Finite Group Databasebeta

← #16856241

Subgroups — #16856241

Order 3968.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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
2#14521677C211yesmaximal, characteristic
3#14521457C212yesmaximal, complemented
4#14521456C212yesmaximal, complemented
5#14521450C212yesmaximal, complemented
6#14521446C212yesmaximal, complemented
7#14521091C211yesmaximal, characteristic
8#14521085C211yesmaximal, characteristic
9C2×#11758396C211yesmaximal, characteristic
10#14521007C211yesmaximal, characteristic
11C31×#428C211yesmaximal, characteristic
12C2×M16×C31C2211yescharacteristic
13C31×#117C2211yescharacteristic
14C2×C4⋊C4×C31C2211yescharacteristic
15#11758503C2211yescharacteristic
16#11758503C2211yescharacteristic
17C2×C4⋊Dic31C2211yescharacteristic
18#11758501—22no—
19C22×C31⋊C8C2211yescharacteristic
20#11758467—21no—
21#11758467C2212yes—
22#11758467C2212yes—
23#11758465—21no—
24#11758465C2212yes—
25#11758465C2212yes—
26#11758455—22no—
27#11758447—21no—
28#11758446—21no—
29C31×#107C2212yes—
30C31×#106C2212yes—
31#11758406C2212yes—
32#11758405C2212yes—
33#11758403C2212yes—
34C8×Dic31C2212yes—
35#11758396C2211yescharacteristic
36#11758396C2212yes—
37#11758396C2211yescharacteristic
38#11758396C2212yes—
39#11758395C2211yescharacteristic
40#11758395C2211yescharacteristic
41C22×C4×C31—21no—
42C22×C4×C31C2311yescharacteristic
43C22×Dic31—21no—
44C2×Dic62—22no—
45C2×Dic62—22no—
46M16×C31Q814yescomplemented
47C2×C8×C31C2312yes—
48C4⋊C4×C31—21no—
49C4⋊C4×C31C2311yescharacteristic
50C4⋊C4×C31C2311yescharacteristic
51C4⋊C4×C31C2312yes—
52C22⋊C4×C31—21no—
53C42×C31—21no—
54C22⋊Dic31—21no—
55C22⋊Dic31D412yes—
56C4⋊Dic31—21no—
57C4⋊Dic31—22no—
58C4⋊Dic31—22no—
59C4⋊Dic31D411yescharacteristic
60C4⋊Dic31D411yescharacteristic
61C4⋊Dic31C2311yescharacteristic
62C4⋊Dic31C2311yescharacteristic
63C4⋊Dic31C2312yes—
64Dic31⋊C4—22no—
65Dic31⋊C4—22no—
66Dic31⋊C4—22no—
67Dic31⋊C4—22no—
68C4×Dic31—21no—
69C4×Dic31C2312yes—
70C2×C31⋊C8Q814yescomplemented
71C2×C31⋊C8C2311yescharacteristic
72C2×C31⋊C8C2311yescharacteristic
73C23×C31C2×D411yescharacteristic
74C2×C4×C31—21no—
75C2×C4×C31—22no—
76C2×C4×C31—21no—
77C2×C4×C31—81no—
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2×Q812yes—
80C2×C4×C31C2×Q812yes—
81C2×C4×C31C2411yescharacteristic
82C2×Dic31—21no—
83C2×Dic31—21no—
84C2×Dic31—22no—
85C2×Dic31—81no—
86C2×Dic31C2×D412yes—
87C2×Dic31C2×D412yes—
88Dic62—82no—
89Dic62—82no—
90C8×C31C2×Q814yes—
91C31⋊C8C2×Q812yes—
92C31⋊C8C2×Q812yes—
93#3163—311nomaximal, Sylow
94C22×C31—21no—
95C22×C31—21no—
96C22×C31C4⋊Q812yes—
97C22×C31C22×D411yescharacteristic
98C4×C31—41no—
99C4×C31—42no—
100C4×C31—41no—
101C4×C31C4⋊Q812yes—
102C4×C31C22×Q811yescharacteristic
103C4×C31C22×Q811yescharacteristic
104Dic31—22no—
105Dic31—22no—
106Dic31—41no—
107Dic31—42no—
108Dic31—41no—
109#544—311no—
110#544—311no—
111#501—312no—
112#501—312no—
113#499—312no—
114#498—312no—
115#428—311no—
116#428D3111yescharacteristic
117#427—311no—
118C2×#106—311no—
119#405—311no—
120C2×C31—22no—
121C2×C31C2×C4⋊Q811yescharacteristic
122C2×C31#57711yescharacteristic
123C2×C31#57711yescharacteristic
124C2×M16C2×D3111yescharacteristic
125C22×C8—311no—
126C4⋊Q8—312no—
127C4⋊Q8—312no—
128C4⋊Q8—621no—
129C4⋊Q8—621no—
130#125—312no—
131#125—312no—
132#125—621no—
133#125—621no—
134C22⋊Q8—622no—
135C22⋊Q8—622no—
136#117—311no—
137#117—311no—
138#117C2×D3111yescharacteristic
139C2×C4⋊C4—311no—
140C2×C4⋊C4C2×D3111yescharacteristic
141#107—312no—
142#107—311no—
143#107—311no—
144#107C2×D3112yes—
145#106—312no—
146#106—312no—
147#106—312no—
148#106—311no—
149#106—311no—
150#106C2×D3112yes—
151#97—312no—
152C4×C8—312no—
153C31#316311yescharacteristic, complemented, Sylow
154C2×Q8—622no—
155C2×Q8—622no—
156C22×C4—21no—
157C22×C4—621no—
158C22×C4C22×D3111yescharacteristic
159M16Dic6214yescomplemented
160C2×C8—314no—
161C2×C8—311no—
162C2×C8—311no—
163C2×C8C22×D3112yes—
164C4⋊C4—21no—
165C4⋊C4—311no—
166C4⋊C4—311no—
167C4⋊C4—312no—
168C4⋊C4—311no—
169C4⋊C4—311no—
170C4⋊C4—622no—
171C4⋊C4—622no—
172C4⋊C4—622no—
173C4⋊C4—622no—
174C4⋊C4—622no—
175C4⋊C4—622no—
176C4⋊C4—621no—
177C4⋊C4C22×D3112yes—
178C4⋊C4C22×D3111yescharacteristic
179C4⋊C4C22×D3111yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—312no—
182C22⋊C4—621no—
183C42—21no—
184C42—312no—
185C42—621no—
186C23D4×D3111yescharacteristic
187Q8—2482no—
188Q8—2482no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—312no—
194C2×C4—312no—
195C2×C4—621no—
196C2×C4—621no—
197C2×C4—622no—
198C2×C4—2481no—
199C2×C4C2×Dic6212yes—
200C2×C4D4×D3111yescharacteristic
201C2×C4Q8×D3112yes—
202C2×C4C23×D3111yescharacteristic
203C8—312no—
204C8—312no—
205C8C2×Dic6214yes—
206C22—21no—
207C22—21no—
208C22#1175846512yes—
209C22C2×D4×D3111yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—622no—
214C4—622no—
215C4—1242no—
216C4—1241no—
217C4—1241no—
218C4#1175846512yes—
219C4C22×Dic6211yescharacteristic
220C4C2×Q8×D3111yescharacteristic
221C2—22no—
222C2C2×#1175846511yescharacteristic
223C2#1452185811yescharacteristic
224C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#428C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenter#428D31
hypocenterC31#3163
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
3. #14521457C2semidirect
4. #14521456C2semidirect
5. #14521450C2semidirect
6. #14521446C2semidirect
46. M16×C31Q8semidirect
70. C2×C31⋊C8Q8semidirect
153. C31#3163semidirect
159. M16Dic62semidirect

Maximal subgroups

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

subgroupindex
#145216802
#145216772
#145214572
#145214562
#145214502
#145214462
#145210912
#145210852
C2×#117583962
#145210072
C31×#4282
#316331