Finite Group Databasebeta

← #30675880

Subgroups — #30675880

Order 16768.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#22553822C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
3#22553649C212yesmaximal, complemented
4#22553643C211yesmaximal, characteristic, complemented
5#22553642C211yesmaximal, characteristic, complemented
6#22553629C212yesmaximal, complemented
7#22553623C211yesmaximal, characteristic, complemented
8#22553353C212yesmaximal, complemented
9#22553310C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic, complemented
11#480×C131C211yesmaximal, characteristic, complemented
12C2×#14523512C2211yescharacteristic
13C22×C4×D131—21no—
14C22×C4×D131C2211yescharacteristic
15C2×#14523496C2212yes—
16#16862228—21no—
17#16862226C2211yescharacteristic
18#16862225C2211yescharacteristic
19#16862217C2211yescharacteristic
20#16862214—21no—
21D4×Dic131—21no—
22(C22⋊D131)⋊C4—21no—
23#16862204C2211yescharacteristic
24C2×C4⋊Dic131C2211yescharacteristic
25C2×Dic131⋊C4—21no—
26C4⋊C4×D131—21no—
27C4⋊C4×D131—21no—
28C4⋊C4×D131C2212yes—
29C4⋊C4×D131C2212yes—
30#16862179C2212yescomplemented
31#16862178C2212yescomplemented
32#16862177C2212yescomplemented
33#16862171C2212yescomplemented
34#16862169C2212yescomplemented
35#16862167C2212yescomplemented
36C22⋊(C131⋊C8)C2211yescharacteristic
37(C2×D131)⋊C8C2212yes—
38#121×C131C2211yescharacteristic
39C2×C4⋊C4×C131C2211yescharacteristic
40#107×C131C2212yescomplemented
41#102×C131C2212yescomplemented
42C22⋊C8×C131C2211yescharacteristic
43C23×D131D411yescharacteristic
44#14523512—24no—
45C2×Dic262C2311yescharacteristic
46C2×C22⋊D131—21no—
47C2×C22⋊D131—21no—
48C22×Dic131—21no—
49C22×Dic131—21no—
50C22×Dic131D411yescharacteristic
51C2×C4×D131—24no—
52C2×C4×D131—22no—
53C2×C4×D131—22no—
54C2×C4×D131—44no—
55C2×C4×D131D411yescharacteristic
56C2×C4×D131D411yescharacteristic
57C2×C4×D131C2311yescharacteristic
58C2×C4×D131C2311yescharacteristic
59#14523496—24no—
60C2×C131⋊C8C2312yes—
61C4⋊Dic131—21no—
62C4⋊Dic131—21no—
63C4⋊Dic131C2311yescharacteristic
64C4⋊Dic131C2312yes—
65Dic131⋊C4—22no—
66Dic131⋊C4—22no—
67Dic131⋊C4—21no—
68C22⋊Dic131—21no—
69C22⋊Dic131—21no—
70(C2×D131)⋊C4—21no—
71(C2×D131)⋊C4—21no—
72C4×Dic131—21no—
73C2×D4×C131—21no—
74C2×D4×C131C2311yescharacteristic
75C22×C4×C131—21no—
76C22×C4×C131C2311yescharacteristic
77C2×C8×C131C2312yes—
78C4⋊C4×C131—21no—
79C4⋊C4×C131C2311yescharacteristic
80C4⋊C4×C131C2312yes—
81C22⋊C4×C131—21no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131—22no—
85C22×D131C2×D411yescharacteristic
86C22×D131C2×D411yescharacteristic
87Dic262—22no—
88C22⋊D131—42no—
89C22⋊D131—42no—
90C2×Dic131—22no—
91C2×Dic131—21no—
92C2×Dic131—21no—
93C2×Dic131—22no—
94C2×Dic131—42no—
95C2×Dic131—42no—
96C2×Dic131C2×D411yescharacteristic
97C2×Dic131C2×D411yescharacteristic
98C4×D131—22no—
99C4×D131—22no—
100C4×D131—22no—
101C4×D131—44no—
102C4×D131—44no—
103C4×D131#4114yes—
104C131⋊C8—22no—
105C23×C131—21no—
106C23×C131C2×D411yescharacteristic
107D4×C131—22no—
108D4×C131—42no—
109C2×C4×C131—22no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—42no—
113C2×C4×C131C2×D411yescharacteristic
114C2×C4×C131C2411yescharacteristic
115C8×C131—22no—
116C22×C131—21no—
117C22×C131—21no—
118C22×C131—41no—
119C22×C131—41no—
120C22×C131—41no—
121C22×C131AGL(1,ℤ/8)12yes—
122C22×C131C22×D411yescharacteristic
123C2×D131—28no—
124C2×D131—21no—
125C2×D131—21no—
126C2×D131—21no—
127C2×D131—21no—
128C2×D131#12312yes—
129C2×D131#13712yes—
130C4×C131—21no—
131C4×C131—42no—
132C4×C131—41no—
133C4×C131C2×#4112yes—
134Dic131—21no—
135Dic131—41no—
136Dic131—42no—
137Dic131—41no—
138Dic131#12312yes—
139C2×C131—22no—
140C2×C131—81no—
141C2×C131C2×#12311yescharacteristic
142C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C131C2×#13711yescharacteristic
144D131—22no—
145D131—22no—
146C131#309211yescharacteristic, complemented, Sylow
147#3092—1311nomaximal, Sylow
148#525—1311no—
149C22×C4⋊C4—1311no—
150#484—1312no—
151#483—1311no—
152#483—1311no—
153#481—1312no—
154#480—1311no—
155#480D13111yescharacteristic, complemented
156#428—1312no—
157#417—1312no—
158#407—1311no—
159C2×#41—1311no—
160C23×C4—2621no—
161C23×C4—1311no—
162C2×M16—1312no—
163#123—2621no—
164C22⋊Q8—1311no—
165C22⋊Q8—1311no—
166#121—1311no—
167#121C2×D13111yescharacteristic
168#120—2621no—
169C4×D4—2621no—
170C4×D4—2621no—
171#117—1311no—
172C2×C4⋊C4—2621no—
173C2×C4⋊C4—2621no—
174C2×C4⋊C4—2621no—
175C2×C4⋊C4—1311no—
176C2×C4⋊C4—1312no—
177C2×C4⋊C4—1312no—
178C2×C4⋊C4C2×D13111yescharacteristic
179#107—1312no—
180#107C2×D13112yescomplemented
181#106—1312no—
182#106—1312no—
183#103—1312no—
184#103—1312no—
185#102—1312no—
186#102C2×D13112yescomplemented
187C22⋊C8—1311no—
188C22⋊C8—1312no—
189C22⋊C8C2×D13111yescharacteristic
190C24—1311no—
191#41—2624no—
192C2×Q8—1311no—
193C2×D4—21no—
194C2×D4—2621no—
195C2×D4—2621no—
196C2×D4C22×D13111yescharacteristic
197C22×C4—21no—
198C22×C4—2624no—
199C22×C4—2622no—
200C22×C4—2622no—
201C22×C4—2621no—
202C22×C4—2621no—
203C22×C4—5244no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4—1311no—
207C22×C4—1311no—
208C22×C4—1311no—
209C22×C4C22×D13111yescharacteristic
210M16—2624no—
211C2×C8—1312no—
212C2×C8C22×D13112yes—
213C4⋊C4—21no—
214C4⋊C4—2622no—
215C4⋊C4—2622no—
216C4⋊C4—2621no—
217C4⋊C4—2621no—
218C4⋊C4—2621no—
219C4⋊C4—1312no—
220C4⋊C4—1311no—
221C4⋊C4C22×D13112yes—
222C4⋊C4C22×D13111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2621no—
225C22⋊C4—2621no—
226C22⋊C4—2621no—
227C22⋊C4—2621no—
228C42—2621no—
229C23—21no—
230C23—2622no—
231C23—2622no—
232C23—2622no—
233C23—1311no—
234C23—1311no—
235C23D4×D13111yescharacteristic
236Q8—2622no—
237D4—22no—
238D4—42no—
239D4—5242no—
240D4—5242no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2622no—
248C2×C4—2622no—
249C2×C4—2621no—
250C2×C4—2621no—
251C2×C4—2622no—
252C2×C4—5244no—
253C2×C4—5244no—
254C2×C4—5242no—
255C2×C4—5242no—
256C2×C4—1311no—
257C2×C4—1311no—
258C2×C4—1314no—
259C2×C4D4×D13111yescharacteristic
260C2×C4C23×D13111yescharacteristic
261C8—22no—
262C8—2622no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2628no—
269C22—2621no—
270C22—2621no—
271C22—2621no—
272C22—2621no—
273C22—1312no—
274C22—1312no—
275C22#1686227012yes—
276C22C2×D4×D13111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—2621no—
281C4—5241no—
282C4—5242no—
283C4—5241no—
284C4—1312no—
285C4#41×D13112yes—
286C2—22no—
287C2—2622no—
288C2—2622no—
289C2—81no—
290C2#123×D13111yescharacteristic
291C2C2×#1686227011yescharacteristic
292C2#137×D13111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#480×C131C2
socleC22×C131C22×D4
prime radical#22553822C2
hypercenter#480D131
hypocenterC131#3092
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D131C2semidirect
3. #22553649C2semidirect
4. #22553643C2semidirect
5. #22553642C2semidirect
6. #22553629C2semidirect
7. #22553623C2semidirect
8. #22553353C2semidirect
9. #22553310C2semidirect
10. #22553274C2semidirect
11. #480×C131C2semidirect
30. #16862179C22semidirect
31. #16862178C22semidirect
32. #16862177C22semidirect
33. #16862171C22semidirect
34. #16862169C22semidirect
35. #16862167C22semidirect
40. #107×C131C22semidirect
41. #102×C131C22semidirect
146. C131#3092semidirect
155. #480D131semidirect
180. #107C2×D131semidirect
186. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538222
C2×C4⋊C4×D1312
#225536492
#225536432
#225536422
#225536292
#225536232
#225533532
#225533102
#225532742
#480×C1312
#3092131