Finite Group Databasebeta

← #16854989

Subgroups — #16854989

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×#11758561C211yesmaximal, characteristic
2C2×#11758486C211yesmaximal, characteristic, complemented
3C22×#92480C211yesmaximal, characteristic, complemented
4#14521625C211yesmaximal, characteristic, complemented
5#14521212C214yesmaximal, complemented
6#14521201C214yesmaximal, complemented
7#14521043C211yesmaximal, characteristic, complemented
8#14521017C211yesmaximal, characteristic, complemented
9C31×#418C211yesmaximal, characteristic, complemented
10C22×D4×C31C2211yescharacteristic
11C22×C22⋊D31—22no—
12#11758561—24no—
13#11758561—24no—
14C2×#92497—21no—
15C2×#92497C2212yes—
16C2×D4×D31—21no—
17C2×D4×D31C2212yes—
18C2×#92495C2211yescharacteristic
19C2×#92495C2211yescharacteristic
20C22×Dic62C2211yescharacteristic, complemented
21C2×M16×C31C2211yescharacteristic, complemented
22C31×#117C2211yescharacteristic, complemented
23C2×#92480—24no—
24C2×#92480C2214yescomplemented
25C22×C31⋊C8C2211yescharacteristic, complemented
26#11758486—24no—
27C2×Dic124C2212yescomplemented
28C2×#92470C2212yescomplemented
29#11758474C2214yescomplemented
30#11758463—22no—
31#11758452—21no—
32#11758446—21no—
33C31×#102C2214yescomplemented
34(C2×D31)⋊C8C2214yescomplemented
35#11758398C2212yescomplemented
36#11758397C2212yescomplemented
37C24×C31—21no—
38C23×D31—22no—
39C2×D4×C31—21no—
40C2×D4×C31—22no—
41C2×D4×C31D414yes—
42C2×D4×C31C2312yes—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—216no—
45C2×C22⋊D31—21no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31D412yes—
49C22×Dic31—22no—
50C22×Dic31—21no—
51#92497—28no—
52#92497—24no—
53#92497—24no—
54D4×D31—28no—
55D4×D31—24no—
56D4×D31—24no—
57#92495—22no—
58#92495—22no—
59#92495—22no—
60#92495D414yes—
61C2×D124D411yescharacteristic
62C2×D124C2311yescharacteristic
63C2×C4×D31—21no—
64C2×C4×D31C2312yes—
65C2×Dic62—22no—
66C2×Dic62—22no—
67C2×Dic62—21no—
68C2×Dic62D411yescharacteristic
69C2×Dic62C2312yes—
70C2×Dic62C2311yescharacteristic
71M16×C31—22no—
72C2×C8×C31C2312yes—
73C4⋊C4×C31C2312yes—
74C22⋊C4×C31—21no—
75C42×C31—21no—
76#92480—28no—
77(C2×D31)⋊C4—24no—
78C4⋊Dic31—24no—
79C2×C31⋊C8—22no—
80C2×C31⋊C8C2312yes—
81Dic124—24no—
82#92470—24no—
83C23×C31—22no—
84C23×C31—24no—
85C23×C31—82no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88C22×D31—28no—
89C22×D31—84no—
90C22×D31C2×D412yes—
91D4×C31—24no—
92D4×C31—22no—
93D4×C31C2×D414yes—
94C2×C4×C31—22no—
95C2×C4×C31C2×D412yes—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2411yescharacteristic
99C22⋊D31—28no—
100C22⋊D31—28no—
101C22⋊D31—24no—
102C22⋊D31—24no—
103C2×Dic31—28no—
104C2×Dic31—21no—
105C2×Dic31—22no—
106C2×Dic31—42no—
107C2×Dic31C2×D412yes—
108D124—22no—
109D124—21no—
110D124C2×D412yes—
111C4×D31—24no—
112C4×D31—22no—
113C4×D31—22no—
114Dic62—22no—
115Dic62—21no—
116Dic62—22no—
117Dic62—22no—
118Dic62—42no—
119Dic62C2×D412yes—
120C8×C31—22no—
121C31⋊C8—22no—
122#3009—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—24no—
125C22×C31—44no—
126C22×C31—84no—
127C22×C31#12012yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—22no—
130C2×D31—22no—
131C2×D31—24no—
132C2×D31—42no—
133C2×D31—88no—
134C4×C31—42no—
135C4×C31#12012yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—22no—
140Dic31—42no—
141C2×H(5,𝔽2)—311no—
142C2×#137—311no—
143C22×SD16—311no—
144#536—311no—
145#452—314no—
146#450—314no—
147#418D3111yescharacteristic, complemented
148#416—311no—
149#408—311no—
150C2×C31—22no—
151C2×C31—44no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—42no—
155D31—42no—
156H(5,𝔽2)—624no—
157H(5,𝔽2)—624no—
158C2×#41—311no—
159C2×#41—312no—
160C2×#41—311no—
161C2×#41—621no—
162C22×Q8—311no—
163C22×D4—312no—
164C22×D4—621no—
165C22×D4—622no—
166C22×D4C2×D3111yescharacteristic
167#137—624no—
168C2×Q16—312no—
169C2×SD16—314no—
170C2×SD16—312no—
171C2×SD16—624no—
172C2×M16C2×D3111yescharacteristic, complemented
173C22×C8—311no—
174C4⋊Q8—621no—
175#124—621no—
176#123—622no—
177C22⋊Q8—314no—
178#117C2×D3111yescharacteristic, complemented
179#103—312no—
180#102—312no—
181#102C2×D3114yescomplemented
182C22⋊C8—314no—
183C31#300911yescharacteristic, complemented, Sylow
184C24—21no—
185C24—622no—
186#41—314no—
187#41—624no—
188#41—628no—
189#41—624no—
190#41—622no—
191#41—622no—
192#41—622no—
193C2×Q8—311no—
194C2×Q8—311no—
195C2×Q8—312no—
196C2×Q8—621no—
197C2×Q8—622no—
198C2×Q8—622no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—312no—
202C2×D4—311no—
203C2×D4—311no—
204C2×D4—622no—
205C2×D4—624no—
206C2×D4—628no—
207C2×D4—622no—
208C2×D4—6216no—
209C2×D4—621no—
210C2×D4—624no—
211C2×D4D12414yes—
212C2×D4C22×D3112yes—
213C22×C4—312no—
214C22×C4—622no—
215C22×C4—621no—
216C22×C4—621no—
217C22×C4C22×D3111yescharacteristic
218Q16—624no—
219SD16—624no—
220SD16—628no—
221M16—22no—
222C2×C8—312no—
223C2×C8—622no—
224C2×C8C22×D3112yes—
225C4⋊C4—624no—
226C4⋊C4C22×D3112yes—
227C22⋊C4—21no—
228C22⋊C4—624no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—312no—
234C23—628no—
235C23—621no—
236C23—2484no—
237C23D4×D3111yescharacteristic
238Q8—312no—
239Q8—622no—
240Q8—621no—
241Q8—622no—
242Q8—622no—
243Q8—1242no—
244D4—24no—
245D4—22no—
246D4—312no—
247D4—628no—
248D4—628no—
249D4—624no—
250D4—624no—
251D4—622no—
252D4—621no—
253D4C2×D12414yes—
254C2×C4—22no—
255C2×C4—312no—
256C2×C4—628no—
257C2×C4—621no—
258C2×C4—624no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4—1242no—
263C2×C4C2×D12412yes—
264C2×C4D4×D3111yescharacteristic
265C2×C4D4×D3112yes—
266C2×C4C23×D3111yescharacteristic
267C8—22no—
268C8—622no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—624no—
274C22—622no—
275C22—622no—
276C22—1242no—
277C22—2488no—
278C22#1175845912yes—
279C22C2×D4×D3111yescharacteristic
280C4—42no—
281C4—622no—
282C4—622no—
283C4—1242no—
284C4#1175845912yes—
285C4C22×D12411yescharacteristic
286C4C2×D4×D3111yescharacteristic
287C2—22no—
288C2—44no—
289C2—1242no—
290C2—1242no—
291C2C2×#1175845911yescharacteristic
292C2#1452185912yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#418C2
socleC22×C31C22×D4
prime radicalC2×#11758561C2
hypercenter#418D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#11758486C2semidirect
3. C22×#92480C2semidirect
4. #14521625C2semidirect
5. #14521212C2semidirect
6. #14521201C2semidirect
7. #14521043C2semidirect
8. #14521017C2semidirect
9. C31×#418C2semidirect
20. C22×Dic62C22semidirect
21. C2×M16×C31C22semidirect
22. C31×#117C22semidirect
24. C2×#92480C22semidirect
25. C22×C31⋊C8C22semidirect
27. C2×Dic124C22semidirect
28. C2×#92470C22semidirect
29. #11758474C22semidirect
33. C31×#102C22semidirect
34. (C2×D31)⋊C8C22semidirect
35. #11758398C22semidirect
36. #11758397C22semidirect
147. #418D31semidirect
172. C2×M16C2×D31semidirect
178. #117C2×D31semidirect
181. #102C2×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×#117585612
C2×#117584862
C22×#924802
#145216252
#145212122
#145212012
#145210432
#145210172
C31×#4182
#300931