Finite Group Databasebeta

← #16846730

Subgroups — #16846730

Order 3968.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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#14521620C211yesmaximal, characteristic, complemented
2C2×#11758513C211yesmaximal, characteristic, complemented
3C2×#11758425C211yesmaximal, characteristic, complemented
4#14521054C211yesmaximal, characteristic, complemented
5C2×#11758412C211yesmaximal, characteristic, complemented
6#14520810C211yesmaximal, characteristic, complemented
7C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
8C2×#41×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic
10C22×Dic62C2211yescharacteristic
11C31×AGL(1,ℤ/8)—22no—
12C31×AGL(1,ℤ/8)—22no—
13C2×SD16×C31—21no—
14C2×D8×C31—21no—
15C2×M16×C31C2211yescharacteristic
16C2×C22⋊Dic31—22no—
17#11758521—21no—
18#11758513—21no—
19#11758513—22no—
20#11758513—22no—
21#11758512—22no—
22#11758503C2211yescharacteristic
23#11758501—22no—
24C2×C4×Dic31C2211yescharacteristic
25C2×#92474C2211yescharacteristic
26#11758425—22no—
27#11758420—22no—
28#11758412—22no—
29C24×C31—21no—
30#41×C31—21no—
31#41×C31—21no—
32#41×C31—21no—
33#41×C31—21no—
34C2×Q8×C31D411yescharacteristic
35C2×D4×C31—21no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31—21no—
39C2×D4×C31—21no—
40C2×D4×C31—21no—
41C2×D4×C31D411yescharacteristic
42C2×D4×C31D412yes—
43C22×C4×C31—21no—
44C22×C4×C31C2311yescharacteristic
45C22×Dic31—21no—
46C22×Dic31D412yes—
47C2×Dic62—21no—
48C2×Dic62—21no—
49C2×Dic62—21no—
50C2×Dic62—21no—
51C2×Dic62—21no—
52C2×Dic62—21no—
53SD16×C31—24no—
54D8×C31—24no—
55M16×C31—21no—
56M16×C31D412yes—
57C2×C8×C31—21no—
58C22⋊Dic31—24no—
59C22⋊Dic31—28no—
60C22⋊Dic31—21no—
61C4⋊Dic31D412yes—
62Dic31⋊C4—24no—
63C4×Dic31—21no—
64C4×Dic31—21no—
65C4×Dic31D412yes—
66#92474—21no—
67#92474—21no—
68C2×C31⋊C8—21no—
69C23×C31—21no—
70C23×C31—22no—
71C23×C31—44no—
72C23×C31—41no—
73C23×C31—41no—
74C23×C31C2×D411yescharacteristic
75Q8×C31—21no—
76Q8×C31—21no—
77D4×C31—21no—
78D4×C31—22no—
79D4×C31—22no—
80D4×C31—21no—
81D4×C31—41no—
82D4×C31—41no—
83D4×C31—41no—
84D4×C31—41no—
85C2×C4×C31—21no—
86C2×C4×C31—41no—
87C2×C4×C31—41no—
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31#4111yescharacteristic
94C2×Dic31—22no—
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31—42no—
100Dic62—41no—
101Dic62—41no—
102Dic62—41no—
103Dic62—41no—
104C8×C31—22no—
105C31⋊C8—41no—
106#2003—311nomaximal, Sylow
107C22×C31—22no—
108C22×C31—44no—
109C22×C31—41no—
110C22×C31—42no—
111C22×C31—42no—
112C22×C31—82no—
113C22×C31—82no—
114C22×C31#12011yescharacteristic
115C22×C31#12011yescharacteristic
116C22×C31#12111yescharacteristic
117C4×C31—41no—
118C4×C31#12011yescharacteristic
119C4×C31#12111yescharacteristic
120C4×C31#12111yescharacteristic
121C4×C31#12711yescharacteristic
122Dic31—42no—
123Dic31—42no—
124Dic31—42no—
125C2×AGL(1,ℤ/8)D3111yescharacteristic
126#536—311no—
127C2×#124—311no—
128C2×#104—311no—
129#418—311no—
130C2×#101—311no—
131#339—311no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—81no—
135C2×C31—82no—
136C2×C31#39211yescharacteristic
137C2×C31#45412yes—
138C2×#41C2×D3111yescharacteristic
139C22×Q8—311no—
140C22×D4C2×D3111yescharacteristic
141AGL(1,ℤ/8)—22no—
142AGL(1,ℤ/8)—22no—
143C2×SD16—21no—
144C2×D8—21no—
145C2×M16—311no—
146C2×M16C2×D3111yescharacteristic
147C4⋊Q8—621no—
148#124—622no—
149#124—622no—
150#124—621no—
151#123—622no—
152C22⋊Q8—622no—
153#117—311no—
154C2×C22⋊C4—622no—
155C2×C42—311no—
156#104—622no—
157#102—622no—
158#101—622no—
159C31#200311yescharacteristic, complemented, Sylow
160C24—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—621no—
166C2×Q8—621no—
167C2×Q8—621no—
168C2×Q8—621no—
169C2×Q8—621no—
170C2×Q8—621no—
171C2×Q8C22⋊D3111yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C22⋊D3111yescharacteristic
179C2×D4C22⋊D3112yes—
180C22×C4—21no—
181C22×C4—312no—
182C22×C4—621no—
183C22×C4C22×D3111yescharacteristic
184SD16—24no—
185D8—24no—
186M16—21no—
187M16—621no—
188M16—621no—
189M16C22⋊D3112yes—
190C2×C8—21no—
191C2×C8—621no—
192C4⋊C4—312no—
193C4⋊C4—624no—
194C22⋊C4—628no—
195C22⋊C4—624no—
196C22⋊C4—621no—
197C42—312no—
198C42—621no—
199C42—621no—
200C23—22no—
201C23—21no—
202C23—44no—
203C23—41no—
204C23—41no—
205C23D4×D3111yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—1241no—
209Q8—1241no—
210Q8—1241no—
211Q8—1241no—
212D4—22no—
213D4—21no—
214D4—21no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—622no—
224C2×C4—622no—
225C2×C4—622no—
226C2×C4—622no—
227C2×C4—622no—
228C2×C4—1242no—
229C2×C4D4×D3111yescharacteristic
230C2×C4D4×D3111yescharacteristic
231C2×C4#9249711yescharacteristic
232C2×C4C2×C22⋊D3111yescharacteristic
233C2×C4C2×C22⋊D3111yescharacteristic
234C2×C4C2×C22⋊D3111yescharacteristic
235C8—22no—
236C8—1241no—
237C22—22no—
238C22—44no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—82no—
244C22#1175851411yescharacteristic
245C22#1175851411yescharacteristic
246C22#1175851511yescharacteristic
247C4—41no—
248C4—1242no—
249C4—1242no—
250C4—1242no—
251C4#1175851611yescharacteristic
252C4#1175851611yescharacteristic
253C4#1175851711yescharacteristic
254C4#1175853011yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—81no—
259C2#1452095711yescharacteristic
260C2#1452122012yes—
subgroupG/H
centerC22#11758514
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
Fitting, prime radicalC2×C31×AGL(1,ℤ/8)C2
socleC22×C31#120
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#2003
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14521620C2semidirect
2. C2×#11758513C2semidirect
3. C2×#11758425C2semidirect
4. #14521054C2semidirect
5. C2×#11758412C2semidirect
6. #14520810C2semidirect
159. C31#2003semidirect

Maximal subgroups

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

subgroupindex
#145216202
C2×#117585132
C2×#117584252
#145210542
C2×#117584122
#145208102
C2×C31×AGL(1,ℤ/8)2
#200331