Finite Group Databasebeta

← #36458922

Subgroups — #36458922

Order 22784.

426 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26025287C211yesmaximal, characteristic
2#26021013C211yesmaximal, characteristic
3#26020379C211yesmaximal, characteristic
4#26018136C211yesmaximal, characteristic
5#26018093C211yesmaximal, characteristic
6#26018091C211yesmaximal, characteristic
7C89×#1948C211yesmaximal, characteristic
8C2×#15668004—21no—
9C2×#15667943—21no—
10C2×C4⋊#10623494C2211yescharacteristic
11C2×C4×#10623494C2211yescharacteristic
12#19151532—22no—
13#19151496—21no—
14#19151494—21no—
15#19151468C2211yescharacteristic
16#19151467—21no—
17C4×C4⋊Dic89C2211yescharacteristic
18#19151360—22no—
19#19151360—21no—
20C2×#106×C89—21no—
21C2×C4⋊C8×C89C2211yescharacteristic
22#384×C89C2211yescharacteristic
23C4×C4⋊C4×C89C2211yescharacteristic
24#340×C89—21no—
25C22×#10623494—21no—
26C22×#10623494—21no—
27#15668004—42no—
28#15668004—42no—
29C2×C4⋊Dic89—21no—
30C2×C4⋊Dic89—21no—
31C2×C4⋊Dic89D411yescharacteristic
32C2×C4⋊Dic89D411yescharacteristic
33C2×C4⋊Dic89D412yes—
34C2×#13083553—22no—
35C2×#13083553—22no—
36C2×#13083553—22no—
37C2×C4×Dic89—21no—
38#15667943—42no—
39C4⋊#10623494—21no—
40C4⋊#10623494Q812yes—
41C4×#10623494—21no—
42C4×#10623494Q812yes—
43#15667919—22no—
44#15667919—21no—
45C22×C8×C89—21no—
46C2×C4⋊C4×C89—21no—
47C2×C4⋊C4×C89—21no—
48C2×C4⋊C4×C89—21no—
49C2×C4⋊C4×C89D411yescharacteristic
50C2×C4⋊C4×C89D411yescharacteristic
51C2×C42×C89—21no—
52C2×C42×C89C2311yescharacteristic
53#106×C89—42no—
54C4⋊C8×C89—21no—
55C4⋊C8×C89Q812yes—
56#95×C89—21no—
57C22×Dic89—21no—
58C22×Dic89—21no—
59C22×Dic89—22no—
60C2×#10623494—22no—
61C2×#10623494—22no—
62C2×#10623494—41no—
63C2×#10623494—41no—
64C2×#10623494—41no—
65C2×#10623494—41no—
66C4⋊Dic89—21no—
67C4⋊Dic89—21no—
68C4⋊Dic89—41no—
69C4⋊Dic89—41no—
70C4⋊Dic89—41no—
71C4⋊Dic89—41no—
72C4⋊Dic89—42no—
73C4⋊Dic89SD1612yes—
74C4⋊Dic89SD1612yes—
75#13083553—44no—
76#13083553—44no—
77#13083553—82no—
78C4×Dic89—41no—
79C4×Dic89—41no—
80C22×C4×C89—21no—
81C22×C4×C89—21no—
82C22×C4×C89—41no—
83C22×C4×C89C2×D411yescharacteristic
84C22×C4×C89C2×D411yescharacteristic
85C22×C4×C89C2×D411yescharacteristic
86C2×C8×C89—22no—
87C2×C8×C89—41no—
88C2×C8×C89—41no—
89C4⋊C4×C89—21no—
90C4⋊C4×C89—21no—
91C4⋊C4×C89—21no—
92C4⋊C4×C89—21no—
93C4⋊C4×C89—42no—
94C4⋊C4×C89—41no—
95C4⋊C4×C89—41no—
96C4⋊C4×C89—41no—
97C4⋊C4×C89—41no—
98C42×C89—41no—
99C42×C89—41no—
100C42×C89C2×Q811yescharacteristic
101C42×C89C2×Q811yescharacteristic
102C42×C89C2×Q811yescharacteristic
103C42×C89#4111yescharacteristic
104C23×C89#12011yescharacteristic
105C2×C4×C89—21no—
106C2×C4×C89—21no—
107C2×C4×C89—41no—
108C2×C4×C89—41no—
109C2×C4×C89—41no—
110C2×C4×C89—41no—
111C2×C4×C89—41no—
112C2×C4×C89—41no—
113C2×C4×C89—81no—
114C2×C4×C89—81no—
115C2×C4×C89—81no—
116C2×C4×C89C22⋊Q811yescharacteristic
117C2×C4×C89C22⋊Q811yescharacteristic
118C2×C4×C89C22⋊Q811yescharacteristic
119C2×C4×C89C4⋊Q811yescharacteristic
120C2×C4×C89C4⋊Q811yescharacteristic
121C2×C4×C89C4⋊Q811yescharacteristic
122C2×C4×C89C2×SD1611yescharacteristic
123C2×C4×C89C2×SD1611yescharacteristic
124C2×C4×C89#13511yescharacteristic
125C2×C4×C89#13511yescharacteristic
126C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C89#13711yescharacteristic
129C2×C4×C89#13711yescharacteristic
130C2×Dic89—22no—
131C2×Dic89—22no—
132C2×Dic89—41no—
133C2×Dic89—41no—
134C2×Dic89—41no—
135C2×Dic89—41no—
136C2×Dic89—42no—
137C2×Dic89—42no—
138C2×Dic89—42no—
139C8×C89—42no—
140#10623494—42no—
141#10623494—42no—
142C22×C89#39511yescharacteristic
143C22×C89#44811yescharacteristic
144C22×C89#44911yescharacteristic
145C22×C89#45011yescharacteristic
146C22×C89#45211yescharacteristic
147C22×C89#45411yescharacteristic
148C22×C89#45511yescharacteristic
149C4×C89—21no—
150C4×C89—21no—
151C4×C89—81no—
152C4×C89—81no—
153C4×C89—161no—
154C4×C89#47511yescharacteristic
155C4×C89#47611yescharacteristic
156C4×C89#47811yescharacteristic
157C4×C89#47911yescharacteristic
158C4×C89#49811yescharacteristic
159C4×C89#49911yescharacteristic
160C4×C89#50111yescharacteristic
161C4×C89#50111yescharacteristic
162Dic89—42no—
163Dic89—42no—
164Dic89—82no—
165#10582—891nomaximal, Sylow
166C2×C89#162611yescharacteristic
167C2×C89#163511yescharacteristic
168C2×C89#164211yescharacteristic
169C2×C89#164311yescharacteristic
170C2×C89#205811yescharacteristic
171C2×C89#205911yescharacteristic
172C2×C89#206111yescharacteristic
173#2552—891no—
174#1948—891no—
175#1948D8911yescharacteristic
176#1834—891no—
177#1296—891no—
178#1279—891no—
179#1279—891no—
180C89#1058211yescharacteristic, complemented, Sylow
181C2×#125—1781no—
182C2×#106—21no—
183C2×#106—1781no—
184C2×C4⋊C8—891no—
185C2×C4⋊C8C2×D8911yescharacteristic
186C2×C4×C8—891no—
187#398—1782no—
188#389—1781no—
189#389—1781no—
190#384—891no—
191#384—1781no—
192#384C2×D8911yescharacteristic
193C4×C4⋊C4—891no—
194C4×C4⋊C4C2×D8911yescharacteristic
195#340—21no—
196#340—1782no—
197#340—1781no—
198C22×C8—21no—
199C22×C8—1781no—
200C22×C8—1781no—
201#125—3562no—
202#125—3562no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—891no—
207C2×C4⋊C4—891no—
208C2×C4⋊C4—892no—
209C2×C4⋊C4—1782no—
210C2×C4⋊C4—1782no—
211C2×C4⋊C4—1781no—
212C2×C4⋊C4—1781no—
213C2×C4⋊C4—1782no—
214C2×C4⋊C4C22⋊D8911yescharacteristic
215C2×C4⋊C4C22⋊D8911yescharacteristic
216C2×C42—21no—
217C2×C42—1781no—
218C2×C42C22×D8911yescharacteristic
219#106—42no—
220#106—3562no—
221C4⋊C8—21no—
222C4⋊C8—892no—
223C4⋊C8—1781no—
224C4⋊C8Dic17812yes—
225C4×C8—892no—
226C4×C8—1781no—
227#95—21no—
228#95—1782no—
229#95—1781no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—1782no—
234C22×C4—1781no—
235C22×C4—1781no—
236C22×C4D4×D8911yescharacteristic
237C22×C4D4×D8911yescharacteristic
238C22×C4C2×C22⋊D8911yescharacteristic
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—3561no—
243C2×C8—3561no—
244C2×C8—3561no—
245C2×C8—3561no—
246C2×C8—1782no—
247C2×C8—1782no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—892no—
258C4⋊C4—892no—
259C4⋊C4—3562no—
260C4⋊C4—3564no—
261C4⋊C4—3564no—
262C4⋊C4—3561no—
263C4⋊C4—3561no—
264C4⋊C4—3561no—
265C4⋊C4—3561no—
266C4⋊C4—1781no—
267C4⋊C4—1781no—
268C4⋊C4—7122no—
269C42—41no—
270C42—41no—
271C42—3561no—
272C42—3561no—
273C42Q8×D8911yescharacteristic
274C42Q8×D8911yescharacteristic
275C42C2×Dic17811yescharacteristic
276C42#1308357411yescharacteristic
277C23#1566798011yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—3561no—
290C2×C4—3561no—
291C2×C4—3561no—
292C2×C4—3561no—
293C2×C4—3562no—
294C2×C4—3562no—
295C2×C4—3562no—
296C2×C4—1782no—
297C2×C4—1782no—
298C2×C4#1566799011yescharacteristic
299C2×C4#1566799011yescharacteristic
300C2×C4#1566799111yescharacteristic
301C2×C4#1566801011yescharacteristic
302C2×C4#1566801111yescharacteristic
303C2×C4#1566801111yescharacteristic
304C2×C4SD16×D8911yescharacteristic
305C2×C4SD16×D8911yescharacteristic
306C2×C4#1566803211yescharacteristic
307C2×C4#1566803211yescharacteristic
308C2×C4#1566803811yescharacteristic
309C2×C4#1566803911yescharacteristic
310C2×C4#1566804111yescharacteristic
311C2×C4#1566804311yescharacteristic
312C8—42no—
313C8—3562no—
314C8—3562no—
315C22#1915151811yescharacteristic
316C22#1915173511yescharacteristic
317C22#1915174111yescharacteristic
318C22#1915174711yescharacteristic
319C22#1915175811yescharacteristic
320C22#1915176911yescharacteristic
321C22#1915177411yescharacteristic
322C4—21no—
323C4—21no—
324C4—81no—
325C4—81no—
326C4—161no—
327C4—3562no—
328C4—3562no—
329C4—7122no—
330C4#1915188211yescharacteristic
331C4#1915188711yescharacteristic
332C4#1915189811yescharacteristic
333C4#1915190511yescharacteristic
334C4#1915199511yescharacteristic
335C4#1915199711yescharacteristic
336C4#1915200411yescharacteristic
337C4#1915200511yescharacteristic
338C2#2601934611yescharacteristic
339C2#2601940211yescharacteristic
340C2#2601945011yescharacteristic
341C2#2601945811yescharacteristic
342C2#2602158311yescharacteristic
343C2#2602159011yescharacteristic
344C2#2602160011yescharacteristic
subgroupG/H
centerC23#15667980
commutator, cosocleC2×C42×C89C23
FrattiniC2×C42C22×D89
FittingC89×#1948C2
socle, prime radicalC23×C89#120
hypercenter#1948D89
hypocenterC89#10582
LCSG ▷ C2×C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
180. C89#10582semidirect

Maximal subgroups

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

subgroupindex
#260252872
#260210132
#260203792
#260181362
#260180932
#260180912
C89×#19482
#1058289