Finite Group Databasebeta

← #16856159

Subgroups — #16856159

Order 3968.

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

normal subgroups only

This group has 656 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
1C2×#11758485C211yesmaximal, characteristic, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3C2×#11758526C211yesmaximal, characteristic, complemented
4#14521595C211yesmaximal, characteristic, complemented
5C2×#11758517C211yesmaximal, characteristic, complemented
6#14521437C212yesmaximal, complemented
7#14521435C212yesmaximal, complemented
8#14521424C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10M16×Dic31C211yesmaximal, characteristic, complemented
11C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×D4×C31C2211yescharacteristic
14C22×C22⋊D31—22no—
15C2×#92495C2211yescharacteristic
16C22×D124C2211yescharacteristic
17C31×AGL(1,ℤ/8)—24no—
18C2×SD16×C31C2212yescomplemented
19C2×D8×C31C2212yescomplemented
20C2×M16×C31C2211yescharacteristic
21#11758526—24no—
22#11758524C2211yescharacteristic, complemented
23#11758521C2211yescharacteristic, complemented
24C2×#92481C2212yescomplemented
25#11758517—22no—
26#11758517C2212yes—
27#11758517C2212yes—
28#11758517C2211yescharacteristic, complemented
29#11758516—22no—
30#11758513C2211yescharacteristic, complemented
31C2×#92480C2212yescomplemented
32#11758509—24no—
33C2×#92479C2212yescomplemented
34C2×#92479C2212yescomplemented
35C4×C22⋊D31—22no—
36C2×C4×Dic31C2211yescharacteristic
37C2×#92474C2211yescharacteristic
38#11758485—24no—
39C2×D248C2212yescomplemented
40C2×#92470C2212yescomplemented
41#11758403C2212yescomplemented
42C8×Dic31C2212yescomplemented
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×C31C2311yescharacteristic
53C2×D4×C31C2312yes—
54C22×C4×C31—21no—
55C22×C4×C31C2311yescharacteristic
56C2×C22⋊D31—21no—
57C2×C22⋊D31—22no—
58C2×C22⋊D31—24no—
59C2×C22⋊D31—24no—
60C2×C22⋊D31—48no—
61C22×Dic31D412yes—
62#92495—22no—
63#92495—22no—
64C2×D124—22no—
65C2×D124—22no—
66C2×D124—21no—
67C2×D124C2312yes—
68C2×D124C2311yescharacteristic
69C2×C4×D31—21no—
70C2×Dic62C2311yescharacteristic
71SD16×C31—24no—
72D8×C31—24no—
73M16×C31D414yescomplemented
74C2×C8×C31C2312yes—
75C22⋊Dic31—22no—
76#92481—24no—
77#92480—24no—
78#92479—24no—
79#92479—24no—
80(C2×D31)⋊C4—22no—
81Dic31⋊C4—22no—
82C4×Dic31D412yes—
83C4×Dic31C2311yescharacteristic
84C4×Dic31C2311yescharacteristic
85#92474D414yescomplemented
86C2×C31⋊C8C2312yes—
87D248—24no—
88#92470—24no—
89C23×C31—21no—
90C23×C31—22no—
91C23×C31—44no—
92C23×C31—42no—
93C23×C31C2×D411yescharacteristic
94C22×D31—21no—
95C22×D31—22no—
96C22×D31—44no—
97C22×D31—42no—
98Q8×C31—21no—
99Q8×C31—21no—
100D4×C31—21no—
101D4×C31—22no—
102D4×C31—21no—
103D4×C31—22no—
104D4×C31—42no—
105D4×C31—42no—
106C2×C4×C31—21no—
107C2×C4×C31—42no—
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2411yescharacteristic
112C22⋊D31—48no—
113C22⋊D31—42no—
114C22⋊D31—44no—
115C22⋊D31—48no—
116C2×Dic31—24no—
117C2×Dic31—21no—
118C2×Dic31C2×D412yes—
119C2×Dic31C2×D412yes—
120D124—22no—
121D124—21no—
122D124—21no—
123D124—22no—
124D124—42no—
125C4×D31—42no—
126Dic62—21no—
127Dic62—21no—
128C8×C31C2×D414yescomplemented
129C31⋊C8C2×D414yescomplemented
130#3151—311nomaximal, Sylow
131C22×C31—22no—
132C22×C31—41no—
133C22×C31—42no—
134C22×C31—44no—
135C22×C31—42no—
136C22×C31—84no—
137C22×C31#12712yes—
138C22×C31C22×D411yescharacteristic
139C2×D31—41no—
140C2×D31—42no—
141C2×D31—44no—
142C2×D31—42no—
143C2×D31—84no—
144C4×C31—41no—
145C4×C31#12712yes—
146C4×C31C22×D411yescharacteristic
147C4×C31C22×D411yescharacteristic
148Dic31—22no—
149Dic31—41no—
150Dic31AGL(1,ℤ/8)14yescomplemented
151C2×AGL(1,ℤ/8)—311no—
152C2×AGL(1,ℤ/8)—311no—
153C2×AGL(1,ℤ/8)—311no—
154C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
155#532—311no—
156C2×#127—311no—
157#496—312no—
158#496—312no—
159#493—312no—
160#492—312no—
161C4×M16—311no—
162C2×C31—22no—
163C2×C31—81no—
164C2×C31—82no—
165C2×C31C2×#12711yescharacteristic
166C2×C31C2×AGL(1,ℤ/8)12yes—
167D31—81no—
168D31—82no—
169C2×#41—311no—
170C2×#41C2×D3111yescharacteristic
171C22×D4—311no—
172C22×D4—622no—
173C22×D4C2×D3111yescharacteristic
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—624no—
176AGL(1,ℤ/8)—624no—
177AGL(1,ℤ/8)—624no—
178C2×SD16—312no—
179C2×SD16—312no—
180C2×SD16—312no—
181C2×SD16C2×D3112yescomplemented
182C2×D8—312no—
183C2×D8—312no—
184C2×D8—312no—
185C2×D8C2×D3112yescomplemented
186C2×M16—311no—
187C2×M16C2×D3111yescharacteristic
188C4⋊Q8—311no—
189#127—312no—
190#127—312no—
191#127—311no—
192#127—622no—
193#124—311no—
194#124—311no—
195#121—622no—
196C4×D4—622no—
197C2×C42—311no—
198#97—312no—
199C4×C8—312no—
200C31#315111yescharacteristic, complemented, Sylow
201C24—21no—
202C24—621no—
203#41—22no—
204#41—22no—
205#41—622no—
206#41—622no—
207C2×Q8—311no—
208C2×Q8C22×D3111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—311no—
214C2×D4—312no—
215C2×D4—622no—
216C2×D4—624no—
217C2×D4—624no—
218C2×D4—621no—
219C2×D4—622no—
220C2×D4—621no—
221C2×D4—622no—
222C2×D4—1248no—
223C2×D4C22×D3111yescharacteristic
224C2×D4C22×D3112yes—
225C22×C4—21no—
226C22×C4—312no—
227C22×C4—621no—
228C22×C4C22×D3111yescharacteristic
229SD16—24no—
230SD16—624no—
231SD16—624no—
232SD16—624no—
233D8—24no—
234D8—624no—
235D8—624no—
236D8—624no—
237M16—314no—
238M16C22⋊D3114yescomplemented
239C2×C8—312no—
240C2×C8C22×D3112yes—
241C4⋊C4—622no—
242C22⋊C4—622no—
243C22⋊C4—622no—
244C42—312no—
245C42—311no—
246C42—311no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—621no—
252C23—622no—
253C23—1244no—
254C23—1242no—
255C23D4×D3111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—621no—
259Q8—621no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—621no—
267D4—622no—
268D4—621no—
269D4—622no—
270D4—1244no—
271D4—1248no—
272D4—1248no—
273D4—1242no—
274D4—1242no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—312no—
278C2×C4—312no—
279C2×C4—624no—
280C2×C4—621no—
281C2×C4—1242no—
282C2×C4D4×D3111yescharacteristic
283C2×C4D4×D3112yes—
284C2×C4C2×C22⋊D3112yes—
285C2×C4C23×D3111yescharacteristic
286C8—314no—
287C8C2×C22⋊D3114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—1244no—
295C22—1242no—
296C22—1241no—
297C22—1242no—
298C22—2484no—
299C22#1175851712yes—
300C22C2×D4×D3111yescharacteristic
301C4—41no—
302C4—314no—
303C4—622no—
304C4—1241no—
305C4#1175851712yes—
306C4C2×D4×D3111yescharacteristic
307C4C22×C22⋊D3111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—2482no—
312C2—2481no—
313C2C2×#1175851711yescharacteristic
314C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3151
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758485C2semidirect
2. C2×#11758509C2semidirect
3. C2×#11758526C2semidirect
4. #14521595C2semidirect
5. C2×#11758517C2semidirect
6. #14521437C2semidirect
7. #14521435C2semidirect
8. #14521424C2semidirect
9. #14521420C2semidirect
10. M16×Dic31C2semidirect
11. C2×C31×AGL(1,ℤ/8)C2semidirect
18. C2×SD16×C31C22semidirect
19. C2×D8×C31C22semidirect
22. #11758524C22semidirect
23. #11758521C22semidirect
24. C2×#92481C22semidirect
28. #11758517C22semidirect
30. #11758513C22semidirect
31. C2×#92480C22semidirect
33. C2×#92479C22semidirect
34. C2×#92479C22semidirect
39. C2×D248C22semidirect
40. C2×#92470C22semidirect
41. #11758403C22semidirect
42. C8×Dic31C22semidirect
73. M16×C31D4semidirect
85. #92474D4semidirect
128. C8×C31C2×D4semidirect
129. C31⋊C8C2×D4semidirect
150. Dic31AGL(1,ℤ/8)semidirect
154. C2×AGL(1,ℤ/8)D31semidirect
181. C2×SD16C2×D31semidirect
185. C2×D8C2×D31semidirect
200. C31#3151semidirect
238. M16C22⋊D31semidirect
287. C8C2×C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
C2×#117585092
C2×#117585262
#145215952
C2×#117585172
#145214372
#145214352
#145214242
#145214202
M16×Dic312
C2×C31×AGL(1,ℤ/8)2
#315131