Finite Group Databasebeta

← #16854990

Subgroups — #16854990

Order 3968.

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

normal subgroups only

This group has 752 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×#11758566C211yesmaximal, characteristic
2C2×#11758486C211yesmaximal, characteristic, complemented
3C22×#92470C211yesmaximal, characteristic, complemented
4#14521620C211yesmaximal, characteristic, complemented
5#14521213C214yesmaximal, complemented
6#14521202C214yesmaximal, complemented
7#14521055C211yesmaximal, characteristic, complemented
8#14521046C211yesmaximal, characteristic, complemented
9C31×#408C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758566—24no—
12#11758566—24no—
13C2×#92499—21no—
14C2×D4×D31—22no—
15C2×D4×D31—21no—
16C2×#92495C2212yes—
17C2×#92495C2211yescharacteristic
18C22×D124C2212yes—
19C22×D124C2211yescharacteristic
20C22×Dic62C2211yescharacteristic, complemented
21C2×M16×C31C2211yescharacteristic, complemented
22C22×C8×C31C2211yescharacteristic, complemented
23#11758521—21no—
24#11758513—21no—
25#11758512—22no—
26#11758503C2211yescharacteristic, complemented
27#11758501C2214yescomplemented
28#11758486—24no—
29C2×Dic124C2212yescomplemented
30C2×#92470—24no—
31C2×#92470C2212yescomplemented
32C2×#92470C2214yescomplemented
33C31×C22⋊C8C2214yescomplemented
34#11758409C2212yescomplemented
35#11758409C2214yescomplemented
36#11758404C2212yescomplemented
37C23×D31—22no—
38C23×D31—21no—
39#41×C31—22no—
40#41×C31—22no—
41C2×Q8×C31D411yescharacteristic
42C2×D4×C31D412yes—
43C2×D4×C31D411yescharacteristic
44C22×C4×C31C2312yes—
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—22no—
47C2×C22⋊D31—21no—
48C22×Dic31—21no—
49#92499—28no—
50D4×D31—216no—
51D4×D31—28no—
52#92495—22no—
53#92495—24no—
54#92495—24no—
55#92495D414yes—
56C2×D124—22no—
57C2×D124—21no—
58C2×D124—22no—
59C2×D124—24no—
60C2×D124—24no—
61C2×D124D414yes—
62C2×D124C2311yescharacteristic
63C2×D124C2312yes—
64C2×C4×D31—22no—
65C2×C4×D31—21no—
66C2×Dic62—22no—
67C2×Dic62—22no—
68C2×Dic62—21no—
69C2×Dic62C2312yes—
70C2×Dic62C2311yescharacteristic
71M16×C31—22no—
72C2×C8×C31—22no—
73C2×C8×C31C2312yes—
74C2×C8×C31C2312yes—
75C22⋊Dic31—24no—
76C22⋊Dic31—21no—
77C4⋊Dic31C2312yes—
78Dic31⋊C4—24no—
79C4×Dic31—21no—
80Dic124—24no—
81#92470—28no—
82#92470—24no—
83C23×C31C2×D412yes—
84C23×C31C2×D411yescharacteristic
85C22×D31—22no—
86C22×D31—21no—
87C22×D31—28no—
88C22×D31—24no—
89C22×D31—82no—
90C22×D31—84no—
91Q8×C31—22no—
92D4×C31—24no—
93D4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2411yescharacteristic
101C22⋊D31—28no—
102C22⋊D31—24no—
103C2×Dic31—21no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—42no—
107D124—28no—
108D124—24no—
109D124—21no—
110D124—22no—
111D124C2×D412yes—
112D124C2×D414yes—
113C4×D31—28no—
114C4×D31—24no—
115Dic62—21no—
116Dic62—22no—
117Dic62—22no—
118Dic62—42no—
119Dic62C2×D412yes—
120C8×C31—22no—
121C8×C31—22no—
122#3009—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—24no—
126C22×C31#12012yes—
127C22×C31C22×D411yescharacteristic
128C2×D31—24no—
129C2×D31—22no—
130C2×D31—44no—
131C2×D31—42no—
132C2×D31—84no—
133C2×D31—88no—
134C4×C31—22no—
135C4×C31#12012yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—42no—
140Dic31—42no—
141C2×H(5,𝔽2)—311no—
142C2×#137—311no—
143C22×SD16—311no—
144#536—311no—
145#452—314no—
146#450—314no—
147#418—311no—
148#416—311no—
149#408D3111yescharacteristic, complemented
150C2×C31—22no—
151C2×C31—42no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—44no—
155D31—42no—
156H(5,𝔽2)—624no—
157H(5,𝔽2)—624no—
158C2×#41—311no—
159C2×#41—312no—
160C2×#41—621no—
161C2×#41C2×D3111yescharacteristic
162C22×Q8—311no—
163C22×D4—311no—
164C22×D4—312no—
165C22×D4—622no—
166C22×D4—621no—
167#137—624no—
168C2×Q16—312no—
169C2×SD16—314no—
170C2×SD16—312no—
171C2×SD16—624no—
172C2×M16C2×D3111yescharacteristic, complemented
173C22×C8C2×D3111yescharacteristic, complemented
174C4⋊Q8—621no—
175#124—621no—
176#123—622no—
177C22⋊Q8—314no—
178#117—311no—
179#103—312no—
180#102—312no—
181#102—314no—
182C22⋊C8C2×D3114yescomplemented
183C31#300911yescharacteristic, complemented, Sylow
184C24—622no—
185C24—621no—
186#41—22no—
187#41—22no—
188#41—314no—
189#41—624no—
190#41—624no—
191#41—628no—
192#41—622no—
193C2×Q8—311no—
194C2×Q8—312no—
195C2×Q8—621no—
196C2×Q8—622no—
197C2×Q8—622no—
198C2×Q8D12411yescharacteristic
199C2×D4—314no—
200C2×D4—312no—
201C2×D4—311no—
202C2×D4—624no—
203C2×D4—624no—
204C2×D4—6216no—
205C2×D4—628no—
206C2×D4—622no—
207C2×D4—621no—
208C2×D4—622no—
209C2×D4—621no—
210C2×D4—622no—
211C2×D4D12412yes—
212C2×D4D12411yescharacteristic
213C22×C4—622no—
214C22×C4—621no—
215C22×C4—621no—
216C22×C4C22×D3112yes—
217C22×C4C22×D3111yescharacteristic
218Q16—624no—
219SD16—628no—
220SD16—624no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D3112yes—
224C2×C8C22×D3112yes—
225C4⋊C4—312no—
226C4⋊C4—624no—
227C22⋊C4—621no—
228C22⋊C4—624no—
229C42—621no—
230C23—628no—
231C23—622no—
232C23—621no—
233C23—624no—
234C23—2482no—
235C23—2484no—
236C23C2×D12412yes—
237C23C2×D12411yescharacteristic
238Q8—22no—
239Q8—312no—
240Q8—621no—
241Q8—622no—
242Q8—622no—
243Q8—1242no—
244D4—24no—
245D4—22no—
246D4—314no—
247D4—312no—
248D4—628no—
249D4—628no—
250D4—624no—
251D4—624no—
252D4—621no—
253D4—622no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—628no—
257C2×C4—621no—
258C2×C4—624no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—1242no—
262C2×C4C2×D12412yes—
263C2×C4C2×D12411yescharacteristic
264C2×C4D4×D3112yes—
265C2×C4D4×D3112yes—
266C2×C4C23×D3111yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—624no—
273C22—622no—
274C22—1244no—
275C22—1242no—
276C22—2484no—
277C22—2488no—
278C22#1175845912yes—
279C22C22×D12411yescharacteristic
280C4—22no—
281C4—622no—
282C4—1242no—
283C4—1242no—
284C4#1175845912yes—
285C4C2×D4×D3111yescharacteristic
286C4C2×D4×D3111yescharacteristic
287C2—22no—
288C2—42no—
289C2—1244no—
290C2—1242no—
291C2C2×#1175845911yescharacteristic
292C2#1452186112yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#408C2
socleC22×C31C22×D4
prime radicalC2×#11758566C2
hypercenter#408D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#11758486C2semidirect
3. C22×#92470C2semidirect
4. #14521620C2semidirect
5. #14521213C2semidirect
6. #14521202C2semidirect
7. #14521055C2semidirect
8. #14521046C2semidirect
9. C31×#408C2semidirect
20. C22×Dic62C22semidirect
21. C2×M16×C31C22semidirect
22. C22×C8×C31C22semidirect
26. #11758503C22semidirect
27. #11758501C22semidirect
29. C2×Dic124C22semidirect
31. C2×#92470C22semidirect
32. C2×#92470C22semidirect
33. C31×C22⋊C8C22semidirect
34. #11758409C22semidirect
35. #11758409C22semidirect
36. #11758404C22semidirect
149. #408D31semidirect
172. C2×M16C2×D31semidirect
173. C22×C8C2×D31semidirect
182. C22⋊C8C2×D31semidirect
183. C31#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584862
C22×#924702
#145216202
#145212132
#145212022
#145210552
#145210462
C31×#4082
#300931