Finite Group Databasebeta

← #36558320

Subgroups — #36558320

Order 22784.

1790 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151636C211yesmaximal, characteristic, complemented
2C2×#19151603C212yesmaximal, complemented
3C2×#19151553C211yesmaximal, characteristic, complemented
4#26022834C212yesmaximal, complemented
5#26021129C214yesmaximal, complemented
6#26021119C214yesmaximal, complemented
7C23×D8×C89C211yesmaximal, characteristic
8C22×#13083558C2211yescharacteristic
9C2×#15667971C2212yes—
10C2×#15667971C2211yescharacteristic
11#19151636—24no—
12C2×#15667946C2212yescomplemented
13C2×#15667944C2212yescomplemented
14#19151603—28no—
15C2×#15667933C2218yescomplemented
16#19151553—24no—
17C2×#15667922C2212yescomplemented
18C2×C8×Dic89C2212yescomplemented
19#19151501—24no—
20#19151501C2218yescomplemented
21#19151454—24no—
22C23×D4×C89C2212yes—
23C22×D8×C89—24no—
24C22×D8×C89—24no—
25C22×D8×C89C418yescomplemented
26C22×D8×C89C2212yes—
27C22×D8×C89C2212yes—
28C23×C8×C89C2211yescharacteristic
29C2×#13083558—22no—
30C2×#13083558—22no—
31C2×#13083558D414yes—
32C22×#10623494C2312yes—
33#15667971—22no—
34#15667971—24no—
35#15667971—24no—
36#15667971—22no—
37C2×C4⋊Dic89D412yes—
38C2×C4⋊Dic89C2314yes—
39C2×C22⋊Dic89—216no—
40C2×C22⋊Dic89—22no—
41C2×C22⋊Dic89D412yes—
42C2×C4×Dic89—22no—
43C2×C4×Dic89C2312yes—
44#15667946—24no—
45#15667944—24no—
46#15667933—216no—
47#15667922—24no—
48C8×Dic89—24no—
49C25×C89—22no—
50C22×D4×C89—24no—
51C22×D4×C89—24no—
52C22×D4×C89—24no—
53C22×D4×C89—22no—
54C22×D4×C89—24no—
55C22×D4×C89—24no—
56C22×D4×C89C2×C418yes—
57C22×D4×C89C2314yes—
58C23×C4×C89C2311yescharacteristic
59C2×D8×C89—232no—
60C2×D8×C89—24no—
61C2×D8×C89—24no—
62C2×D8×C89—24no—
63C2×D8×C89—24no—
64C2×D8×C89C2×C418yes—
65C2×D8×C89C2×C418yes—
66C22×C8×C89C2×C414yes—
67C22×C8×C89D414yes—
68C22×C8×C89D414yes—
69C22×C8×C89C2311yescharacteristic
70C22×C8×C89C2311yescharacteristic
71C22×Dic89—24no—
72C22×Dic89C2×D414yes—
73#13083558—28no—
74C2×#10623494—22no—
75C2×#10623494—22no—
76C2×#10623494C2×D414yes—
77C4⋊Dic89—24no—
78C4⋊Dic89—24no—
79C4⋊Dic89—24no—
80C22⋊Dic89—24no—
81C22⋊Dic89—44no—
82C22⋊Dic89—816no—
83C4×Dic89—22no—
84C4×Dic89—22no—
85C4×Dic89—44no—
86C24×C89—24no—
87C24×C89—28no—
88C24×C89—44no—
89C24×C89—44no—
90C24×C89—88no—
91C24×C89C2×D411yescharacteristic
92C2×D4×C89—216no—
93C2×D4×C89—216no—
94C2×D4×C89—28no—
95C2×D4×C89—24no—
96C2×D4×C89—24no—
97C2×D4×C89—24no—
98C2×D4×C89—24no—
99C2×D4×C89—24no—
100C2×D4×C89—48no—
101C2×D4×C89—44no—
102C2×D4×C89—48no—
103C2×D4×C89—44no—
104C2×D4×C89C22×C418yes—
105C22×C4×C89C22×C412yes—
106C22×C4×C89C2×D412yes—
107C22×C4×C89C2×D412yes—
108C22×C4×C89C2×D411yescharacteristic
109C22×C4×C89C2×D412yes—
110C22×C4×C89#4112yes—
111C22×C4×C89#4112yes—
112C22×C4×C89C2411yescharacteristic
113D8×C89—216no—
114D8×C89—216no—
115C2×C8×C89C22⋊C4116yescomplemented
116C2×C8×C89C22×C412yes—
117C2×C8×C89C22×C412yes—
118C2×C8×C89C2×D412yes—
119C2×C8×C89C2×D412yes—
120C2×C8×C89C2×D412yes—
121C2×C8×C89C2×D412yes—
122C23×C89—24no—
123C23×C89—28no—
124C23×C89—44no—
125C23×C89—44no—
126C23×C89—416no—
127C23×C89—416no—
128C23×C89—816no—
129C23×C89—816no—
130C23×C89—816no—
131C23×C89C4×D412yes—
132C23×C89#12112yes—
133C23×C89#12712yes—
134C23×C89C22×D411yescharacteristic
135D4×C89—28no—
136D4×C89—28no—
137D4×C89—44no—
138D4×C89—44no—
139D4×C89—416no—
140C2×C4×C89C2×C22⋊C414yes—
141C2×C4×C89C4×D414yes—
142C2×C4×C89C4×D412yes—
143C2×C4×C89#12114yes—
144C2×C4×C89#12112yes—
145C2×C4×C89#12414yes—
146C2×C4×C89#12712yes—
147C2×C4×C89C23×C411yescharacteristic
148C2×C4×C89C22×D411yescharacteristic
149C2×C4×C89C22×D411yescharacteristic
150C2×C4×C89C22×D411yescharacteristic
151C2×C4×C89C2×#4111yescharacteristic
152C2×C4×C89C2×#4111yescharacteristic
153C2×Dic89—24no—
154C2×Dic89—28no—
155C2×Dic89—44no—
156C2×Dic89—48no—
157C8×C89C2×C22⋊C414yes—
158C8×C89C2×C22⋊C414yes—
159#10623494—24no—
160C22×C89—24no—
161C22×C89—24no—
162C22×C89—24no—
163C22×C89—416no—
164C22×C89—48no—
165C22×C89—832no—
166C22×C89—88no—
167C22×C89—88no—
168C22×C89#39014yes—
169C22×C89C2×C4×D411yescharacteristic
170C22×C89C2×#12111yescharacteristic
171C22×C89C2×#12711yescharacteristic
172C22×C89#57614yes—
173C4×C89#39014yes—
174C4×C89C22×C22⋊C411yescharacteristic
175C4×C89C2×C4×D411yescharacteristic
176C4×C89C2×#12111yescharacteristic
177C4×C89C2×#12411yescharacteristic
178Dic89—44no—
179Dic89—84no—
180#21233—891nomaximal, Sylow
181C2×C89—24no—
182C2×C89—88no—
183C2×C89C2×#39011yescharacteristic
184C2×C89#294812yes—
185C2×C89#305812yes—
186C2×C89#315512yes—
187C23×D8D8911yescharacteristic
188C2×#427—891no—
189C2×#418—892no—
190C2×#405—891no—
191#2309—892no—
192#1972—894no—
193#1970—894no—
194C89#2123311yescharacteristic, complemented, Sylow
195C23×D4C2×D8912yes—
196C22×D8—24no—
197C22×D8—24no—
198C22×D8Dic8918yescomplemented
199C22×D8C2×D8912yes—
200C22×D8C2×D8912yes—
201C22×M16—891no—
202C23×C8C2×D8911yescharacteristic
203C2×#117—891no—
204C2×#117—892no—
205#427—1784no—
206C2×#107—892no—
207C2×#106—892no—
208#418—1788no—
209C2×#102—898no—
210#405—1784no—
211C2×#97—892no—
212C2×C4×C8—892no—
213#390—898no—
214#390—1784no—
215#379—1784no—
216C25—22no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4C2×Dic8918yes—
224C22×D4C22×D8914yes—
225C23×C4C22×D8911yescharacteristic
226C2×D8—232no—
227C2×D8—24no—
228C2×D8—24no—
229C2×D8—24no—
230C2×D8—24no—
231C2×D8C2×Dic8918yes—
232C2×D8C2×Dic8918yes—
233C2×M16—894no—
234C2×M16—1782no—
235C2×M16—1782no—
236C22×C8—892no—
237C22×C8C2×Dic8914yes—
238C22×C8C22⋊D8914yes—
239C22×C8C22⋊D8914yes—
240C22×C8C22×D8911yescharacteristic
241C22×C8C22×D8911yescharacteristic
242#117—1784no—
243#117—1784no—
244#117—1782no—
245#117—1782no—
246C2×C4⋊C4—892no—
247C2×C4⋊C4—894no—
248C2×C22⋊C4—892no—
249C2×C22⋊C4—1782no—
250C2×C22⋊C4—17816no—
251C2×C42—892no—
252C2×C42—1782no—
253#107—1784no—
254#106—1784no—
255#102—17816no—
256#97—1784no—
257C4×C8—1784no—
258C24—24no—
259C24—28no—
260C24—44no—
261C24—44no—
262C24—88no—
263C24D4×D8911yescharacteristic
264C2×D4—216no—
265C2×D4—28no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—216no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4C22×Dic8918yes—
277C22×C4—894no—
278C22×C4—1784no—
279C22×C4C22×Dic8912yes—
280C22×C4D4×D8911yescharacteristic
281C22×C4D4×D8912yes—
282C22×C4C2×C22⋊D8912yes—
283C22×C4C2×C22⋊D8912yes—
284C22×C4#1308357312yes—
285C22×C4#1308357312yes—
286C22×C4C23×D8911yescharacteristic
287D8—216no—
288D8—216no—
289M16—1788no—
290C2×C8—894no—
291C2×C8—1782no—
292C2×C8—1782no—
293C2×C8C22⋊Dic89116yescomplemented
294C2×C8C22×Dic8912yes—
295C2×C8C22×Dic8912yes—
296C2×C8C2×C22⋊D8912yes—
297C2×C8C2×C22⋊D8912yes—
298C2×C8C2×C22⋊D8912yes—
299C2×C8C2×C22⋊D8912yes—
300C4⋊C4—1784no—
301C4⋊C4—1784no—
302C4⋊C4—1784no—
303C22⋊C4—3564no—
304C22⋊C4—1784no—
305C22⋊C4—71216no—
306C42—3564no—
307C42—1782no—
308C42—1782no—
309C23—24no—
310C23—28no—
311C23—44no—
312C23—416no—
313C23—416no—
314C23—44no—
315C23—816no—
316C23—816no—
317C23—816no—
318C23D4×Dic8912yes—
319C23#1566798312yes—
320C23#1566800812yes—
321C23C2×D4×D8911yescharacteristic
322D4—28no—
323D4—28no—
324D4—416no—
325D4—44no—
326D4—44no—
327C2×C4—3564no—
328C2×C4—3568no—
329C2×C4—1788no—
330C2×C4—1784no—
331C2×C4C2×C22⋊Dic8914yes—
332C2×C4D4×Dic8912yes—
333C2×C4D4×Dic8914yes—
334C2×C4#1566798312yes—
335C2×C4#1566798314yes—
336C2×C4#1566799914yes—
337C2×C4#1566800812yes—
338C2×C4C23×Dic8911yescharacteristic
339C2×C4C2×D4×D8911yescharacteristic
340C2×C4C22×C22⋊D8911yescharacteristic
341C2×C4C22×C22⋊D8911yescharacteristic
342C2×C4C2×#1308357311yescharacteristic
343C2×C4C2×#1308357311yescharacteristic
344C8—1784no—
345C8C2×C22⋊Dic8914yes—
346C8C2×C22⋊Dic8914yes—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—416no—
351C22—48no—
352C22—832no—
353C22—88no—
354C22—88no—
355C22#1915150114yes—
356C22C2×D4×Dic8911yescharacteristic
357C22C2×#1566798311yescharacteristic
358C22C2×#1566800811yescharacteristic
359C22#1915239914yes—
360C4—3564no—
361C4—7124no—
362C4#1915150114yes—
363C4C22×C22⋊Dic8911yescharacteristic
364C4C2×D4×Dic8911yescharacteristic
365C4C2×#1566798311yescharacteristic
366C4C2×#1566799911yescharacteristic
367C2—24no—
368C2—88no—
369C2C2×#1915150111yescharacteristic
370C2#2602910412yes—
371C2#2602989612yes—
372C2#2603073312yes—
subgroupG/H
centerC23C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
Fitting, prime radicalC23×D8×C89C2
socleC23×C89C22×D4
cosocleC22×C4×C89C24
hypercenterC23×D8D89
hypocenterC89#21233
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19151636C2semidirect
2. C2×#19151603C2semidirect
3. C2×#19151553C2semidirect
4. #26022834C2semidirect
5. #26021129C2semidirect
6. #26021119C2semidirect
12. C2×#15667946C22semidirect
13. C2×#15667944C22semidirect
15. C2×#15667933C22semidirect
17. C2×#15667922C22semidirect
18. C2×C8×Dic89C22semidirect
20. #19151501C22semidirect
25. C22×D8×C89C4semidirect
115. C2×C8×C89C22⋊C4semidirect
194. C89#21233semidirect
198. C22×D8Dic89semidirect
293. C2×C8C22⋊Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191516362
C2×#191516032
C2×#191515532
#260228342
#260211292
#260211192
C23×D8×C892
#2123389