Finite Group Databasebeta

← #36452834

Subgroups — #36452834

Order 22784.

386 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#26021214C211yesmaximal, characteristic
2#26021176C211yesmaximal, characteristic
3#26021011C211yesmaximal, characteristic
4#26020122C211yesmaximal, characteristic
5#26018136C211yesmaximal, characteristic
6#26018086C211yesmaximal, characteristic
7C2×#332×C89C211yesmaximal, characteristic
8C2×#15667946—21no—
9C2×C4⋊#10623494C2211yescharacteristic
10C2×#15667936—21no—
11C2×#15667936—21no—
12C2×#15667922—21no—
13C2×C4×#10623494C2211yescharacteristic
14#19151468C2211yescharacteristic
15C4×C4⋊Dic89C2211yescharacteristic
16#19151361—21no—
17#19151360—21no—
18#19151360—21no—
19#19151352—21no—
20C2×C4⋊Q8×C89C2211yescharacteristic
21C2×C4⋊C8×C89C2211yescharacteristic
22C2×C4⋊C8×C89C2211yescharacteristic
23#332×C89C414yescomplemented
24C22×#10623494—21no—
25C22×#10623494D411yescharacteristic
26C22×#10623494D411yescharacteristic
27C2×C4⋊Dic89—21no—
28C2×C4⋊Dic89—21no—
29C2×C4⋊Dic89—21no—
30C2×C4⋊Dic89D411yescharacteristic
31C2×C4⋊Dic89D411yescharacteristic
32C2×C4×Dic89—21no—
33#15667946—42no—
34C4⋊#10623494—21no—
35C4⋊#10623494—21no—
36#15667936—42no—
37#15667936—42no—
38#15667922—42no—
39C4×#10623494—21no—
40C4×#10623494—21no—
41#15667919—21no—
42C22×Q8×C89—21no—
43C22×C8×C89—21no—
44C22×C8×C89—21no—
45C4⋊Q8×C89—21no—
46C4⋊Q8×C89—21no—
47C4⋊Q8×C89—21no—
48C4⋊Q8×C89C2×C412yes—
49C2×C4⋊C4×C89—21no—
50C2×C4⋊C4×C89—21no—
51C2×C42×C89C2311yescharacteristic
52C4⋊C8×C89—21no—
53C4⋊C8×C89—21no—
54C4⋊C8×C89C2×C412yes—
55C4⋊C8×C89C2×C412yes—
56C22×Dic89—21no—
57C22×Dic89—21no—
58C22×Dic89—41no—
59C2×#10623494—21no—
60C2×#10623494—21no—
61C2×#10623494—21no—
62C2×#10623494—21no—
63C2×#10623494—21no—
64C2×#10623494—41no—
65C2×#10623494—41no—
66C2×#10623494—41no—
67C2×#10623494Q1612yes—
68C4⋊Dic89—21no—
69C4⋊Dic89—21no—
70C4⋊Dic89—21no—
71C4⋊Dic89—21no—
72C4⋊Dic89—41no—
73C4⋊Dic89—41no—
74C4⋊Dic89—41no—
75C4⋊Dic89—41no—
76C4⋊Dic89—81no—
77C4×Dic89—41no—
78C4×Dic89—41no—
79C2×Q8×C89—22no—
80C2×Q8×C89—41no—
81C2×Q8×C89—41no—
82C2×Q8×C89—41no—
83C2×Q8×C89—41no—
84C2×Q8×C89—41no—
85C22×C4×C89—41no—
86C22×C4×C89C2×D411yescharacteristic
87C22×C4×C89C2×D411yescharacteristic
88C22×C4×C89#4111yescharacteristic
89C2×C8×C89—22no—
90C2×C8×C89—22no—
91C2×C8×C89—41no—
92C2×C8×C89—41no—
93C2×C8×C89—41no—
94C2×C8×C89—41no—
95C4⋊C4×C89—22no—
96C4⋊C4×C89—22no—
97C4⋊C4×C89—41no—
98C4⋊C4×C89—41no—
99C42×C89C22×C411yescharacteristic
100C42×C89#4111yescharacteristic
101C42×C89#4111yescharacteristic
102C42×C89#4111yescharacteristic
103C23×C89#12111yescharacteristic
104Q8×C89—42no—
105Q8×C89—42no—
106Q8×C89—81no—
107Q8×C89—81no—
108C2×C4×C89—21no—
109C2×C4×C89—21no—
110C2×C4×C89—42no—
111C2×C4×C89—81no—
112C2×C4×C89—81no—
113C2×C4×C89#11711yescharacteristic
114C2×C4×C89C4×D411yescharacteristic
115C2×C4×C89C4×D411yescharacteristic
116C2×C4×C89#12311yescharacteristic
117C2×C4×C89#12411yescharacteristic
118C2×C4×C89#12611yescharacteristic
119C2×C4×C89C2×Q1611yescharacteristic
120C2×C4×C89#13511yescharacteristic
121C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C89#13711yescharacteristic
126C2×C4×C89#13711yescharacteristic
127C2×Dic89—41no—
128C2×Dic89—41no—
129C2×Dic89—41no—
130C2×Dic89—41no—
131C2×Dic89—41no—
132C2×Dic89—41no—
133C2×Dic89—81no—
134C2×Dic89—81no—
135C2×Dic89—81no—
136C8×C89—42no—
137C8×C89—42no—
138#10623494—22no—
139#10623494—41no—
140#10623494—81no—
141C22×C89#38811yescharacteristic
142C22×C89#46211yescharacteristic
143C22×C89#46311yescharacteristic
144C22×C89#46811yescharacteristic
145C22×C89#46911yescharacteristic
146C22×C89#47111yescharacteristic
147C22×C89#47311yescharacteristic
148C4×C89—21no—
149C4×C89—21no—
150C4×C89—82no—
151C4×C89C4×Q1611yescharacteristic
152C4×C89#44011yescharacteristic
153C4×C89#44011yescharacteristic
154C4×C89#44211yescharacteristic
155C4×C89#48411yescharacteristic
156C4×C89#48511yescharacteristic
157C4×C89#48911yescharacteristic
158C4×C89#49011yescharacteristic
159Dic89—81no—
160Dic89—81no—
161Dic89—161no—
162#9574—891nomaximal, Sylow
163C2×C89#169211yescharacteristic
164C2×C89#169411yescharacteristic
165C2×C89#169911yescharacteristic
166C2×C89#170111yescharacteristic
167C2×C89#192811yescharacteristic
168C2×C89#193011yescharacteristic
169C2×C89#193611yescharacteristic
170#1990—891no—
171#1981—891no—
172#1947—891no—
173#1770—891no—
174C2×#332D8911yescharacteristic
175#1296—891no—
176#1277—891no—
177C89#957411yescharacteristic, complemented, Sylow
178C2×C4⋊Q8C2×D8911yescharacteristic
179C2×#107—1781no—
180C2×C4⋊C8—891no—
181C2×C4⋊C8C2×D8911yescharacteristic
182C2×C4⋊C8C2×D8911yescharacteristic
183C2×#103—1781no—
184C2×#103—1781no—
185C2×#97—1781no—
186C2×C4×C8—891no—
187#384—891no—
188C4×C4⋊C4—891no—
189#340—1781no—
190#340—1781no—
191#340—1781no—
192#336—1781no—
193#332Dic8914yescomplemented
194C22×Q8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—891no—
198C22×C8—891no—
199C22×C8—1781no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C2×Dic8912yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—891no—
207C2×C4⋊C4—891no—
208C2×C4⋊C4—1781no—
209C2×C4⋊C4—1781no—
210C2×C4⋊C4—1781no—
211C2×C42—1781no—
212C2×C42C22×D8911yescharacteristic
213#107—3562no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—1781no—
217C4⋊C8—1781no—
218C4⋊C8C2×Dic8912yes—
219C4⋊C8C2×Dic8912yes—
220#103—3562no—
221#103—3562no—
222#97—3562no—
223C4×C8—1781no—
224C4×C8—1781no—
225#95—1781no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—41no—
233C22×C4—3561no—
234C22×C4—1781no—
235C22×C4—1781no—
236C22×C4D4×D8911yescharacteristic
237C22×C4D4×D8911yescharacteristic
238C22×C4#1308357411yescharacteristic
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—892no—
246C2×C8—3561no—
247C2×C8—3561no—
248C2×C8—3561no—
249C2×C8—1781no—
250C2×C8—1781no—
251C2×C8—1781no—
252C2×C8—1781no—
253C2×C8—1781no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—3561no—
259C4⋊C4—3561no—
260C4⋊C4—3561no—
261C4⋊C4—3561no—
262C4⋊C4—1781no—
263C4⋊C4—1781no—
264C4⋊C4—1781no—
265C4⋊C4—1781no—
266C4⋊C4—7121no—
267C42—3561no—
268C42—3561no—
269C42C22×Dic8911yescharacteristic
270C42#1308357311yescharacteristic
271C42#1308357311yescharacteristic
272C42#1308357411yescharacteristic
273C23#1566798511yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—81no—
277Q8—81no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—3561no—
284C2×C4—3561no—
285C2×C4—3561no—
286C2×C4—3561no—
287C2×C4—3561no—
288C2×C4—3561no—
289C2×C4—7121no—
290C2×C4—7121no—
291C2×C4—7121no—
292C2×C4#1566797111yescharacteristic
293C2×C4D4×Dic8911yescharacteristic
294C2×C4D4×Dic8911yescharacteristic
295C2×C4#1566799411yescharacteristic
296C2×C4#1566799811yescharacteristic
297C2×C4#1566800511yescharacteristic
298C2×C4Q16×D8911yescharacteristic
299C2×C4#1566803511yescharacteristic
300C2×C4#1566803611yescharacteristic
301C2×C4#1566803611yescharacteristic
302C2×C4#1566803911yescharacteristic
303C2×C4#1566803911yescharacteristic
304C2×C4#1566804111yescharacteristic
305C2×C4#1566804111yescharacteristic
306C8—42no—
307C8—42no—
308C8—3561no—
309C8—1782no—
310C8—7121no—
311C22#1915149111yescharacteristic
312C22#1915180711yescharacteristic
313C22#1915181611yescharacteristic
314C22#1915184311yescharacteristic
315C22#1915185011yescharacteristic
316C22#1915186411yescharacteristic
317C22#1915187211yescharacteristic
318C4—21no—
319C4—21no—
320C4—82no—
321C4—14241no—
322C4—7121no—
323C4—7121no—
324C4Q16×Dic8911yescharacteristic
325C4#1915169911yescharacteristic
326C4#1915169911yescharacteristic
327C4#1915171011yescharacteristic
328C4#1915193211yescharacteristic
329C4#1915193811yescharacteristic
330C4#1915195611yescharacteristic
331C4#1915196111yescharacteristic
332C2#2601975711yescharacteristic
333C2#2601976911yescharacteristic
334C2#2601980211yescharacteristic
335C2#2601981611yescharacteristic
336C2#2602089511yescharacteristic
337C2#2602090911yescharacteristic
338C2#2602095111yescharacteristic
subgroupG/H
centerC23#15667985
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC2×#332×C89C2
socle, prime radicalC23×C89#121
cosocleC2×C42×C89C23
hypercenterC2×#332D89
hypocenterC89#9574
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
23. #332×C89C4semidirect
177. C89#9574semidirect
193. #332Dic89semidirect

Maximal subgroups

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

subgroupindex
#260212142
#260211762
#260210112
#260201222
#260181362
#260180862
C2×#332×C892
#957489