Finite Group Databasebeta

← #36552654

Subgroups — #36552654

Order 22784.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151659C211yesmaximal, characteristic
2#26022750C211yesmaximal, characteristic
3#26020124C212yesmaximal, complemented
4C8×C4⋊Dic89C212yesmaximal, complemented
5#26020033C212yesmaximal
6C8×C22⋊Dic89C212yesmaximal
7#26020005C211yesmaximal, characteristic, complemented
8#26020000C211yesmaximal, characteristic, complemented
9#26019938C211yesmaximal, characteristic, complemented
10C4×C8×Dic89C211yesmaximal, characteristic, complemented
11C2×#405×C89C211yesmaximal, characteristic
12#19151659C418yescomplemented
13C2×C4⋊#10623494C2212yes—
14C2×C22⋊#10623494C2212yes—
15C2×#15667923C2211yescharacteristic
16C2×#15667922C2212yes—
17C2×C4×#10623494C2211yescharacteristic
18C2×C8×Dic89C2212yes—
19C2×C8×Dic89C2212yes—
20C2×C8×Dic89C2212yes—
21#19151491C414yes—
22#19151484C414yes—
23C4×C4⋊Dic89C2212yes—
24C4×C22⋊Dic89C2212yes—
25#19151444C2211yescharacteristic
26C42×Dic89C2211yescharacteristic
27#19151352C2212yes—
28#19151352C2212yes—
29#19151352C2212yes—
30#19151352C2212yes—
31C22×M16×C89C2211yescharacteristic
32C23×C8×C89C2211yescharacteristic
33C2×#117×C89C2211yescharacteristic
34#405×C89C418yescomplemented
35C2×#97×C89C2211yescharacteristic
36C2×#97×C89C2211yescharacteristic
37C2×C4×C8×C89C2211yescharacteristic
38C2×C4×C8×C89C2211yescharacteristic
39C22×#10623494C2312yes—
40C22×#10623494C2312yes—
41C2×C4⋊Dic89C2×C414yes—
42C2×C22⋊Dic89C2×C414yes—
43C2×C4×Dic89C2×C414yes—
44C2×C4×Dic89C2312yes—
45C2×C4×Dic89C2312yes—
46C4⋊#10623494C2×C418yes—
47C22⋊#10623494C2×C418yes—
48#15667923C2×C414yes—
49#15667922—24no—
50C4×#10623494C2×C414yes—
51C8×Dic89—24no—
52C8×Dic89—24no—
53C8×Dic89—24no—
54#15667919C2×C414yes—
55#15667919C2×C414yes—
56#15667919C2×C414yes—
57C23×C4×C89C2311yescharacteristic
58C2×M16×C89—24no—
59C2×M16×C89C2×C414yes—
60C22×C8×C89—24no—
61C22×C8×C89C2×C414yes—
62C22×C8×C89C2311yescharacteristic
63C22×C8×C89C2311yescharacteristic
64C22×C8×C89C2311yescharacteristic
65C22×C8×C89C2311yescharacteristic
66#117×C89C2×C414yes—
67#117×C89C2×C414yes—
68C2×C4⋊C4×C89C2×C412yes—
69C2×C22⋊C4×C89C2×C412yes—
70C2×C42×C89C2311yescharacteristic
71C2×C42×C89C2311yescharacteristic
72#97×C89C2×C414yes—
73#97×C89C2×C414yes—
74C4×C8×C89C2×C414yes—
75C4×C8×C89C2×C414yes—
76C22×Dic89C22×C414yes—
77C22×Dic89C22×C414yes—
78C2×#10623494—22no—
79C2×#10623494—22no—
80C2×#10623494—22no—
81C2×#10623494—22no—
82C2×#10623494C22×C414yes—
83C2×#10623494C22×C414yes—
84C4⋊Dic89—28no—
85C22⋊Dic89—28no—
86C4×Dic89—22no—
87C4×Dic89—22no—
88C4×Dic89—22no—
89C4×Dic89—22no—
90C4×Dic89—28no—
91C24×C89C22×C411yescharacteristic
92C22×C4×C89—22no—
93C22×C4×C89—22no—
94C22×C4×C89C22×C412yes—
95C22×C4×C89C22×C412yes—
96C22×C4×C89C22×C411yescharacteristic
97C22×C4×C89C22×C412yes—
98C22×C4×C89C22×C412yes—
99C22×C4×C89C2411yescharacteristic
100M16×C89—28no—
101C2×C8×C89—22no—
102C2×C8×C89—22no—
103C2×C8×C89—28no—
104C2×C8×C89C22×C412yes—
105C2×C8×C89C22×C412yes—
106C2×C8×C89C22×C412yes—
107C2×C8×C89C22×C412yes—
108C2×C8×C89#4114yes—
109C2×C8×C89#4114yes—
110C4⋊C4×C89C4218yes—
111C22⋊C4×C89C4218yes—
112C42×C89C22×C412yes—
113C42×C89C22×C412yes—
114C42×C89C22×C412yes—
115C42×C89C22×C412yes—
116C23×C89—22no—
117C23×C89—24no—
118C23×C89C2×C4212yes—
119C23×C89C23×C411yescharacteristic
120C2×C4×C89—22no—
121C2×C4×C89—24no—
122C2×C4×C89—24no—
123C2×C4×C89—22no—
124C2×C4×C89—22no—
125C2×C4×C89—24no—
126C2×C4×C89C2×C4214yes—
127C2×C4×C89C2×C4214yes—
128C2×C4×C89C2×C4212yes—
129C2×C4×C89C23×C411yescharacteristic
130C2×C4×C89C23×C411yescharacteristic
131C2×C4×C89C2×#4112yes—
132C2×C4×C89C2×#4112yes—
133C2×Dic89—24no—
134C2×Dic89—24no—
135C2×Dic89—24no—
136C2×Dic89—24no—
137C2×Dic89—24no—
138C2×Dic89#13118yes—
139C8×C89—22no—
140C8×C89—22no—
141C8×C89#11714yes—
142C8×C89#11714yes—
143#10623494—24no—
144#10623494—24no—
145C22×C89—24no—
146C22×C89—22no—
147C22×C89—44no—
148C22×C89C22×C4211yescharacteristic
149C22×C89C4×#4112yes—
150C22×C89C2×#13114yes—
151C4×C89—24no—
152C4×C89—24no—
153C4×C89—22no—
154C4×C89C2×#11711yescharacteristic
155C4×C89C2×#11711yescharacteristic
156C4×C89C4×#4112yes—
157Dic89—28no—
158Dic89—44no—
159#20474—891nomaximal, Sylow
160C2×C89—42no—
161C2×C89C4×#11711yescharacteristic
162C2×C89C4×#13112yes—
163C2×C89#296614yes—
164C2×#433—891no—
165C2×#405D8911yescharacteristic
166#2298—891no—
167#1771—892no—
168C8×C4⋊C4—892no—
169#1753—892no—
170C8×C22⋊C4—892no—
171#1746—891no—
172#1745—891no—
173#1726—891no—
174C42×C8—891no—
175C89#2047411yescharacteristic, complemented, Sylow
176C22×M16C2×D8911yescharacteristic
177C23×C8C2×D8911yescharacteristic
178C2×#117C2×D8911yescharacteristic
179#433—898no—
180C2×C4⋊C8—892no—
181C2×C22⋊C8—892no—
182#405Dic8918yescomplemented
183C2×#97—891no—
184C2×#97—892no—
185C2×#97C2×D8911yescharacteristic
186C2×#97C2×D8911yescharacteristic
187C2×C4×C8—891no—
188C2×C4×C8—892no—
189C2×C4×C8—892no—
190C2×C4×C8—892no—
191C2×C4×C8C2×D8911yescharacteristic
192C2×C4×C8C2×D8911yescharacteristic
193#388—894no—
194#387—894no—
195C4×C4⋊C4—892no—
196C4×C22⋊C4—892no—
197#376—891no—
198C43—891no—
199#336—892no—
200#336—892no—
201#336—892no—
202#336—892no—
203C23×C4C22×D8911yescharacteristic
204C2×M16—24no—
205C2×M16C2×Dic8914yes—
206C22×C8—24no—
207C22×C8—892no—
208C22×C8—892no—
209C22×C8C2×Dic8914yes—
210C22×C8C22×D8911yescharacteristic
211C22×C8C22×D8911yescharacteristic
212C22×C8C22×D8911yescharacteristic
213C22×C8C22×D8911yescharacteristic
214#117C4×D8914yes—
215#117C2×Dic8914yes—
216C2×C4⋊C4—894no—
217C2×C4⋊C4C4×D8912yes—
218C2×C22⋊C4—894no—
219C2×C22⋊C4C4×D8912yes—
220C2×C42—894no—
221C2×C42—892no—
222C2×C42—892no—
223C2×C42C22×D8911yescharacteristic
224C2×C42C22×D8911yescharacteristic
225C4⋊C8—898no—
226C22⋊C8—898no—
227#97—894no—
228#97—1784no—
229#97C2×Dic8914yes—
230#97C2×Dic8914yes—
231C4×C8—894no—
232C4×C8—1784no—
233C4×C8—1784no—
234C4×C8—1784no—
235C4×C8C2×Dic8914yes—
236C4×C8C2×Dic8914yes—
237#95—894no—
238#95—894no—
239#95—894no—
240C24C2×C4×D8911yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—894no—
244C22×C4—894no—
245C22×C4C2×C4×D8912yes—
246C22×C4C2×C4×D8911yescharacteristic
247C22×C4C2×C4×D8912yes—
248C22×C4C2×C4×D8912yes—
249C22×C4C22×Dic8912yes—
250C22×C4C23×D8911yescharacteristic
251M16—28no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—28no—
255C2×C8—894no—
256C2×C8—894no—
257C2×C8—1782no—
258C2×C8—1782no—
259C2×C8—1782no—
260C2×C8—1782no—
261C2×C8C22×Dic8912yes—
262C2×C8C22×Dic8912yes—
263C2×C8C22×Dic8912yes—
264C2×C8C22×Dic8912yes—
265C2×C8#1308357414yes—
266C2×C8#1308357414yes—
267C4⋊C4—1788no—
268C4⋊C4C4×Dic8918yes—
269C22⋊C4—1788no—
270C22⋊C4C4×Dic8918yes—
271C42—1788no—
272C42—1782no—
273C42—1782no—
274C42—1782no—
275C42—1782no—
276C42C2×C4×D8912yes—
277C42C2×C4×D8912yes—
278C42C22×Dic8912yes—
279C42C22×Dic8912yes—
280C23—22no—
281C23—24no—
282C23C2×C4×Dic8912yes—
283C23C22×C4×D8911yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—898no—
291C2×C4—1784no—
292C2×C4—1784no—
293C2×C4—1784no—
294C2×C4—1784no—
295C2×C4—1784no—
296C2×C4C2×C4×Dic8914yes—
297C2×C4C2×C4×Dic8914yes—
298C2×C4C2×C4×Dic8912yes—
299C2×C4C22×C4×D8911yescharacteristic
300C2×C4C23×Dic8911yescharacteristic
301C2×C4C2×#1308357412yes—
302C2×C4C2×#1308357412yes—
303C8—22no—
304C8—22no—
305C8—1784no—
306C8—1784no—
307C8#1566797114yes—
308C8#1566797114yes—
309C22—24no—
310C22—22no—
311C22—44no—
312C22C22×C4×Dic8911yescharacteristic
313C22C4×#1308357412yes—
314C22#131×D8914yes—
315C4—24no—
316C4—24no—
317C4—22no—
318C4—3564no—
319C4—1788no—
320C4C2×#1566797111yescharacteristic
321C4C2×#1566797111yescharacteristic
322C4C4×#1308357412yes—
323C2—42no—
324C2C4×#1566797111yescharacteristic
325C2#131×Dic8912yes—
326C2#2602922314yes—
subgroupG/H
centerC22×C8C22×D89
commutatorC22×C89C22×C42
FrattiniC22×C4C23×D89
FittingC2×#405×C89C2
socleC23×C89C23×C4
prime radicalC24×C89C22×C4
cosocleC22×C4×C89C24
hypercenterC2×#405D89
hypocenterC89#20474
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22×C8 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #26020124C2semidirect
4. C8×C4⋊Dic89C2semidirect
7. #26020005C2semidirect
8. #26020000C2semidirect
9. #26019938C2semidirect
10. C4×C8×Dic89C2semidirect
12. #19151659C4semidirect
34. #405×C89C4semidirect
175. C89#20474semidirect
182. #405Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191516592
#260227502
#260201242
C8×C4⋊Dic892
#260200332
C8×C22⋊Dic892
#260200052
#260200002
#260199382
C4×C8×Dic892
C2×#405×C892
#2047489