Finite Group Databasebeta

← #16846213

Subgroups — #16846213

Order 3968.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#14521849C211yesmaximal, characteristic
2#14521517C211yesmaximal, characteristic, complemented
3#14521065C211yesmaximal, characteristic, complemented
4C2×#11758413C211yesmaximal, characteristic, complemented
5#14521022C211yesmaximal, characteristic, complemented
6#14520810C211yesmaximal, characteristic, complemented
7C31×#344C211yesmaximal, characteristic, complemented
8C2×#92495C2211yescharacteristic
9C2×#92495C2211yescharacteristic
10C2×M16×C31C2211yescharacteristic
11C2×M16×C31C2211yescharacteristic
12C31×#117C2211yescharacteristic
13#11758503C2211yescharacteristic
14C2×C4×Dic31C2211yescharacteristic
15#11758486—21no—
16#11758486C412yescomplemented
17#11758485—21no—
18#11758485C412yescomplemented
19#11758481—22no—
20#11758470—21no—
21#11758464—21no—
22(C22⋊D31)⋊C4—22no—
23#11758449—21no—
24C42×D31—21no—
25#11758413—22no—
26#11758413—21no—
27#11758413—21no—
28#11758394—22no—
29C22×C4×C31C2311yescharacteristic
30C2×C22⋊D31—21no—
31C2×C22⋊D31—21no—
32C22×Dic31—21no—
33#92495—21no—
34#92495—21no—
35#92495—21no—
36#92495—21no—
37#92495—21no—
38#92495—21no—
39#92495C2×C412yes—
40#92495D412yes—
41C2×D124C2×C411yescharacteristic, complemented
42C2×D124D411yescharacteristic
43C2×C4×D31—21no—
44C2×C4×D31—22no—
45C2×C4×D31—21no—
46C2×Dic62C2×C411yescharacteristic, complemented
47C2×Dic62D411yescharacteristic
48M16×C31—21no—
49M16×C31—21no—
50M16×C31—21no—
51M16×C31C2×C412yescomplemented
52C2×C8×C31—21no—
53C2×C8×C31—21no—
54C4⋊C4×C31—21no—
55C22⋊C4×C31D412yes—
56C42×C31—21no—
57C22⋊Dic31—21no—
58(C2×D31)⋊C4—22no—
59C4⋊Dic31—21no—
60Dic31⋊C4—22no—
61C4×Dic31—22no—
62C4×Dic31—21no—
63C4×Dic31—21no—
64C4×Dic31D411yescharacteristic
65C4×Dic31D411yescharacteristic
66Dic124—22no—
67D248—22no—
68#92470—22no—
69#92470—22no—
70C23×C31C2×D411yescharacteristic
71C22×D31—21no—
72C22×D31—21no—
73C2×C4×C31—22no—
74C2×C4×C31C22×C411yescharacteristic
75C2×C4×C31C2×D411yescharacteristic
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31#4111yescharacteristic
79C2×C4×C31#4111yescharacteristic
80C22⋊D31—22no—
81C22⋊D31—22no—
82C22⋊D31—41no—
83C22⋊D31—41no—
84C2×Dic31—21no—
85C2×Dic31—21no—
86C2×Dic31—21no—
87C2×Dic31—21no—
88C2×Dic31—42no—
89C2×Dic31—41no—
90D124—21no—
91D124—21no—
92D124—21no—
93D124C22⋊C412yes—
94C4×D31—22no—
95C4×D31—22no—
96C4×D31—44no—
97C4×D31—41no—
98C4×D31—41no—
99Dic62—21no—
100Dic62—21no—
101Dic62—21no—
102Dic62C22⋊C412yes—
103C8×C31—22no—
104C8×C31—41no—
105#1906—311nomaximal, Sylow
106C22×C31—41no—
107C22×C31C4×D411yescharacteristic
108C22×C31#12011yescharacteristic
109C22×C31#12111yescharacteristic
110C2×D31—22no—
111C2×D31—42no—
112C2×D31—41no—
113C2×D31—81no—
114C4×C31—42no—
115C4×C31C2×C22⋊C411yescharacteristic
116C4×C31C4×D411yescharacteristic
117C4×C31#12111yescharacteristic
118C4×C31#12311yescharacteristic
119Dic31—22no—
120Dic31—22no—
121Dic31—41no—
122Dic31—41no—
123Dic31—81no—
124#575—311no—
125C4×#41—311no—
126#421—311no—
127C2×#104—311no—
128#410—311no—
129#344D3111yescharacteristic, complemented
130#339—311no—
131C2×C31—21no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31#38611yescharacteristic
135D31—42no—
136D31—81no—
137C2×#41—311no—
138C2×#41—311no—
139#137—312no—
140#137—621no—
141AGL(1,ℤ/8)—312no—
142AGL(1,ℤ/8)—621no—
143#135—622no—
144C2×M16C2×D3111yescharacteristic
145C2×M16C2×D3111yescharacteristic
146C4×Q8—621no—
147C4×D4—621no—
148C4×D4—622no—
149#117—311no—
150#117—621no—
151#117C2×D3111yescharacteristic
152C2×C42—311no—
153C2×C42—621no—
154#104—621no—
155#104—621no—
156#104—622no—
157#99—622no—
158C31#190611yescharacteristic, complemented, Sylow
159#41—312no—
160#41—312no—
161#41—621no—
162#41—621no—
163#41—621no—
164#41—621no—
165#41—621no—
166#41—621no—
167C2×Q8—311no—
168C2×Q8—311no—
169C2×D4—311no—
170C2×D4—311no—
171C2×D4—621no—
172C2×D4—621no—
173C22×C4—622no—
174C22×C4—621no—
175C22×C4—621no—
176C22×C4—621no—
177C22×C4C22×D3111yescharacteristic
178Q16—622no—
179SD16—622no—
180SD16—622no—
181D8—622no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D3112yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—622no—
190C4⋊C4—621no—
191C22⋊C4—622no—
192C22⋊C4—621no—
193C22⋊C4D12412yes—
194C42—21no—
195C42—311no—
196C42—311no—
197C42—622no—
198C42—621no—
199C42—621no—
200C23—621no—
201C23—621no—
202C23C2×D12411yescharacteristic
203Q8—312no—
204Q8—621no—
205Q8—621no—
206Q8—621no—
207D4—312no—
208D4—622no—
209D4—622no—
210D4—621no—
211D4—621no—
212D4—621no—
213D4—1241no—
214D4—1241no—
215C2×C4—22no—
216C2×C4—621no—
217C2×C4—621no—
218C2×C4—622no—
219C2×C4—622no—
220C2×C4—621no—
221C2×C4—621no—
222C2×C4—1244no—
223C2×C4—1242no—
224C2×C4—1241no—
225C2×C4—1241no—
226C2×C4—1241no—
227C2×C4C2×C4×D3111yescharacteristic
228C2×C4#9249511yescharacteristic
229C2×C4D4×D3111yescharacteristic
230C2×C4D4×D3111yescharacteristic
231C2×C4D4×D3111yescharacteristic
232C2×C4#9249911yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—622no—
237C22—1242no—
238C22—1241no—
239C22—2481no—
240C22C4×D12411yescharacteristic
241C22#1175845911yescharacteristic
242C22#1175847211yescharacteristic
243C4—42no—
244C4—622no—
245C4—622no—
246C4—1241no—
247C4—1241no—
248C4—2481no—
249C4C22⋊C4×D3111yescharacteristic
250C4#1175846111yescharacteristic
251C4#1175847011yescharacteristic
252C4#1175847111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—1242no—
257C2—2481no—
258C2#1452092711yescharacteristic
subgroupG/H
centerC4#11758461
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#344C2
socleC2×C31#386
prime radical#14521849C2
cosocleC22×C4×C31C23
hypercenter#344D31
hypocenterC31#1906
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #14521517C2semidirect
3. #14521065C2semidirect
4. C2×#11758413C2semidirect
5. #14521022C2semidirect
6. #14520810C2semidirect
7. C31×#344C2semidirect
16. #11758486C4semidirect
18. #11758485C4semidirect
41. C2×D124C2×C4semidirect
46. C2×Dic62C2×C4semidirect
51. M16×C31C2×C4semidirect
129. #344D31semidirect
158. C31#1906semidirect
185. M16C4×D31semidirect

Maximal subgroups

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

subgroupindex
#145218492
#145215172
#145210652
C2×#117584132
#145210222
#145208102
C31×#3442
#190631