Finite Group Databasebeta

← #16856194

Subgroups — #16856194

Order 3968.

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

normal subgroups only

This group has 596 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×#11758509C211yesmaximal, characteristic, complemented
2C2×#11758526C211yesmaximal, characteristic, complemented
3C2×#11758527C211yesmaximal, characteristic, complemented
4C22×#92479C211yesmaximal, characteristic, complemented
5#14521618C211yesmaximal, characteristic, complemented
6#14521439C212yesmaximal, complemented
7#14521436C212yesmaximal, complemented
8#14521431C212yesmaximal, complemented
9#14521425C212yesmaximal, complemented
10#14521005C211yesmaximal, characteristic, complemented
11C31×#535C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×D4×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C22×D124C2211yescharacteristic
16C31×#127C2212yescomplemented
17C31×#124C2212yescomplemented
18C31×#121—22no—
19C31×#120—22no—
20C31×#117C2211yescharacteristic
21#11758527—24no—
22#11758526—24no—
23C2×#92482C2211yescharacteristic, complemented
24C2×#92481C2212yescomplemented
25C2×#92481C2211yescharacteristic, complemented
26C2×#92480C2212yescomplemented
27C2×#92480C2211yescharacteristic, complemented
28#11758509—24no—
29C2×#92479—24no—
30C2×#92479C2212yes—
31C2×#92479C2212yes—
32C2×#92479C2211yescharacteristic, complemented
33C2×#92479C2212yescomplemented
34C2×#92479C2212yescomplemented
35C2×#92474C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37#11758472—22no—
38#11758459—22no—
39#11758452C2212yescomplemented
40#11758451C2212yescomplemented
41#11758391C2212yescomplemented
42C4×C31⋊C8C2212yescomplemented
43C24×C31—21no—
44C23×D31—21no—
45#41×C31—22no—
46#41×C31—22no—
47C2×Q8×C31C2311yescharacteristic
48C2×D4×C31—21no—
49C2×D4×C31—22no—
50C2×D4×C31—22no—
51C2×D4×C31—21no—
52C2×D4×C31—24no—
53C2×D4×C31C2312yes—
54C2×D4×C31C2311yescharacteristic
55C22×C4×C31—21no—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—21no—
58#92495—22no—
59#92495—22no—
60C2×D124—22no—
61C2×D124—22no—
62C2×D124—21no—
63C2×D124—24no—
64C2×D124C2311yescharacteristic
65C2×D124C2312yes—
66C2×C4×D31—21no—
67C2×Dic62C2311yescharacteristic
68C4⋊C4×C31D412yes—
69C22⋊C4×C31—24no—
70C22⋊C4×C31D412yes—
71C42×C31C2312yes—
72#92482—22no—
73#92481—24no—
74#92481—22no—
75#92480—24no—
76#92480—22no—
77#92479—24no—
78#92479—24no—
79#92479—24no—
80#92479—24no—
81#92479—22no—
82(C2×D31)⋊C4—24no—
83#92474D414yescomplemented
84C2×C31⋊C8D414yescomplemented
85C2×C31⋊C8C2311yescharacteristic
86C2×C31⋊C8C2312yes—
87C2×C31⋊C8C2311yescharacteristic
88C23×C31—22no—
89C23×C31—21no—
90C23×C31—42no—
91C23×C31—82no—
92C23×C31C2×D411yescharacteristic
93C22×D31—22no—
94C22×D31—21no—
95C22×D31—42no—
96C22×D31—82no—
97Q8×C31—21no—
98Q8×C31—21no—
99D4×C31—22no—
100D4×C31—21no—
101D4×C31—22no—
102D4×C31—21no—
103D4×C31—42no—
104D4×C31—42no—
105D4×C31—84no—
106C2×C4×C31—21no—
107C2×C4×C31—42no—
108C2×C4×C31C2×D412yes—
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D414yes—
111C2×C4×C31C2×D411yescharacteristic
112C2×C4×C31C2411yescharacteristic
113C22⋊D31—42no—
114C2×Dic31—21no—
115D124—22no—
116D124—21no—
117D124—22no—
118D124—21no—
119D124—42no—
120D124—84no—
121C4×D31—42no—
122Dic62—21no—
123Dic62—21no—
124C31⋊C8C2×D412yes—
125C31⋊C8C2×D412yes—
126C31⋊C8C2×D414yes—
127#3155—311nomaximal, Sylow
128C22×C31—22no—
129C22×C31—41no—
130C22×C31—42no—
131C22×C31—42no—
132C22×C31—44no—
133C22×C31—84no—
134C22×C31#12712yes—
135C22×C31C22×D411yescharacteristic
136C2×D31—41no—
137C2×D31—42no—
138C2×D31—42no—
139C2×D31—44no—
140C2×D31—84no—
141C4×C31—24no—
142C4×C31—41no—
143C4×C31#12712yes—
144C4×C31C22×D411yescharacteristic
145C4×C31C22×D411yescharacteristic
146Dic31—41no—
147C2×AGL(1,ℤ/8)—311no—
148C2×AGL(1,ℤ/8)—311no—
149C2×#135—311no—
150C22×D8—311no—
151#535—311no—
152#535D3111yescharacteristic, complemented
153#496—312no—
154#496—312no—
155#495—312no—
156#493—312no—
157#405—311no—
158C2×C31—22no—
159C2×C31—81no—
160C2×C31—82no—
161C2×C31C2×#12711yescharacteristic
162C2×C31#57612yes—
163D31—81no—
164D31—82no—
165C2×#41—311no—
166C2×#41C2×D3111yescharacteristic
167C22×D4—311no—
168C22×D4C2×D3111yescharacteristic
169AGL(1,ℤ/8)—624no—
170AGL(1,ℤ/8)—624no—
171#135—624no—
172C2×Q16—311no—
173C2×SD16—311no—
174C2×SD16—312no—
175C2×SD16—311no—
176C2×SD16—312no—
177C2×D8—312no—
178C2×D8—312no—
179C2×D8—311no—
180C2×D8—312no—
181C2×D8—312no—
182C2×D8—624no—
183C2×M16—311no—
184C22×C8—311no—
185#127—312no—
186#127C2×D3112yescomplemented
187#124—312no—
188#124C2×D3112yescomplemented
189#121—22no—
190#121—622no—
191#120—22no—
192#120—622no—
193#117C2×D3111yescharacteristic
194#97—312no—
195C4×C8—312no—
196C31#315511yescharacteristic, complemented, Sylow
197C24—21no—
198C24—621no—
199#41—22no—
200#41—22no—
201#41—622no—
202#41—622no—
203C2×Q8—311no—
204C2×Q8C22×D3111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—312no—
211C2×D4—311no—
212C2×D4—621no—
213C2×D4—622no—
214C2×D4—622no—
215C2×D4—621no—
216C2×D4—624no—
217C2×D4C22×D3112yes—
218C2×D4C22×D3111yescharacteristic
219C22×C4—21no—
220C22×C4—621no—
221C22×C4C22×D3111yescharacteristic
222Q16—622no—
223SD16—622no—
224SD16—624no—
225SD16—622no—
226SD16—624no—
227D8—624no—
228D8—624no—
229D8—622no—
230D8—624no—
231D8—624no—
232M16—314no—
233C2×C8—314no—
234C2×C8—311no—
235C2×C8—312no—
236C2×C8—311no—
237C4⋊C4C22⋊D3112yes—
238C22⋊C4—24no—
239C22⋊C4—624no—
240C22⋊C4C22⋊D3112yes—
241C42C22×D3112yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—622no—
247C23—621no—
248C23—1242no—
249C23—2482no—
250C23C2×C22⋊D3111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—621no—
254Q8—621no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—622no—
263D4—621no—
264D4—622no—
265D4—621no—
266D4—1242no—
267D4—1242no—
268D4—2484no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—621no—
272C2×C4—1242no—
273C2×C4D4×D3112yes—
274C2×C4D4×D3112yes—
275C2×C4C2×C22⋊D3114yes—
276C2×C4C2×C22⋊D3111yescharacteristic
277C2×C4C23×D3111yescharacteristic
278C8—312no—
279C8—312no—
280C8—314no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—1241no—
288C22—1242no—
289C22—1242no—
290C22—1244no—
291C22—2484no—
292C22#1175851712yes—
293C22C22×C22⋊D3111yescharacteristic
294C4—24no—
295C4—41no—
296C4—1241no—
297C4#1175851712yes—
298C4C2×D4×D3111yescharacteristic
299C4C2×D4×D3111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—2482no—
304C2—2481no—
305C2C2×#1175851711yescharacteristic
306C2#1452185212yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#535C2
socleC22×C31C22×D4
hypercenter#535D31
hypocenterC31#3155
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11758509C2semidirect
2. C2×#11758526C2semidirect
3. C2×#11758527C2semidirect
4. C22×#92479C2semidirect
5. #14521618C2semidirect
6. #14521439C2semidirect
7. #14521436C2semidirect
8. #14521431C2semidirect
9. #14521425C2semidirect
10. #14521005C2semidirect
11. C31×#535C2semidirect
16. C31×#127C22semidirect
17. C31×#124C22semidirect
23. C2×#92482C22semidirect
24. C2×#92481C22semidirect
25. C2×#92481C22semidirect
26. C2×#92480C22semidirect
27. C2×#92480C22semidirect
32. C2×#92479C22semidirect
33. C2×#92479C22semidirect
34. C2×#92479C22semidirect
39. #11758452C22semidirect
40. #11758451C22semidirect
41. #11758391C22semidirect
42. C4×C31⋊C8C22semidirect
83. #92474D4semidirect
84. C2×C31⋊C8D4semidirect
152. #535D31semidirect
186. #127C2×D31semidirect
188. #124C2×D31semidirect
196. C31#3155semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
C2×#117585262
C2×#117585272
C22×#924792
#145216182
#145214392
#145214362
#145214312
#145214252
#145210052
C31×#5352
#315531