Finite Group Databasebeta

← #16845679

Subgroups — #16845679

Order 3968.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#11758566C211yesmaximal, characteristic
2#14521067C212yesmaximal, complemented
3#14521053C212yesmaximal, complemented
4#14521033C211yesmaximal, characteristic, complemented
5C31×#410C211yesmaximal, characteristic, complemented
6C2×#41×C31C2211yescharacteristic
7#11758566—24no—
8#11758566—24no—
9C2×#92499—21no—
10C2×D4×D31—22no—
11C2×D4×D31—21no—
12C2×#92495C412yescomplemented
13C2×#92495C2211yescharacteristic
14C22×D124C412yescomplemented
15C22×D124C2211yescharacteristic
16C31×#117C2212yescomplemented
17C2×#92474C2212yescomplemented
18C31×#99—22no—
19#11758424—21no—
20#11758421—21no—
21#11758397—24no—
22#11758393—24no—
23C23×D31—22no—
24C23×D31—21no—
25#41×C31—22no—
26#41×C31—22no—
27C2×Q8×C31D411yescharacteristic
28C2×D4×C31D412yes—
29C2×D4×C31D411yescharacteristic
30C22×C4×C31C2×C412yes—
31C22×C4×C31C2311yescharacteristic
32C2×C22⋊D31—22no—
33C2×C22⋊D31—21no—
34#92499—28no—
35D4×D31—216no—
36D4×D31—28no—
37#92495—22no—
38#92495—24no—
39#92495—24no—
40#92495D414yes—
41C2×D124—22no—
42C2×D124—21no—
43C2×D124—22no—
44C2×D124—24no—
45C2×D124—24no—
46C2×D124C2×C411yescharacteristic
47C2×D124C2×C412yes—
48C2×D124D414yes—
49C2×C4×D31—22no—
50C2×C4×D31—21no—
51C2×Dic62C2×C411yescharacteristic
52C4⋊C4×C31—22no—
53C22⋊C4×C31—22no—
54C42×C31—22no—
55#92474—22no—
56#92474—22no—
57C2×C31⋊C8—22no—
58C23×C31C22⋊C412yes—
59C23×C31C2×D411yescharacteristic
60C22×D31—22no—
61C22×D31—21no—
62C22×D31—28no—
63C22×D31—24no—
64C22×D31—82no—
65C22×D31—84no—
66Q8×C31—22no—
67D4×C31—24no—
68D4×C31—22no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31—42no—
72C2×C4×C31C22⋊C412yes—
73C2×C4×C31C22×C411yescharacteristic
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2×D412yes—
76C2×C4×C31C2×D412yes—
77C22⋊D31—28no—
78C22⋊D31—24no—
79C2×Dic31—21no—
80D124—28no—
81D124—21no—
82D124—22no—
83D124—24no—
84D124C22⋊C412yes—
85D124C22⋊C414yes—
86C4×D31—28no—
87C4×D31—24no—
88Dic62—21no—
89Dic62C22⋊C412yes—
90C31⋊C8—42no—
91#1795—311nomaximal, Sylow
92C22×C31—22no—
93C22×C31—41no—
94C22×C31—42no—
95C22×C31C2×C22⋊C411yescharacteristic
96C22×C31#12012yes—
97C2×D31—24no—
98C2×D31—22no—
99C2×D31—44no—
100C2×D31—42no—
101C2×D31—84no—
102C2×D31—88no—
103C4×C31—22no—
104C4×C31—82no—
105C4×C31C2×C22⋊C411yescharacteristic
106C4×C31C2×C22⋊C411yescharacteristic
107C4×C31#12012yes—
108Dic31—22no—
109C2×H(5,𝔽2)—311no—
110#421—312no—
111#418—312no—
112#413—311no—
113#410D3111yescharacteristic, complemented
114C2×C31—22no—
115C2×C31—21no—
116C2×C31—42no—
117C2×C31#37911yescharacteristic
118D31—44no—
119D31—42no—
120H(5,𝔽2)—624no—
121H(5,𝔽2)—624no—
122C2×#41—312no—
123C2×#41—311no—
124C2×#41—621no—
125C2×#41C2×D3111yescharacteristic
126C22×D4—312no—
127C22×D4—311no—
128C22×D4—621no—
129C22×D4—622no—
130C2×M16—312no—
131#117C2×D3112yescomplemented
132#104—624no—
133#102—624no—
134#101—621no—
135#100—621no—
136#99—22no—
137C31#179511yescharacteristic, complemented, Sylow
138C24—622no—
139C24—621no—
140#41—22no—
141#41—22no—
142#41—314no—
143#41—624no—
144#41—624no—
145#41—628no—
146#41—622no—
147C2×Q8—311no—
148C2×Q8D12411yescharacteristic
149C2×D4—312no—
150C2×D4—311no—
151C2×D4—314no—
152C2×D4—624no—
153C2×D4—624no—
154C2×D4—6216no—
155C2×D4—628no—
156C2×D4—622no—
157C2×D4—621no—
158C2×D4—622no—
159C2×D4—621no—
160C2×D4—622no—
161C2×D4D12411yescharacteristic
162C2×D4C22⋊D3112yes—
163C22×C4—622no—
164C22×C4—621no—
165C22×C4C4×D3112yes—
166C22×C4C22×D3111yescharacteristic
167M16—622no—
168M16—622no—
169C2×C8—622no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—628no—
174C23—622no—
175C23—621no—
176C23—624no—
177C23—2482no—
178C23—2484no—
179C23(C2×D31)⋊C412yes—
180C23C2×C22⋊D3111yescharacteristic
181Q8—22no—
182Q8—312no—
183Q8—621no—
184D4—24no—
185D4—22no—
186D4—314no—
187D4—312no—
188D4—628no—
189D4—628no—
190D4—624no—
191D4—621no—
192D4—622no—
193D4—624no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—628no—
198C2×C4—621no—
199C2×C4—624no—
200C2×C4(C2×D31)⋊C412yes—
201C2×C4C2×C4×D3111yescharacteristic
202C2×C4C2×D12411yescharacteristic
203C2×C4D4×D3112yes—
204C2×C4D4×D3112yes—
205C8—1242no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—624no—
210C22—622no—
211C22—1244no—
212C22—1242no—
213C22—2484no—
214C22—2488no—
215C22C2×(C2×D31)⋊C411yescharacteristic
216C22#1175851412yes—
217C4—22no—
218C4—82no—
219C4—622no—
220C4C22⋊C4×D3111yescharacteristic
221C4C22⋊C4×D3111yescharacteristic
222C4#1175845912yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—1244no—
227C2—1242no—
228C2#1452090411yescharacteristic
subgroupG/H
centerC2#14520904
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#410C2
socleC2×C31#379
prime radicalC2×#11758566C2
cosocleC22×C4×C31C23
hypercenter#410D31
hypocenterC31#1795
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #14521067C2semidirect
3. #14521053C2semidirect
4. #14521033C2semidirect
5. C31×#410C2semidirect
12. C2×#92495C4semidirect
14. C22×D124C4semidirect
16. C31×#117C22semidirect
17. C2×#92474C22semidirect
113. #410D31semidirect
131. #117C2×D31semidirect
137. C31#1795semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
#145210672
#145210532
#145210332
C31×#4102
#179531