Finite Group Databasebeta

← #16855075

Subgroups — #16855075

Order 3968.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#14521847C212yesmaximal, complemented
2#14521623C211yesmaximal, characteristic, complemented
3#14521232C214yesmaximal, complemented
4#14521222C214yesmaximal, complemented
5#14521066C212yesmaximal, complemented
6C2×#11758424C211yesmaximal, characteristic, complemented
7C2×C31×2−1+4C211yesmaximal, characteristic, complemented
8C31×2−1+4—24no—
9C31×2−1+4—22no—
10C31×2−1+4C2214yes—
11C2×#41×C31—22no—
12C2×#41×C31—22no—
13C2×#41×C31C2212yes—
14C22×Q8×C31—22no—
15C22×Q8×C31C2211yescharacteristic
16C2×#92495C2211yescharacteristic
17#11758528C2214yescomplemented
18#11758527—24no—
19#11758526C2214yescomplemented
20#11758524C2212yescomplemented
21#11758523—22no—
22#11758521C2212yescomplemented
23#11758519—22no—
24#11758509—22no—
25#11758506—22no—
26#11758503C2211yescharacteristic
27C2×#92474C2212yes—
28#11758425C2218yescomplemented
29#11758424C2214yescomplemented
30#41×C31—28no—
31#41×C31—28no—
32#41×C31—28no—
33#41×C31—28no—
34#41×C31—22no—
35#41×C31—22no—
36#41×C31D414yes—
37#41×C31C2314yes—
38C2×Q8×C31—22no—
39C2×Q8×C31—28no—
40C2×Q8×C31—28no—
41C2×Q8×C31—22no—
42C2×Q8×C31—22no—
43C2×Q8×C31D412yes—
44C2×Q8×C31D412yes—
45C2×Q8×C31C2312yes—
46C2×D4×C31—22no—
47C2×D4×C31—22no—
48C2×D4×C31D412yes—
49C22×C4×C31—24no—
50C22×C4×C31—22no—
51C22×C4×C31D412yes—
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—21no—
54#92495—22no—
55#92495—22no—
56C2×D124C2311yescharacteristic
57C2×C4×D31—21no—
58C2×Dic62C2311yescharacteristic
59C22⋊Dic31—21no—
60#92482—24no—
61#92481—24no—
62#92480—24no—
63#92479—24no—
64(C2×D31)⋊C4—24no—
65C4⋊Dic31—21no—
66Dic31⋊C4—24no—
67C4×Dic31C2312yes—
68#92474—22no—
69#92474C2314yes—
70C2×C31⋊C8—22no—
71C23×C31—22no—
72C23×C31C2×D411yescharacteristic
73C22×D31—21no—
74Q8×C31—28no—
75Q8×C31—22no—
76Q8×C31—24no—
77Q8×C31—24no—
78Q8×C31C2×D414yes—
79D4×C31—28no—
80D4×C31—24no—
81D4×C31—24no—
82D4×C31—22no—
83D4×C31C2×D414yes—
84C2×C4×C31—216no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31—24no—
89C2×C4×C31—24no—
90C2×C4×C31C2×D414yes—
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2×D412yes—
94C2×C4×C31C2411yescharacteristic
95C22⋊D31—41no—
96C22⋊D31—41no—
97C2×Dic31—22no—
98C2×Dic31—21no—
99D124—22no—
100C4×D31—42no—
101Dic62—22no—
102C31⋊C8—24no—
103#3020—311nomaximal, Sylow
104C22×C31—24no—
105C22×C31—41no—
106C22×C31—44no—
107C22×C31#12011yescharacteristic
108C22×C31#12011yescharacteristic
109C22×C31C22×D411yescharacteristic
110C2×D31—42no—
111C2×D31—41no—
112C4×C31—24no—
113C4×C31—24no—
114C4×C31#12012yes—
115C4×C31C22×D412yes—
116Dic31—42no—
117Dic31—41no—
118C2×2−1+4D3111yescharacteristic, complemented
119#575—312no—
120#536—311no—
121#456—314no—
122#454—314no—
123#421—312no—
124C2×#101—311no—
125C2×C31—21no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31—44no—
129C2×C31C2×#12011yescharacteristic
130D31—81no—
1312−1+4—24no—
1322−1+4—22no—
1332−1+4C2×D3114yes—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—311no—
137C2×#41C2×D3112yes—
138C22×Q8—22no—
139C22×Q8C2×D3111yescharacteristic
140#137—314no—
141#137—622no—
142AGL(1,ℤ/8)—314no—
143AGL(1,ℤ/8)—622no—
144#135—624no—
145C2×M16—312no—
146C4⋊Q8—312no—
147#124—312no—
148#123—622no—
149C22⋊Q8—622no—
150#117—311no—
151#104—318no—
152#101—314no—
153C31#302011yescharacteristic, complemented, Sylow
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—622no—
161#41—622no—
162#41C22⋊D3114yes—
163#41C22×D3114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—311no—
170C2×Q8C22⋊D3112yes—
171C2×Q8C22⋊D3112yes—
172C2×Q8C22×D3112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—311no—
176C2×D4—621no—
177C2×D4C22⋊D3112yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—621no—
181C22×C4C22⋊D3112yes—
182C22×C4C22×D3111yescharacteristic
183Q16—624no—
184SD16—624no—
185SD16—624no—
186D8—624no—
187M16—314no—
188M16—622no—
189C2×C8—622no—
190C4⋊C4—624no—
191C4⋊C4—621no—
192C22⋊C4—621no—
193C22⋊C4—624no—
194C42—312no—
195C23—22no—
196C23—621no—
197C23C2×C22⋊D3111yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—622no—
203Q8C2×C22⋊D3114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—622no—
209D4—1241no—
210D4—1241no—
211D4C2×C22⋊D3114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—622no—
219C2×C4—621no—
220C2×C4—1242no—
221C2×C4C2×C22⋊D3114yes—
222C2×C4C2×C22⋊D3111yescharacteristic
223C2×C4C2×C22⋊D3112yes—
224C2×C4C2×C22⋊D3112yes—
225C2×C4C23×D3111yescharacteristic
226C8—624no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—1242no—
231C22—1241no—
232C22#1175853011yescharacteristic
233C22#1175853011yescharacteristic
234C22C22×C22⋊D3111yescharacteristic
235C4—24no—
236C4—24no—
237C4—1242no—
238C4—1241no—
239C4#1175853012yes—
240C4C22×C22⋊D3112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—2481no—
246C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×2−1+4C2
socleC2×C31C2×#120
hypercenterC2×2−1+4D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #14521847C2semidirect
2. #14521623C2semidirect
3. #14521232C2semidirect
4. #14521222C2semidirect
5. #14521066C2semidirect
6. C2×#11758424C2semidirect
7. C2×C31×2−1+4C2semidirect
17. #11758528C22semidirect
19. #11758526C22semidirect
20. #11758524C22semidirect
22. #11758521C22semidirect
28. #11758425C22semidirect
29. #11758424C22semidirect
118. C2×2−1+4D31semidirect
153. C31#3020semidirect

Maximal subgroups

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

subgroupindex
#145218472
#145216232
#145212322
#145212222
#145210662
C2×#117584242
C2×C31×2−1+42
#302031