Finite Group Databasebeta

← #16846153

Subgroups — #16846153

Order 3968.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#14521561C211yesmaximal, characteristic
2C2×C42×D31C211yesmaximal, characteristic, complemented
3C2×#11758393C211yesmaximal, characteristic, complemented
4C2×#11758413C211yesmaximal, characteristic, complemented
5#14521012C211yesmaximal, characteristic, complemented
6#14520808C211yesmaximal, characteristic, complemented
7C31×#339C211yesmaximal, characteristic, complemented
8C2×#92495C2211yescharacteristic
9C22×C4×D31—21no—
10C22×C4×D31C2211yescharacteristic
11C2×M16×C31C2211yescharacteristic
12C31×#117C2211yescharacteristic
13C2×C42×C31C2211yescharacteristic
14C2×C4×Dic31—21no—
15C2×C4×Dic31C2211yescharacteristic
16C2×#92474C2211yescharacteristic
17#11758474C412yescomplemented
18#11758472C412yescomplemented
19#11758463—21no—
20#11758459—21no—
21C4×D124—22no—
22C42×D31—22no—
23C42×D31—22no—
24#11758413—22no—
25(C2×D31)⋊C8—22no—
26#11758393—22no—
27C23×D31D411yescharacteristic
28C22×C4×C31—21no—
29C22×C4×C31C2311yescharacteristic
30C2×C22⋊D31—21no—
31C22×Dic31—21no—
32C22×Dic31D411yescharacteristic
33#92495—21no—
34#92495—21no—
35#92495—21no—
36#92495—21no—
37C2×D124—22no—
38C2×D124C2×C411yescharacteristic, complemented
39C2×C4×D31—24no—
40C2×C4×D31—28no—
41C2×C4×D31—21no—
42C2×C4×D31—21no—
43C2×C4×D31—21no—
44C2×C4×D31—21no—
45C2×C4×D31—21no—
46C2×C4×D31C2×C412yes—
47C2×C4×D31D412yes—
48C2×Dic62C2×C411yescharacteristic, complemented
49M16×C31—21no—
50M16×C31—21no—
51C2×C8×C31—21no—
52C4⋊C4×C31C2×C412yescomplemented
53C22⋊C4×C31—21no—
54C42×C31—21no—
55C42×C31—21no—
56C42×C31D412yescomplemented
57(C2×D31)⋊C4—22no—
58(C2×D31)⋊C4—22no—
59C4⋊Dic31—22no—
60C4×Dic31—24no—
61C4×Dic31—21no—
62C4×Dic31D412yescomplemented
63#92474—21no—
64#92474—21no—
65C2×C31⋊C8—21no—
66C23×C31C2×D411yescharacteristic
67C22×D31—21no—
68C22×D31—21no—
69C22×D31—24no—
70C22×D31—21no—
71C22×D31C22⋊C412yes—
72C2×C4×C31—24no—
73C2×C4×C31—22no—
74C2×C4×C31—22no—
75C2×C4×C31C22×C411yescharacteristic
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31#4111yescharacteristic
80C2×C4×C31#4111yescharacteristic
81C22⋊D31—41no—
82C22⋊D31—41no—
83C2×Dic31—24no—
84C2×Dic31—21no—
85C2×Dic31—22no—
86C2×Dic31—21no—
87C2×Dic31—21no—
88C2×Dic31C22⋊C412yes—
89D124—21no—
90D124—21no—
91D124—44no—
92C4×D31—216no—
93C4×D31—22no—
94C4×D31—22no—
95C4×D31—22no—
96C4×D31—22no—
97C4×D31—41no—
98C4×D31—41no—
99Dic62—21no—
100Dic62—21no—
101C8×C31—41no—
102C31⋊C8—41no—
103#1896—311nomaximal, Sylow
104C22×C31—22no—
105C22×C31C2×C22⋊C411yescharacteristic
106C22×C31#12011yescharacteristic
107C22×C31#12311yescharacteristic
108C2×D31—24no—
109C2×D31—24no—
110C2×D31—22no—
111C2×D31—22no—
112C2×D31—41no—
113C2×D31—42no—
114C2×D31#10414yescomplemented
115C4×C31—24no—
116C4×C31—42no—
117C4×C31C4×D411yescharacteristic
118C4×C31C4×D411yescharacteristic
119C4×C31#12111yescharacteristic
120C4×C31#12111yescharacteristic
121Dic31—24no—
122Dic31—22no—
123Dic31—41no—
124#525—311no—
125C22×C42—311no—
126C2×#104—311no—
127C2×#104—311no—
128#407—311no—
129#339—311no—
130#339D3111yescharacteristic, complemented
131C2×C31—21no—
132C2×C31—21no—
133C2×C31#38611yescharacteristic
134C2×C31C2×#10412yes—
135D31—24no—
136D31—81no—
137C2×#41—311no—
138C23×C4—311no—
139C23×C4—621no—
140C2×M16—311no—
141C2×M16C2×D3111yescharacteristic
142#123—621no—
143C22⋊Q8—312no—
144#121—312no—
145#120—621no—
146C4×D4—622no—
147#117C2×D3111yescharacteristic
148C2×C42—311no—
149C2×C42—622no—
150C2×C42—621no—
151C2×C42—622no—
152C2×C42C2×D3111yescharacteristic
153#104—622no—
154#104—622no—
155C22⋊C8—622no—
156C31#189611yescharacteristic, complemented, Sylow
157C24—311no—
158#41—621no—
159#41—621no—
160#41—621no—
161#41—621no—
162C2×Q8—311no—
163C2×D4—311no—
164C2×D4—622no—
165C2×D4—621no—
166C22×C4—21no—
167C22×C4—311no—
168C22×C4—312no—
169C22×C4—312no—
170C22×C4—624no—
171C22×C4—628no—
172C22×C4—621no—
173C22×C4—621no—
174C22×C4—621no—
175C22×C4—621no—
176C22×C4—621no—
177C22×C4—621no—
178C22×C4C22×D3111yescharacteristic
179M16—21no—
180M16—21no—
181M16—621no—
182M16—621no—
183C2×C8—21no—
184C2×C8—621no—
185C4⋊C4—622no—
186C4⋊C4C4×D3112yescomplemented
187C22⋊C4—21no—
188C22⋊C4—622no—
189C22⋊C4—622no—
190C42—21no—
191C42—21no—
192C42—312no—
193C42—624no—
194C42—621no—
195C42D12412yescomplemented
196C23—312no—
197C23—624no—
198C23—621no—
199C23—621no—
200C23—621no—
201C23D4×D3111yescharacteristic
202Q8—621no—
203Q8—621no—
204D4—621no—
205D4—621no—
206D4—1244no—
207D4—1241no—
208D4—1241no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—312no—
213C2×C4—6216no—
214C2×C4—624no—
215C2×C4—622no—
216C2×C4—621no—
217C2×C4—621no—
218C2×C4—622no—
219C2×C4—622no—
220C2×C4—622no—
221C2×C4—622no—
222C2×C4—621no—
223C2×C4—1241no—
224C2×C4—1241no—
225C2×C4C2×C4×D3111yescharacteristic
226C2×C4C2×D12411yescharacteristic
227C2×C4#9249511yescharacteristic
228C2×C4D4×D3111yescharacteristic
229C2×C4D4×D3111yescharacteristic
230C2×C4#9249911yescharacteristic
231C8—41no—
232C8—1241no—
233C22—22no—
234C22—314no—
235C22—624no—
236C22—624no—
237C22—622no—
238C22—622no—
239C22—1241no—
240C22—1242no—
241C22C22⋊C4×D3111yescharacteristic
242C22#1175845911yescharacteristic
243C22#1175847111yescharacteristic
244C4—24no—
245C4—42no—
246C4—624no—
247C4—622no—
248C4—1241no—
249C4C4×D12411yescharacteristic
250C4#1175846111yescharacteristic
251C4#1175847011yescharacteristic
252C4#1175847211yescharacteristic
253C2—21no—
254C2—21no—
255C2—624no—
256C2—2481no—
257C2#1452092711yescharacteristic
258C2#104×D3112yes—
subgroupG/H
centerC2×C4D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#339C2
socleC22×C31C2×C22⋊C4
prime radical#14521561C2
cosocleC22×C4×C31C23
hypercenter#339D31
hypocenterC31#1896
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×C42×D31C2semidirect
3. C2×#11758393C2semidirect
4. C2×#11758413C2semidirect
5. #14521012C2semidirect
6. #14520808C2semidirect
7. C31×#339C2semidirect
17. #11758474C4semidirect
18. #11758472C4semidirect
38. C2×D124C2×C4semidirect
48. C2×Dic62C2×C4semidirect
52. C4⋊C4×C31C2×C4semidirect
56. C42×C31D4semidirect
62. C4×Dic31D4semidirect
114. C2×D31#104semidirect
130. #339D31semidirect
156. C31#1896semidirect
186. C4⋊C4C4×D31semidirect
195. C42D124semidirect

Maximal subgroups

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

subgroupindex
#145215612
C2×C42×D312
C2×#117583932
C2×#117584132
#145210122
#145208082
C31×#3392
#189631