Finite Group Databasebeta

← #36553817

Subgroups — #36553817

Order 22784.

1498 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
1C2×#19151625C212yesmaximal, complemented
2C2×#19151565C212yesmaximal
3#26020236C214yesmaximal, complemented
4#26020232C214yesmaximal, complemented
5#26019946C211yesmaximal, characteristic, complemented
6C2×#19151353C211yesmaximal, characteristic, complemented
7C2×C4×#41×C89C211yesmaximal, characteristic
8C22×#13083558C2212yes—
9C23×#10623494C2212yes—
10#19151625—28no—
11C2×C4⋊#10623494C2218yes—
12#19151565—28no—
13C2×C22⋊#10623494C2218yes—
14#19151353C2212yes—
15#19151353C2212yes—
16#19151353C2212yes—
17#19151353C2212yes—
18C22×#41×C89C2211yescharacteristic
19C4×#41×C89—28no—
20C2×C4×Q8×C89C412yes—
21C2×C4×D4×C89C412yes—
22C2×C4×D4×C89C414yes—
23C2×#117×C89C2212yes—
24C2×#117×C89C2211yescharacteristic
25C22×C42×C89C2211yescharacteristic
26C22×C42×C89C2212yes—
27C2×#13083558—24no—
28C2×#13083558—24no—
29C2×#13083558—24no—
30C22×#10623494—24no—
31C22×#10623494—24no—
32C22×#10623494—24no—
33C22×#10623494C2314yes—
34C22×#10623494C2314yes—
35C4⋊#10623494—216no—
36C22⋊#10623494—216no—
37C2×#41×C89—24no—
38C2×#41×C89—24no—
39C2×#41×C89D414yes—
40C2×#41×C89Q814yes—
41C22×Q8×C89C2×C411yescharacteristic
42C22×D4×C89C2×C412yes—
43C22×D4×C89C2×C411yescharacteristic
44C23×C4×C89C2×C412yes—
45C23×C4×C89C2×C414yes—
46C23×C4×C89C2312yes—
47C23×C4×C89C2311yescharacteristic
48C4×Q8×C89—28no—
49C4×D4×C89—28no—
50C4×D4×C89—216no—
51#117×C89—28no—
52#117×C89D418yes—
53C2×C4⋊C4×C89C2×C414yes—
54C2×C4⋊C4×C89C2×C412yes—
55C2×C22⋊C4×C89C2×C414yes—
56C2×C22⋊C4×C89C2×C412yes—
57C2×C42×C89—24no—
58C2×C42×C89—28no—
59C2×C42×C89C2×C412yes—
60C2×C42×C89C2×C414yes—
61C2×C42×C89C2311yescharacteristic
62C2×C42×C89C2311yescharacteristic
63C2×C42×C89C2312yes—
64#13083558—48no—
65C2×#10623494—24no—
66C2×#10623494—24no—
67C2×#10623494—24no—
68C2×#10623494—24no—
69C2×#10623494—24no—
70C2×#10623494—24no—
71C2×#10623494—48no—
72C24×C89C22×C412yes—
73C24×C89C22×C411yescharacteristic
74#41×C89—216no—
75#41×C89—216no—
76C2×Q8×C89—22no—
77C2×Q8×C89—22no—
78C2×Q8×C89C4⋊C414yes—
79C2×D4×C89—24no—
80C2×D4×C89—24no—
81C2×D4×C89—22no—
82C2×D4×C89—22no—
83C2×D4×C89C4⋊C418yes—
84C2×D4×C89C4⋊C414yes—
85C22×C4×C89—24no—
86C22×C4×C89—24no—
87C22×C4×C89—24no—
88C22×C4×C89—28no—
89C22×C4×C89—28no—
90C22×C4×C89—24no—
91C22×C4×C89—44no—
92C22×C4×C89—42no—
93C22×C4×C89C22×C412yes—
94C22×C4×C89C22×C414yes—
95C22×C4×C89C22×C412yes—
96C22×C4×C89C22×C412yes—
97C22×C4×C89C22×C411yescharacteristic
98C22×C4×C89C2×D414yes—
99C22×C4×C89C2×D414yes—
100C22×C4×C89C2×D412yes—
101C22×C4×C89C2×Q814yes—
102C22×C4×C89C2×Q812yes—
103C22×C4×C89#4114yes—
104C22×C4×C89C2411yescharacteristic
105C4⋊C4×C89—28no—
106C4⋊C4×C89C22⋊C418yes—
107C22⋊C4×C89—28no—
108C22⋊C4×C89C22⋊C418yes—
109C42×C89—28no—
110C42×C89—24no—
111C42×C89—22no—
112C42×C89—22no—
113C42×C89C2×D418yes—
114C23×C89—22no—
115C23×C89—22no—
116C23×C89—22no—
117C23×C89—22no—
118C23×C89—24no—
119C23×C89—24no—
120C23×C89C2×C22⋊C412yes—
121C23×C89C2×C4⋊C414yes—
122C23×C89C2×C4⋊C412yes—
123C23×C89#11712yes—
124C23×C89C23×C411yescharacteristic
125Q8×C89—28no—
126D4×C89—216no—
127D4×C89—28no—
128C2×C4×C89—22no—
129C2×C4×C89—22no—
130C2×C4×C89—22no—
131C2×C4×C89—24no—
132C2×C4×C89—22no—
133C2×C4×C89—24no—
134C2×C4×C89—28no—
135C2×C4×C89—28no—
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—416no—
140C2×C4×C89—48no—
141C2×C4×C89C2×C22⋊C418yes—
142C2×C4×C89C2×C22⋊C412yes—
143C2×C4×C89C2×C4⋊C414yes—
144C2×C4×C89C2×C4⋊C412yes—
145C2×C4×C89#11712yes—
146C2×C4×C89#12118yes—
147C2×C4×C89C22⋊Q818yes—
148C2×C4×C89#13114yes—
149C2×C4×C89C22×D412yes—
150C2×C4×C89C22×D411yescharacteristic
151C2×C4×C89C22×Q811yescharacteristic
152C2×C4×C89C2×#4112yes—
153#10623494—44no—
154#10623494—44no—
155C22×C89—24no—
156C22×C89—24no—
157C22×C89—22no—
158C22×C89—22no—
159C22×C89—28no—
160C22×C89—24no—
161C22×C89—44no—
162C22×C89—42no—
163C22×C89—42no—
164C22×C89#38014yes—
165C22×C89C22×C22⋊C411yescharacteristic
166C22×C89C22×C4⋊C411yescharacteristic
167C22×C89C2×#11711yescharacteristic
168C22×C89C2×#13112yes—
169C22×C89#56812yes—
170C4×C89—24no—
171C4×C89—28no—
172C4×C89—24no—
173C4×C89—42no—
174C4×C89#38014yes—
175C4×C89C2×#12112yes—
176C4×C89C2×C22⋊Q812yes—
177#20630—891nomaximal, Sylow
178C2×C89—24no—
179C2×C89—44no—
180C2×C89C2×#38011yescharacteristic
181C2×C89#288012yes—
182C2×C89#290612yes—
183C2×C89#292211yescharacteristic
184C2×C89#292311yescharacteristic
185C2×C4×#41D8911yescharacteristic
186C2×#424—892no—
187C2×#408—892no—
188#1798—894no—
189#1797—894no—
190#1728—891no—
191C2×#336—891no—
192C89#2063011yescharacteristic, complemented, Sylow
193C22×#41C2×D8911yescharacteristic
194C22×M16—892no—
195C23×C8—892no—
196C4×#41—28no—
197C2×C4×Q8Dic8912yes—
198C2×C4×D4Dic8912yes—
199C2×C4×D4Dic8914yes—
200C2×#117C2×D8912yes—
201C2×#117C2×D8911yescharacteristic
202C22×C42C2×D8912yes—
203C22×C42C2×D8911yescharacteristic
204#424—1788no—
205C2×C4⋊C8—898no—
206#408—1788no—
207C2×C22⋊C8—898no—
208#336—892no—
209#336—892no—
210#336—892no—
211#336—892no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41Dic17814yes—
215C2×#41D35614yes—
216C22×Q8C2×Dic8911yescharacteristic
217C22×D4C2×Dic8912yes—
218C22×D4C2×Dic8911yescharacteristic
219C23×C4C2×Dic8912yes—
220C23×C4C2×Dic8914yes—
221C23×C4C22×D8912yes—
222C23×C4C22×D8911yescharacteristic
223C2×M16—1784no—
224C2×M16—1784no—
225C2×M16—1784no—
226C22×C8—894no—
227C22×C8—894no—
228C22×C8—1784no—
229C22×C8—1784no—
230C22×C8—1784no—
231C4×Q8—28no—
232C4×D4—28no—
233C4×D4—216no—
234#117—28no—
235#117C22⋊D8918yes—
236C2×C4⋊C4C2×Dic8914yes—
237C2×C4⋊C4C2×Dic8912yes—
238C2×C22⋊C4C2×Dic8914yes—
239C2×C22⋊C4C2×Dic8912yes—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42C2×Dic8912yes—
243C2×C42C2×Dic8914yes—
244C2×C42C22×D8911yescharacteristic
245C2×C42C22×D8911yescharacteristic
246C2×C42C22×D8912yes—
247C4⋊C8—17816no—
248C22⋊C8—17816no—
249C24C22×Dic8912yes—
250C24C22×Dic8911yescharacteristic
251#41—216no—
252#41—216no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8C4⋊Dic8914yes—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4C4⋊Dic8918yes—
261C2×D4C4⋊Dic8914yes—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4C22×Dic8912yes—
271C22×C4C22×Dic8912yes—
272C22×C4C22×Dic8914yes—
273C22×C4C22×Dic8912yes—
274C22×C4C22×Dic8911yescharacteristic
275C22×C4C2×C22⋊D8914yes—
276C22×C4C2×D35614yes—
277C22×C4C2×D35612yes—
278C22×C4C2×Dic17814yes—
279C22×C4C2×Dic17812yes—
280C22×C4#1308357414yes—
281C22×C4C23×D8911yescharacteristic
282M16—3568no—
283C2×C8—3568no—
284C2×C8—1784no—
285C2×C8—1784no—
286C2×C8—1784no—
287C2×C8—1784no—
288C2×C8—1784no—
289C2×C8—1784no—
290C4⋊C4—28no—
291C4⋊C4C22⋊Dic8918yes—
292C22⋊C4—28no—
293C22⋊C4C22⋊Dic8918yes—
294C42—22no—
295C42—22no—
296C42—28no—
297C42—24no—
298C42C2×C22⋊D8918yes—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—24no—
305C23C2×C22⋊Dic8912yes—
306C23C2×C4⋊Dic8914yes—
307C23C2×C4⋊Dic8912yes—
308C23#1566797112yes—
309C23C23×Dic8911yescharacteristic
310Q8—28no—
311D4—216no—
312D4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—416no—
325C2×C4—48no—
326C2×C4C2×C22⋊Dic8918yes—
327C2×C4C2×C22⋊Dic8912yes—
328C2×C4C2×C4⋊Dic8914yes—
329C2×C4C2×C4⋊Dic8912yes—
330C2×C4#1566797112yes—
331C2×C4#1566798618yes—
332C2×C4#1566799118yes—
333C2×C4#1566801914yes—
334C2×C4C22×C22⋊D8912yes—
335C2×C4C22×D35611yescharacteristic
336C2×C4C22×Dic17811yescharacteristic
337C2×C4C2×#1308357412yes—
338C8—3564no—
339C8—3564no—
340C22—24no—
341C22—22no—
342C22—22no—
343C22—24no—
344C22—28no—
345C22—24no—
346C22—42no—
347C22—42no—
348C22—44no—
349C22#1915145814yes—
350C22C22×C22⋊Dic8911yescharacteristic
351C22C22×C4⋊Dic8911yescharacteristic
352C22C2×#1566797111yescharacteristic
353C22C2×#1566801912yes—
354C22#1915236512yes—
355C4—24no—
356C4—28no—
357C4—24no—
358C4—42no—
359C4#1915145814yes—
360C4C2×#1566798612yes—
361C4C2×#1566799112yes—
362C2—24no—
363C2—44no—
364C2C2×#1915145811yescharacteristic
365C2#2602870112yes—
366C2#2602882012yes—
367C2#2602889611yescharacteristic
368C2#2602890311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC2×C4×#41×C89C2
prime radicalC22×#41×C89C22
cosocleC22×C4×C89C24
hypercenterC2×C4×#41D89
hypocenterC89#20630
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19151625C2semidirect
3. #26020236C2semidirect
4. #26020232C2semidirect
5. #26019946C2semidirect
6. C2×#19151353C2semidirect
192. C89#20630semidirect

Maximal subgroups

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

subgroupindex
C2×#191516252
C2×#191515652
#260202362
#260202322
#260199462
C2×#191513532
C2×C4×#41×C892
#2063089