Finite Group Databasebeta

← #30675903

Subgroups — #30675903

Order 16768.

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

normal subgroups only

This group has 462 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×#16862221C211yesmaximal, characteristic
2#117×D131C211yesmaximal, characteristic, complemented
3#22553639C212yesmaximal, complemented
4#22553638C212yesmaximal, complemented
5#22553632C212yesmaximal, complemented
6#22553627C212yesmaximal, complemented
7#22553352C211yesmaximal, characteristic, complemented
8#22553318C211yesmaximal, characteristic, complemented
9#22553316C211yesmaximal, characteristic, complemented
10#22553275C211yesmaximal, characteristic, complemented
11#428×C131C211yesmaximal, characteristic, complemented
12C22×D524—21no—
13C22×D524C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15C2×#14523497C2211yescharacteristic
16#16862221—21no—
17#16862221—21no—
18#16862221C2212yes—
19#16862221C2212yes—
20#16862205C2211yescharacteristic
21#16862204C2211yescharacteristic, complemented
22#16862204C2211yescharacteristic, complemented
23#16862203—21no—
24#16862202—21no—
25C4⋊C4×D131C2211yescharacteristic, complemented
26C4⋊C4×D131C2211yescharacteristic, complemented
27C2×(C2×D131)⋊C4—21no—
28C22⋊C4×D131—21no—
29C42×D131—21no—
30#16862179C2212yescomplemented
31#16862177C2212yescomplemented
32#16862168C2212yescomplemented
33#16862168C2212yescomplemented
34#16862166C2212yescomplemented
35#16862166C2212yescomplemented
36(C2×D131)⋊C8C2212yescomplemented
37(C2×D131)⋊C8C2212yescomplemented
38C2×M16×C131C2211yescharacteristic
39#117×C131C2211yescharacteristic
40C2×C4⋊C4×C131C2211yescharacteristic
41#107×C131C2212yescomplemented
42#106×C131C2212yescomplemented
43C23×D131—21no—
44C23×D131D411yescharacteristic
45C2×D524—22no—
46C2×D524—22no—
47C2×D524—21no—
48C2×D524—22no—
49C2×D524—22no—
50C2×D524—44no—
51C2×D524C2312yes—
52C22×Dic131D411yescharacteristic
53C2×C4×D131—21no—
54C2×C4×D131—21no—
55C2×C4×D131—22no—
56C2×C4×D131—22no—
57C2×C4×D131D411yescharacteristic
58C2×C4×D131D411yescharacteristic
59C2×C4×D131C2311yescharacteristic
60C2×C4×D131C2311yescharacteristic
61#14523497—22no—
62C2×C131⋊C8C2312yes—
63C4⋊Dic131C2311yescharacteristic
64C4⋊Dic131C2311yescharacteristic
65Dic131⋊C4—21no—
66Dic131⋊C4—21no—
67C22⋊Dic131—21no—
68(C2×D131)⋊C4—21no—
69(C2×D131)⋊C4—21no—
70(C2×D131)⋊C4—22no—
71(C2×D131)⋊C4—22no—
72C4×Dic131—21no—
73C4×Dic131—21no—
74C4×Dic131—21no—
75C22×C4×C131—21no—
76C22×C4×C131C2311yescharacteristic
77M16×C131—22no—
78C2×C8×C131C2312yes—
79C4⋊C4×C131—21no—
80C4⋊C4×C131C2312yes—
81C4⋊C4×C131C2311yescharacteristic
82C4⋊C4×C131C2311yescharacteristic
83C22⋊C4×C131—21no—
84C42×C131—21no—
85C22×D131—22no—
86C22×D131—22no—
87C22×D131—22no—
88C22×D131—41no—
89C22×D131—42no—
90C22×D131—42no—
91C22×D131—42no—
92C22×D131C2×D411yescharacteristic
93C22×D131C2×D411yescharacteristic
94D524—22no—
95D524—22no—
96D524—44no—
97D524—44no—
98D524—42no—
99C2×Dic131—22no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131C2×D411yescharacteristic
103C2×Dic131C2×D411yescharacteristic
104C4×D131—22no—
105C4×D131—22no—
106C4×D131—21no—
107C4×D131—21no—
108C4×D131—21no—
109C4×D131—21no—
110C4×D131—42no—
111C4×D131—42no—
112C131⋊C8—22no—
113C23×C131C2×D411yescharacteristic
114C2×C4×C131—21no—
115C2×C4×C131—21no—
116C2×C4×C131—22no—
117C2×C4×C131—42no—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131#4112yes—
120C2×C4×C131#4112yes—
121C2×C4×C131C2411yescharacteristic
122C8×C131—22no—
123C22×C131—21no—
124C22×C131—21no—
125C22×C131#12312yes—
126C22×C131C22×D411yescharacteristic
127C2×D131—21no—
128C2×D131—21no—
129C2×D131—21no—
130C2×D131—21no—
131C2×D131—21no—
132C2×D131—44no—
133C2×D131—42no—
134C2×D131—42no—
135C2×D131—42no—
136C2×D131—84no—
137C2×D131AGL(1,ℤ/8)12yes—
138C4×C131—42no—
139C4×C131—41no—
140C4×C131—41no—
141C4×C131#12312yes—
142C4×C131C2×#4111yescharacteristic
143C4×C131C2×#4111yescharacteristic
144Dic131—21no—
145Dic131—21no—
146Dic131—41no—
147Dic131—41no—
148C2×C131—22no—
149C2×C131C2×#12311yescharacteristic
150C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C131#57511yescharacteristic
152D131—22no—
153D131—41no—
154D131—82no—
155C131#309411yescharacteristic, complemented, Sylow
156#3094—1311nomaximal, Sylow
157C2×#121—1311no—
158C2×#117—1311no—
159#482—1312no—
160#482—1312no—
161#481—1312no—
162#480—1312no—
163#428—1311no—
164#428D13111yescharacteristic, complemented
165#418—1311no—
166#418—1311no—
167#407—1311no—
168C22×D4—2621no—
169C22×D4—1311no—
170C23×C4—1311no—
171C2×M16—1311no—
172C2×M16C2×D13111yescharacteristic
173#121—2621no—
174#121—2621no—
175#121—1312no—
176#121—1312no—
177#117—2621no—
178#117—2621no—
179#117—1311no—
180#117—1311no—
181#117—1311no—
182#117C2×D13111yescharacteristic
183C2×C4⋊C4—1311no—
184C2×C4⋊C4—1311no—
185C2×C4⋊C4C2×D13111yescharacteristic
186C2×C22⋊C4—2621no—
187C2×C22⋊C4—2621no—
188C2×C42—2621no—
189#107—1312no—
190#107C2×D13112yescomplemented
191#106—1312no—
192#106C2×D13112yescomplemented
193#102—1312no—
194#102—1312no—
195#102—1312no—
196#102—1312no—
197C22⋊C8—1312no—
198C22⋊C8—1312no—
199C24—2621no—
200C24—1311no—
201C2×D4—2622no—
202C2×D4—2622no—
203C2×D4—2621no—
204C2×D4—2622no—
205C2×D4—2622no—
206C2×D4—5244no—
207C2×D4—1312no—
208C22×C4—21no—
209C22×C4—2621no—
210C22×C4—2621no—
211C22×C4—2622no—
212C22×C4—2622no—
213C22×C4—1311no—
214C22×C4—1311no—
215C22×C4—1311no—
216C22×C4—1311no—
217C22×C4—1311no—
218C22×C4C22×D13111yescharacteristic
219M16—22no—
220M16—2622no—
221C2×C8—1312no—
222C2×C8C22×D13112yes—
223C4⋊C4—21no—
224C4⋊C4—2621no—
225C4⋊C4—2621no—
226C4⋊C4—1311no—
227C4⋊C4—1311no—
228C4⋊C4C22×D13111yescharacteristic
229C4⋊C4C22×D13111yescharacteristic
230C4⋊C4C22×D13112yes—
231C22⋊C4—21no—
232C22⋊C4—2621no—
233C22⋊C4—2621no—
234C22⋊C4—2622no—
235C22⋊C4—2622no—
236C22⋊C4—2621no—
237C42—21no—
238C42—2621no—
239C42—2621no—
240C42—2621no—
241C23—2622no—
242C23—2622no—
243C23—2622no—
244C23—5241no—
245C23—5242no—
246C23—5242no—
247C23—5242no—
248C23—1311no—
249C23—1311no—
250C23D4×D13111yescharacteristic
251D4—2622no—
252D4—2622no—
253D4—5244no—
254D4—5244no—
255D4—5242no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2621no—
264C2×C4—2621no—
265C2×C4—2621no—
266C2×C4—2621no—
267C2×C4—2621no—
268C2×C4—2621no—
269C2×C4—5242no—
270C2×C4—5242no—
271C2×C4—1311no—
272C2×C4—1311no—
273C2×C4D4×D13111yescharacteristic
274C2×C4#1452351312yes—
275C2×C4#1452351412yes—
276C2×C4C23×D13111yescharacteristic
277C8—22no—
278C8—2622no—
279C22—21no—
280C22—21no—
281C22—2621no—
282C22—2621no—
283C22—2621no—
284C22—2621no—
285C22—2621no—
286C22—5244no—
287C22—5242no—
288C22—5242no—
289C22—5242no—
290C22—10484no—
291C22—1312no—
292C22#1686223112yes—
293C22C2×D4×D13111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—2621no—
298C4—2621no—
299C4—5241no—
300C4—5241no—
301C4#1686223112yes—
302C4C2×#1452351311yescharacteristic
303C4C2×#1452351411yescharacteristic
304C2—22no—
305C2—2622no—
306C2—5241no—
307C2—10482no—
308C2C2×#1686223111yescharacteristic
309C2AGL(1,ℤ/8)×D13111yescharacteristic
310C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#428×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862221C2
hypercenter#428D131
hypocenterC131#3094
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
2. #117×D131C2semidirect
3. #22553639C2semidirect
4. #22553638C2semidirect
5. #22553632C2semidirect
6. #22553627C2semidirect
7. #22553352C2semidirect
8. #22553318C2semidirect
9. #22553316C2semidirect
10. #22553275C2semidirect
11. #428×C131C2semidirect
21. #16862204C22semidirect
22. #16862204C22semidirect
25. C4⋊C4×D131C22semidirect
26. C4⋊C4×D131C22semidirect
30. #16862179C22semidirect
31. #16862177C22semidirect
32. #16862168C22semidirect
33. #16862168C22semidirect
34. #16862166C22semidirect
35. #16862166C22semidirect
36. (C2×D131)⋊C8C22semidirect
37. (C2×D131)⋊C8C22semidirect
41. #107×C131C22semidirect
42. #106×C131C22semidirect
155. C131#3094semidirect
164. #428D131semidirect
190. #107C2×D131semidirect
192. #106C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622212
#117×D1312
#225536392
#225536382
#225536322
#225536272
#225533522
#225533182
#225533162
#225532752
#428×C1312
#3094131