Finite Group Databasebeta

← #16846733

Subgroups — #16846733

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
1C2×#11758509C211yesmaximal, characteristic
2#14521623C211yesmaximal, characteristic, complemented
3C2×#11758393C211yesmaximal, characteristic, complemented
4#14521054C211yesmaximal, characteristic, complemented
5C2×#11758424C211yesmaximal, characteristic, complemented
6#14520809C211yesmaximal, characteristic, complemented
7C2×C31×#124C211yesmaximal, characteristic, complemented
8C22×Q8×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic
10C2×#92495C2211yescharacteristic
11C31×#124—22no—
12C31×#124—22no—
13C2×C22⋊C4×C31—22no—
14C2×C42×C31C2211yescharacteristic
15#11758524—21no—
16#11758523—22no—
17#11758521—21no—
18C2×#92480—21no—
19#11758509—22no—
20#11758509—22no—
21C2×#92479—21no—
22#11758506—22no—
23#11758503C2211yescharacteristic
24C2×#92474C2211yescharacteristic
25C2×#92474C2211yescharacteristic
26#11758424—22no—
27#11758420—22no—
28#11758393—22no—
29C24×C31—21no—
30C2×Q8×C31—21no—
31C2×Q8×C31—21no—
32C2×Q8×C31—21no—
33C2×Q8×C31—21no—
34C2×Q8×C31—21no—
35C2×Q8×C31—21no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31—21no—
39C2×D4×C31—21no—
40C2×D4×C31—21no—
41C2×D4×C31D412yes—
42C22×C4×C31—21no—
43C22×C4×C31D412yes—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—21no—
46#92495—21no—
47#92495—21no—
48#92495—21no—
49#92495—21no—
50C2×D124D411yescharacteristic
51C2×C4×D31—21no—
52C2×Dic62D411yescharacteristic
53C22⋊C4×C31—28no—
54C42×C31—21no—
55C42×C31D412yes—
56C22⋊Dic31—21no—
57#92480—24no—
58#92479—24no—
59(C2×D31)⋊C4—24no—
60C4⋊Dic31D412yes—
61Dic31⋊C4—24no—
62C4×Dic31—21no—
63#92474—21no—
64#92474—21no—
65#92474—21no—
66#92474D412yes—
67C2×C31⋊C8—21no—
68C2×C31⋊C8—21no—
69C23×C31—22no—
70C23×C31—44no—
71C23×C31—41no—
72C23×C31—41no—
73C23×C31C2×D411yescharacteristic
74C22×D31—21no—
75Q8×C31—41no—
76Q8×C31—41no—
77Q8×C31—41no—
78Q8×C31—41no—
79D4×C31—22no—
80D4×C31—22no—
81D4×C31—41no—
82D4×C31—41no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—42no—
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31#4111yescharacteristic
94C22⋊D31—41no—
95C22⋊D31—41no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98D124—21no—
99D124—21no—
100C4×D31—41no—
101C4×D31—41no—
102Dic62—21no—
103Dic62—21no—
104C31⋊C8—22no—
105C31⋊C8—41no—
106#2003—311nomaximal, Sylow
107C22×C31—22no—
108C22×C31—44no—
109C22×C31—42no—
110C22×C31—82no—
111C22×C31—82no—
112C22×C31#12011yescharacteristic
113C22×C31#12011yescharacteristic
114C22×C31#12111yescharacteristic
115C2×D31—41no—
116C2×D31—42no—
117C4×C31—42no—
118C4×C31—42no—
119C4×C31#12011yescharacteristic
120C4×C31#12111yescharacteristic
121C4×C31#12111yescharacteristic
122C4×C31#12711yescharacteristic
123Dic31—41no—
124Dic31—42no—
125C2×AGL(1,ℤ/8)—311no—
126#536—311no—
127C2×#124D3111yescharacteristic, complemented
128C2×#104—311no—
129#418—311no—
130C2×#101—311no—
131#339—311no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—82no—
135C2×C31#39211yescharacteristic
136C2×C31#45412yes—
137D31—81no—
138C2×#41—311no—
139C22×Q8C2×D3111yescharacteristic
140C22×D4C2×D3111yescharacteristic
141AGL(1,ℤ/8)—622no—
142AGL(1,ℤ/8)—622no—
143C2×SD16—621no—
144C2×D8—621no—
145C2×M16—311no—
146C2×M16—311no—
147C4⋊Q8—621no—
148#124—22no—
149#124—22no—
150#124—621no—
151#123—622no—
152C22⋊Q8—622no—
153#117—311no—
154C2×C22⋊C4—22no—
155C2×C42C2×D3111yescharacteristic
156#104—622no—
157#102—622no—
158#101—622no—
159C31#200311yescharacteristic, complemented, Sylow
160C24—21no—
161#41—621no—
162#41—621no—
163#41—621no—
164#41—621no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—311no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—311no—
178C2×D4—621no—
179C2×D4C22⋊D3112yes—
180C22×C4—21no—
181C22×C4—621no—
182C22×C4C22⋊D3112yes—
183C22×C4C22×D3111yescharacteristic
184SD16—624no—
185D8—624no—
186M16—312no—
187M16—621no—
188M16—621no—
189M16—621no—
190C2×C8—621no—
191C2×C8—621no—
192C4⋊C4—312no—
193C4⋊C4—624no—
194C22⋊C4—28no—
195C22⋊C4—624no—
196C22⋊C4—621no—
197C42—21no—
198C42—621no—
199C42C22⋊D3112yes—
200C23—22no—
201C23—44no—
202C23—41no—
203C23—41no—
204C23—621no—
205C23C2×C22⋊D3111yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—621no—
211Q8—621no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—621no—
217D4—621no—
218D4—1241no—
219D4—1241no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—622no—
226C2×C4—621no—
227C2×C4—1241no—
228C2×C4—1241no—
229C2×C4D4×D3111yescharacteristic
230C2×C4D4×D3111yescharacteristic
231C2×C4D4×D3111yescharacteristic
232C2×C4#9249711yescharacteristic
233C2×C4C2×C22⋊D3111yescharacteristic
234C2×C4C2×C22⋊D3111yescharacteristic
235C8—622no—
236C8—1241no—
237C22—22no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—82no—
242C22—1241no—
243C22—1242no—
244C22#1175851411yescharacteristic
245C22#1175851611yescharacteristic
246C22#1175853011yescharacteristic
247C4—42no—
248C4—42no—
249C4—1241no—
250C4—1242no—
251C4#1175851411yescharacteristic
252C4#1175851511yescharacteristic
253C4#1175851611yescharacteristic
254C4#1175851711yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—2481no—
259C2#1452095711yescharacteristic
260C2#1452122112yes—
subgroupG/H
centerC22#11758514
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC2×C31×#124C2
socleC22×C31#120
prime radicalC2×#11758509C2
hypercenterC2×#124D31
hypocenterC31#2003
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #14521623C2semidirect
3. C2×#11758393C2semidirect
4. #14521054C2semidirect
5. C2×#11758424C2semidirect
6. #14520809C2semidirect
7. C2×C31×#124C2semidirect
127. C2×#124D31semidirect
159. C31#2003semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
#145216232
C2×#117583932
#145210542
C2×#117584242
#145208092
C2×C31×#1242
#200331