Finite Group Databasebeta

← #16858401

Subgroups — #16858401

Order 3968.

336 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 115 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#14521774C211yesmaximal, characteristic, complemented
2#14521773C211yesmaximal, characteristic
3#14521405C212yesmaximal
4#14521354C212yesmaximal, complemented
5#14521326C211yesmaximal, characteristic
6#14521274C212yesmaximal
7#14521256C211yesmaximal, characteristic, complemented
8C8×Dic62C211yesmaximal, characteristic, complemented
9#14521141C212yesmaximal, complemented
10C4×#92470C211yesmaximal, characteristic
11C31×#487C211yesmaximal, characteristic, complemented
12C31×C4⋊Q8C2211yescharacteristic
13C31×C4⋊Q8C2211yescharacteristic
14#11758524C2212yes—
15Q8×Dic31—21no—
16Q8×Dic31C2212yes—
17#11758521—21no—
18C2×#92482C2212yes—
19C2×#92470C2211yescharacteristic
20#11758475—22no—
21#11758474—21no—
22#11758469—21no—
23#11758467C2212yes—
24#11758466—22no—
25#11758464C2212yes—
26C4×D124C2211yescharacteristic
27C4×Dic62C2211yescharacteristic
28C4×Dic62C2211yescharacteristic
29C31×#103C2212yes—
30C31×#103C2212yes—
31C4×C8×C31C2211yescharacteristic
32#11758423C2212yes—
33#11758409C2211yescharacteristic
34#11758405C2211yescharacteristic
35#11758404C2211yescharacteristic
36Dic31⋊C8C2212yes—
37C8×Dic31C2212yes—
38#11758397C2212yes—
39#11758395C2212yes—
40C4⋊(C31⋊C8)C2211yescharacteristic
41C2×Q8×C31—21no—
42C2×Q8×C31C2312yes—
43C2×D124C2311yescharacteristic
44C2×C4×D31—21no—
45C2×Dic62D411yescharacteristic
46C2×Dic62C2311yescharacteristic
47C2×C8×C31C2311yescharacteristic
48C2×C8×C31C2311yescharacteristic
49C4⋊C4×C31—21no—
50C4⋊C4×C31—21no—
51C4⋊C4×C31—21no—
52C4⋊C4×C31C2312yes—
53C42×C31C2311yescharacteristic
54#92482—24no—
55(C2×D31)⋊C4—21no—
56(C2×D31)⋊C4—22no—
57(C2×D31)⋊C4—22no—
58C4⋊Dic31—22no—
59C4⋊Dic31—22no—
60C4⋊Dic31D411yescharacteristic
61C4⋊Dic31C2311yescharacteristic
62C4⋊Dic31C2311yescharacteristic
63Dic31⋊C4—21no—
64Dic31⋊C4—22no—
65Dic31⋊C4—22no—
66Dic31⋊C4D412yes—
67C4×Dic31—21no—
68C4×Dic31—22no—
69C4×Dic31—22no—
70C4×Dic31C2312yes—
71C2×C31⋊C8C2312yes—
72#92470—22no—
73C22×D31—21no—
74Q8×C31—22no—
75Q8×C31—22no—
76Q8×C31—42no—
77C2×C4×C31—22no—
78C2×C4×C31—22no—
79C2×C4×C31C2×D411yescharacteristic
80C2×C4×C31C2×D411yescharacteristic
81C2×C4×C31C2411yescharacteristic
82C2×Dic31—21no—
83C2×Dic31—21no—
84C2×Dic31—21no—
85C2×Dic31C2×D412yes—
86C2×Dic31C2×D412yes—
87D124—21no—
88D124—21no—
89C4×D31—42no—
90Dic62—22no—
91Dic62—21no—
92Dic62#4112yes—
93C8×C31—21no—
94C8×C31—21no—
95C31⋊C8—22no—
96#3364—311nomaximal, Sylow
97C22×C31C22×D411yescharacteristic
98C2×D31—41no—
99C2×D31—41no—
100C2×D31—41no—
101C4×C31—21no—
102C4×C31—42no—
103C4×C31—42no—
104C4×C31#13512yes—
105C4×C31C2×#4111yescharacteristic
106C4×C312−1+411yescharacteristic
107Dic31—22no—
108Dic31—22no—
109Dic31—22no—
110Dic31—41no—
111Dic31—41no—
112#557—311no—
113#556—311no—
114#488—312no—
115#487D3111yescharacteristic, complemented
116#479—312no—
117#474—311no—
118#464—312no—
119#462—311no—
120C8×Q8—311no—
121C4×Q16—312no—
122C4×SD16—311no—
123C2×C31#54811yescharacteristic
124C2×C31C2×#13511yescharacteristic
125C2×C31#57811yescharacteristic
126D31—81no—
127C2×Q16—312no—
128C2×SD16—311no—
129C4⋊Q8—621no—
130C4⋊Q8C2×D3111yescharacteristic
131C4⋊Q8C2×D3111yescharacteristic
132#126—622no—
133#125—312no—
134#125—622no—
135#124—312no—
136C22⋊Q8—621no—
137C4×Q8—311no—
138C4×Q8—312no—
139C4×Q8—312no—
140C4×Q8—311no—
141C4×Q8—621no—
142C4×D4—311no—
143#117—621no—
144#107—312no—
145#106—311no—
146C4⋊C8—312no—
147C4⋊C8—311no—
148#103—312no—
149#103—311no—
150#103C2×D3112yes—
151#103C2×D3112yes—
152#102—311no—
153#102—312no—
154C4×C8—312no—
155C4×C8C2×D3111yescharacteristic
156C31#336411yescharacteristic, complemented, Sylow
157C2×Q8—21no—
158C2×Q8—311no—
159C2×Q8—311no—
160C2×Q8C22×D3112yes—
161C2×D4—311no—
162C22×C4—621no—
163Q16—624no—
164SD16—622no—
165C2×C8—312no—
166C2×C8C22×D3111yescharacteristic
167C2×C8C22×D3111yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—312no—
172C4⋊C4—311no—
173C4⋊C4—311no—
174C4⋊C4—311no—
175C4⋊C4—622no—
176C4⋊C4—622no—
177C4⋊C4—622no—
178C4⋊C4—622no—
179C4⋊C4—621no—
180C4⋊C4C22×D3112yes—
181C22⋊C4—622no—
182C22⋊C4—622no—
183C22⋊C4—621no—
184C42—312no—
185C42—622no—
186C42—622no—
187C42—621no—
188C42C22×D3111yescharacteristic
189C23—621no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—312no—
194Q8—622no—
195Q8—621no—
196D4—621no—
197D4—621no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—312no—
201C2×C4—312no—
202C2×C4—621no—
203C2×C4—621no—
204C2×C4—621no—
205C2×C4—1242no—
206C2×C4D4×D3111yescharacteristic
207C2×C4D4×D3111yescharacteristic
208C2×C4C23×D3111yescharacteristic
209C8—21no—
210C8—21no—
211C8—622no—
212C22—1241no—
213C22—1241no—
214C22—1241no—
215C22C2×D4×D3111yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—622no—
220C4—622no—
221C4—622no—
222C4—1241no—
223C4—1241no—
224C4#1175849312yes—
225C4#41×D3111yescharacteristic
226C4#1175856711yescharacteristic
227C2—2481no—
228C2#1452170711yescharacteristic
229C2C2×#1175849311yescharacteristic
230C2#1452186211yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#487C2
socleC22×C31C22×D4
prime radicalC2×D124C23
hypercenter#487D31
hypocenterC31#3364
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #14521774C2semidirect
4. #14521354C2semidirect
7. #14521256C2semidirect
8. C8×Dic62C2semidirect
9. #14521141C2semidirect
11. C31×#487C2semidirect
115. #487D31semidirect
156. C31#3364semidirect

Maximal subgroups

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

subgroupindex
#145217742
#145217732
#145214052
#145213542
#145213262
#145212742
#145212562
C8×Dic622
#145211412
C4×#924702
C31×#4872
#336431