Finite Group Databasebeta

← #30675355

Subgroups — #30675355

Order 16768.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#22554148C211yesmaximal, characteristic, complemented
2#22553824C212yesmaximal, complemented
3#22553819C211yesmaximal, characteristic, complemented
4#22553503C214yesmaximal, complemented
5#22553500C214yesmaximal, complemented
6#22553285C212yesmaximal, complemented
7#410×C131C211yesmaximal, characteristic, complemented
8#16862294—21no—
9#16862293—22no—
10#16862292—22no—
11#16862291—21no—
12#16862291C2214yes—
13C2×#14523513—22no—
14C2×#14523513C2212yes—
15#41×D131—24no—
16C22×C4×D131C2211yescharacteristic, complemented
17#16862230C2214yescomplemented
18#16862227C2212yescomplemented
19#16862226—22no—
20#16862222—21no—
21#16862221—22no—
22#16862217—21no—
23#16862216C2214yescomplemented
24#16862214C2212yescomplemented
25C4×D524—24no—
26C4×C22⋊D131—22no—
27#16862205C2211yescharacteristic, complemented
28#16862160C2218yescomplemented
29C2×#41×C131C2211yescharacteristic
30#117×C131C2212yescomplemented
31#99×C131C2214yescomplemented
32C23×D131C2311yescharacteristic
33#14523514—24no—
34#14523513—24no—
35#14523513—24no—
36#14523513—22no—
37#14523513—22no—
38#14523513—24no—
39#14523513—22no—
40#14523513—22no—
41#14523512—24no—
42#14523512—28no—
43C2×Dic262—22no—
44C2×Dic262D412yescomplemented
45Q8×D131—24no—
46C2×D524—22no—
47C2×D524—24no—
48C2×D524D412yescomplemented
49C2×C22⋊D131—24no—
50C2×C22⋊D131—22no—
51C2×C22⋊D131C2314yes—
52D4×D131—24no—
53D4×D131—28no—
54C22×Dic131C2311yescharacteristic
55C2×C4×D131—24no—
56C2×C4×D131—22no—
57C2×C4×D131—21no—
58C2×C4×D131—22no—
59C2×C4×D131—21no—
60C2×C4×D131D414yescomplemented
61C4⋊Dic131—24no—
62C4⋊Dic131—21no—
63Dic131⋊C4—22no—
64C22⋊Dic131—22no—
65C22⋊Dic131C2312yes—
66(C2×D131)⋊C4—24no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—22no—
69C4×Dic131—21no—
70#41×C131—22no—
71#41×C131—22no—
72C2×Q8×C131D411yescharacteristic, complemented
73C2×D4×C131D411yescharacteristic, complemented
74C2×D4×C131C2312yes—
75C22×C4×C131D412yescomplemented
76C22×C4×C131C2311yescharacteristic
77C4⋊C4×C131—22no—
78C22⋊C4×C131C2314yes—
79C42×C131—22no—
80C22×D131—21no—
81C22×D131—22no—
82C22×D131—24no—
83C22×D131—42no—
84C22×D131C2×D414yes—
85Dic262—24no—
86Dic262—22no—
87Dic262—22no—
88D524—24no—
89D524—22no—
90D524—22no—
91D524—48no—
92C22⋊D131—28no—
93C22⋊D131—24no—
94C22⋊D131—24no—
95C22⋊D131—44no—
96C2×Dic131—24no—
97C2×Dic131—22no—
98C2×Dic131—22no—
99C2×Dic131—21no—
100C2×Dic131C2×D414yes—
101C4×D131—28no—
102C4×D131—24no—
103C4×D131—24no—
104C4×D131—41no—
105C4×D131—42no—
106C4×D131—41no—
107C23×C131C2×D412yes—
108C23×C131C2411yescharacteristic
109Q8×C131—22no—
110D4×C131—24no—
111D4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—24no—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131C2×D411yescharacteristic
120C22×C131—22no—
121C22×C131—41no—
122C22×C131—42no—
123C22×C131C22×D412yes—
124C22×C131C22×D411yescharacteristic
125C2×D131—22no—
126C2×D131—24no—
127C2×D131—24no—
128C2×D131—41no—
129C2×D131—42no—
130C2×D131—41no—
131C2×D131—44no—
132C2×D131—84no—
133C4×C131—22no—
134C4×C131—44no—
135C4×C131#12011yescharacteristic
136C4×C131#12012yes—
137C4×C131#12011yescharacteristic
138Dic131—24no—
139Dic131—42no—
140Dic131—41no—
141C2×C131—21no—
142C2×C131—22no—
143C2×C131—42no—
144C2×C131C2×#12011yescharacteristic
145D131—42no—
146D131—44no—
147C131#302511yescharacteristic, complemented, Sylow
148#3025—1311nomaximal, Sylow
149#585—1311no—
150#525—1311no—
151#525—1312no—
152#458—1314no—
153B(4,𝔽2)—1314no—
154#410—1312no—
155#410D13111yescharacteristic, complemented
1562−1+4—2622no—
1572−1+4—2621no—
158H(5,𝔽2)—2622no—
159H(5,𝔽2)—2621no—
160H(5,𝔽2)—1314no—
161C2×#41—2624no—
162C2×#41—2622no—
163C2×#41—1312no—
164C2×#41C2×D13111yescharacteristic
165C23×C4—1311no—
166#123—1312no—
167#123—1314no—
168C22⋊Q8—2622no—
169C22⋊Q8—2621no—
170#121—2622no—
171#121—2621no—
172#120—1312no—
173#120—1314no—
174C4×D4—2624no—
175C4×D4—2622no—
176#117—1311no—
177#117C2×D13112yescomplemented
178#99—1318no—
179#99C2×D13114yescomplemented
180C24—1311no—
181#41—22no—
182#41—22no—
183#41—2624no—
184#41—2624no—
185#41—2624no—
186#41—2624no—
187#41—2628no—
188#41—2622no—
189#41—2622no—
190#41—2622no—
191#41—2624no—
192#41—2622no—
193C2×Q8—2622no—
194C2×Q8—2624no—
195C2×Q8—1312no—
196C2×Q8D52411yescharacteristic, complemented
197C2×D4—2622no—
198C2×D4—2624no—
199C2×D4—2624no—
200C2×D4—2628no—
201C2×D4—2624no—
202C2×D4—2622no—
203C2×D4—1314no—
204C2×D4—1312no—
205C2×D4D52411yescharacteristic, complemented
206C2×D4C22×D13112yes—
207C22×C4—2624no—
208C22×C4—2622no—
209C22×C4—2622no—
210C22×C4—2621no—
211C22×C4—2621no—
212C22×C4—1311no—
213C22×C4—1314no—
214C22×C4D52412yescomplemented
215C22×C4C22×D13111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—2624no—
218C4⋊C4—2622no—
219C4⋊C4—2621no—
220C22⋊C4—2624no—
221C22⋊C4—2624no—
222C22⋊C4—2622no—
223C22⋊C4—2622no—
224C22⋊C4—1312no—
225C22⋊C4C22×D13114yes—
226C42—22no—
227C42—2621no—
228C23—2624no—
229C23—2622no—
230C23—2621no—
231C23—5242no—
232C23—1314no—
233C23C2×D52412yes—
234C23C23×D13111yescharacteristic
235Q8—22no—
236Q8—2624no—
237Q8—2622no—
238Q8—2622no—
239D4—24no—
240D4—22no—
241D4—2624no—
242D4—2628no—
243D4—2624no—
244D4—2624no—
245D4—2622no—
246D4—2622no—
247D4—5248no—
248D4—5244no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—2628no—
253C2×C4—2624no—
254C2×C4—2624no—
255C2×C4—2624no—
256C2×C4—2622no—
257C2×C4—2622no—
258C2×C4—2621no—
259C2×C4—5241no—
260C2×C4—5242no—
261C2×C4—5241no—
262C2×C4—1314no—
263C2×C4D4×D13112yes—
264C2×C4D4×D13112yes—
265C2×C4C2×D52412yes—
266C2×C4C2×D52411yescharacteristic
267C2×C4C2×D52411yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—2624no—
272C22—2622no—
273C22—2624no—
274C22—5244no—
275C22—5242no—
276C22—5241no—
277C22—5241no—
278C22—10484no—
279C22C2×D4×D13112yes—
280C22C22×D52411yescharacteristic
281C4—22no—
282C4—44no—
283C4—2624no—
284C4—5242no—
285C4—5241no—
286C4#1686221611yescharacteristic
287C4#1686221612yes—
288C4#1686221611yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—5242no—
293C2—5244no—
294C2C2×#1686221611yescharacteristic
subgroupG/H
centerC4#16862216
commutator, cosocleC23×C131C24
FrattiniC23C23×D131
Fitting#410×C131C2
socleC2×C131C2×#120
hypercenter#410D131
hypocenterC131#3025
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22554148C2semidirect
2. #22553824C2semidirect
3. #22553819C2semidirect
4. #22553503C2semidirect
5. #22553500C2semidirect
6. #22553285C2semidirect
7. #410×C131C2semidirect
16. C22×C4×D131C22semidirect
17. #16862230C22semidirect
18. #16862227C22semidirect
23. #16862216C22semidirect
24. #16862214C22semidirect
27. #16862205C22semidirect
28. #16862160C22semidirect
30. #117×C131C22semidirect
31. #99×C131C22semidirect
44. C2×Dic262D4semidirect
48. C2×D524D4semidirect
60. C2×C4×D131D4semidirect
72. C2×Q8×C131D4semidirect
73. C2×D4×C131D4semidirect
75. C22×C4×C131D4semidirect
147. C131#3025semidirect
155. #410D131semidirect
177. #117C2×D131semidirect
179. #99C2×D131semidirect
196. C2×Q8D524semidirect
205. C2×D4D524semidirect
214. C22×C4D524semidirect

Maximal subgroups

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

subgroupindex
#225541482
#225538242
#225538192
#225535032
#225535002
#225532852
#410×C1312
#3025131