Finite Group Databasebeta

← #36524206

Subgroups — #36524206

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029219C212yesmaximal, complemented
2#26019166C212yesmaximal, complemented
3#26019161C212yesmaximal, complemented
4#26019155C214yesmaximal, complemented
5#26019000C211yesmaximal, characteristic, complemented
6#26018634C212yesmaximal, complemented
7C2×#19151319C211yesmaximal, characteristic, complemented
8C4×#135×C89C211yesmaximal, characteristic
9#19151729C2214yes—
10#19151676C2214yes—
11#19151670—24no—
12#19151651—22no—
13#19151651C2212yes—
14C4×#13083558—22no—
15C2×C4×#10623494C2211yescharacteristic
16C2×C4×#10623494C2211yescharacteristic
17#19151351C2212yes—
18#19151349C2212yes—
19Q8⋊#10623494C2214yes—
20D4⋊#10623494C2214yes—
21#19151319C2212yes—
22#19151319C2212yes—
23C2×#135×C89C411yescharacteristic
24C4×#41×C89C2212yes—
25C4×Q16×C89C411yescharacteristic
26C4×Q16×C89C2211yescharacteristic
27C4×SD16×C89C412yes—
28C4×SD16×C89C2212yes—
29C4×D8×C89C411yescharacteristic
30C4×D8×C89C2211yescharacteristic
31#427×C89C411yescharacteristic
32#416×C89C412yes—
33C2×C4×C8×C89C2211yescharacteristic
34C2×#13083558—24no—
35C22×#10623494—21no—
36C22×#10623494D412yes—
37C4⋊#10623494—24no—
38C4⋊#10623494C2314yes—
39C22⋊#10623494—24no—
40C22⋊#10623494—22no—
41#15667923—24no—
42C4×#10623494—22no—
43C4×#10623494—21no—
44C4×#10623494D412yes—
45C4×#10623494C2312yes—
46C4×#10623494C2312yes—
47C2×#41×C89C2×C412yes—
48#135×C89—24no—
49C2×Q16×C89C2×C411yescharacteristic
50C2×SD16×C89C2×C412yes—
51C2×D8×C89C2×C411yescharacteristic
52C22×C8×C89C2×C411yescharacteristic
53C22×C8×C89C2×C411yescharacteristic
54C4×Q8×C89C2×C412yes—
55C4×Q8×C89C2312yes—
56C4×D4×C89—22no—
57C4×D4×C89—22no—
58C4×D4×C89C2×C412yes—
59C4×D4×C89C2312yes—
60#117×C89—22no—
61#117×C89C2×C412yes—
62C2×C42×C89—22no—
63C2×C42×C89C2311yescharacteristic
64#107×C89C2×C411yescharacteristic
65#107×C89C2×C411yescharacteristic
66#106×C89C2×C412yes—
67#103×C89C2×C412yes—
68#103×C89C2×C412yes—
69#102×C89C2×C412yes—
70#102×C89C2×C412yes—
71C4×C8×C89C2×C411yescharacteristic
72C4×C8×C89C2×C411yescharacteristic
73C4×C8×C89C2311yescharacteristic
74C4×C8×C89C2311yescharacteristic
75#13083558—48no—
76C2×#10623494—24no—
77C2×#10623494—22no—
78C2×#10623494—42no—
79C2×#10623494—42no—
80C2×#10623494C2×D414yes—
81#41×C89—24no—
82#41×C89—24no—
83C2×Q8×C89C22×C412yes—
84C2×D4×C89—22no—
85C2×D4×C89C22×C412yes—
86C22×C4×C89—22no—
87C22×C4×C89—22no—
88C22×C4×C89—22no—
89C22×C4×C89C22×C411yescharacteristic
90C22×C4×C89C2×D411yescharacteristic
91C22×C4×C89#4111yescharacteristic
92Q16×C89—22no—
93SD16×C89—24no—
94D8×C89—22no—
95C2×C8×C89—22no—
96C2×C8×C89M1614yescomplemented
97C2×C8×C89C22×C411yescharacteristic
98C2×C8×C89C22×C411yescharacteristic
99C2×C8×C89C22×C411yescharacteristic
100C2×C8×C89C22×C411yescharacteristic
101C4⋊C4×C89—22no—
102C4⋊C4×C89—22no—
103C4⋊C4×C89C22×C412yes—
104C4⋊C4×C89C22×C412yes—
105C22⋊C4×C89—22no—
106C22⋊C4×C89—22no—
107C22⋊C4×C89—22no—
108C42×C89—22no—
109C42×C89—22no—
110C42×C89—22no—
111C42×C89C22×C411yescharacteristic
112C42×C89C2×D411yescharacteristic
113C42×C89#4111yescharacteristic
114C42×C89C2411yescharacteristic
115C23×C89—22no—
116C23×C89C4×D411yescharacteristic
117Q8×C89—22no—
118Q8×C89—22no—
119D4×C89—22no—
120D4×C89—22no—
121D4×C89—44no—
122C2×C4×C89—22no—
123C2×C4×C89—22no—
124C2×C4×C89—22no—
125C2×C4×C89—22no—
126C2×C4×C89—22no—
127C2×C4×C89—44no—
128C2×C4×C89—44no—
129C2×C4×C89—44no—
130C2×C4×C89C4×D411yescharacteristic
131C2×C4×C89C4×D411yescharacteristic
132C2×C4×C89C4×D411yescharacteristic
133C2×C4×C89C2×M1612yes—
134C2×C4×C89#13112yes—
135C2×C4×C89C23×C411yescharacteristic
136C2×C4×C89C22×D411yescharacteristic
137C2×C4×C89C2×#4111yescharacteristic
138C8×C89—21no—
139C8×C89—21no—
140C8×C89C2×M1612yes—
141C8×C89C2×M1612yes—
142#10623494—24no—
143#10623494—42no—
144#10623494—44no—
145C22×C89—21no—
146C22×C89—21no—
147C22×C89—42no—
148C22×C89—42no—
149C22×C89—42no—
150C22×C89#43612yes—
151C22×C89C2×C4×D411yescharacteristic
152C4×C89—21no—
153C4×C89—21no—
154C4×C89—42no—
155C4×C89—42no—
156C4×C89—42no—
157C4×C89#43612yes—
158C4×C89#44412yes—
159C4×C89C22×M1611yescharacteristic
160C4×C89C2×#13111yescharacteristic
161C4×C89#57512yes—
162#17890—891nomaximal, Sylow
163C2×C89—22no—
164C2×C89—82no—
165C2×C89C2×#43611yescharacteristic
166C2×C89#294311yescharacteristic
167C2×C89C2×#44411yescharacteristic
168#2965—892no—
169C4×#135D8911yescharacteristic
170#1594—892no—
171#1593—892no—
172#1592—894no—
173#1564—891no—
174#1476—892no—
175C2×#322—891no—
176C89#1789011yescharacteristic, complemented, Sylow
177C2×#135Dic8911yescharacteristic
178C4×#41C2×D8912yes—
179#446—894no—
180C4×Q16Dic8911yescharacteristic
181C4×Q16C2×D8911yescharacteristic
182C4×SD16Dic8912yes—
183C4×SD16C2×D8912yes—
184C4×D8Dic8911yescharacteristic
185C4×D8C2×D8911yescharacteristic
186#436—894no—
187#435—1784no—
188#431—892no—
189#431—1782no—
190#427Dic8911yescharacteristic
191#416Dic8912yes—
192C4×M16—1782no—
193C2×C4×C8—891no—
194C2×C4×C8—891no—
195C2×C4×C8C2×D8911yescharacteristic
196#335—892no—
197#334—892no—
198Q8⋊C8—894no—
199D4⋊C8—894no—
200#322—892no—
201#322—892no—
202C2×#41C2×Dic8912yes—
203#135—24no—
204C2×Q16C2×Dic8911yescharacteristic
205C2×SD16C2×Dic8912yes—
206C2×D8C2×Dic8911yescharacteristic
207C2×M16—1784no—
208C22×C8—892no—
209C22×C8—1781no—
210C22×C8C2×Dic8911yescharacteristic
211C22×C8C2×Dic8911yescharacteristic
212C4×Q8C2×Dic8912yes—
213C4×Q8C22×D8912yes—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4C2×Dic8912yes—
217C4×D4C22×D8912yes—
218#117—22no—
219#117C2×Dic8912yes—
220C2×C42—22no—
221C2×C42C22×D8911yescharacteristic
222#107C2×Dic8911yescharacteristic
223#107C2×Dic8911yescharacteristic
224#106C2×Dic8912yes—
225C4⋊C8—894no—
226C4⋊C8—1784no—
227#103C2×Dic8912yes—
228#103C2×Dic8912yes—
229#102C2×Dic8912yes—
230#102C2×Dic8912yes—
231C22⋊C8—1784no—
232C22⋊C8—1782no—
233#97—1784no—
234C4×C8—892no—
235C4×C8—892no—
236C4×C8—892no—
237C4×C8—1782no—
238C4×C8—1781no—
239C4×C8C2×Dic8911yescharacteristic
240C4×C8C2×Dic8911yescharacteristic
241C4×C8C22×D8911yescharacteristic
242C4×C8C22×D8911yescharacteristic
243#41—24no—
244#41—24no—
245C2×Q8C22×Dic8912yes—
246C2×D4—22no—
247C2×D4C22×Dic8912yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4C22×Dic8911yescharacteristic
252C22×C4D4×D8911yescharacteristic
253C22×C4#1308357311yescharacteristic
254Q16—22no—
255SD16—24no—
256D8—22no—
257M16—3568no—
258C2×C8—22no—
259C2×C8—894no—
260C2×C8—3562no—
261C2×C8—3562no—
262C2×C8—1784no—
263C2×C8—1782no—
264C2×C8#1308355814yescomplemented
265C2×C8C22×Dic8911yescharacteristic
266C2×C8C22×Dic8911yescharacteristic
267C2×C8C22×Dic8911yescharacteristic
268C2×C8C22×Dic8911yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4C22×Dic8912yes—
272C4⋊C4C22×Dic8912yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42C22×Dic8911yescharacteristic
280C42D4×D8911yescharacteristic
281C42#1308357311yescharacteristic
282C42C23×D8911yescharacteristic
283C23—22no—
284C23D4×Dic8911yescharacteristic
285Q8—22no—
286Q8—22no—
287D4—22no—
288D4—22no—
289D4—44no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4D4×Dic8911yescharacteristic
299C2×C4D4×Dic8911yescharacteristic
300C2×C4D4×Dic8911yescharacteristic
301C2×C4C2×#1308355812yes—
302C2×C4#1566801912yes—
303C2×C4C23×Dic8911yescharacteristic
304C2×C4C2×D4×D8911yescharacteristic
305C2×C4C2×#1308357311yescharacteristic
306C8—21no—
307C8—21no—
308C8—3562no—
309C8—3564no—
310C8—1784no—
311C8C2×#1308355812yes—
312C8C2×#1308355812yes—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22#1915167612yes—
319C22C2×D4×Dic8911yescharacteristic
320C4—21no—
321C4—21no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4#1915167612yes—
326C4#1915172012yes—
327C4C22×#1308355811yescharacteristic
328C4C2×#1566801911yescharacteristic
329C4#1915239512yes—
330C2—22no—
331C2—82no—
332C2C2×#1915167611yescharacteristic
333C2#2602905311yescharacteristic
334C2C2×#1915172011yescharacteristic
subgroupG/H
centerC42D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC4×#135×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×#135×C89C4
cosocleC42×C89C24
hypercenterC4×#135D89
hypocenterC89#17890
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #26029219C2semidirect
2. #26019166C2semidirect
3. #26019161C2semidirect
4. #26019155C2semidirect
5. #26019000C2semidirect
6. #26018634C2semidirect
7. C2×#19151319C2semidirect
96. C2×C8×C89M16semidirect
176. C89#17890semidirect
264. C2×C8#13083558semidirect

Maximal subgroups

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

subgroupindex
#260292192
#260191662
#260191612
#260191552
#260190002
#260186342
C2×#191513192
C4×#135×C892
#1789089