Finite Group Databasebeta

← #36447410

Subgroups — #36447410

Order 22784.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26020290C211yesmaximal, characteristic, complemented
2#26020281C211yesmaximal, characteristic, complemented
3#26018795C212yesmaximal
4#26018157C212yesmaximal, complemented
5C89×#2929C211yesmaximal, characteristic
6C2×#15667948—21no—
7C2×#15667946—21no—
8#19151562C2212yes—
9#19151458C2212yes—
10#19151363—21no—
11#19151361—21no—
12#19151335—22no—
13#19151332—22no—
14C2×#131×C89C411yescharacteristic, complemented
15C23×C8×C89C2211yescharacteristic
16#525×C89C2211yescharacteristic
17#435×C89—21no—
18#435×C89—21no—
19C8×D4×C89—21no—
20C8×D4×C89—21no—
21#424×C89C411yescharacteristic, complemented
22#408×C89C411yescharacteristic, complemented
23#407×C89C2211yescharacteristic
24#405×C89C411yescharacteristic, complemented
25C2×#13083558—22no—
26C2×C4⋊Dic89—22no—
27C2×C22⋊Dic89—22no—
28#15667948—42no—
29#15667946—42no—
30C22⋊#10623494—22no—
31C22⋊#10623494—22no—
32#15667919—22no—
33C2×#41×C89C2×C411yescharacteristic
34C23×C4×C89C2311yescharacteristic
35#131×C89—42no—
36C2×M16×C89—21no—
37C2×M16×C89C2×C411yescharacteristic
38C2×M16×C89C2×C411yescharacteristic
39C22×C8×C89—21no—
40C22×C8×C89—42no—
41C22×C8×C89C2×C411yescharacteristic
42C22×C8×C89C2×C411yescharacteristic
43C22×C8×C89D411yescharacteristic
44C22×C8×C89D411yescharacteristic
45C22×C8×C89D411yescharacteristic
46C22×C8×C89D411yescharacteristic
47#123×C89—21no—
48C22⋊Q8×C89D411yescharacteristic
49C22⋊Q8×C89Q811yescharacteristic
50#121×C89D411yescharacteristic
51#121×C89Q811yescharacteristic
52#120×C89—21no—
53C4×D4×C89—21no—
54C4×D4×C89—21no—
55#117×C89C2×C411yescharacteristic
56C4⋊C8×C89—21no—
57C4⋊C8×C89—21no—
58C22⋊C8×C89—21no—
59C22⋊C8×C89—21no—
60C22⋊C8×C89—21no—
61C22⋊C8×C89—21no—
62#97×C89—21no—
63C4×C8×C89—21no—
64C22×Dic89—42no—
65#13083558—42no—
66#13083558—42no—
67C2×#10623494—42no—
68C4⋊Dic89—42no—
69C4⋊Dic89—42no—
70C22⋊Dic89—82no—
71C24×C89#4111yescharacteristic
72#41×C89—41no—
73#41×C89—41no—
74C2×Q8×C89C4⋊C411yescharacteristic
75C2×D4×C89—21no—
76C2×D4×C89—21no—
77C2×D4×C89—21no—
78C2×D4×C89C4⋊C411yescharacteristic
79C22×C4×C89—21no—
80C22×C4×C89—41no—
81C22×C4×C89—41no—
82C22×C4×C89C22×C411yescharacteristic
83C22×C4×C89C2×D411yescharacteristic
84C22×C4×C89C2×D411yescharacteristic
85C22×C4×C89C2×D411yescharacteristic
86C22×C4×C89C2×Q811yescharacteristic
87C22×C4×C89#4111yescharacteristic
88M16×C89—42no—
89M16×C89—41no—
90M16×C89—41no—
91C2×C8×C89—22no—
92C2×C8×C89—22no—
93C2×C8×C89—22no—
94C2×C8×C89—21no—
95C2×C8×C89—21no—
96C2×C8×C89—41no—
97C2×C8×C89—41no—
98C2×C8×C89—41no—
99C2×C8×C89—42no—
100C2×C8×C89C22⋊C411yescharacteristic
101C2×C8×C89C22⋊C411yescharacteristic
102C2×C8×C89C22⋊C411yescharacteristic
103C2×C8×C89C22⋊C411yescharacteristic
104C4⋊C4×C89—21no—
105C4⋊C4×C89—21no—
106C4⋊C4×C89C4⋊C411yescharacteristic
107C4⋊C4×C89C4⋊C411yescharacteristic
108C22⋊C4×C89—21no—
109C22⋊C4×C89—21no—
110C22⋊C4×C89—21no—
111C22⋊C4×C89—21no—
112C22⋊C4×C89—21no—
113C42×C89—21no—
114C23×C89—21no—
115C23×C89—21no—
116C23×C89—21no—
117C23×C89—41no—
118C23×C89—41no—
119C23×C89#11711yescharacteristic
120C23×C89#12111yescharacteristic
121C23×C89C22⋊Q811yescharacteristic
122Q8×C89—41no—
123D4×C89—41no—
124D4×C89—42no—
125D4×C89—42no—
126D4×C89—41no—
127D4×C89—41no—
128C2×C4×C89—21no—
129C2×C4×C89—21no—
130C2×C4×C89—21no—
131C2×C4×C89—21no—
132C2×C4×C89—21no—
133C2×C4×C89—21no—
134C2×C4×C89—21no—
135C2×C4×C89—21no—
136C2×C4×C89—21no—
137C2×C4×C89—41no—
138C2×C4×C89—41no—
139C2×C4×C89—41no—
140C2×C4×C89—41no—
141C2×C4×C89—41no—
142C2×C4×C89C2×C22⋊C411yescharacteristic
143C2×C4×C89C2×C4⋊C411yescharacteristic
144C2×C4×C89#12111yescharacteristic
145C2×C4×C89C22⋊Q811yescharacteristic
146C2×Dic89—82no—
147C2×Dic89—82no—
148C2×Dic89—82no—
149C8×C89—21no—
150C8×C89—21no—
151C8×C89—21no—
152C8×C89—41no—
153C8×C89—41no—
154C8×C89#9912yes—
155#10623494—82no—
156C22×C89—21no—
157C22×C89—21no—
158C22×C89—21no—
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—21no—
162C22×C89—21no—
163C22×C89—21no—
164C22×C89—42no—
165C22×C89—42no—
166C22×C89—41no—
167C22×C89—41no—
168C22×C89—41no—
169C22×C89#38011yescharacteristic
170C22×C89#47211yescharacteristic
171C22×C89#47311yescharacteristic
172C4×C89—21no—
173C4×C89—21no—
174C4×C89—21no—
175C4×C89—41no—
176C4×C89—41no—
177C4×C89—41no—
178C4×C89C2×#9911yescharacteristic
179C4×C89#41011yescharacteristic
180Dic89—162no—
181#8628—891nomaximal, Sylow
182C2×C89—21no—
183C2×C89—21no—
184C2×C89—41no—
185C2×C89—41no—
186C2×C89—81no—
187C2×C89#179911yescharacteristic
188C2×C89#181411yescharacteristic
189C2×C89#181611yescharacteristic
190#2929D8911yescharacteristic
191#1810—891no—
192#1808—891no—
193#1513—892no—
194#1305—892no—
195C89#862811yescharacteristic, complemented, Sylow
196C2×#131Dic8911yescharacteristic, complemented
197C23×C8C2×D8911yescharacteristic
198#525C2×D8911yescharacteristic
199#435—21no—
200#435—21no—
201C8×D4—21no—
202C8×D4—21no—
203C2×#108—1781no—
204C2×#107—1781no—
205#424Dic8911yescharacteristic, complemented
206#408Dic8911yescharacteristic, complemented
207#407—892no—
208#407C2×D8911yescharacteristic
209#405Dic8911yescharacteristic, complemented
210#380—892no—
211#341—1781no—
212#340—1781no—
213#328—1782no—
214#327—1782no—
215C2×#41C2×Dic8911yescharacteristic
216C23×C4C22×D8911yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—1782no—
220C2×M16C2×Dic8911yescharacteristic
221C2×M16C2×Dic8911yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×Dic8911yescharacteristic
225C22×C8C2×Dic8911yescharacteristic
226C22×C8C22⋊D8911yescharacteristic
227C22×C8C22⋊D8911yescharacteristic
228C22×C8C22⋊D8911yescharacteristic
229C22×C8C22⋊D8911yescharacteristic
230#123—21no—
231C22⋊Q8Dic17811yescharacteristic
232C22⋊Q8D35611yescharacteristic
233#121Dic17811yescharacteristic
234#121D35611yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×Dic8911yescharacteristic
239C2×C4⋊C4—1782no—
240C2×C22⋊C4—1782no—
241#108—3562no—
242#107—3562no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1782no—
250C22⋊C8—1782no—
251#97—21no—
252C4×C8—21no—
253#95—1782no—
254C24#1308357411yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊Dic8911yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊Dic8911yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3562no—
266C22×C4C22×Dic8911yescharacteristic
267C22×C4C2×C22⋊D8911yescharacteristic
268C22×C4C2×C22⋊D8911yescharacteristic
269C22×C4C2×D35611yescharacteristic
270C22×C4C2×Dic17811yescharacteristic
271C22×C4#1308357411yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—3562no—
276M16—3562no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—3562no—
287C2×C8C22⋊Dic8911yescharacteristic
288C2×C8C22⋊Dic8911yescharacteristic
289C2×C8C22⋊Dic8911yescharacteristic
290C2×C8C22⋊Dic8911yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—3562no—
294C4⋊C4—3562no—
295C4⋊C4C4⋊Dic8911yescharacteristic
296C4⋊C4C4⋊Dic8911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—7122no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#1566797111yescharacteristic
310C23#1566798611yescharacteristic
311C23#1566799111yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—7122no—
333C2×C4—7122no—
334C2×C4—7122no—
335C2×C4C2×C22⋊Dic8911yescharacteristic
336C2×C4C2×C4⋊Dic8911yescharacteristic
337C2×C4#1566798611yescharacteristic
338C2×C4#1566799111yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—7122no—
345C8#1566792712yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1915145811yescharacteristic
360C22#1915186811yescharacteristic
361C22#1915187311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—14242no—
369C4C2×#1566792711yescharacteristic
370C4#1915157211yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#2602024011yescharacteristic
377C2#2602030611yescharacteristic
378C2#2602031611yescharacteristic
subgroupG/H
centerC22#19151458
commutatorC22×C4×C89C22×C4
FrattiniC23×C4C22×D89
FittingC89×#2929C2
socleC22×C89#380
prime radical#525×C89C22
cosocleC23×C4×C89C23
hypercenter#2929D89
hypocenterC89#8628
LCSG ▷ C22×C4×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #26020290C2semidirect
2. #26020281C2semidirect
4. #26018157C2semidirect
14. C2×#131×C89C4semidirect
21. #424×C89C4semidirect
22. #408×C89C4semidirect
24. #405×C89C4semidirect
195. C89#8628semidirect
196. C2×#131Dic89semidirect
205. #424Dic89semidirect
206. #408Dic89semidirect
209. #405Dic89semidirect

Maximal subgroups

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

subgroupindex
#260202902
#260202812
#260187952
#260181572
C89×#29292
#862889