Finite Group Databasebeta

← #16856188

Subgroups — #16856188

Order 3968.

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

normal subgroups only

This group has 564 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×#11758528C211yesmaximal, characteristic, complemented
2C2×#11758526C211yesmaximal, characteristic, complemented
3C22×#92470C211yesmaximal, characteristic, complemented
4#14521620C211yesmaximal, characteristic, complemented
5#14521613C211yesmaximal, characteristic, complemented
6#14521442C212yesmaximal, complemented
7#14521437C212yesmaximal, complemented
8#14521430C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10#14521004C211yesmaximal, characteristic, complemented
11C2×C31×#135C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#41×C31C2211yescharacteristic
14C22×D124C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C31×#135—24no—
17C2×Q16×C31C2211yescharacteristic, complemented
18C2×SD16×C31C2211yescharacteristic, complemented
19C2×SD16×C31C2211yescharacteristic, complemented
20C2×D8×C31C2211yescharacteristic, complemented
21C22×C8×C31C2211yescharacteristic
22#11758528—24no—
23#11758526—24no—
24#11758524C2212yes—
25#11758521C2212yescomplemented
26C2×#92482C2212yescomplemented
27C2×#92481C2212yes—
28#11758517C2212yes—
29#11758514—22no—
30#11758513C2212yescomplemented
31#11758512—22no—
32C2×#92480C2212yescomplemented
33C2×#92479C2212yes—
34#11758507—22no—
35#11758503C2211yescharacteristic
36#11758501—22no—
37C2×#92474C2211yescharacteristic
38C2×#92470—24no—
39C2×#92470C2212yes—
40C2×#92470C2212yes—
41#11758403C2212yescomplemented
42C8×Dic31C2212yescomplemented
43C23×D31—21no—
44#41×C31—22no—
45#41×C31—22no—
46#41×C31—22no—
47#41×C31—22no—
48C2×Q8×C31C2311yescharacteristic
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31C2311yescharacteristic
53C2×D4×C31C2311yescharacteristic
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—24no—
58C22×Dic31—21no—
59C2×D124—22no—
60C2×D124—22no—
61C2×D124—21no—
62C2×D124C2312yes—
63C2×Dic62—22no—
64C2×Dic62—22no—
65C2×Dic62—21no—
66C2×Dic62C2312yes—
67Q16×C31—22no—
68SD16×C31—22no—
69SD16×C31—22no—
70D8×C31—22no—
71C2×C8×C31D414yescomplemented
72C2×C8×C31C2311yescharacteristic
73C2×C8×C31C2311yescharacteristic
74C22⋊Dic31—24no—
75C22⋊Dic31D412yes—
76#92482—24no—
77#92481—24no—
78#92480—24no—
79#92479—24no—
80(C2×D31)⋊C4—24no—
81C4⋊Dic31D412yes—
82Dic31⋊C4—24no—
83C4×Dic31C2312yes—
84#92474D414yescomplemented
85C2×C31⋊C8C2312yes—
86#92470—24no—
87#92470—24no—
88C23×C31—21no—
89C23×C31—21no—
90C23×C31C2×D411yescharacteristic
91C22×D31—22no—
92C22×D31—42no—
93C22×D31—82no—
94Q8×C31—21no—
95Q8×C31—21no—
96Q8×C31—21no—
97Q8×C31—21no—
98D4×C31—21no—
99D4×C31—21no—
100D4×C31—21no—
101D4×C31—21no—
102D4×C31—42no—
103D4×C31—42no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2411yescharacteristic
112C22⋊D31—84no—
113C2×Dic31—22no—
114C2×Dic31—42no—
115C2×Dic31C2×D414yes—
116D124—22no—
117D124—22no—
118D124—42no—
119Dic62—22no—
120Dic62—22no—
121Dic62—42no—
122C8×C31C2×D412yes—
123C8×C31C2×D412yes—
124C31⋊C8C2×D414yes—
125#3154—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—41no—
128C22×C31—41no—
129C22×C31—42no—
130C22×C31—42no—
131C22×C31#12712yes—
132C22×C31C22×D411yescharacteristic
133C2×D31—42no—
134C2×D31—44no—
135C2×D31—84no—
136C4×C31—41no—
137C4×C31—41no—
138C4×C31#12712yes—
139C4×C31C22×D411yescharacteristic
140C4×C31C22×D411yescharacteristic
141Dic31—24no—
142Dic31—42no—
143C2×#137—311no—
144C2×AGL(1,ℤ/8)—311no—
145C2×#135D3111yescharacteristic, complemented
146C22×SD16—311no—
147#536—311no—
148#535—311no—
149#497—312no—
150#496—312no—
151#495—312no—
152#492—312no—
153#405—311no—
154C2×C31—22no—
155C2×C31—81no—
156C2×C31—81no—
157C2×C31C2×#12711yescharacteristic
158C2×C31#57712yes—
159D31—82no—
160C2×#41C2×D3111yescharacteristic
161C2×#41C2×D3111yescharacteristic
162C22×Q8—311no—
163C22×D4—311no—
164#137—624no—
165AGL(1,ℤ/8)—624no—
166#135—24no—
167C2×Q16—312no—
168C2×Q16C2×D3111yescharacteristic, complemented
169C2×SD16—312no—
170C2×SD16—312no—
171C2×SD16—312no—
172C2×SD16—312no—
173C2×SD16—624no—
174C2×SD16C2×D3111yescharacteristic, complemented
175C2×SD16C2×D3111yescharacteristic, complemented
176C2×D8—312no—
177C2×D8C2×D3111yescharacteristic, complemented
178C2×M16—311no—
179C22×C8C2×D3111yescharacteristic
180C4⋊Q8—312no—
181#127—312no—
182#124—312no—
183#124—312no—
184#123—622no—
185C22⋊Q8—622no—
186#121—622no—
187#120—622no—
188#117—311no—
189#97—312no—
190C4×C8—312no—
191C31#315411yescharacteristic, complemented, Sylow
192C24—621no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—312no—
198C2×Q8—621no—
199C2×Q8—622no—
200C2×Q8—622no—
201C2×Q8C22×D3111yescharacteristic
202C2×Q8C22×D3111yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—312no—
206C2×D4—624no—
207C2×D4—622no—
208C2×D4—621no—
209C2×D4—622no—
210C2×D4C22×D3111yescharacteristic
211C2×D4C22×D3111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—621no—
215C22×C4C22×D3111yescharacteristic
216Q16—22no—
217Q16—624no—
218SD16—22no—
219SD16—22no—
220SD16—624no—
221SD16—624no—
222SD16—624no—
223SD16—624no—
224D8—22no—
225D8—624no—
226M16—314no—
227C2×C8—312no—
228C2×C8C22⋊D3114yescomplemented
229C2×C8C22×D3111yescharacteristic
230C2×C8C22×D3111yescharacteristic
231C4⋊C4—312no—
232C4⋊C4—624no—
233C22⋊C4—312no—
234C22⋊C4—624no—
235C22⋊C4—624no—
236C42—312no—
237C23—21no—
238C23—21no—
239C23—622no—
240C23—1242no—
241C23—2482no—
242C23D4×D3111yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—622no—
248Q8—622no—
249Q8—1242no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—622no—
257D4—622no—
258D4—1242no—
259D4—2484no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—314no—
265C2×C4—622no—
266C2×C4—1242no—
267C2×C4D4×D3111yescharacteristic
268C2×C4D4×D3112yes—
269C2×C4C2×C22⋊D3112yes—
270C2×C4C23×D3111yescharacteristic
271C8—314no—
272C8C2×C22⋊D3112yes—
273C8C2×C22⋊D3112yes—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—1244no—
280C22—1242no—
281C22—2484no—
282C22#1175851712yes—
283C22C2×D4×D3111yescharacteristic
284C4—41no—
285C4—41no—
286C4—624no—
287C4—1242no—
288C4#1175851712yes—
289C4C2×D4×D3111yescharacteristic
290C4C22×C22⋊D3111yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—2482no—
295C2C2×#1175851711yescharacteristic
296C2#1452185612yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#135C2
socleC22×C31C22×D4
hypercenterC2×#135D31
hypocenterC31#3154
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C2×#11758526C2semidirect
3. C22×#92470C2semidirect
4. #14521620C2semidirect
5. #14521613C2semidirect
6. #14521442C2semidirect
7. #14521437C2semidirect
8. #14521430C2semidirect
9. #14521420C2semidirect
10. #14521004C2semidirect
11. C2×C31×#135C2semidirect
17. C2×Q16×C31C22semidirect
18. C2×SD16×C31C22semidirect
19. C2×SD16×C31C22semidirect
20. C2×D8×C31C22semidirect
25. #11758521C22semidirect
26. C2×#92482C22semidirect
30. #11758513C22semidirect
32. C2×#92480C22semidirect
41. #11758403C22semidirect
42. C8×Dic31C22semidirect
71. C2×C8×C31D4semidirect
84. #92474D4semidirect
145. C2×#135D31semidirect
168. C2×Q16C2×D31semidirect
174. C2×SD16C2×D31semidirect
175. C2×SD16C2×D31semidirect
177. C2×D8C2×D31semidirect
191. C31#3154semidirect
228. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C2×#117585262
C22×#924702
#145216202
#145216132
#145214422
#145214372
#145214302
#145214202
#145210042
C2×C31×#1352
#315431