Finite Group Databasebeta

← #30675911

Subgroups — #30675911

Order 16768.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16862226C211yesmaximal, characteristic
2#117×D131C211yesmaximal, characteristic
3#22553656C212yesmaximal, complemented
4#22553655C212yesmaximal, complemented
5#22553649C212yesmaximal, complemented
6#22553644C212yesmaximal, complemented
7#22553352C211yesmaximal, characteristic, complemented
8#22553322C211yesmaximal, characteristic, complemented
9#22553320C211yesmaximal, characteristic, complemented
10#22553275C211yesmaximal, characteristic
11#428×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C2×#14523497C2211yescharacteristic
15#16862226—21no—
16#16862226—21no—
17#16862226C2212yes—
18#16862226C2212yes—
19#16862205C2211yescharacteristic
20#16862204C2211yescharacteristic
21#16862204C2211yescharacteristic
22#16862203—21no—
23#16862202—21no—
24C2×C4⋊Dic131—21no—
25C4⋊C4×D131C2211yescharacteristic
26C4⋊C4×D131C2211yescharacteristic
27C2×(C2×D131)⋊C4—21no—
28C22⋊C4×D131—21no—
29C42×D131—21no—
30#16862179C2212yescomplemented
31#16862177C2212yescomplemented
32#16862171C2212yescomplemented
33#16862171C2212yescomplemented
34#16862169C2212yescomplemented
35#16862169C2212yescomplemented
36(C2×D131)⋊C8C2212yes—
37(C2×D131)⋊C8C2212yes—
38C2×M16×C131C2211yescharacteristic
39#117×C131C2211yescharacteristic
40C2×C4⋊C4×C131C2211yescharacteristic
41#107×C131C2212yescomplemented
42#106×C131C2212yescomplemented
43C23×D131D411yescharacteristic
44C2×Dic262—21no—
45C2×Dic262—22no—
46C2×Dic262—22no—
47C2×Dic262C2312yes—
48C22×Dic131—21no—
49C22×Dic131D411yescharacteristic
50C2×C4×D131—21no—
51C2×C4×D131—21no—
52C2×C4×D131—22no—
53C2×C4×D131—22no—
54C2×C4×D131D411yescharacteristic
55C2×C4×D131D411yescharacteristic
56C2×C4×D131C2311yescharacteristic
57C2×C4×D131C2311yescharacteristic
58#14523497—22no—
59C2×C131⋊C8C2312yes—
60C4⋊Dic131—22no—
61C4⋊Dic131—22no—
62C4⋊Dic131C2311yescharacteristic
63C4⋊Dic131C2311yescharacteristic
64Dic131⋊C4—21no—
65Dic131⋊C4—21no—
66C22⋊Dic131—21no—
67(C2×D131)⋊C4—22no—
68(C2×D131)⋊C4—22no—
69(C2×D131)⋊C4—21no—
70(C2×D131)⋊C4—21no—
71C4×Dic131—21no—
72C4×Dic131—21no—
73C4×Dic131—21no—
74C22×C4×C131—21no—
75C22×C4×C131C2311yescharacteristic
76M16×C131—22no—
77C2×C8×C131C2312yes—
78C4⋊C4×C131—21no—
79C4⋊C4×C131C2312yes—
80C4⋊C4×C131C2311yescharacteristic
81C4⋊C4×C131C2311yescharacteristic
82C22⋊C4×C131—21no—
83C42×C131—21no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—41no—
87C22×D131C2×D411yescharacteristic
88C22×D131C2×D411yescharacteristic
89Dic262—22no—
90Dic262—22no—
91Dic262—42no—
92C2×Dic131—22no—
93C2×Dic131—21no—
94C2×Dic131—21no—
95C2×Dic131—22no—
96C2×Dic131—42no—
97C2×Dic131C2×D411yescharacteristic
98C2×Dic131C2×D411yescharacteristic
99C4×D131—22no—
100C4×D131—22no—
101C4×D131—21no—
102C4×D131—21no—
103C4×D131—21no—
104C4×D131—21no—
105C4×D131—42no—
106C4×D131—42no—
107C131⋊C8—22no—
108C23×C131C2×D411yescharacteristic
109C2×C4×C131—21no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—42no—
113C2×C4×C131C2×D411yescharacteristic
114C2×C4×C131#4112yes—
115C2×C4×C131#4112yes—
116C2×C4×C131C2411yescharacteristic
117C8×C131—22no—
118C22×C131—21no—
119C22×C131—21no—
120C22×C131#12312yes—
121C22×C131C22×D411yescharacteristic
122C2×D131—21no—
123C2×D131—21no—
124C2×D131—21no—
125C2×D131—21no—
126C2×D131—21no—
127C2×D131—44no—
128C2×D131#13712yes—
129C4×C131—41no—
130C4×C131—42no—
131C4×C131—41no—
132C4×C131#12312yes—
133C4×C131C2×#4111yescharacteristic
134C4×C131C2×#4111yescharacteristic
135Dic131—21no—
136Dic131—21no—
137Dic131—42no—
138Dic131—41no—
139Dic131—41no—
140C2×C131—22no—
141C2×C131C2×#12311yescharacteristic
142C2×C131C2×#13711yescharacteristic
143C2×C131#57511yescharacteristic
144D131—22no—
145D131—41no—
146C131#309511yescharacteristic, complemented, Sylow
147#3095—1311nomaximal, Sylow
148C2×C22⋊Q8—1311no—
149C2×#117—1311no—
150#485—1312no—
151#485—1312no—
152#484—1312no—
153#483—1312no—
154#428—1311no—
155#428D13111yescharacteristic, complemented
156#419—1311no—
157#419—1311no—
158#407—1311no—
159C22×Q8—1311no—
160C23×C4—1311no—
161C2×M16—1311no—
162C2×M16C2×D13111yescharacteristic
163C22⋊Q8—2621no—
164C22⋊Q8—2621no—
165C22⋊Q8—1312no—
166C22⋊Q8—1312no—
167#117—2621no—
168#117—2621no—
169#117—1311no—
170#117—1311no—
171#117—1311no—
172#117C2×D13111yescharacteristic
173C2×C4⋊C4—2621no—
174C2×C4⋊C4—1311no—
175C2×C4⋊C4—1311no—
176C2×C4⋊C4C2×D13111yescharacteristic
177C2×C22⋊C4—2621no—
178C2×C22⋊C4—2621no—
179C2×C42—2621no—
180#107—1312no—
181#107C2×D13112yescomplemented
182#106—1312no—
183#106C2×D13112yescomplemented
184#103—1312no—
185#103—1312no—
186#103—1312no—
187#103—1312no—
188C22⋊C8—1312no—
189C22⋊C8—1312no—
190C24—1311no—
191C2×Q8—2621no—
192C2×Q8—2622no—
193C2×Q8—2622no—
194C2×Q8—1312no—
195C22×C4—21no—
196C22×C4—2621no—
197C22×C4—2621no—
198C22×C4—2622no—
199C22×C4—2622no—
200C22×C4—2621no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4C22×D13111yescharacteristic
207M16—22no—
208M16—2622no—
209C2×C8—1312no—
210C2×C8C22×D13112yes—
211C4⋊C4—21no—
212C4⋊C4—2621no—
213C4⋊C4—2621no—
214C4⋊C4—2622no—
215C4⋊C4—2622no—
216C4⋊C4—1311no—
217C4⋊C4—1311no—
218C4⋊C4C22×D13111yescharacteristic
219C4⋊C4C22×D13111yescharacteristic
220C4⋊C4C22×D13112yes—
221C22⋊C4—21no—
222C22⋊C4—2621no—
223C22⋊C4—2621no—
224C22⋊C4—2622no—
225C22⋊C4—2622no—
226C22⋊C4—2621no—
227C42—21no—
228C42—2621no—
229C42—2621no—
230C42—2621no—
231C23—2622no—
232C23—2622no—
233C23—5241no—
234C23—1311no—
235C23—1311no—
236C23D4×D13111yescharacteristic
237Q8—2622no—
238Q8—2622no—
239Q8—5242no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2622no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2621no—
248C2×C4—2621no—
249C2×C4—2621no—
250C2×C4—2621no—
251C2×C4—2621no—
252C2×C4—2621no—
253C2×C4—2622no—
254C2×C4—5242no—
255C2×C4—5242no—
256C2×C4—5242no—
257C2×C4—1311no—
258C2×C4—1311no—
259C2×C4D4×D13111yescharacteristic
260C2×C4#1452351312yes—
261C2×C4#1452351412yes—
262C2×C4C23×D13111yescharacteristic
263C8—22no—
264C8—2622no—
265C22—21no—
266C22—21no—
267C22—2621no—
268C22—2621no—
269C22—2621no—
270C22—2621no—
271C22—2621no—
272C22—5244no—
273C22—1312no—
274C22#1686223112yes—
275C22C2×D4×D13111yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2621no—
280C4—2621no—
281C4—5242no—
282C4—5241no—
283C4—5241no—
284C4#1686223112yes—
285C4C2×#1452351311yescharacteristic
286C4C2×#1452351411yescharacteristic
287C2—22no—
288C2—2622no—
289C2—5241no—
290C2C2×#1686223111yescharacteristic
291C2#137×D13111yescharacteristic
292C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#428×C131C2
socleC22×C131C22×D4
prime radicalC23×D131D4
hypercenter#428D131
hypocenterC131#3095
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22553656C2semidirect
4. #22553655C2semidirect
5. #22553649C2semidirect
6. #22553644C2semidirect
7. #22553352C2semidirect
8. #22553322C2semidirect
9. #22553320C2semidirect
11. #428×C131C2semidirect
30. #16862179C22semidirect
31. #16862177C22semidirect
32. #16862171C22semidirect
33. #16862171C22semidirect
34. #16862169C22semidirect
35. #16862169C22semidirect
41. #107×C131C22semidirect
42. #106×C131C22semidirect
146. C131#3095semidirect
155. #428D131semidirect
181. #107C2×D131semidirect
183. #106C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622262
#117×D1312
#225536562
#225536552
#225536492
#225536442
#225533522
#225533222
#225533202
#225532752
#428×C1312
#3095131