Finite Group Databasebeta

← #36524205

Subgroups — #36524205

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#26029213C212yesmaximal, complemented
2C4×#15668034C211yesmaximal, characteristic
3#26019165C212yesmaximal, complemented
4#26019160C212yesmaximal, complemented
5#26019153C214yesmaximal, complemented
6#26018632C212yesmaximal, complemented
7C2×#19151319C211yesmaximal, characteristic, complemented
8C89×#1564C211yesmaximal, characteristic, complemented
9C2×#15668034C411yescharacteristic
10C4×#13083574C2212yes—
11#19151725C2214yes—
12C4×Dic356C411yescharacteristic
13C4×Dic356C2211yescharacteristic
14C4×#13083563C412yes—
15C4×#13083563C2212yes—
16C4×D712C411yescharacteristic
17C4×D712C2211yescharacteristic
18#19151675C2214yes—
19#19151666—24no—
20#19151649—22no—
21#19151636C411yescharacteristic
22#19151595C412yes—
23C4×#13083557—22no—
24C2×C4×#10623494C2211yescharacteristic
25C2×C4×#10623494C2211yescharacteristic
26#19151328C2214yes—
27#19151325C2214yes—
28#19151319C2212yes—
29#19151319C2212yes—
30#431×C89C2212yes—
31C2×C4×C8×C89C2211yescharacteristic
32#335×C89C2212yes—
33#334×C89C2212yes—
34C2×#13083574C2×C412yes—
35#15668034—24no—
36C2×Dic356C2×C411yescharacteristic
37C2×#13083563C2×C412yes—
38C2×D712C2×C411yescharacteristic
39C2×#13083557—24no—
40C22×#10623494—21no—
41C22×#10623494D412yes—
42C4×Dic178C2×C412yes—
43C4×Dic178C2312yes—
44C4×D356C2×C412yes—
45C4×D356C2312yes—
46C4×C22⋊D89—22no—
47C4×C22⋊D89—22no—
48#15667971C2×C412yes—
49#15667969—22no—
50C42×D89—22no—
51#15667946C2×C411yescharacteristic
52#15667946C2×C411yescharacteristic
53#15667944C2×C412yes—
54#15667941—24no—
55#15667937C2×C412yes—
56#15667937C2×C412yes—
57#15667934C2×C412yes—
58#15667934C2×C412yes—
59#15667924—24no—
60#15667922—24no—
61C4×#10623494—21no—
62C4×#10623494D412yes—
63C4×#10623494C2312yes—
64C4×#10623494C2312yes—
65C22×C8×C89C2×C411yescharacteristic
66C22×C8×C89C2×C411yescharacteristic
67C2×C42×C89C2311yescharacteristic
68C4⋊C8×C89C2314yes—
69C22⋊C8×C89—22no—
70C4×C8×C89—22no—
71C4×C8×C89C2×C411yescharacteristic
72C4×C8×C89C2×C411yescharacteristic
73C4×C8×C89C2311yescharacteristic
74C4×C8×C89C2311yescharacteristic
75#13083574—24no—
76#13083574—24no—
77C2×Dic178C22×C412yes—
78C2×D356C22×C412yes—
79C2×C22⋊D89—22no—
80C2×C4×D89—22no—
81C2×C4×D89—22no—
82C2×C4×D89—22no—
83Dic356—22no—
84#13083563—24no—
85D712—22no—
86#13083557—48no—
87C2×#10623494—22no—
88C2×#10623494—42no—
89C2×#10623494—42no—
90C2×#10623494C2×D414yes—
91C4⋊Dic89C22×C412yes—
92C4⋊Dic89C22×C412yes—
93#13083553—22no—
94#13083553—22no—
95C22⋊Dic89—22no—
96#13083551—22no—
97#13083551—22no—
98C4×Dic89—22no—
99C4×Dic89—22no—
100C4×Dic89—22no—
101C22×C4×C89C22×C411yescharacteristic
102C22×C4×C89C2×D411yescharacteristic
103C22×C4×C89#4111yescharacteristic
104C2×C8×C89—22no—
105C2×C8×C89—24no—
106C2×C8×C89M1614yescomplemented
107C2×C8×C89C22×C411yescharacteristic
108C2×C8×C89C22×C411yescharacteristic
109C2×C8×C89C22×C411yescharacteristic
110C2×C8×C89C22×C411yescharacteristic
111C42×C89C22×C411yescharacteristic
112C42×C89C2×D411yescharacteristic
113C42×C89#4111yescharacteristic
114C42×C89C2411yescharacteristic
115C23×C89C4×D411yescharacteristic
116C22×D89—22no—
117C2×C4×C89—22no—
118C2×C4×C89—22no—
119C2×C4×C89C4×D411yescharacteristic
120C2×C4×C89C4×D411yescharacteristic
121C2×C4×C89C4×D411yescharacteristic
122C2×C4×C89C2×M1612yes—
123C2×C4×C89#13112yes—
124C2×C4×C89C23×C411yescharacteristic
125C2×C4×C89C22×D411yescharacteristic
126C2×C4×C89C2×#4111yescharacteristic
127C22⋊D89—44no—
128C2×Dic89—22no—
129C2×Dic89—22no—
130C2×Dic89—22no—
131D356—22no—
132D356—22no—
133C4×D89—44no—
134C4×D89—44no—
135C4×D89—44no—
136Dic178—22no—
137Dic178—22no—
138C8×C89—21no—
139C8×C89—21no—
140C8×C89—44no—
141C8×C89C2×M1612yes—
142C8×C89C2×M1612yes—
143#10623494—24no—
144#10623494—42no—
145C22×C89—21no—
146C22×C89—21no—
147C22×C89#43612yes—
148C22×C89C2×C4×D411yescharacteristic
149C2×D89—42no—
150C2×D89—42no—
151C2×D89—42no—
152C4×C89—21no—
153C4×C89—21no—
154C4×C89#43612yes—
155C4×C89#44412yes—
156C4×C89C22×M1611yescharacteristic
157C4×C89C2×#13111yescharacteristic
158C4×C89#57512yes—
159Dic89—42no—
160Dic89—42no—
161Dic89—42no—
162#17890—891nomaximal, Sylow
163C2×C89—22no—
164C2×C89C2×#43611yescharacteristic
165C2×C89#294311yescharacteristic
166C2×C89C2×#44411yescharacteristic
167D89—82no—
168#2965—892no—
169C4×#135—891no—
170#1594—892no—
171#1593—892no—
172#1592—894no—
173#1564D8911yescharacteristic, complemented
174#1476—892no—
175C2×#322—891no—
176C89#1789011yescharacteristic, complemented, Sylow
177C2×#135—891no—
178C4×#41—892no—
179#446—894no—
180C4×Q16—891no—
181C4×Q16—891no—
182C4×SD16—892no—
183C4×SD16—892no—
184C4×D8—891no—
185C4×D8—891no—
186#436—894no—
187#435—1784no—
188#431—1782no—
189#431C2×D8912yes—
190#427—891no—
191#416—892no—
192C4×M16—1782no—
193C2×C4×C8—891no—
194C2×C4×C8—891no—
195C2×C4×C8C2×D8911yescharacteristic
196#335C2×D8912yes—
197#334C2×D8912yes—
198Q8⋊C8—894no—
199D4⋊C8—894no—
200#322—892no—
201#322—892no—
202C2×#41—892no—
203#135—1784no—
204C2×Q16—891no—
205C2×SD16—892no—
206C2×D8—891no—
207C2×M16—1784no—
208C22×C8—892no—
209C22×C8—1781no—
210C22×C8C4×D8911yescharacteristic
211C22×C8C4×D8911yescharacteristic
212C4×Q8—892no—
213C4×Q8—892no—
214C4×D4—892no—
215C4×D4—892no—
216C4×D4—1782no—
217C4×D4—1782no—
218#117—892no—
219#117—1782no—
220C2×C42—1782no—
221C2×C42C22×D8911yescharacteristic
222#107—891no—
223#107—891no—
224#106—892no—
225C4⋊C8—1784no—
226C4⋊C8C22×D8914yes—
227#103—892no—
228#103—892no—
229#102—892no—
230#102—892no—
231C22⋊C8—22no—
232C22⋊C8—1784no—
233#97—1784no—
234C4×C8—22no—
235C4×C8—892no—
236C4×C8—892no—
237C4×C8—892no—
238C4×C8—1781no—
239C4×C8C4×D8911yescharacteristic
240C4×C8C4×D8911yescharacteristic
241C4×C8C22×D8911yescharacteristic
242C4×C8C22×D8911yescharacteristic
243#41—1784no—
244#41—1784no—
245C2×Q8—892no—
246C2×D4—892no—
247C2×D4—1782no—
248C22×C4—1782no—
249C22×C4—1782no—
250C22×C4—1782no—
251C22×C4C2×C4×D8911yescharacteristic
252C22×C4D4×D8911yescharacteristic
253C22×C4#1308357511yescharacteristic
254Q16—1782no—
255SD16—1784no—
256D8—1782no—
257M16—3568no—
258C2×C8—22no—
259C2×C8—24no—
260C2×C8—894no—
261C2×C8—3562no—
262C2×C8—3562no—
263C2×C8—1782no—
264C2×C8#1308355714yescomplemented
265C2×C8C2×C4×D8911yescharacteristic
266C2×C8C2×C4×D8911yescharacteristic
267C2×C8C2×C4×D8911yescharacteristic
268C2×C8C2×C4×D8911yescharacteristic
269C4⋊C4—892no—
270C4⋊C4—892no—
271C4⋊C4—1782no—
272C4⋊C4—1782no—
273C22⋊C4—1782no—
274C22⋊C4—1782no—
275C22⋊C4—1782no—
276C42—1782no—
277C42—1782no—
278C42—1782no—
279C42C2×C4×D8911yescharacteristic
280C42D4×D8911yescharacteristic
281C42#1308357511yescharacteristic
282C42C23×D8911yescharacteristic
283C23—1782no—
284C23#1566797411yescharacteristic
285Q8—1782no—
286Q8—1782no—
287D4—3564no—
288D4—1782no—
289D4—1782no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—3564no—
293C2×C4—3564no—
294C2×C4—3564no—
295C2×C4—1782no—
296C2×C4—1782no—
297C2×C4—1782no—
298C2×C4#1566797411yescharacteristic
299C2×C4#1566797411yescharacteristic
300C2×C4#1566797411yescharacteristic
301C2×C4C2×#1308355712yes—
302C2×C4#1566802012yes—
303C2×C4C22×C4×D8911yescharacteristic
304C2×C4C2×D4×D8911yescharacteristic
305C2×C4C2×#1308357511yescharacteristic
306C8—21no—
307C8—21no—
308C8—44no—
309C8—3562no—
310C8—1784no—
311C8C2×#1308355712yes—
312C8C2×#1308355712yes—
313C22—21no—
314C22—21no—
315C22—3562no—
316C22—3562no—
317C22—3562no—
318C22#1915167512yes—
319C22C2×#1566797411yescharacteristic
320C4—21no—
321C4—21no—
322C4—3562no—
323C4—3562no—
324C4—3562no—
325C4#1915167512yes—
326C4#1915171912yes—
327C4C22×#1308355711yescharacteristic
328C4C2×#1566802011yescharacteristic
329C4#1915239512yes—
330C2—22no—
331C2—7122no—
332C2C2×#1915167511yescharacteristic
333C2#2602905211yescharacteristic
334C2C2×#1915171911yescharacteristic
subgroupG/H
centerC42D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1564C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668034C4
cosocleC42×C89C24
hypercenter#1564D89
hypocenterC89#17890
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C42 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #26029213C2semidirect
3. #26019165C2semidirect
4. #26019160C2semidirect
5. #26019153C2semidirect
6. #26018632C2semidirect
7. C2×#19151319C2semidirect
8. C89×#1564C2semidirect
106. C2×C8×C89M16semidirect
173. #1564D89semidirect
176. C89#17890semidirect
264. C2×C8#13083557semidirect

Maximal subgroups

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

subgroupindex
#260292132
C4×#156680342
#260191652
#260191602
#260191532
#260186322
C2×#191513192
C89×#15642
#1789089