Finite Group Databasebeta

← #36525898

Subgroups — #36525898

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029270C212yesmaximal, complemented
2C4×#15668034C211yesmaximal, characteristic
3#26019138C212yesmaximal, complemented
4#26019097C214yesmaximal, complemented
5C8×Dic356C212yesmaximal, complemented
6#26019015C211yesmaximal, characteristic, complemented
7#26018687C212yesmaximal, complemented
8C8×M16×C89C211yesmaximal, characteristic, complemented
9C2×#15668034C411yescharacteristic
10C4×#13083574C2212yes—
11C8×Dic178C2214yes—
12C4×Dic356C411yescharacteristic
13C4×Dic356C2211yescharacteristic
14C4×#13083563C412yes—
15C4×#13083563C2212yes—
16C4×D712C411yescharacteristic
17C4×D712C2211yescharacteristic
18#19151677C2214yes—
19#19151674—24no—
20#19151655C2212yes—
21#19151653—22no—
22#19151636C411yescharacteristic
23#19151595C412yes—
24#19151550—22no—
25#19151351C2212yes—
26#19151349C2212yes—
27#19151330C2214yes—
28#19151327C2214yes—
29#431×C89C2211yescharacteristic
30C4×M16×C89C2211yescharacteristic
31C2×C4×C8×C89C2211yescharacteristic
32#322×C89C2212yes—
33C82×C89C2212yes—
34C2×#13083574C2×C412yes—
35#15668034—24no—
36C2×Dic356C2×C411yescharacteristic
37C2×#13083563C2×C412yes—
38C2×D712C2×C411yescharacteristic
39C2×#13083557—24no—
40C4×Dic178C2×C412yes—
41C4×Dic178C2312yes—
42C4×D356C2×C412yes—
43C4×D356C2312yes—
44C4×C22⋊D89—22no—
45C4×C22⋊D89—22no—
46#15667971C2×C412yes—
47#15667969—22no—
48C42×D89—22no—
49#15667946C2×C411yescharacteristic
50#15667946C2×C411yescharacteristic
51#15667944C2×C412yes—
52C4⋊#10623494C2314yes—
53#15667941—24no—
54#15667937C2×C412yes—
55#15667937C2×C412yes—
56#15667934C2×C412yes—
57#15667934C2×C412yes—
58C22⋊#10623494—22no—
59#15667924—24no—
60#15667923—22no—
61C8×Dic89—24no—
62C2×M16×C89D412yes—
63C22×C8×C89C2×C411yescharacteristic
64C22×C8×C89C2×C411yescharacteristic
65C2×C42×C89C2311yescharacteristic
66C4⋊C8×C89—21no—
67C22⋊C8×C89—21no—
68#97×C89D411yescharacteristic
69#97×C89D411yescharacteristic
70C4×C8×C89C2×C411yescharacteristic
71C4×C8×C89C2×C411yescharacteristic
72C4×C8×C89C2312yes—
73C4×C8×C89C2311yescharacteristic
74C4×C8×C89C2311yescharacteristic
75C4×C8×C89C2312yes—
76#13083574—24no—
77#13083574—42no—
78C2×Dic178C22×C412yes—
79C2×D356C22×C412yes—
80C2×C22⋊D89—22no—
81C2×C4×D89—22no—
82C2×C4×D89—22no—
83C2×C4×D89—22no—
84Dic356M1614yescomplemented
85#13083563—24no—
86D712—22no—
87#13083557—48no—
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—24no—
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×D412yes—
111C2×C8×C89C2×D412yes—
112C42×C89C22×C411yescharacteristic
113C42×C89C2×D411yescharacteristic
114C42×C89#4111yescharacteristic
115C42×C89C2411yescharacteristic
116C23×C89C4×D411yescharacteristic
117C22×D89—22no—
118C2×C4×C89—21no—
119C2×C4×C89—21no—
120C2×C4×C89—22no—
121C2×C4×C89—41no—
122C2×C4×C89C4×D411yescharacteristic
123C2×C4×C89C4×D411yescharacteristic
124C2×C4×C89C4×D411yescharacteristic
125C2×C4×C89C23×C411yescharacteristic
126C2×C4×C89C22×D411yescharacteristic
127C2×C4×C89C2×#4111yescharacteristic
128C22⋊D89—44no—
129C2×Dic89—22no—
130C2×Dic89—22no—
131C2×Dic89—22no—
132D356—22no—
133D356—22no—
134C4×D89—44no—
135C4×D89—44no—
136C4×D89—82no—
137Dic178—22no—
138Dic178C2×M1614yes—
139C8×C89—21no—
140C8×C89—21no—
141C8×C89—21no—
142C8×C89—22no—
143C8×C89—42no—
144C8×C89C2×M1612yes—
145C8×C89#13514yescomplemented
146#10623494—44no—
147C22×C89—21no—
148C22×C89—21no—
149C22×C89—21no—
150C22×C89C2×C4×D411yescharacteristic
151C2×D89—42no—
152C2×D89—42no—
153C2×D89—42no—
154C4×C89—21no—
155C4×C89—21no—
156C4×C89—21no—
157C4×C89—21no—
158C4×C89C22×M1611yescharacteristic
159C4×C89#56811yescharacteristic
160C4×C89C2×#13512yes—
161Dic89—24no—
162Dic89—42no—
163Dic89—42no—
164#18039—891nomaximal, Sylow
165C2×C89—41no—
166C2×C89D4×M1611yescharacteristic
167C2×C89C4×#13511yescharacteristic
168C2×C89#295711yescharacteristic
169D89—82no—
170#2971—892no—
171C4×#135—891no—
172#1589—892no—
173#1583—894no—
174C8×Q16—892no—
175#1568—891no—
176#1488—892no—
177C8×M16D8911yescharacteristic, complemented
178C89#1803911yescharacteristic, complemented, Sylow
179C2×#135—891no—
180C4×#41—892no—
181C8×Q8—894no—
182C4×Q16—891no—
183C4×Q16—891no—
184C4×SD16—892no—
185C4×SD16—892no—
186C4×D8—891no—
187C4×D8—891no—
188#436—894no—
189#436—1784no—
190#432—892no—
191#432—1782no—
192#431C2×D8911yescharacteristic
193#427—891no—
194#416—892no—
195C4×M16—1782no—
196C4×M16C2×D8911yescharacteristic
197C2×C4×C8C2×D8911yescharacteristic
198#335—892no—
199#334—892no—
200Q8⋊C8—894no—
201D4⋊C8—894no—
202#322C2×D8912yes—
203C82C2×D8912yes—
204C2×#41—892no—
205#135—1784no—
206C2×Q16—891no—
207C2×SD16—892no—
208C2×D8—891no—
209C2×M16—1784no—
210C2×M16D35612yes—
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—21no—
227C4⋊C8—894no—
228C4⋊C8—1784no—
229#103—892no—
230#103—892no—
231#102—892no—
232#102—892no—
233C22⋊C8—21no—
234C22⋊C8—1782no—
235C22⋊C8—1784no—
236#97—1782no—
237#97D35611yescharacteristic
238#97D35611yescharacteristic
239C4×C8—1784no—
240C4×C8C4×D8911yescharacteristic
241C4×C8C4×D8911yescharacteristic
242C4×C8C22×D8912yes—
243C4×C8C22×D8911yescharacteristic
244C4×C8C22×D8911yescharacteristic
245C4×C8C22×D8912yes—
246#41—3562no—
247#41—1784no—
248C2×Q8—892no—
249C2×D4—892no—
250C2×D4—1782no—
251C22×C4—1782no—
252C22×C4—1782no—
253C22×C4—1782no—
254C22×C4C2×C4×D8911yescharacteristic
255C22×C4C2×D35611yescharacteristic
256C22×C4#1308357411yescharacteristic
257Q16—894no—
258SD16—1784no—
259D8—1782no—
260M16—24no—
261M16—3568no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—1784no—
266C2×C8C2×C4×D8911yescharacteristic
267C2×C8C2×C4×D8911yescharacteristic
268C2×C8C2×C4×D8911yescharacteristic
269C2×C8C2×C4×D8911yescharacteristic
270C2×C8C2×D35612yes—
271C2×C8C2×D35612yes—
272C4⋊C4—892no—
273C4⋊C4—892no—
274C4⋊C4—1782no—
275C4⋊C4—1782no—
276C22⋊C4—1782no—
277C22⋊C4—1782no—
278C22⋊C4—1782no—
279C42—1782no—
280C42—1782no—
281C42—1782no—
282C42C2×C4×D8911yescharacteristic
283C42C2×D35611yescharacteristic
284C42#1308357411yescharacteristic
285C42C23×D8911yescharacteristic
286C23—1782no—
287C23C4×D35611yescharacteristic
288Q8—894no—
289Q8—1782no—
290D4—3564no—
291D4—1782no—
292D4—1782no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—3564no—
298C2×C4—3564no—
299C2×C4—1782no—
300C2×C4—1782no—
301C2×C4—1782no—
302C2×C4—7122no—
303C2×C4C4×D35611yescharacteristic
304C2×C4C4×D35611yescharacteristic
305C2×C4C4×D35611yescharacteristic
306C2×C4C22×C4×D8911yescharacteristic
307C2×C4C22×D35611yescharacteristic
308C2×C4C2×#1308357411yescharacteristic
309C8—21no—
310C8—21no—
311C8—21no—
312C8—22no—
313C8—42no—
314C8—3564no—
315C8M16×D8912yes—
316C8#1566803414yescomplemented
317C22—21no—
318C22—21no—
319C22—21no—
320C22—3562no—
321C22—3562no—
322C22—3562no—
323C22C2×C4×D35611yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—3562no—
329C4—3562no—
330C4—1784no—
331C4C2×M16×D8911yescharacteristic
332C4#1915236411yescharacteristic
333C4C2×#1566803412yes—
334C2—41no—
335C2—7122no—
336C2M16×D35611yescharacteristic
337C2C4×#1566803411yescharacteristic
338C2#2602917011yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC8×M16×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668034C4
cosocleC42×C89C24
hypercenterC8×M16D89
hypocenterC89#18039
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
1. #26029270C2semidirect
3. #26019138C2semidirect
4. #26019097C2semidirect
5. C8×Dic356C2semidirect
6. #26019015C2semidirect
7. #26018687C2semidirect
8. C8×M16×C89C2semidirect
84. Dic356M16semidirect
145. C8×C89#135semidirect
177. C8×M16D89semidirect
178. C89#18039semidirect
316. C8#15668034semidirect

Maximal subgroups

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

subgroupindex
#260292702
C4×#156680342
#260191382
#260190972
C8×Dic3562
#260190152
#260186872
C8×M16×C892
#1803989