Finite Group Databasebeta

← #16855074

Subgroups — #16855074

Order 3968.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11758564C211yesmaximal, characteristic, complemented
2#14521847C212yesmaximal, complemented
3#14521231C214yesmaximal, complemented
4#14521221C214yesmaximal, complemented
5#14521067C212yesmaximal, complemented
6C2×#11758424C211yesmaximal, characteristic, complemented
7C31×#536C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C22×Q8×C31C2211yescharacteristic
10#11758564—22no—
11#11758564—24no—
12#11758564C2214yes—
13C2×#92499—22no—
14C2×Q8×D31—22no—
15C2×#92495—22no—
16C2×#92495C2212yes—
17C31×C4⋊Q8C2212yescomplemented
18C31×#124C2212yescomplemented
19C31×#123—22no—
20C31×C22⋊Q8—22no—
21C31×#117C2211yescharacteristic
22#11758528—22no—
23#11758527—24no—
24#11758526—22no—
25#11758519C2214yescomplemented
26#11758509C2214yescomplemented
27C2×#92474C2212yes—
28#11758424C2214yescomplemented
29#11758393C2218yescomplemented
30#41×C31—22no—
31#41×C31—22no—
32C2×Q8×C31—22no—
33C2×Q8×C31—22no—
34C2×Q8×C31D412yes—
35C2×Q8×C31C2311yescharacteristic
36C2×Q8×C31C2312yes—
37C2×D4×C31—21no—
38C2×D4×C31C2311yescharacteristic
39C22×C4×C31—21no—
40C22×C4×C31D412yes—
41C22×C4×C31C2311yescharacteristic
42C2×C22⋊D31—22no—
43#92499—28no—
44#92499—28no—
45Q8×D31—28no—
46Q8×D31—28no—
47#92495—28no—
48#92495—28no—
49#92495—22no—
50#92495—22no—
51#92495D414yes—
52#92495C2314yes—
53C2×D124—22no—
54C2×D124D412yes—
55C2×C4×D31—24no—
56C2×C4×D31—22no—
57C2×Dic62—22no—
58C2×Dic62D412yes—
59C4⋊C4×C31—21no—
60C4⋊C4×C31—24no—
61C22⋊C4×C31—21no—
62C22⋊C4×C31—24no—
63C42×C31C2312yes—
64#92482—24no—
65#92481—24no—
66#92480—24no—
67#92479—24no—
68#92474—22no—
69#92474C2314yes—
70C2×C31⋊C8—22no—
71C23×C31—21no—
72C23×C31C2×D411yescharacteristic
73C22×D31—22no—
74Q8×C31—24no—
75Q8×C31—24no—
76Q8×C31—22no—
77D4×C31—22no—
78D4×C31—41no—
79D4×C31—41no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—21no—
84C2×C4×C31—42no—
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D412yes—
87C2×C4×C31C2×D414yes—
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2411yescharacteristic
90C22⋊D31—24no—
91C22⋊D31—24no—
92C2×Dic31—22no—
93D124—28no—
94D124—22no—
95D124C2×D414yes—
96C4×D31—216no—
97C4×D31—24no—
98C4×D31—24no—
99Dic62—28no—
100Dic62—22no—
101Dic62C2×D414yes—
102C31⋊C8—24no—
103#3020—311nomaximal, Sylow
104C22×C31—41no—
105C22×C31—42no—
106C22×C31—41no—
107C22×C31#12011yescharacteristic
108C22×C31#12011yescharacteristic
109C22×C31C22×D411yescharacteristic
110C2×D31—24no—
111C2×D31—44no—
112C4×C31—24no—
113C4×C31—42no—
114C4×C31—41no—
115C4×C31#12012yes—
116C4×C31C22×D412yes—
117Dic31—24no—
118C2×2−1+4—311no—
119#575—312no—
120#536D3111yescharacteristic, complemented
121#456—314no—
122#454—314no—
123#421—312no—
124C2×#101—311no—
125C2×C31—21no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31—81no—
129C2×C31C2×#12011yescharacteristic
130D31—44no—
1312−1+4—314no—
1322−1+4—624no—
1332−1+4—622no—
134C2×#41—312no—
135C2×#41—622no—
136C2×#41—622no—
137C2×#41C2×D3111yescharacteristic
138C22×Q8—622no—
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⋊Q8C2×D3112yescomplemented
147#124C2×D3112yescomplemented
148#123—22no—
149C22⋊Q8—22no—
150#117C2×D3111yescharacteristic
151#104—318no—
152#101—314no—
153C31#302011yescharacteristic, complemented, Sylow
154#41—22no—
155#41—22no—
156#41—314no—
157#41—314no—
158#41—628no—
159#41—628no—
160#41—628no—
161#41—622no—
162#41—622no—
163#41—628no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—312no—
167C2×Q8—622no—
168C2×Q8—628no—
169C2×Q8—628no—
170C2×Q8C22⋊D3112yes—
171C2×Q8C22×D3111yescharacteristic
172C2×Q8C22×D3112yes—
173C2×D4—21no—
174C2×D4—312no—
175C2×D4—622no—
176C2×D4—622no—
177C2×D4C22×D3111yescharacteristic
178C22×C4—21no—
179C22×C4—624no—
180C22×C4—622no—
181C22×C4C22⋊D3112yes—
182C22×C4C22×D3111yescharacteristic
183Q16—624no—
184SD16—624no—
185SD16—624no—
186D8—624no—
187M16—314no—
188M16—622no—
189C2×C8—622no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D3112yes—
195C23—21no—
196C23—622no—
197C23C2×C22⋊D3111yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—314no—
202Q8—628no—
203Q8—622no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—314no—
208D4—628no—
209D4—624no—
210D4—624no—
211D4—622no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—6216no—
218C2×C4—622no—
219C2×C4—624no—
220C2×C4—624no—
221C2×C4D4×D3112yes—
222C2×C4D4×D3112yes—
223C2×C4C2×C22⋊D3114yes—
224C2×C4C2×C22⋊D3111yescharacteristic
225C2×C4C23×D3111yescharacteristic
226C8—624no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—624no—
231C22—1244no—
232C22#1175851411yescharacteristic
233C22#1175851411yescharacteristic
234C22C22×C22⋊D3111yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—624no—
239C4#1175851412yes—
240C4C2×D4×D3112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—1244no—
246C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#536C2
socleC2×C31C2×#120
hypercenter#536D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. #14521847C2semidirect
3. #14521231C2semidirect
4. #14521221C2semidirect
5. #14521067C2semidirect
6. C2×#11758424C2semidirect
7. C31×#536C2semidirect
17. C31×C4⋊Q8C22semidirect
18. C31×#124C22semidirect
25. #11758519C22semidirect
26. #11758509C22semidirect
28. #11758424C22semidirect
29. #11758393C22semidirect
120. #536D31semidirect
146. C4⋊Q8C2×D31semidirect
147. #124C2×D31semidirect
153. C31#3020semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
#145218472
#145212312
#145212212
#145210672
C2×#117584242
C31×#5362
#302031