Finite Group Databasebeta

← #16854940

Subgroups — #16854940

Order 3968.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D31C211yesmaximal, characteristic, complemented
2C2×#11758485C211yesmaximal, characteristic, complemented
3C2×#11758509C211yesmaximal, characteristic, complemented
4#14521561C211yesmaximal, characteristic, complemented
5#14521201C214yesmaximal, complemented
6#14521184C214yesmaximal, complemented
7#14521053C211yesmaximal, characteristic, complemented
8#14521012C211yesmaximal, characteristic, complemented
9C31×#418C211yesmaximal, characteristic, complemented
10C24×D31—21no—
11C22×D4×C31C2211yescharacteristic
12C22×C22⋊D31—21no—
13C2×D4×D31—24no—
14C2×D4×D31—24no—
15C2×D4×D31—22no—
16C2×D4×D31C2214yes—
17C2×#92495C2211yescharacteristic, complemented
18C22×D124C2211yescharacteristic
19C22×C4×D31C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic, complemented
21C31×#117C2211yescharacteristic, complemented
22C2×#92480C2212yescomplemented
23#11758509—24no—
24C2×#92479C2212yescomplemented
25C2×#92474C2211yescharacteristic, complemented
26#11758485—24no—
27C2×D248C2212yescomplemented
28C2×#92470C2212yescomplemented
29#11758474C2212yescomplemented
30#11758472C2212yescomplemented
31#11758463—21no—
32#11758459—21no—
33C4×D124—22no—
34C31×#102C2214yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758397C2214yescomplemented
37C24×C31—21no—
38C23×D31—28no—
39C23×D31—21no—
40C23×D31—24no—
41C23×D31—48no—
42C23×D31D411yescharacteristic
43C2×D4×C31—21no—
44C2×D4×C31—22no—
45C2×D4×C31D414yescomplemented
46C2×D4×C31C2312yes—
47C22×C4×C31C2311yescharacteristic
48C2×C22⋊D31—28no—
49C2×C22⋊D31—21no—
50C2×C22⋊D31—24no—
51C22×Dic31D411yescharacteristic
52D4×D31—216no—
53D4×D31—28no—
54D4×D31—28no—
55#92495—22no—
56#92495—22no—
57C2×D124—21no—
58C2×D124—22no—
59C2×D124—22no—
60C2×D124D414yescomplemented
61C2×D124C2312yes—
62C2×D124C2311yescharacteristic, complemented
63C2×C4×D31—22no—
64C2×C4×D31—22no—
65C2×C4×D31—21no—
66C2×C4×D31D412yes—
67C2×C4×D31C2312yes—
68C2×Dic62C2311yescharacteristic, complemented
69M16×C31—22no—
70C2×C8×C31C2312yescomplemented
71C4⋊C4×C31C2312yescomplemented
72C22⋊C4×C31—21no—
73C42×C31—21no—
74#92480—24no—
75#92479—24no—
76(C2×D31)⋊C4—22no—
77(C2×D31)⋊C4—22no—
78C4⋊Dic31—22no—
79#92474—22no—
80C2×C31⋊C8C2312yescomplemented
81D248—24no—
82#92470—24no—
83C23×C31—24no—
84C23×C31—22no—
85C23×C31—44no—
86C23×C31C2×D411yescharacteristic
87C22×D31—22no—
88C22×D31—24no—
89C22×D31—216no—
90C22×D31—24no—
91C22×D31—22no—
92C22×D31—21no—
93C22×D31—432no—
94C22×D31—48no—
95C22×D31—44no—
96C22×D31—48no—
97C22×D31C2×D412yes—
98D4×C31—24no—
99D4×C31—22no—
100D4×C31C2×D414yes—
101C2×C4×C31—22no—
102C2×C4×C31C2×D412yes—
103C2×C4×C31C2×D411yescharacteristic
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2411yescharacteristic
106C22⋊D31—216no—
107C22⋊D31—42no—
108C2×Dic31—22no—
109C2×Dic31—21no—
110C2×Dic31C2×D412yes—
111D124—24no—
112D124—22no—
113D124—21no—
114D124—21no—
115D124—44no—
116D124C2×D414yes—
117C4×D31—24no—
118C4×D31—22no—
119C4×D31—22no—
120C4×D31—42no—
121Dic62—21no—
122Dic62—21no—
123C8×C31—22no—
124C31⋊C8—22no—
125#3003—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—24no—
128C22×C31—48no—
129C22×C31—44no—
130C22×C31#12012yes—
131C22×C31C22×D411yescharacteristic
132C2×D31—28no—
133C2×D31—22no—
134C2×D31—22no—
135C2×D31—24no—
136C2×D31—432no—
137C2×D31—48no—
138C2×D31—44no—
139C2×D31—41no—
140C2×D31—42no—
141C2×D31AGL(1,ℤ/8)14yescomplemented
142C4×C31—42no—
143C4×C31#12012yes—
144C4×C31C22×D411yescharacteristic
145C4×C31C22×D411yescharacteristic
146Dic31—22no—
147Dic31—41no—
148C23×D4—311no—
149C2×AGL(1,ℤ/8)—311no—
150C2×AGL(1,ℤ/8)—311no—
151#525—311no—
152#450—314no—
153#447—314no—
154#418—311no—
155#418D3111yescharacteristic, complemented
156#407—311no—
157C2×C31—22no—
158C2×C31—44no—
159C2×C31C2×#12011yescharacteristic
160C2×C31C2×AGL(1,ℤ/8)12yes—
161D31—24no—
162D31—44no—
163D31—81no—
164C25—621no—
165C2×#41—311no—
166C22×D4—314no—
167C22×D4—311no—
168C22×D4—622no—
169C22×D4—621no—
170C22×D4—624no—
171C22×D4—624no—
172C22×D4C2×D3111yescharacteristic
173C23×C4—311no—
174AGL(1,ℤ/8)—624no—
175AGL(1,ℤ/8)—624no—
176C2×SD16—312no—
177C2×SD16—312no—
178C2×D8—312no—
179C2×D8—312no—
180C2×M16—311no—
181C2×M16C2×D3111yescharacteristic, complemented
182#123—621no—
183C22⋊Q8—312no—
184#121—312no—
185#120—621no—
186C4×D4—622no—
187#117C2×D3111yescharacteristic, complemented
188#102—314no—
189#102C2×D3114yescomplemented
190C22⋊C8—314no—
191C31#300311yescharacteristic, complemented, Sylow
192C24—21no—
193C24—311no—
194C24—624no—
195C24—628no—
196C24—621no—
197C24—1248no—
198#41—622no—
199#41—622no—
200C2×Q8—311no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—314no—
204C2×D4—312no—
205C2×D4—311no—
206C2×D4—622no—
207C2×D4—624no—
208C2×D4—628no—
209C2×D4—6216no—
210C2×D4—628no—
211C2×D4—628no—
212C2×D4—621no—
213C2×D4—622no—
214C2×D4—621no—
215C2×D4D12414yescomplemented
216C2×D4C22×D3112yes—
217C22×C4—311no—
218C22×C4—312no—
219C22×C4—312no—
220C22×C4—622no—
221C22×C4—622no—
222C22×C4—621no—
223C22×C4C22×D3111yescharacteristic
224SD16—624no—
225SD16—624no—
226D8—624no—
227D8—624no—
228M16—22no—
229M16—622no—
230C2×C8—312no—
231C2×C8C22×D3112yescomplemented
232C4⋊C4—622no—
233C4⋊C4C22×D3112yescomplemented
234C22⋊C4—21no—
235C22⋊C4—622no—
236C22⋊C4—622no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—312no—
242C23—624no—
243C23—6216no—
244C23—622no—
245C23—624no—
246C23—622no—
247C23—621no—
248C23—1244no—
249C23—1248no—
250C23—12432no—
251C23—1248no—
252C23D4×D3111yescharacteristic
253Q8—621no—
254Q8—621no—
255D4—24no—
256D4—22no—
257D4—314no—
258D4—6216no—
259D4—624no—
260D4—622no—
261D4—621no—
262D4—621no—
263D4—1244no—
264D4—1242no—
265D4C2×D12414yes—
266C2×C4—22no—
267C2×C4—312no—
268C2×C4—622no—
269C2×C4—624no—
270C2×C4—622no—
271C2×C4—622no—
272C2×C4—621no—
273C2×C4—1242no—
274C2×C4C2×D12412yes—
275C2×C4D4×D3111yescharacteristic
276C2×C4D4×D3112yes—
277C2×C4C23×D3111yescharacteristic
278C8—22no—
279C8—622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—314no—
285C22—628no—
286C22—622no—
287C22—622no—
288C22—624no—
289C22—1248no—
290C22—12432no—
291C22—1244no—
292C22—1241no—
293C22—1242no—
294C22#1175845912yes—
295C22C2×D4×D3111yescharacteristic
296C4—42no—
297C4—622no—
298C4—1241no—
299C4#1175845912yes—
300C4C22×D12411yescharacteristic
301C4C2×D4×D3111yescharacteristic
302C2—22no—
303C2—44no—
304C2—624no—
305C2—1244no—
306C2—2481no—
307C2C2×#1175845911yescharacteristic
308C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#418C2
socleC22×C31C22×D4
hypercenter#418D31
hypocenterC31#3003
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
1. C22×D4×D31C2semidirect
2. C2×#11758485C2semidirect
3. C2×#11758509C2semidirect
4. #14521561C2semidirect
5. #14521201C2semidirect
6. #14521184C2semidirect
7. #14521053C2semidirect
8. #14521012C2semidirect
9. C31×#418C2semidirect
17. C2×#92495C22semidirect
20. C2×M16×C31C22semidirect
21. C31×#117C22semidirect
22. C2×#92480C22semidirect
24. C2×#92479C22semidirect
25. C2×#92474C22semidirect
27. C2×D248C22semidirect
28. C2×#92470C22semidirect
29. #11758474C22semidirect
30. #11758472C22semidirect
34. C31×#102C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758397C22semidirect
45. C2×D4×C31D4semidirect
60. C2×D124D4semidirect
62. C2×D124C23semidirect
68. C2×Dic62C23semidirect
70. C2×C8×C31C23semidirect
71. C4⋊C4×C31C23semidirect
80. C2×C31⋊C8C23semidirect
141. C2×D31AGL(1,ℤ/8)semidirect
155. #418D31semidirect
181. C2×M16C2×D31semidirect
187. #117C2×D31semidirect
189. #102C2×D31semidirect
191. C31#3003semidirect
215. C2×D4D124semidirect
231. C2×C8C22×D31semidirect
233. C4⋊C4C22×D31semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584852
C2×#117585092
#145215612
#145212012
#145211842
#145210532
#145210122
C31×#4182
#300331