Finite Group Databasebeta

← #36552869

Subgroups — #36552869

Order 22784.

1234 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151676C212yesmaximal
2C2×C4×#13083558C212yesmaximal, complemented
3C4⋊C4×#10623494C212yesmaximal, complemented
4#26020105C211yesmaximal, characteristic, complemented
5#26020036C214yesmaximal, complemented
6#26020014C212yesmaximal, complemented
7C4×#15667923C211yesmaximal, characteristic, complemented
8C42×D4×C89C211yesmaximal, characteristic
9C22×#13083558C2214yes—
10#19151676C4116yescomplemented
11C2×C4⋊#10623494C2212yes—
12C2×C22⋊#10623494C2214yes—
13C4×#13083558—28no—
14C2×#15667923C2212yes—
15C2×#15667923C2211yescharacteristic
16C2×#15667923C2212yes—
17C2×#15667923C2211yescharacteristic
18C2×C4×#10623494C2214yes—
19C2×C4×#10623494C2212yes—
20#19151353C2212yes—
21#19151353C2214yes—
22C2×C4×D4×C89C412yes—
23C2×C4×D4×C89C2211yescharacteristic
24C22×C42×C89C2212yes—
25C4×C4⋊C4×C89C412yes—
26C4×C4⋊C4×C89C2211yescharacteristic
27C4×C22⋊C4×C89C414yes—
28C4×C22⋊C4×C89C2212yes—
29C43×C89C2211yescharacteristic
30C2×#13083558—28no—
31C2×#13083558—28no—
32C2×#13083558C2×C4116yescomplemented
33C22×#10623494C2314yes—
34C22×#10623494C2312yes—
35C22×#10623494C2312yes—
36C4⋊#10623494C2×C418yescomplemented
37C22⋊#10623494C2×C4116yescomplemented
38#15667923—24no—
39#15667923—24no—
40#15667923—22no—
41#15667923D414yescomplemented
42C4×#10623494—28no—
43C4×#10623494C2×C418yescomplemented
44C22×D4×C89C2×C411yescharacteristic
45C23×C4×C89C2×C414yes—
46C23×C4×C89C2312yes—
47C4×D4×C89—28no—
48C4×D4×C89C2×C418yes—
49C2×C4⋊C4×C89C2×C411yescharacteristic
50C2×C4⋊C4×C89C2×C412yes—
51C2×C22⋊C4×C89C2×C414yes—
52C2×C22⋊C4×C89C2×C412yes—
53C2×C42×C89—24no—
54C2×C42×C89—24no—
55C2×C42×C89C2×C412yes—
56C2×C42×C89C2×C414yes—
57C2×C42×C89C2×C412yes—
58C2×C42×C89C2311yescharacteristic
59C2×C42×C89C2312yes—
60C2×C42×C89C2311yescharacteristic
61C2×C42×C89C2311yescharacteristic
62#95×C89C2×C412yes—
63#95×C89C2×C414yes—
64#13083558—232no—
65C2×#10623494—24no—
66C2×#10623494—24no—
67C2×#10623494—22no—
68C2×#10623494—22no—
69C2×#10623494C22×C418yes—
70C2×#10623494C22×C414yes—
71C2×#10623494C22×C414yes—
72C2×#10623494C2×D414yes—
73C2×#10623494#4114yes—
74C24×C89C22×C412yes—
75C2×D4×C89—24no—
76C2×D4×C89C4214yes—
77C22×C4×C89—22no—
78C22×C4×C89—24no—
79C22×C4×C89—22no—
80C22×C4×C89—28no—
81C22×C4×C89—28no—
82C22×C4×C89—44no—
83C22×C4×C89C22×C412yes—
84C22×C4×C89C22×C411yescharacteristic
85C22×C4×C89C22×C412yes—
86C22×C4×C89C22×C418yes—
87C22×C4×C89C22×C412yes—
88C22×C4×C89C22×C414yes—
89C22×C4×C89C22×C411yescharacteristic
90C22×C4×C89C2411yescharacteristic
91C4⋊C4×C89C4214yes—
92C4⋊C4×C89M1618yescomplemented
93C22⋊C4×C89—28no—
94C22⋊C4×C89C4218yes—
95C42×C89—28no—
96C42×C89—24no—
97C42×C89—22no—
98C42×C89—22no—
99C42×C89—22no—
100C42×C89—42no—
101C42×C89C22×C414yes—
102C42×C89C2×D412yes—
103C42×C89#4112yes—
104C23×C89—22no—
105C23×C89—22no—
106C23×C89—24no—
107C23×C89—24no—
108C23×C89C2×C4214yes—
109C23×C89C23×C411yescharacteristic
110D4×C89—28no—
111C2×C4×C89—24no—
112C2×C4×C89—22no—
113C2×C4×C89—22no—
114C2×C4×C89—22no—
115C2×C4×C89—22no—
116C2×C4×C89—22no—
117C2×C4×C89—24no—
118C2×C4×C89—28no—
119C2×C4×C89—28no—
120C2×C4×C89—22no—
121C2×C4×C89—22no—
122C2×C4×C89—22no—
123C2×C4×C89—416no—
124C2×C4×C89C2×C4212yes—
125C2×C4×C89C2×C4214yes—
126C2×C4×C89C2×C4212yes—
127C2×C4×C89C4×D414yes—
128C2×C4×C89C2×M1618yes—
129C2×C4×C89C2×M1614yes—
130C2×C4×C89C23×C412yes—
131C2×C4×C89C22×D411yescharacteristic
132C2×C4×C89C2×#4112yes—
133C2×C4×C89C2×#4111yescharacteristic
134#10623494—28no—
135#10623494—24no—
136#10623494C4×D418yescomplemented
137C22×C89—24no—
138C22×C89—24no—
139C22×C89—24no—
140C22×C89—44no—
141C22×C89—44no—
142C22×C89C22×C4211yescharacteristic
143C22×C89C2×C4×D411yescharacteristic
144C22×C89C4×#4111yescharacteristic
145C22×C89C22×M1612yes—
146C22×C89#56812yes—
147C4×C89—22no—
148C4×C89—24no—
149C4×C89—28no—
150C4×C89—24no—
151C4×C89—42no—
152C4×C89C4×M1614yescomplemented
153C4×C89C2×C4×D412yes—
154C4×C89C4×#4112yes—
155#20494—891nomaximal, Sylow
156C2×C89—44no—
157C2×C89C42×D411yescharacteristic
158C2×C89C2×C4×M1611yescharacteristic
159C2×C89#287511yescharacteristic
160C2×C89D4×M1612yes—
161C2×C89#297112yes—
162C2×#436—892no—
163C2×C4×M16—892no—
164C42×D4D8911yescharacteristic
165C8×C4⋊C4—892no—
166#1767—891no—
167#1753—894no—
168#1749—892no—
169C4×#97—891no—
170C89#2049411yescharacteristic, complemented, Sylow
171C22×M16—894no—
172C2×C4×D4Dic8912yes—
173C2×C4×D4C2×D8911yescharacteristic
174C22×C42C2×D8912yes—
175#436—8916no—
176C2×C4⋊C8—892no—
177C2×C22⋊C8—894no—
178C4×M16—1788no—
179C2×#97—891no—
180C2×#97—892no—
181C2×#97—891no—
182C2×#97—892no—
183C2×C4×C8—892no—
184C2×C4×C8—894no—
185C4×C4⋊C4Dic8912yes—
186C4×C4⋊C4C2×D8911yescharacteristic
187C4×C22⋊C4Dic8914yes—
188C4×C22⋊C4C2×D8912yes—
189C43C2×D8911yescharacteristic
190#336—894no—
191#336—892no—
192C22×D4C2×Dic8911yescharacteristic
193C23×C4C2×Dic8914yes—
194C23×C4C22×D8912yes—
195C2×M16—8916no—
196C2×M16—1788no—
197C2×M16—1788no—
198C22×C8—892no—
199C22×C8—894no—
200C22×C8—892no—
201C4×D4—28no—
202C4×D4C4×D8918yes—
203C2×C4⋊C4C2×Dic8911yescharacteristic
204C2×C4⋊C4C2×Dic8912yes—
205C2×C22⋊C4C2×Dic8914yes—
206C2×C22⋊C4C2×Dic8912yes—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42C2×Dic8912yes—
210C2×C42C2×Dic8914yes—
211C2×C42C2×Dic8912yes—
212C2×C42C22×D8911yescharacteristic
213C2×C42C22×D8912yes—
214C2×C42C22×D8911yescharacteristic
215C2×C42C22×D8911yescharacteristic
216C4⋊C8—898no—
217C22⋊C8—8916no—
218#97—894no—
219#97—1784no—
220#97—1784no—
221#97—1782no—
222C4×C8—898no—
223C4×C8—1788no—
224#95C2×Dic8912yes—
225#95C2×Dic8914yes—
226C24C22×Dic8912yes—
227C2×D4—24no—
228C2×D4C4×Dic8914yes—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—44no—
235C22×C4C2×C4×D8918yes—
236C22×C4C22×Dic8912yes—
237C22×C4C22×Dic8912yes—
238C22×C4C22×Dic8914yes—
239C22×C4C22×Dic8911yescharacteristic
240C22×C4C22×Dic8912yes—
241C22×C4C22×Dic8911yescharacteristic
242C22×C4C23×D8911yescharacteristic
243M16—17832no—
244C2×C8—894no—
245C2×C8—894no—
246C2×C8—894no—
247C2×C8—894no—
248C2×C8—898no—
249C2×C8—1784no—
250C2×C8—1782no—
251C2×C8—1784no—
252C2×C8—1782no—
253C4⋊C4C4×Dic8914yes—
254C4⋊C4#1308355818yescomplemented
255C22⋊C4—28no—
256C22⋊C4C4×Dic8918yes—
257C42—28no—
258C42—24no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42C2×C4×D8914yes—
264C42D4×D8912yes—
265C42#1308357312yes—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23C2×C4×Dic8914yes—
271C23C23×Dic8911yescharacteristic
272D4—28no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—416no—
286C2×C4C2×C4×Dic8912yes—
287C2×C4C2×C4×Dic8914yes—
288C2×C4C2×C4×Dic8912yes—
289C2×C4D4×Dic8914yes—
290C2×C4C2×#1308355818yes—
291C2×C4C2×#1308355814yes—
292C2×C4C22×C4×D8912yes—
293C2×C4C2×D4×D8911yescharacteristic
294C2×C4#41×D8912yes—
295C2×C4C2×#1308357311yescharacteristic
296C8—898no—
297C8—1788no—
298C8—1784no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—44no—
303C22—44no—
304C22C22×C4×Dic8911yescharacteristic
305C22C2×D4×Dic8911yescharacteristic
306C22#41×Dic8911yescharacteristic
307C22C22×#1308355812yes—
308C22#1915236512yes—
309C4—22no—
310C4—24no—
311C4—28no—
312C4—24no—
313C4—42no—
314C4C4×#1308355814yescomplemented
315C4C4×D4×D8912yes—
316C4C4×#1308357312yes—
317C2—44no—
318C2C4×D4×Dic8911yescharacteristic
319C2C2×C4×#1308355811yescharacteristic
320C2#2602868011yescharacteristic
321C2D4×#1308355812yes—
322C2#2602927212yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutatorC22×C89C22×C42
FittingC42×D4×C89C2
socleC23×C89C23×C4
prime radicalC22×D4×C89C2×C4
cosocleC22×C4×C89C24
hypercenterC42×D4D89
hypocenterC89#20494
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13083558C2semidirect
3. C4⋊C4×#10623494C2semidirect
4. #26020105C2semidirect
5. #26020036C2semidirect
6. #26020014C2semidirect
7. C4×#15667923C2semidirect
10. #19151676C4semidirect
32. C2×#13083558C2×C4semidirect
36. C4⋊#10623494C2×C4semidirect
37. C22⋊#10623494C2×C4semidirect
41. #15667923D4semidirect
43. C4×#10623494C2×C4semidirect
92. C4⋊C4×C89M16semidirect
136. #10623494C4×D4semidirect
152. C4×C89C4×M16semidirect
170. C89#20494semidirect
254. C4⋊C4#13083558semidirect
314. C4C4×#13083558semidirect

Maximal subgroups

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

subgroupindex
C2×#191516762
C2×C4×#130835582
C4⋊C4×#106234942
#260201052
#260200362
#260200142
C4×#156679232
C42×D4×C892
#2049489