(C2×D31)⋊C4—22no—
63C4⋊Dic31—22no—
64C4⋊Dic31D411yescharacteristic
65C4⋊Dic31D411yescharacteristic
66C4⋊Dic31C2311yescharacteristic
67C4⋊Dic31C2311yescharacteristic
68Dic31⋊C4—22no—
69Dic31⋊C4—22no—
70C4×Dic31—21no—
71C4×Dic31C2312yes—
72C2×C31⋊C8—22no—
73C2×C31⋊C8C2311yescharacteristic
74C2×C31⋊C8C2311yescharacteristic
75C23×C31C2×D411yescharacteristic
76C22×D31—22no—
77C22×D31—24no—
78C22×D31—81no—
79C22×D31—81no—
80C2×C4×C31—22no—
81C2×C4×C31—21no—
82C2×C4×C31—21no—
83C2×C4×C31—81no—
84C2×C4×C31C2×D411yescharacteristic
85C2×C4×C31C2×Q812yes—
86C2×C4×C31#4112yes—
87C2×C4×C31C2411yescharacteristic
88C2×Dic31—21no—
89C2×Dic31—21no—
90C2×Dic31C2×D412yes—
91C2×Dic31C2×D412yes—
92D124—24no—
93D124—22no—
94D124C2×Q814yes—
95C4×D31—82no—
96C4×D31—82no—
97Dic62—82no—
98C8×C31—22no—
99C31⋊C8—21no—
100C31⋊C8—21no—
101#3081—311nomaximal, Sylow
102C22×C31—21no—
103C22×C31—21no—
104C22×C31C22⋊Q812yes—
105C22×C31C22×D411yescharacteristic
106C2×D31—24no—
107C2×D31—42no—
108C2×D31—42no—
109C2×D31—84no—
110C4×C31—42no—
111C4×C31—41no—
112C4×C31—41no—
113C4×C31C22⋊Q812yes—
114C4×C31C22×Q811yescharacteristic
115C4×C31C2×#4111yescharacteristic
116Dic31—22no—
117Dic31—22no—
118Dic31—41no—
119Dic31—41no—
120#544—311no—
121#518—311no—
122#478—312no—
123#478—312no—
124#476—312no—
125#474—312no—
126#428D3111yescharacteristic, complemented
127#427—311no—
128#424—311no—
129#418—311no—
130C2×#102—311no—
131C2×C31—22no—
132C2×C31C2×C22⋊Q811yescharacteristic
133C2×C31#57611yescharacteristic
134C2×C31#57711yescharacteristic
135D31—44no—
136C22×D4—311no—
137C2×M16C2×D3111yescharacteristic
138C22×C8—311no—
139C4⋊Q8—312no—
140C4⋊Q8—621no—
141#125—312no—
142#125—621no—
143C22⋊Q8—622no—
144C4×D4—312no—
145C4×D4—312no—
146C4×D4—621no—
147C4×D4—621no—
148#117—311no—
149#117—311no—
150#117C2×D3111yescharacteristic
151C2×C4⋊C4C2×D3111yescharacteristic
152C2×C22⋊C4—622no—
153#107—311no—
154#107—311no—
155#107C2×D3112yescomplemented
156#106—311no—
157#106—311no—
158#106C2×D3112yescomplemented
159C4⋊C8—312no—
160C4⋊C8—312no—
161#102—312no—
162#102—312no—
163#102—312no—
164#102—312no—
165C31#308111yescharacteristic, complemented, Sylow
166C24—621no—
167C2×Q8—622no—
168C2×D4—314no—
169C2×D4—312no—
170C2×D4—622no—
171C2×D4—621no—
172C22×C4—21no—
173C22×C4—622no—
174C22×C4—622no—
175C22×C4C22×D3111yescharacteristic
176M16—22no—
177C2×C8—311no—
178C2×C8—311no—
179C2×C8—622no—
180C2×C8C22×D3112yes—
181C4⋊C4—21no—
182C4⋊C4—311no—
183C4⋊C4—311no—
184C4⋊C4—311no—
185C4⋊C4—311no—
186C4⋊C4—622no—
187C4⋊C4—622no—
188C4⋊C4—622no—
189C4⋊C4C22×D3112yes—
190C4⋊C4C22×D3111yescharacteristic
191C4⋊C4C22×D3111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—312no—
194C22⋊C4—622no—
195C22⋊C4—622no—
196C22⋊C4—621no—
197C42—21no—
198C42—312no—
199C42—621no—
200C23—622no—
201C23—624no—
202C23—2481no—
203C23—2481no—
204C23D4×D3111yescharacteristic
205Q8—2482no—
206D4—314no—
207D4—624no—
208D4—622no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—312no—
214C2×C4—312no—
215C2×C4—621no—
216C2×C4—621no—
217C2×C4—2482no—
218C2×C4—2482no—
219C2×C4#9249512yes—
220C2×C4D4×D3111yescharacteristic
221C2×C4Q8×D3112yes—
222C2×C4C23×D3111yescharacteristic
223C8—22no—
224C8—621no—
225C8—621no—
226C22—21no—
227C22—21no—
228C22—624no—
229C22—1242no—
230C22—1242no—
231C22—2484no—
232C22#1175847312yes—
233C22C2×D4×D3111yescharacteristic
234C4—41no—
235C4—42no—
236C4—41no—
237C4—622no—
238C4—622no—
239C4—1241no—
240C4—1241no—
241C4#1175847312yes—
242C4C2×#9249511yescharacteristic
243C4C2×Q8×D3111yescharacteristic
244C2—22no—
245C2—1244no—
246C2C2×#1175847311yescharacteristic
247C2#1452185311yescharacteristic
248C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#428C2
socleC22×C31C22×D4
prime radicalC22×D124C22
hypercenter#428D31
hypocenterC31#3081
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #14521680C2semidirect
3. #14521352C2semidirect
4. #14521348C2semidirect
5. #14521338C2semidirect
6. #14521328C2semidirect
7. #14521085C2semidirect
8. #14521077C2semidirect
11. C31×#428C2semidirect
23. #11758467C22semidirect
25. #11758465C22semidirect
29. C31×#107C22semidirect
30. C31×#106C22semidirect
33. Dic31⋊C8C22semidirect
34. Dic31⋊C8C22semidirect
37. #11758396C22semidirect
38. #11758396C22semidirect
39. #11758395C22semidirect
40. #11758395C22semidirect
46. C2×D124Q8semidirect
126. #428D31semidirect
155. #107C2×D31semidirect
158. #106C2×D31semidirect
165. C31#3081semidirect

Maximal subgroups

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

subgroupindex
#145216802
#145215232
#145213522
#145213482
#145213382
#145213282
#145210852
#145210772
#145210552
C2×#117583972
C31×#4282
#308131