Finite Group Databasebeta

← #36447415

Subgroups — #36447415

Order 22784.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26020290C211yesmaximal, characteristic, complemented
2#26020277C211yesmaximal, characteristic, complemented
3#26018795C212yesmaximal
4#26018157C212yesmaximal, complemented
5C89×#2929C211yesmaximal, characteristic
6C2×#15667948—21no—
7C2×#15667944—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#15667944—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—22no—
115C23×C89—21no—
116C23×C89—41no—
117C23×C89—41no—
118C23×C89#11711yescharacteristic
119C23×C89#12111yescharacteristic
120C23×C89C22⋊Q811yescharacteristic
121Q8×C89—41no—
122D4×C89—41no—
123D4×C89—42no—
124D4×C89—42no—
125D4×C89—42no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
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—41no—
136C2×C4×C89—41no—
137C2×C4×C89—41no—
138C2×C4×C89—41no—
139C2×C4×C89—41no—
140C2×C4×C89C2×C22⋊C411yescharacteristic
141C2×C4×C89C2×C4⋊C411yescharacteristic
142C2×C4×C89#12111yescharacteristic
143C2×C4×C89C22⋊Q811yescharacteristic
144C2×Dic89—82no—
145C2×Dic89—82no—
146C2×Dic89—82no—
147C8×C89—21no—
148C8×C89—21no—
149C8×C89—21no—
150C8×C89—41no—
151C8×C89—41no—
152C8×C89#9912yes—
153#10623494—82no—
154C22×C89—21no—
155C22×C89—21no—
156C22×C89—21no—
157C22×C89—21no—
158C22×C89—21no—
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—21no—
162C22×C89—44no—
163C22×C89—41no—
164C22×C89—41no—
165C22×C89—41no—
166C22×C89#38011yescharacteristic
167C22×C89#47112yes—
168C4×C89—21no—
169C4×C89—21no—
170C4×C89—21no—
171C4×C89—41no—
172C4×C89—41no—
173C4×C89—41no—
174C4×C89C2×#9911yescharacteristic
175C4×C89#41011yescharacteristic
176Dic89—162no—
177#8629—891nomaximal, Sylow
178C2×C89—22no—
179C2×C89—41no—
180C2×C89—41no—
181C2×C89—81no—
182C2×C89#179911yescharacteristic
183C2×C89#181411yescharacteristic
184C2×C89#181511yescharacteristic
185#2929D8911yescharacteristic
186#1810—891no—
187#1807—891no—
188#1513—892no—
189#1305—892no—
190C89#862911yescharacteristic, complemented, Sylow
191C2×#131Dic8911yescharacteristic, complemented
192C23×C8C2×D8911yescharacteristic
193#525C2×D8911yescharacteristic
194#435—21no—
195#435—21no—
196C8×D4—21no—
197C8×D4—21no—
198C2×#108—1781no—
199C2×#106—1781no—
200#424Dic8911yescharacteristic, complemented
201#408Dic8911yescharacteristic, complemented
202#407—892no—
203#407C2×D8911yescharacteristic
204#405Dic8911yescharacteristic, complemented
205#380—892no—
206#341—1781no—
207#340—1781no—
208#328—1782no—
209#327—1782no—
210C2×#41C2×Dic8911yescharacteristic
211C23×C4C22×D8911yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—1782no—
215C2×M16C2×Dic8911yescharacteristic
216C2×M16C2×Dic8911yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×Dic8911yescharacteristic
220C22×C8C2×Dic8911yescharacteristic
221C22×C8C22⋊D8911yescharacteristic
222C22×C8C22⋊D8911yescharacteristic
223C22×C8C22⋊D8911yescharacteristic
224C22×C8C22⋊D8911yescharacteristic
225#123—21no—
226C22⋊Q8Dic17811yescharacteristic
227C22⋊Q8D35611yescharacteristic
228#121Dic17811yescharacteristic
229#121D35611yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×Dic8911yescharacteristic
234C2×C4⋊C4—1782no—
235C2×C22⋊C4—1782no—
236#108—3562no—
237#106—3562no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—1782no—
245C22⋊C8—1782no—
246#97—21no—
247C4×C8—21no—
248#95—1782no—
249C24#1308357411yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊Dic8911yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊Dic8911yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—3562no—
261C22×C4C22×Dic8911yescharacteristic
262C22×C4C2×C22⋊D8911yescharacteristic
263C22×C4C2×C22⋊D8911yescharacteristic
264C22×C4C2×D35611yescharacteristic
265C22×C4C2×Dic17811yescharacteristic
266C22×C4#1308357411yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—3562no—
271M16—3562no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—3562no—
282C2×C8C22⋊Dic8911yescharacteristic
283C2×C8C22⋊Dic8911yescharacteristic
284C2×C8C22⋊Dic8911yescharacteristic
285C2×C8C22⋊Dic8911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—3562no—
289C4⋊C4—3562no—
290C4⋊C4C4⋊Dic8911yescharacteristic
291C4⋊C4C4⋊Dic8911yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—7122no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#1566797111yescharacteristic
304C23#1566798611yescharacteristic
305C23#1566799111yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—7122no—
326C2×C4—7122no—
327C2×C4—7122no—
328C2×C4C2×C22⋊Dic8911yescharacteristic
329C2×C4C2×C4⋊Dic8911yescharacteristic
330C2×C4#1566798611yescharacteristic
331C2×C4#1566799111yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—7122no—
338C8#1566792712yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#1915145811yescharacteristic
352C22#1915186112yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—14242no—
360C4C2×#1566792711yescharacteristic
361C4#1915157211yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#2602024011yescharacteristic
367C2#2602030611yescharacteristic
368C2#2602031111yescharacteristic
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#8629
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. #26020277C2semidirect
4. #26018157C2semidirect
14. C2×#131×C89C4semidirect
21. #424×C89C4semidirect
22. #408×C89C4semidirect
24. #405×C89C4semidirect
190. C89#8629semidirect
191. C2×#131Dic89semidirect
200. #424Dic89semidirect
201. #408Dic89semidirect
204. #405Dic89semidirect

Maximal subgroups

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

subgroupindex
#260202902
#260202772
#260187952
#260181572
C89×#29292
#862989