Finite Group Databasebeta

← #16854971

Subgroups — #16854971

Order 3968.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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×#11758528C212yesmaximal, complemented
2C2×#11758501C211yesmaximal, characteristic, complemented
3#14521211C214yesmaximal, complemented
4#14521205C212yesmaximal, complemented
5#14521200C212yesmaximal, complemented
6#14521050C212yesmaximal, complemented
7#14521013C211yesmaximal, characteristic, complemented
8C22×#41×C31C211yesmaximal, characteristic
9C2×#41×C31—28no—
10C2×#41×C31—21no—
11C2×#41×C31—21no—
12C2×#41×C31C2212yes—
13C2×#41×C31C2212yes—
14C22×Q8×C31C2211yescharacteristic
15C22×D4×C31—21no—
16C22×D4×C31C2211yescharacteristic
17C23×C4×C31—21no—
18C23×C4×C31C2211yescharacteristic
19C22×Dic62C2211yescharacteristic, complemented
20C2×C22⋊Dic31—21no—
21#11758528—28no—
22C2×#92482C2214yescomplemented
23C2×#92480C2214yescomplemented
24C2×C4⋊Dic31C2211yescharacteristic, complemented
25#11758501—21no—
26#11758501—21no—
27#11758501C2212yes—
28#11758501C2212yescomplemented
29C2×Dic31⋊C4—21no—
30C2×#92474C2212yescomplemented
31#11758423C2214yescomplemented
32#11758420C2214yescomplemented
33C22⋊(C31⋊C8)C2212yes—
34C22⋊(C31⋊C8)C2212yescomplemented
35C24×C31—21no—
36C24×C31D411yescharacteristic
37#41×C31—28no—
38#41×C31—28no—
39#41×C31—24no—
40#41×C31—24no—
41#41×C31—24no—
42#41×C31D418yes—
43C2×Q8×C31—24no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2312yes—
46C2×D4×C31—28no—
47C2×D4×C31—22no—
48C2×D4×C31—22no—
49C2×D4×C31—24no—
50C2×D4×C31—21no—
51C2×D4×C31C2312yes—
52C22×C4×C31—28no—
53C22×C4×C31—21no—
54C22×C4×C31—22no—
55C22×C4×C31—22no—
56C22×C4×C31—24no—
57C22×C4×C31D411yescharacteristic
58C22×C4×C31D411yescharacteristic
59C22×C4×C31D411yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×Dic31—21no—
64C22×Dic31—21no—
65C2×Dic62—24no—
66C2×Dic62—21no—
67C2×Dic62C2312yes—
68C22⋊Dic31—22no—
69C22⋊Dic31—22no—
70#92482—28no—
71#92480—28no—
72C4⋊Dic31—21no—
73C4⋊Dic31C2312yes—
74Dic31⋊C4—22no—
75Dic31⋊C4—22no—
76#92474—24no—
77C2×C31⋊C8C2314yes—
78C23×C31—24no—
79C23×C31—22no—
80C23×C31—22no—
81C23×C31—22no—
82C23×C31—41no—
83C23×C31—42no—
84C23×C31—42no—
85C23×C31C2×D411yescharacteristic
86C23×C31C2×D411yescharacteristic
87C23×C31C2×D411yescharacteristic
88Q8×C31—24no—
89Q8×C31—22no—
90Q8×C31C2×D414yes—
91D4×C31—28no—
92D4×C31—24no—
93D4×C31—28no—
94D4×C31—22no—
95D4×C31C2×D414yes—
96C2×C4×C31—24no—
97C2×C4×C31—28no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—22no—
101C2×C4×C31—28no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31C2×D411yescharacteristic
106C2×C4×C31C2×D411yescharacteristic
107C2×C4×C31C2×D411yescharacteristic
108C2×C4×C31C2×D414yes—
109C2×C4×C31C2411yescharacteristic
110C2×Dic31—22no—
111C2×Dic31—22no—
112C2×Dic31—42no—
113C2×Dic31—42no—
114Dic62—24no—
115Dic62—42no—
116C31⋊C8—24no—
117#3007—311nomaximal, Sylow
118C22×C31—21no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—24no—
124C22×C31—21no—
125C22×C31—21no—
126C22×C31—44no—
127C22×C31—48no—
128C22×C31—42no—
129C22×C31—42no—
130C22×C31#12012yes—
131C22×C31#13712yescomplemented
132C22×C31C22×D411yescharacteristic
133C4×C31—21no—
134C4×C31—24no—
135C4×C31—21no—
136C4×C31#12012yes—
137C4×C31C22×D412yes—
138Dic31—42no—
139Dic31—42no—
140C22×#41D3111yescharacteristic
141C2×#137—312no—
142C2×C22⋊Q8—311no—
143#452—314no—
144#451—312no—
145#450—312no—
146#417—312no—
147#407—311no—
148C2×C31—22no—
149C2×C31—22no—
150C2×C31—41no—
151C2×C31—44no—
152C2×C31C2×#12011yescharacteristic
153C2×C31C2×#13711yescharacteristic
154C2×C31#57511yescharacteristic
155C2×#41—28no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41C2×D3112yes—
159C2×#41C2×D3112yes—
160C22×Q8—311no—
161C22×Q8C2×D3111yescharacteristic
162C22×D4—21no—
163C22×D4C2×D3111yescharacteristic
164C23×C4—21no—
165C23×C4C2×D3111yescharacteristic
166#137—628no—
167C2×Q16—314no—
168C2×SD16—314no—
169C2×M16—312no—
170C22⋊Q8—312no—
171C22⋊Q8—312no—
172C22⋊Q8—621no—
173C22⋊Q8—621no—
174C2×C4⋊C4—311no—
175C2×C4⋊C4—621no—
176C2×C22⋊C4—621no—
177#103—314no—
178#102—314no—
179C22⋊C8—312no—
180C22⋊C8—312no—
181C31#300711yescharacteristic, complemented, Sylow
182C24—21no—
183C24C22⋊D3111yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D3118yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—312no—
193C2×Q8—621no—
194C2×Q8—624no—
195C2×Q8C22×D3112yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D3112yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—621no—
208C22×C4—621no—
209C22×C4C22⋊D3111yescharacteristic
210C22×C4C22⋊D3111yescharacteristic
211C22×C4C22⋊D3111yescharacteristic
212C22×C4C22×D3111yescharacteristic
213C22×C4C22×D3111yescharacteristic
214C22×C4C22×D3111yescharacteristic
215Q16—628no—
216SD16—628no—
217M16—624no—
218C2×C8—314no—
219C4⋊C4—312no—
220C4⋊C4—622no—
221C4⋊C4—622no—
222C4⋊C4—621no—
223C22⋊C4—622no—
224C22⋊C4—622no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D3111yescharacteristic
233C23C2×C22⋊D3111yescharacteristic
234C23C2×C22⋊D3111yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—624no—
238Q8—1242no—
239Q8C2×C22⋊D3114yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D3114yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—1242no—
257C2×C4—1242no—
258C2×C4C2×C22⋊D3111yescharacteristic
259C2×C4C2×C22⋊D3111yescharacteristic
260C2×C4C2×C22⋊D3111yescharacteristic
261C2×C4C2×C22⋊D3114yes—
262C2×C4C23×D3111yescharacteristic
263C8—624no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1175852812yescomplemented
277C22#1175853012yes—
278C22C22×C22⋊D3111yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—1242no—
283C4—1242no—
284C4#1175853012yes—
285C4C22×C22⋊D3112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1175853011yescharacteristic
291C2C2×#1175852811yescharacteristic
292C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC22×#41×C31C2
socleC22×C31C22×D4
hypercenterC22×#41D31
hypocenterC31#3007
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C2×#11758501C2semidirect
3. #14521211C2semidirect
4. #14521205C2semidirect
5. #14521200C2semidirect
6. #14521050C2semidirect
7. #14521013C2semidirect
19. C22×Dic62C22semidirect
22. C2×#92482C22semidirect
23. C2×#92480C22semidirect
24. C2×C4⋊Dic31C22semidirect
28. #11758501C22semidirect
30. C2×#92474C22semidirect
31. #11758423C22semidirect
32. #11758420C22semidirect
34. C22⋊(C31⋊C8)C22semidirect
131. C22×C31#137semidirect
181. C31#3007semidirect
276. C22#11758528semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C2×#117585012
#145212112
#145212052
#145212002
#145210502
#145210132
C22×#41×C312
#300731