Finite Group Databasebeta

← #36556533

Subgroups — #36556533

Order 22784.

906 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151625C211yesmaximal, characteristic
2C2×#19151556C211yesmaximal, characteristic
3#26022655C211yesmaximal, characteristic
4#26020819C214yesmaximal, complemented
5#26020812C214yesmaximal, complemented
6#26019948C211yesmaximal, characteristic
7#26019945C211yesmaximal, characteristic
8C2×#19151352C211yesmaximal, characteristic
9C89×#1729C211yesmaximal, characteristic
10C22×#13083558C2211yescharacteristic
11C23×#10623494C2211yescharacteristic
12C2×#15667971C2211yescharacteristic
13C22×C4×Dic89C2211yescharacteristic
14#19151625—24no—
15C2×C4⋊#10623494C2212yes—
16C2×C4⋊#10623494C2212yes—
17#19151556—24no—
18C2×#15667922C2212yes—
19C2×C8×Dic89C2212yes—
20#19151467C411yescharacteristic
21#19151467C411yescharacteristic
22#19151463C411yescharacteristic
23#19151463C411yescharacteristic
24#19151456C414yes—
25C4×#13083553C2214yes—
26#19151352C2212yes—
27#19151352C2212yes—
28#19151352C2214yes—
29#19151352C2214yes—
30C22×M16×C89C2211yescharacteristic
31C23×C8×C89C2211yescharacteristic
32C2×#117×C89C2211yescharacteristic
33#336×C89C2214yes—
34C23×Dic89C2×C412yes—
35C2×#13083558—22no—
36C2×#13083558—22no—
37C2×#13083558—22no—
38C22×#10623494—22no—
39C22×#10623494—22no—
40C22×#10623494D414yes—
41C22×#10623494C2312yes—
42C22×#10623494C2311yescharacteristic
43C22×#10623494C2311yescharacteristic
44#15667971—22no—
45#15667971—22no—
46C2×C4⋊Dic89C2×C411yescharacteristic
47C2×C4⋊Dic89C2×C411yescharacteristic
48C2×#13083553C2×C414yes—
49C2×#13083553C2×C414yes—
50C2×C22⋊Dic89C2×C412yes—
51C2×C4×Dic89—22no—
52C2×C4×Dic89—22no—
53C2×C4×Dic89C2×C411yescharacteristic
54C2×C4×Dic89C2×C411yescharacteristic
55C2×C4×Dic89C2312yes—
56C2×C4×Dic89C2312yes—
57C4⋊#10623494—24no—
58C4⋊#10623494—24no—
59#15667922—24no—
60C8×Dic89—24no—
61#15667919C2×C414yes—
62#15667919C2×C414yes—
63C23×C4×C89C2311yescharacteristic
64C2×M16×C89—22no—
65C2×M16×C89—22no—
66C2×M16×C89Q814yes—
67C22×C8×C89—22no—
68C22×C8×C89—22no—
69C22×C8×C89—22no—
70C22×C8×C89C2312yes—
71C22×C8×C89C2312yes—
72#117×C89—22no—
73#117×C89—22no—
74C2×C4⋊C4×C89C2×C411yescharacteristic
75C2×C4⋊C4×C89C2×C411yescharacteristic
76C2×C22⋊C4×C89C2×C412yes—
77C2×C42×C89C2312yes—
78C22×Dic89—24no—
79C22×Dic89—24no—
80C22×Dic89—42no—
81C22×Dic89C22×C412yes—
82C22×Dic89C22×C412yes—
83C22×Dic89C22×C412yes—
84C22×Dic89C22×C412yes—
85#13083558—44no—
86C2×#10623494—28no—
87C2×#10623494—21no—
88C2×#10623494—21no—
89C2×#10623494—21no—
90C2×#10623494—21no—
91C2×#10623494—22no—
92C2×#10623494—22no—
93C2×#10623494—22no—
94C2×#10623494C2×D412yes—
95C2×#10623494C2×D412yes—
96C4⋊Dic89—22no—
97C4⋊Dic89—22no—
98#13083553—28no—
99#13083553—44no—
100C22⋊Dic89—24no—
101C4×Dic89—22no—
102C4×Dic89—22no—
103C4×Dic89—22no—
104C4×Dic89—22no—
105C4×Dic89—22no—
106C4×Dic89—22no—
107C24×C89C22×C411yescharacteristic
108C22×C4×C89C22×C412yes—
109C22×C4×C89C22×C412yes—
110C22×C4×C89C22×C411yescharacteristic
111C22×C4×C89C2×D412yes—
112C22×C4×C89C2×Q812yes—
113C22×C4×C89#4112yes—
114C22×C4×C89#4112yes—
115C22×C4×C89#4112yes—
116C22×C4×C89#4112yes—
117C22×C4×C89C2411yescharacteristic
118M16×C89—28no—
119C2×C8×C89—22no—
120C2×C8×C89—22no—
121C2×C8×C89—22no—
122C2×C8×C89—22no—
123C2×C8×C89—22no—
124C2×C8×C89—44no—
125C2×C8×C89C2×Q814yes—
126C4⋊C4×C89—22no—
127C4⋊C4×C89—22no—
128C22⋊C4×C89—24no—
129C42×C89—22no—
130C42×C89—22no—
131C23×C89—22no—
132C23×C89—22no—
133C23×C89#11712yes—
134C23×C89C4×D412yes—
135C23×C89C4×Q812yes—
136C23×C89C23×C411yescharacteristic
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—24no—
140C2×C4×C89—24no—
141C2×C4×C89#11712yes—
142C2×C4×C89C4×D412yes—
143C2×C4×C89C4×Q812yes—
144C2×C4×C89C22⋊Q814yes—
145C2×C4×C89#12314yes—
146C2×C4×C89#12514yes—
147C2×C4×C89#12614yes—
148C2×C4×C89C22×D411yescharacteristic
149C2×C4×C89C22×Q811yescharacteristic
150C2×C4×C89C2×#4111yescharacteristic
151C2×C4×C89C2×#4111yescharacteristic
152C2×C4×C89C2×#4111yescharacteristic
153C2×C4×C89C2×#4111yescharacteristic
154C2×Dic89—22no—
155C2×Dic89—22no—
156C2×Dic89—22no—
157C2×Dic89—22no—
158C2×Dic89—22no—
159C2×Dic89—24no—
160C2×Dic89—22no—
161C2×Dic89—24no—
162C2×Dic89—22no—
163C2×Dic89—48no—
164C2×Dic89#13114yes—
165C8×C89—24no—
166C8×C89—42no—
167#10623494—22no—
168#10623494—22no—
169#10623494—42no—
170C22×C89—24no—
171C22×C89—22no—
172C22×C89—22no—
173C22×C89—24no—
174C22×C89#38714yes—
175C22×C89C2×#11711yescharacteristic
176C22×C89C2×C4×D411yescharacteristic
177C22×C89C2×C4×Q811yescharacteristic
178C22×C89C2×#13112yes—
179C22×C89#56812yes—
180C4×C89—42no—
181C4×C89—42no—
182C4×C89#38714yes—
183C4×C89C2×C22⋊Q811yescharacteristic
184C4×C89C2×#12311yescharacteristic
185C4×C89C2×#12511yescharacteristic
186C4×C89C2×#12611yescharacteristic
187Dic89—24no—
188Dic89—42no—
189Dic89—42no—
190Dic89—42no—
191#21025—891nomaximal, Sylow
192C2×C89—24no—
193C2×C89C2×#38711yescharacteristic
194C2×C89#292412yes—
195C2×C89#293111yescharacteristic
196C2×C89#293211yescharacteristic
197C2×C89#296212yes—
198C2×#424—891no—
199C2×#405—891no—
200#2285—891no—
201#1917—894no—
202#1916—894no—
203#1729—891no—
204#1729D8911yescharacteristic
205#1728—891no—
206C2×#336—891no—
207C89#2102511yescharacteristic, complemented, Sylow
208C22×M16—891no—
209C22×M16C2×D8911yescharacteristic
210C23×C8—891no—
211C23×C8C2×D8911yescharacteristic
212C2×#117—891no—
213C2×#117C2×D8911yescharacteristic
214C22×C42—891no—
215#424—1784no—
216C2×C4⋊C8—892no—
217C2×C4⋊C8—892no—
218#405—1784no—
219C2×#97—892no—
220C2×C4×C8—892no—
221#384—891no—
222#384—891no—
223#382—891no—
224#382—891no—
225#380—894no—
226C4×C4⋊C4—894no—
227#336—892no—
228#336—894no—
229#336—894no—
230#336—892no—
231#336C2×D8914yes—
232C23×C4—892no—
233C23×C4C22×D8911yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—1782no—
237C2×M16—1782no—
238C2×M16—1782no—
239C2×M16Dic17814yes—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—894no—
244C22×C8—891no—
245C22×C8—891no—
246C22×C8—892no—
247C22×C8—1782no—
248C22×C8—1782no—
249C22×C8C22×D8912yes—
250C22×C8C22×D8912yes—
251#117—22no—
252#117—22no—
253#117—1782no—
254#117—1782no—
255C2×C4⋊C4—894no—
256C2×C4⋊C4—891no—
257C2×C4⋊C4—891no—
258C2×C4⋊C4—894no—
259C2×C4⋊C4C4×D8911yescharacteristic
260C2×C4⋊C4C4×D8911yescharacteristic
261C2×C22⋊C4—892no—
262C2×C22⋊C4C4×D8912yes—
263C2×C42—891no—
264C2×C42—891no—
265C2×C42—892no—
266C2×C42—892no—
267C2×C42—1782no—
268C2×C42—1782no—
269C2×C42C22×D8912yes—
270C4⋊C8—1784no—
271C4⋊C8—1784no—
272#97—1784no—
273C4×C8—1784no—
274#95—894no—
275#95—894no—
276C24C2×C4×D8911yescharacteristic
277C22×C4—892no—
278C22×C4—892no—
279C22×C4—892no—
280C22×C4—892no—
281C22×C4—3562no—
282C22×C4—1784no—
283C22×C4—1784no—
284C22×C4C2×C4×D8912yes—
285C22×C4C2×C4×D8912yes—
286C22×C4C2×C4×D8911yescharacteristic
287C22×C4D4×D8912yes—
288C22×C4C2×Dic17812yes—
289C22×C4#1308357312yes—
290C22×C4#1308357312yes—
291C22×C4#1308357412yes—
292C22×C4#1308357512yes—
293C22×C4C23×D8911yescharacteristic
294M16—28no—
295M16—3564no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—44no—
302C2×C8—892no—
303C2×C8—892no—
304C2×C8—1782no—
305C2×C8—1782no—
306C2×C8—1782no—
307C2×C8—1788no—
308C2×C8—1781no—
309C2×C8—1781no—
310C2×C8—1781no—
311C2×C8—1781no—
312C2×C8C2×Dic17814yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—3564no—
316C4⋊C4—1788no—
317C4⋊C4—1782no—
318C4⋊C4—1782no—
319C22⋊C4—24no—
320C22⋊C4—1784no—
321C42—22no—
322C42—22no—
323C42—1782no—
324C42—1782no—
325C42—1782no—
326C42—1782no—
327C42—1782no—
328C42—1782no—
329C23—22no—
330C23—22no—
331C23#1566797012yes—
332C23#1566797212yes—
333C23C4×Dic17812yes—
334C23C22×C4×D8911yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—894no—
340C2×C4—3568no—
341C2×C4—1782no—
342C2×C4—1782no—
343C2×C4—1782no—
344C2×C4—1784no—
345C2×C4—1782no—
346C2×C4—1784no—
347C2×C4—1782no—
348C2×C4—1782no—
349C2×C4—1782no—
350C2×C4#1566797012yes—
351C2×C4#1566797212yes—
352C2×C4C4×Dic17812yes—
353C2×C4#1566798914yes—
354C2×C4#1566799714yes—
355C2×C4#1566800314yes—
356C2×C4#1566800514yes—
357C2×C4C2×D4×D8911yescharacteristic
358C2×C4C22×Dic17811yescharacteristic
359C2×C4C2×#1308357311yescharacteristic
360C2×C4C2×#1308357311yescharacteristic
361C2×C4C2×#1308357411yescharacteristic
362C2×C4C2×#1308357511yescharacteristic
363C8—24no—
364C8—42no—
365C8—3562no—
366C8—1782no—
367C8—1782no—
368C22—24no—
369C22—24no—
370C22—22no—
371C22—22no—
372C22#1915148514yes—
373C22C2×#1566797011yescharacteristic
374C22C2×#1566797211yescharacteristic
375C22C2×C4×Dic17811yescharacteristic
376C22#131×D8912yes—
377C22#1915236312yes—
378C4—42no—
379C4—42no—
380C4—3562no—
381C4—3562no—
382C4—3562no—
383C4—1784no—
384C4#1915148514yes—
385C4C2×#1566798911yescharacteristic
386C4C2×#1566799711yescharacteristic
387C4C2×#1566800311yescharacteristic
388C4C2×#1566800511yescharacteristic
389C2—24no—
390C2C2×#1915148511yescharacteristic
391C2#2602890912yes—
392C2#2602896011yescharacteristic
393C2#2602896711yescharacteristic
394C2#2602919612yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC89×#1729C2
prime radicalC24×C89C22×C4
cosocleC22×C4×C89C24
hypercenter#1729D89
hypocenterC89#21025
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #26020819C2semidirect
5. #26020812C2semidirect
207. C89#21025semidirect

Maximal subgroups

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

subgroupindex
C2×#191516252
C2×#191515562
#260226552
#260208192
#260208122
#260199482
#260199452
C2×#191513522
C89×#17292
#2102589