Finite Group Databasebeta

← #16856096

Subgroups — #16856096

Order 3968.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521604C211yesmaximal, characteristic, complemented
2#124×D31C211yesmaximal, characteristic
3#14521411C211yesmaximal, characteristic, complemented
4#14521409C211yesmaximal, characteristic, complemented
5#14521408C211yesmaximal, characteristic, complemented
6#14521405C212yesmaximal, complemented
7#14521402C212yesmaximal, complemented
8#14521056C212yesmaximal, complemented
9#14521052C212yesmaximal
10#14521001C211yesmaximal, characteristic, complemented
11C31×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D31C2211yescharacteristic
13C2×D4×D31C2211yescharacteristic
14C31×C4⋊Q8C2211yescharacteristic, complemented
15C31×#124C2211yescharacteristic
16#11758524C2211yescharacteristic
17#11758523—21no—
18Q8×Dic31—21no—
19#11758513C2211yescharacteristic
20C2×#92469C2212yes—
21#11758474—21no—
22#11758469C2212yes—
23#11758467C2212yescomplemented
24#11758464—21no—
25#11758462—21no—
26#11758462—21no—
27C22⋊C4×D31—21no—
28C22⋊C4×D31—21no—
29#11758452C2211yescharacteristic
30C42×D31C2211yescharacteristic
31#11758446C2211yescharacteristic, complemented
32C31×#103C2212yescomplemented
33C31×#102C2212yes—
34C31×#97C2211yescharacteristic, complemented
35#11758423C2212yescomplemented
36#11758420C2212yes—
37#11758409C2212yes—
38#11758404C2212yescomplemented
39C8×Dic31C2212yescomplemented
40#11758398C2212yescomplemented
41#11758397C2212yes—
42#11758391C2211yescharacteristic, complemented
43C23×D31—21no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2311yescharacteristic
46C2×D4×C31C2311yescharacteristic
47C2×C22⋊D31—21no—
48Q8×D31—24no—
49D4×D31—24no—
50C2×D124C2311yescharacteristic
51C2×C4×D31—21no—
52C2×C4×D31D411yescharacteristic
53C2×C4×D31D411yescharacteristic
54C2×C4×D31C2311yescharacteristic
55C2×Dic62—21no—
56C2×Dic62—21no—
57C2×Dic62C2311yescharacteristic
58C2×C8×C31C2312yes—
59C4⋊C4×C31—21no—
60C4⋊C4×C31C2312yes—
61C22⋊C4×C31—21no—
62C22⋊C4×C31—21no—
63C42×C31C2311yescharacteristic
64C22⋊Dic31—21no—
65C22⋊Dic31—21no—
66(C2×D31)⋊C4—22no—
67(C2×D31)⋊C4—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—21no—
70(C2×D31)⋊C4—21no—
71C4⋊Dic31—21no—
72C4⋊Dic31—22no—
73C4⋊Dic31C2312yes—
74Dic31⋊C4—22no—
75C4×Dic31—22no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C2×C31⋊C8C2312yes—
80#92469—24no—
81C23×C31—21no—
82C22×D31—21no—
83C22×D31—42no—
84C22×D31—42no—
85C22×D31—42no—
86C22×D31C2×D411yescharacteristic
87Q8×C31—22no—
88Q8×C31—42no—
89D4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—21no—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2411yescharacteristic
95C22⋊D31—42no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98C2×Dic31C2×D411yescharacteristic
99C2×Dic31C2×D411yescharacteristic
100C2×Dic31C2×D411yescharacteristic
101D124—22no—
102C4×D31—41no—
103C4×D31—41no—
104C4×D31—42no—
105C4×D31#4114yes—
106Dic62—22no—
107Dic62—42no—
108Dic62—42no—
109C8×C31—22no—
110C31⋊C8—22no—
111#3140—311nomaximal, Sylow
112C22×C31—41no—
113C22×C31—41no—
114C22×C31—41no—
115C22×C31C22×D411yescharacteristic
116C2×D31—21no—
117C2×D31—21no—
118C2×D31—41no—
119C2×D31—41no—
120C2×D31—41no—
121C2×D31—84no—
122C2×D31#12412yes—
123C4×C31—21no—
124C4×C31—21no—
125C4×C31—41no—
126C4×C31—42no—
127C4×C31C2×#4112yes—
128Dic31—21no—
129Dic31—21no—
130Dic31—41no—
131Dic31—42no—
132Dic31#12412yes—
133#533—311no—
134C2×#124—311no—
135#489—311no—
136#489—311no—
137#489—311no—
138#489D3111yescharacteristic, complemented
139#488—312no—
140#488—312no—
141#419—312no—
142#418—312no—
143C4×M16—311no—
144C2×C31—81no—
145C2×C31C2×#12411yescharacteristic
146C2×C31#57511yescharacteristic
147C2×C31#57511yescharacteristic
148D31—22no—
149D31—81no—
150C22×Q8—311no—
151C22×D4—311no—
152C2×M16—312no—
153C4⋊Q8—311no—
154C4⋊Q8C2×D3111yescharacteristic, complemented
155#125—312no—
156#124—311no—
157#124—311no—
158#124—311no—
159#124—621no—
160#124—621no—
161#124C2×D3111yescharacteristic
162C22⋊Q8—621no—
163C22⋊Q8—621no—
164C4×Q8—621no—
165C4×Q8—621no—
166#117—312no—
167C2×C22⋊C4—621no—
168C2×C22⋊C4—621no—
169C2×C42—311no—
170#103—312no—
171#103—312no—
172#103—312no—
173#103C2×D3112yescomplemented
174#102—312no—
175#102—312no—
176#102—312no—
177#102C2×D3112yes—
178#97—311no—
179#97C2×D3111yescharacteristic, complemented
180C4×C8—312no—
181C31#314011yescharacteristic, complemented, Sylow
182C24—621no—
183C2×Q8—21no—
184C2×Q8—311no—
185C2×Q8—621no—
186C2×Q8—624no—
187C2×Q8—621no—
188C2×Q8C22×D3111yescharacteristic
189C2×D4—311no—
190C2×D4—624no—
191C2×D4—621no—
192C2×D4C22×D3111yescharacteristic
193C22×C4—311no—
194C22×C4—311no—
195C22×C4—311no—
196C22×C4—621no—
197M16—624no—
198C2×C8—312no—
199C2×C8C22×D3112yes—
200C4⋊C4—21no—
201C4⋊C4—312no—
202C4⋊C4—622no—
203C4⋊C4—622no—
204C4⋊C4—621no—
205C4⋊C4C22×D3112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—621no—
209C22⋊C4—621no—
210C22⋊C4—622no—
211C22⋊C4—621no—
212C22⋊C4—621no—
213C22⋊C4—621no—
214C22⋊C4—621no—
215C42—311no—
216C42—311no—
217C42—311no—
218C42—622no—
219C42C22×D3111yescharacteristic
220C23—21no—
221C23—311no—
222C23—621no—
223C23—1242no—
224C23—1242no—
225C23—1242no—
226Q8—22no—
227Q8—42no—
228Q8—622no—
229Q8—1242no—
230Q8—1242no—
231D4—22no—
232D4—622no—
233D4—1242no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—311no—
237C2×C4—314no—
238C2×C4—311no—
239C2×C4—311no—
240C2×C4—622no—
241C2×C4—621no—
242C2×C4—1241no—
243C2×C4—1241no—
244C2×C4—1242no—
245C2×C4D4×D3111yescharacteristic
246C2×C4D4×D3111yescharacteristic
247C2×C4C23×D3111yescharacteristic
248C8—22no—
249C8—622no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—312no—
254C22—621no—
255C22—621no—
256C22—1241no—
257C22—1241no—
258C22—1241no—
259C22—2484no—
260C22C2×D4×D3111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—312no—
266C4—621no—
267C4—621no—
268C4—1241no—
269C4—1242no—
270C4#41×D3112yes—
271C2—81no—
272C2—622no—
273C2—2481no—
274C2#124×D3111yescharacteristic
275C2#1452184511yescharacteristic
276C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#489C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
hypercenter#489D31
hypocenterC31#3140
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #14521604C2semidirect
3. #14521411C2semidirect
4. #14521409C2semidirect
5. #14521408C2semidirect
6. #14521405C2semidirect
7. #14521402C2semidirect
8. #14521056C2semidirect
10. #14521001C2semidirect
11. C31×#489C2semidirect
14. C31×C4⋊Q8C22semidirect
23. #11758467C22semidirect
31. #11758446C22semidirect
32. C31×#103C22semidirect
34. C31×#97C22semidirect
35. #11758423C22semidirect
38. #11758404C22semidirect
39. C8×Dic31C22semidirect
40. #11758398C22semidirect
42. #11758391C22semidirect
138. #489D31semidirect
154. C4⋊Q8C2×D31semidirect
173. #103C2×D31semidirect
179. #97C2×D31semidirect
181. C31#3140semidirect

Maximal subgroups

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

subgroupindex
#145216042
#124×D312
#145214112
#145214092
#145214082
#145214052
#145214022
#145210562
#145210522
#145210012
C31×#4892
#314031