Finite Group Databasebeta

← #16855002

Subgroups — #16855002

Order 3968.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758564C211yesmaximal, characteristic
2C2×#11758519C211yesmaximal, characteristic, complemented
3#14521623C211yesmaximal, characteristic, complemented
4#14521210C214yesmaximal, complemented
5#14521203C214yesmaximal, complemented
6#14521058C211yesmaximal, characteristic, complemented
7#14521046C211yesmaximal, characteristic, complemented
8#14521015C211yesmaximal, characteristic, complemented
9C22×Q16×C31C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×Q8×C31C2211yescharacteristic
12#11758564—24no—
13#11758564—24no—
14C2×#92499—21no—
15C2×#92499C2212yes—
16C2×Q8×D31—21no—
17C2×Q8×D31C2212yes—
18C2×#92495—22no—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic
21C2×Q16×C31—24no—
22C2×Q16×C31C2214yes—
23C22×C8×C31C2211yescharacteristic
24#11758524—21no—
25#11758523C2214yes—
26#11758521—21no—
27C2×#92482C2212yes—
28#11758519—24no—
29C2×#92481C2212yes—
30#11758506—22no—
31#11758503C2211yescharacteristic
32C2×#92474C2211yescharacteristic
33#11758423C2214yes—
34#11758409C2212yes—
35(C2×D31)⋊C8C2214yes—
36#11758404C2212yes—
37C2×Q8×C31—21no—
38C2×Q8×C31—22no—
39C2×Q8×C31—21no—
40C2×Q8×C31—22no—
41C2×Q8×C31—22no—
42C2×Q8×C31D414yes—
43C2×Q8×C31C2312yes—
44C2×Q8×C31C2312yes—
45C22×C4×C31—21no—
46C22×C4×C31—21no—
47C22×C4×C31C2311yescharacteristic
48C2×C22⋊D31—21no—
49C2×C22⋊D31D412yes—
50#92499—28no—
51#92499—24no—
52#92499—24no—
53Q8×D31—28no—
54Q8×D31—24no—
55Q8×D31—24no—
56#92495—216no—
57#92495—22no—
58#92495—22no—
59#92495—22no—
60#92495D414yes—
61C2×D124—22no—
62C2×D124D411yescharacteristic
63C2×D124C2311yescharacteristic
64C2×C4×D31—22no—
65C2×C4×D31—21no—
66C2×C4×D31—22no—
67C2×C4×D31C2312yes—
68C2×Dic62—22no—
69C2×Dic62D411yescharacteristic
70C2×Dic62C2311yescharacteristic
71Q16×C31—28no—
72C2×C8×C31—22no—
73C2×C8×C31C2312yes—
74C22⋊Dic31—21no—
75#92482—24no—
76#92481—24no—
77(C2×D31)⋊C4—24no—
78C4⋊Dic31C2312yes—
79Dic31⋊C4—24no—
80C4×Dic31—21no—
81#92474—22no—
82C2×C31⋊C8C2312yes—
83C23×C31C2×D411yescharacteristic
84C22×D31—21no—
85C22×D31C2×D412yes—
86Q8×C31—24no—
87Q8×C31—22no—
88Q8×C31—22no—
89Q8×C31—22no—
90Q8×C31—42no—
91Q8×C31C2×D414yes—
92C2×C4×C31—24no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—42no—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—24no—
101C22⋊D31—24no—
102C2×Dic31—21no—
103C2×Dic31—22no—
104C2×Dic31C2×D412yes—
105D124—28no—
106D124—22no—
107D124—21no—
108D124C2×D412yes—
109C4×D31—28no—
110C4×D31—28no—
111C4×D31—24no—
112C4×D31—22no—
113C4×D31—22no—
114Dic62—28no—
115Dic62—22no—
116Dic62—21no—
117Dic62C2×D412yes—
118C8×C31—22no—
119C31⋊C8—22no—
120#3011—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31#12012yes—
123C22×C31C22×D411yescharacteristic
124C2×D31—22no—
125C2×D31—22no—
126C2×D31—24no—
127C2×D31—42no—
128C4×C31—24no—
129C4×C31—42no—
130C4×C31#12012yes—
131C4×C31C22×D411yescharacteristic
132C4×C31C22×D411yescharacteristic
133Dic31—22no—
134Dic31—22no—
135Dic31—42no—
136C2×2−1+4—311no—
137C2×#137—311no—
138C22×Q16D3111yescharacteristic, complemented
139#536—311no—
140#452—314no—
141#451—314no—
142#419—311no—
143#416—311no—
144#408—311no—
145C2×C31—22no—
146C2×C31C2×#12011yescharacteristic
147C2×C31#57812yes—
148D31—42no—
149D31—42no—
1502−1+4—624no—
1512−1+4—624no—
152C2×#41—311no—
153C2×#41—312no—
154C2×#41—311no—
155C2×#41—622no—
156C2×#41—621no—
157C22×Q8—312no—
158C22×Q8—621no—
159C22×Q8C2×D3111yescharacteristic
160C22×Q8C2×D3111yescharacteristic
161#137—624no—
162C2×Q16—24no—
163C2×Q16—312no—
164C2×Q16C2×D3114yes—
165C2×SD16—312no—
166C2×M16—311no—
167C22×C8C2×D3111yescharacteristic
168C4⋊Q8—621no—
169#124—621no—
170#123—622no—
171C22⋊Q8—314no—
172#117—311no—
173#103—314no—
174#103—312no—
175#102—312no—
176C22⋊C8—314no—
177C31#301111yescharacteristic, complemented, Sylow
178#41—314no—
179#41—6216no—
180#41—624no—
181#41—628no—
182#41—624no—
183#41—622no—
184#41—622no—
185#41—622no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—311no—
192C2×Q8—311no—
193C2×Q8—622no—
194C2×Q8—624no—
195C2×Q8—628no—
196C2×Q8—624no—
197C2×Q8C22⋊D3114yes—
198C2×Q8C22×D3112yes—
199C2×Q8C22×D3112yes—
200C2×D4—312no—
201C2×D4—311no—
202C2×D4—311no—
203C2×D4—622no—
204C2×D4—621no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—312no—
208C22×C4—622no—
209C22×C4—622no—
210C22×C4—621no—
211C22×C4C22×D3111yescharacteristic
212Q16—28no—
213Q16—624no—
214SD16—624no—
215M16—622no—
216C2×C8—22no—
217C2×C8—312no—
218C2×C8C22×D3112yes—
219C4⋊C4—312no—
220C4⋊C4—624no—
221C22⋊C4—624no—
222C22⋊C4—621no—
223C42—621no—
224C23—312no—
225C23—621no—
226C23D4×D3111yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—312no—
233Q8—628no—
234Q8—622no—
235Q8—621no—
236Q8C2×C22⋊D3114yes—
237D4—312no—
238D4—628no—
239D4—624no—
240D4—624no—
241D4—622no—
242D4—621no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—312no—
248C2×C4—628no—
249C2×C4—628no—
250C2×C4—621no—
251C2×C4—624no—
252C2×C4—622no—
253C2×C4—622no—
254C2×C4—622no—
255C2×C4D4×D3111yescharacteristic
256C2×C4D4×D3112yes—
257C2×C4C2×C22⋊D3112yes—
258C2×C4C23×D3111yescharacteristic
259C8—22no—
260C8—622no—
261C22—22no—
262C22—624no—
263C22—622no—
264C22—622no—
265C22—1242no—
266C22#1175851412yes—
267C22C2×D4×D3111yescharacteristic
268C4—24no—
269C4—42no—
270C4—622no—
271C4—622no—
272C4—1242no—
273C4#1175851412yes—
274C4C2×D4×D3111yescharacteristic
275C4C22×C22⋊D3111yescharacteristic
276C2—22no—
277C2—1242no—
278C2—1242no—
279C2C2×#1175851411yescharacteristic
280C2#1452186212yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×Q16×C31C2
socleC22×C31C22×D4
prime radicalC2×#11758564C2
hypercenterC22×Q16D31
hypocenterC31#3011
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521623C2semidirect
4. #14521210C2semidirect
5. #14521203C2semidirect
6. #14521058C2semidirect
7. #14521046C2semidirect
8. #14521015C2semidirect
9. C22×Q16×C31C2semidirect
138. C22×Q16D31semidirect
177. C31#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#117585192
#145216232
#145212102
#145212032
#145210582
#145210462
#145210152
C22×Q16×C312
#301131