Finite Group Databasebeta

← #30677775

Subgroups — #30677775

Order 16768.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#22553966C211yesmaximal, characteristic, complemented
2D4×D524C211yesmaximal, characteristic, complemented
3#22553659C211yesmaximal, characteristic, complemented
4#22553626C212yesmaximal, complemented
5#22553593C211yesmaximal, characteristic, complemented
6#22553548C212yesmaximal, complemented
7#22553508C211yesmaximal, characteristic, complemented
8#22553448C212yesmaximal, complemented
9C4×D1048C211yesmaximal, characteristic, complemented
10C2×#16862168C212yesmaximal, complemented
11C8×D4×C131C211yesmaximal, characteristic, complemented
12C2×#14523512—21no—
13C22×D524C2212yes—
14C2×D4×D131—21no—
15C2×D1048C2211yescharacteristic
16C2×D1048C2212yescomplemented
17#16862246C2211yescharacteristic, complemented
18#16862243C2211yescharacteristic
19#16862230—22no—
20#16862226—21no—
21#16862221—21no—
22#16862220C2212yes—
23#16862220C2212yescomplemented
24#16862216—22no—
25C4×D524C2211yescharacteristic
26C4×D524C2211yescharacteristic, complemented
27C2×C4⋊Dic131C2212yescomplemented
28#16862180C2211yescharacteristic, complemented
29#16862180C2212yescomplemented
30#16862168C2211yescharacteristic
31#16862168C2212yes—
32#16862168C2212yescomplemented
33#16862168C2212yescomplemented
34#16862168C2211yescharacteristic, complemented
35#16862168C2212yescomplemented
36C22×C8×C131C2212yescomplemented
37C4×D4×C131C2211yescharacteristic
38C4⋊C8×C131C2211yescharacteristic, complemented
39C22⋊C8×C131C2212yescomplemented
40C4×C8×C131C2211yescharacteristic, complemented
41C23×D131—22no—
42#14523512—44no—
43C2×Dic262—21no—
44C2×D524—24no—
45C2×D524—22no—
46C2×D524—21no—
47C2×D524—21no—
48C2×D524—42no—
49C2×D524C2311yescharacteristic
50C2×D524C2312yes—
51C2×D524C2311yescharacteristic
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54D4×D131—44no—
55C22×Dic131—22no—
56C2×C4×D131—21no—
57C2×C4×D131—21no—
58D1048—22no—
59D1048—42no—
60C4⋊Dic131—22no—
61C4⋊Dic131—21no—
62C4⋊Dic131C2311yescharacteristic
63C4⋊Dic131C2311yescharacteristic
64C4⋊Dic131C2312yes—
65(C2×D131)⋊C4—21no—
66(C2×D131)⋊C4—21no—
67(C2×D131)⋊C4—22no—
68(C2×D131)⋊C4—22no—
69C2×D4×C131D411yescharacteristic
70C22×C4×C131C2312yes—
71C2×C8×C131—42no—
72C2×C8×C131C2311yescharacteristic
73C2×C8×C131C2312yes—
74C2×C8×C131C2311yescharacteristic
75C4⋊C4×C131D411yescharacteristic
76C22⋊C4×C131D412yes—
77C42×C131C2311yescharacteristic
78C22×D131—21no—
79C22×D131—21no—
80C22×D131—22no—
81C22×D131—44no—
82C22×D131—44no—
83C22×D131—82no—
84Dic262—42no—
85D524—21no—
86D524—22no—
87D524—21no—
88D524—21no—
89D524—22no—
90D524—44no—
91D524—42no—
92D524—81no—
93D524#4112yescomplemented
94C22⋊D131—44no—
95C22⋊D131—44no—
96C2×Dic131—21no—
97C2×Dic131—21no—
98C2×Dic131—22no—
99C2×Dic131—44no—
100C4×D131—42no—
101C4×D131—42no—
102C23×C131C2×D412yes—
103D4×C131D814yescomplemented
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131C2×D411yescharacteristic
107C2×C4×C131C2×D412yes—
108C2×C4×C131C2×D411yescharacteristic
109C2×C4×C131C2411yescharacteristic
110C8×C131—21no—
111C8×C131—21no—
112C8×C131—22no—
113C22×C131—22no—
114C22×C131—22no—
115C22×C131C2×D814yes—
116C22×C131C22×D411yescharacteristic
117C2×D131—22no—
118C2×D131—41no—
119C2×D131—41no—
120C2×D131—42no—
121C2×D131—41no—
122C2×D131—41no—
123C2×D131—42no—
124C2×D131—42no—
125C2×D131—41no—
126C2×D131—88no—
127C4×C131—22no—
128C4×C131—21no—
129C4×C131C2×D812yes—
130C4×C131C2×#4111yescharacteristic
131C4×C131H(5,𝔽2)11yescharacteristic
132Dic131—41no—
133Dic131—41no—
134Dic131—42no—
135C2×C131—24no—
136C2×C131#54611yescharacteristic
137C2×C131C22×D811yescharacteristic
138C2×C131#57711yescharacteristic
139D131—42no—
140D131—81no—
141D131—82no—
142C131#329411yescharacteristic, complemented, Sylow
143#3294—1311nomaximal, Sylow
144#547—1311no—
145D42—1311no—
146#486—1311no—
147#480—1312no—
148#474—1311no—
149#466—1312no—
150#459—1311no—
151#447—1312no—
152C4×D8—1311no—
153C8×D4D13111yescharacteristic, complemented
154C2×#102—1312no—
155C2×#41—2621no—
156C22×D4—2621no—
157C22×D4—1312no—
158C2×D8—1311no—
159C2×D8—1312no—
160C22×C8C2×D13112yescomplemented
161C4⋊Q8—1311no—
162#127—1311no—
163#123—2622no—
164C22⋊Q8—2621no—
165#121—2621no—
166#121—1312no—
167#121—1312no—
168#120—2622no—
169C4×D4—1311no—
170C4×D4—1311no—
171C4×D4C2×D13111yescharacteristic
172C2×C4⋊C4—1312no—
173#107—1311no—
174#107—1312no—
175C4⋊C8C2×D13111yescharacteristic, complemented
176#102—1311no—
177#102—1312no—
178#102—1312no—
179#102—1312no—
180#102—1311no—
181#102—1312no—
182C22⋊C8C2×D13112yescomplemented
183C4×C8C2×D13111yescharacteristic, complemented
184C24—2622no—
185#41—5244no—
186C2×Q8—2621no—
187C2×D4—2624no—
188C2×D4—2622no—
189C2×D4—2622no—
190C2×D4—2622no—
191C2×D4—2621no—
192C2×D4—2621no—
193C2×D4—5242no—
194C2×D4—5244no—
195C2×D4—1312no—
196C2×D4—1311no—
197C2×D4—1311no—
198C2×D4D52411yescharacteristic
199C22×C4—2622no—
200C22×C4—2621no—
201C22×C4—2621no—
202C22×C4C22×D13112yes—
203D8—2622no—
204D8—5242no—
205C2×C8—42no—
206C2×C8C22×D13111yescharacteristic
207C2×C8C22×D13111yescharacteristic
208C2×C8C22×D13112yes—
209C4⋊C4—2622no—
210C4⋊C4—2621no—
211C4⋊C4—1311no—
212C4⋊C4—1312no—
213C4⋊C4—1311no—
214C4⋊C4D52411yescharacteristic
215C22⋊C4—2621no—
216C22⋊C4—2621no—
217C22⋊C4—2622no—
218C22⋊C4—2622no—
219C22⋊C4D52412yes—
220C42C22×D13111yescharacteristic
221C23—2621no—
222C23—2621no—
223C23—2622no—
224C23—5244no—
225C23—5244no—
226C23—10482no—
227C23C2×D52412yes—
228Q8—5242no—
229D4—2621no—
230D4—2622no—
231D4—2621no—
232D4—2621no—
233D4—2622no—
234D4—5244no—
235D4—5244no—
236D4—5244no—
237D4—5242no—
238D4—10481no—
239D4—1312no—
240D4D104814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—2621no—
244C2×C4—2621no—
245C2×C4—2622no—
246C2×C4—5244no—
247C2×C4—5242no—
248C2×C4—5242no—
249C2×C4C2×D52411yescharacteristic
250C2×C4C2×D52412yes—
251C2×C4C2×D52411yescharacteristic
252C2×C4C23×D13111yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—2622no—
259C22—5241no—
260C22—5241no—
261C22—5242no—
262C22—5241no—
263C22—5242no—
264C22—5241no—
265C22—5241no—
266C22—5242no—
267C22—10488no—
268C22C2×D104814yes—
269C22C22×D52411yescharacteristic
270C4—21no—
271C4—22no—
272C4—5241no—
273C4—5241no—
274C4—5242no—
275C4C2×D104812yes—
276C4#41×D13111yescharacteristic
277C4#1686229111yescharacteristic
278C2—24no—
279C2—5242no—
280C2—10482no—
281C2—10481no—
282C2#2255395711yescharacteristic
283C2C22×D104811yescharacteristic
284C2#2255412411yescharacteristic
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC8×D4×C131C2
socleC22×C131C22×D4
hypercenterC8×D4D131
hypocenterC131#3294
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22553966C2semidirect
2. D4×D524C2semidirect
3. #22553659C2semidirect
4. #22553626C2semidirect
5. #22553593C2semidirect
6. #22553548C2semidirect
7. #22553508C2semidirect
8. #22553448C2semidirect
9. C4×D1048C2semidirect
10. C2×#16862168C2semidirect
11. C8×D4×C131C2semidirect
16. C2×D1048C22semidirect
17. #16862246C22semidirect
23. #16862220C22semidirect
26. C4×D524C22semidirect
27. C2×C4⋊Dic131C22semidirect
28. #16862180C22semidirect
29. #16862180C22semidirect
32. #16862168C22semidirect
33. #16862168C22semidirect
34. #16862168C22semidirect
35. #16862168C22semidirect
36. C22×C8×C131C22semidirect
38. C4⋊C8×C131C22semidirect
39. C22⋊C8×C131C22semidirect
40. C4×C8×C131C22semidirect
93. D524#41semidirect
103. D4×C131D8semidirect
142. C131#3294semidirect
153. C8×D4D131semidirect
160. C22×C8C2×D131semidirect
175. C4⋊C8C2×D131semidirect
182. C22⋊C8C2×D131semidirect
183. C4×C8C2×D131semidirect
240. D4D1048semidirect

Maximal subgroups

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

subgroupindex
#225539662
D4×D5242
#225536592
#225536262
#225535932
#225535482
#225535082
#225534482
C4×D10482
C2×#168621682
C8×D4×C1312
#3294131