Finite Group Databasebeta

← #36526164

Subgroups — #36526164

Order 22784.

648 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029381C212yesmaximal, complemented
2C4×#15668034C211yesmaximal, characteristic
3#26019167C212yesmaximal, complemented
4#26019097C214yesmaximal, complemented
5#26019072C212yesmaximal, complemented
6#26019008C211yesmaximal, characteristic, complemented
7#26018692C212yesmaximal, complemented
8C89×#1451C211yesmaximal, characteristic, complemented
9C2×#15668034C411yescharacteristic
10C4×#13083574C2212yes—
11C8×Dic178C2214yes—
12C4×Dic356C411yescharacteristic
13C4×Dic356C2211yescharacteristic
14C4×#13083563C412yes—
15C4×#13083563C2212yes—
16C4×D712C411yescharacteristic
17C4×D712C2211yescharacteristic
18#19151677C2214yes—
19#19151673—24no—
20#19151650—22no—
21#19151636C411yescharacteristic
22#19151622C2212yes—
23#19151620—22no—
24#19151595C412yes—
25#19151349C2212yes—
26#19151349C2212yes—
27#19151330C2214yes—
28#19151327C2214yes—
29#431×C89C2211yescharacteristic
30C4×M16×C89C2211yescharacteristic
31C2×C4×C8×C89C2211yescharacteristic
32#322×C89C2212yes—
33#322×C89C2212yes—
34C2×#13083574C2×C412yes—
35#15668034—24no—
36C2×Dic356C2×C411yescharacteristic
37C2×#13083563C2×C412yes—
38C2×D712C2×C411yescharacteristic
39C2×#13083558—22no—
40C2×#13083557—24no—
41C4×Dic178C2×C412yes—
42C4×Dic178C2312yes—
43C4×D356C2×C412yes—
44C4×D356C2312yes—
45C4×C22⋊D89—22no—
46C4×C22⋊D89—22no—
47#15667971C2×C412yes—
48#15667969—22no—
49C42×D89—22no—
50#15667946C2×C411yescharacteristic
51#15667946C2×C411yescharacteristic
52#15667944C2×C412yes—
53C4⋊#10623494—22no—
54C4⋊#10623494C2314yes—
55#15667941—24no—
56#15667937C2×C412yes—
57#15667937C2×C412yes—
58#15667934C2×C412yes—
59#15667934C2×C412yes—
60#15667924—24no—
61C8×Dic89—24no—
62C2×M16×C89—21no—
63C22×C8×C89C2×C411yescharacteristic
64C22×C8×C89C2×C411yescharacteristic
65C2×C42×C89C2311yescharacteristic
66C4⋊C8×C89D412yes—
67C22⋊C8×C89D412yes—
68#97×C89—21no—
69C4×C8×C89C2×C411yescharacteristic
70C4×C8×C89C2×C411yescharacteristic
71C4×C8×C89C2312yes—
72C4×C8×C89C2311yescharacteristic
73C4×C8×C89C2311yescharacteristic
74C4×C8×C89C2312yes—
75#13083574—24no—
76#13083574—42no—
77C2×Dic178C22×C412yes—
78C2×D356C22×C412yes—
79C2×C22⋊D89—22no—
80C2×C4×D89—22no—
81C2×C4×D89—22no—
82C2×C4×D89—22no—
83Dic356M1614yescomplemented
84#13083563—24no—
85D712—22no—
86#13083558—82no—
87#13083557—84no—
88C2×#10623494—24no—
89C4⋊Dic89C22×C412yes—
90C4⋊Dic89C22×C412yes—
91#13083553—22no—
92#13083553—22no—
93C22⋊Dic89—22no—
94#13083551—22no—
95#13083551—22no—
96C4×Dic89—22no—
97C4×Dic89—22no—
98C4×Dic89—22no—
99C22×C4×C89C22×C411yescharacteristic
100C22×C4×C89C2×D411yescharacteristic
101C22×C4×C89#4111yescharacteristic
102M16×C89—42no—
103C2×C8×C89—22no—
104C2×C8×C89—22no—
105C2×C8×C89—41no—
106C2×C8×C89C22×C411yescharacteristic
107C2×C8×C89C22×C411yescharacteristic
108C2×C8×C89C22×C411yescharacteristic
109C2×C8×C89C22×C411yescharacteristic
110C2×C8×C89C2×D414yes—
111C42×C89C22×C411yescharacteristic
112C42×C89C2×D411yescharacteristic
113C42×C89#4111yescharacteristic
114C42×C89C2411yescharacteristic
115C23×C89C4×D411yescharacteristic
116C22×D89—22no—
117C2×C4×C89—21no—
118C2×C4×C89—21no—
119C2×C4×C89—22no—
120C2×C4×C89—22no—
121C2×C4×C89C4×D411yescharacteristic
122C2×C4×C89C4×D411yescharacteristic
123C2×C4×C89C4×D411yescharacteristic
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—42no—
135C4×D89—42no—
136C4×D89—44no—
137Dic178—22no—
138Dic178C2×M1614yes—
139C8×C89—21no—
140C8×C89—21no—
141C8×C89—21no—
142C8×C89—24no—
143C8×C89—42no—
144C8×C89C2×M1612yes—
145#10623494—44no—
146C22×C89—21no—
147C22×C89—21no—
148C22×C89—21no—
149C22×C89C2×C4×D411yescharacteristic
150C2×D89—42no—
151C2×D89—42no—
152C2×D89—42no—
153C4×C89—21no—
154C4×C89—21no—
155C4×C89—21no—
156C4×C89#44412yes—
157C4×C89C22×M1611yescharacteristic
158C4×C89#56811yescharacteristic
159C4×C89#57712yes—
160Dic89—24no—
161Dic89—42no—
162Dic89—42no—
163#18063—891nomaximal, Sylow
164C2×C89—41no—
165C2×C89D4×M1611yescharacteristic
166C2×C89#295211yescharacteristic
167C2×C89C2×#44411yescharacteristic
168D89—82no—
169#2991—892no—
170C4×#135—891no—
171#1594—892no—
172#1583—894no—
173#1579—892no—
174#1566—891no—
175#1489—892no—
176#1451D8911yescharacteristic, complemented
177C89#1806311yescharacteristic, complemented, Sylow
178C2×#135—891no—
179C4×#41—892no—
180C8×Q8—894no—
181C4×Q16—891no—
182C4×Q16—891no—
183C4×SD16—892no—
184C4×SD16—892no—
185C4×D8—891no—
186C4×D8—891no—
187#436—894no—
188#436—1784no—
189#431—1782no—
190#431C2×D8911yescharacteristic
191#427—891no—
192#423—892no—
193#423—1782no—
194#416—892no—
195C4×M16C2×D8911yescharacteristic
196C2×C4×C8C2×D8911yescharacteristic
197#334—892no—
198#334—892no—
199Q8⋊C8—894no—
200D4⋊C8—894no—
201#322C2×D8912yes—
202#322C2×D8912yes—
203C2×#41—892no—
204#135—1784no—
205C2×Q16—891no—
206C2×SD16—892no—
207C2×D8—891no—
208C2×M16—21no—
209C2×M16—1782no—
210C2×M16—1784no—
211C22×C8C4×D8911yescharacteristic
212C22×C8C4×D8911yescharacteristic
213C4×Q8—892no—
214C4×Q8—892no—
215C4×D4—892no—
216C4×D4—892no—
217C4×D4—1782no—
218C4×D4—1782no—
219#117—892no—
220#117—1782no—
221C2×C42—1782no—
222C2×C42C22×D8911yescharacteristic
223#107—891no—
224#107—891no—
225#106—892no—
226C4⋊C8—894no—
227C4⋊C8—1784no—
228C4⋊C8—1782no—
229C4⋊C8D35612yes—
230#103—892no—
231#103—892no—
232#102—892no—
233#102—892no—
234C22⋊C8—1784no—
235C22⋊C8D35612yes—
236#97—21no—
237C4×C8—1784no—
238C4×C8C4×D8911yescharacteristic
239C4×C8C4×D8911yescharacteristic
240C4×C8C22×D8912yes—
241C4×C8C22×D8911yescharacteristic
242C4×C8C22×D8911yescharacteristic
243C4×C8C22×D8912yes—
244#41—3562no—
245#41—1784no—
246C2×Q8—892no—
247C2×D4—892no—
248C2×D4—1782no—
249C22×C4—1782no—
250C22×C4—1782no—
251C22×C4—1782no—
252C22×C4C2×C4×D8911yescharacteristic
253C22×C4C2×D35611yescharacteristic
254C22×C4#1308357411yescharacteristic
255Q16—894no—
256SD16—1784no—
257D8—1782no—
258M16—42no—
259M16—7124no—
260M16—7122no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—1784no—
265C2×C8C2×C4×D8911yescharacteristic
266C2×C8C2×C4×D8911yescharacteristic
267C2×C8C2×C4×D8911yescharacteristic
268C2×C8C2×C4×D8911yescharacteristic
269C2×C8C2×D35614yes—
270C4⋊C4—892no—
271C4⋊C4—892no—
272C4⋊C4—1782no—
273C4⋊C4—1782no—
274C22⋊C4—1782no—
275C22⋊C4—1782no—
276C22⋊C4—1782no—
277C42—1782no—
278C42—1782no—
279C42—1782no—
280C42C2×C4×D8911yescharacteristic
281C42C2×D35611yescharacteristic
282C42#1308357411yescharacteristic
283C42C23×D8911yescharacteristic
284C23—1782no—
285C23C4×D35611yescharacteristic
286Q8—894no—
287Q8—1782no—
288D4—3564no—
289D4—1782no—
290D4—1782no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3564no—
296C2×C4—3562no—
297C2×C4—3562no—
298C2×C4—3564no—
299C2×C4—1782no—
300C2×C4—1782no—
301C2×C4—1782no—
302C2×C4C4×D35611yescharacteristic
303C2×C4C4×D35611yescharacteristic
304C2×C4C4×D35611yescharacteristic
305C2×C4C22×C4×D8911yescharacteristic
306C2×C4C22×D35611yescharacteristic
307C2×C4C2×#1308357411yescharacteristic
308C8—24no—
309C8—21no—
310C8—21no—
311C8—21no—
312C8—42no—
313C8—3564no—
314C8M16×D8912yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—3562no—
319C22—3562no—
320C22—3562no—
321C22C2×C4×D35611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—3562no—
326C4—3562no—
327C4—1784no—
328C4#1915172112yes—
329C4C2×M16×D8911yescharacteristic
330C4#1915236411yescharacteristic
331C4#1915241012yes—
332C2—41no—
333C2—7122no—
334C2M16×D35611yescharacteristic
335C2#2602913711yescharacteristic
336C2C2×#1915172111yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1451C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668034C4
cosocleC42×C89C24
hypercenter#1451D89
hypocenterC89#18063
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1451

Complements

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

NH ≅ G/Ntype
1. #26029381C2semidirect
3. #26019167C2semidirect
4. #26019097C2semidirect
5. #26019072C2semidirect
6. #26019008C2semidirect
7. #26018692C2semidirect
8. C89×#1451C2semidirect
83. Dic356M16semidirect
176. #1451D89semidirect
177. C89#18063semidirect

Maximal subgroups

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

subgroupindex
#260293812
C4×#156680342
#260191672
#260190972
#260190722
#260190082
#260186922
C89×#14512
#1806389