Finite Group Databasebeta

← #36447595

Subgroups — #36447595

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#10623494C211yesmaximal, characteristic
2#26020272C211yesmaximal, characteristic
3C4×#15667945C211yesmaximal, characteristic, complemented
4#26019010C211yesmaximal, characteristic, complemented
5#26018155C211yesmaximal, characteristic, complemented
6#26018153C211yesmaximal, characteristic, complemented
7C89×#1806C211yesmaximal, characteristic
8Q8×#10623494C411yescharacteristic, complemented
9Q8×#10623494C411yescharacteristic, complemented
10D4×#10623494—21no—
11D4×#10623494—21no—
12D4×#10623494C411yescharacteristic, complemented
13D4×#10623494C411yescharacteristic, complemented
14#19151651—21no—
15#19151651C2211yescharacteristic
16C2×#15667945—21no—
17#19151622C2211yescharacteristic
18C2×#15667939—21no—
19C2×C4×#10623494—21no—
20C2×C4×#10623494C2211yescharacteristic
21C4×C4⋊Dic89C2211yescharacteristic
22#19151360—21no—
23#19151357—21no—
24#19151350—21no—
25#19151350—21no—
26C4×#41×C89C2211yescharacteristic
27#423×C89C2211yescharacteristic
28C2×#104×C89—21no—
29C4×C4⋊C4×C89C2211yescharacteristic
30#339×C89—21no—
31C2×#13083558—21no—
32C22×#10623494—21no—
33C22×#10623494—21no—
34C22×#10623494D411yescharacteristic
35C22×#10623494D411yescharacteristic
36C2×C4⋊Dic89—21no—
37C2×C4×Dic89—21no—
38#15667945—24no—
39C4⋊#10623494—21no—
40C4⋊#10623494—21no—
41C4⋊#10623494—21no—
42C4⋊#10623494—21no—
43C4⋊#10623494C2×C411yescharacteristic
44C4⋊#10623494C2×C411yescharacteristic
45#15667939—42no—
46C22⋊#10623494—21no—
47C22⋊#10623494—21no—
48C22⋊#10623494—21no—
49C4×#10623494—21no—
50C4×#10623494—21no—
51C4×#10623494—21no—
52C4×#10623494C2×C411yescharacteristic
53C4×#10623494C2×C411yescharacteristic
54C4×#10623494D411yescharacteristic
55C4×#10623494D411yescharacteristic
56#15667919—21no—
57C2×#41×C89D411yescharacteristic
58C2×M16×C89—21no—
59C4×Q8×C89C2×C411yescharacteristic
60C4×Q8×C89Q811yescharacteristic
61C4×D4×C89—21no—
62C4×D4×C89—21no—
63C4×D4×C89C2×C411yescharacteristic
64C4×D4×C89Q811yescharacteristic
65#117×C89—21no—
66#117×C89D411yescharacteristic
67C2×C4⋊C4×C89—21no—
68C2×C42×C89—21no—
69C2×C42×C89—21no—
70C2×C42×C89C2311yescharacteristic
71C4⋊C8×C89—21no—
72C4⋊C8×C89—21no—
73#104×C89—42no—
74#95×C89—21no—
75C22×Dic89—41no—
76#13083558—41no—
77#13083558—41no—
78C2×#10623494—22no—
79C2×#10623494—21no—
80C2×#10623494—21no—
81C2×#10623494—41no—
82C2×#10623494—42no—
83C2×#10623494—42no—
84C2×#10623494—41no—
85C2×#10623494C22⋊C411yescharacteristic
86C2×#10623494C22⋊C411yescharacteristic
87C2×#10623494C22⋊C411yescharacteristic
88C2×#10623494C22⋊C411yescharacteristic
89C4⋊Dic89—22no—
90C4⋊Dic89—41no—
91C4×Dic89—41no—
92C4×Dic89—41no—
93#41×C89—21no—
94#41×C89—21no—
95#41×C89D812yes—
96#41×C89Q1612yes—
97C2×Q8×C89C4⋊C411yescharacteristic
98C2×D4×C89—21no—
99C2×D4×C89C4⋊C411yescharacteristic
100C22×C4×C89—21no—
101C22×C4×C89—21no—
102C22×C4×C89—21no—
103C22×C4×C89—41no—
104C22×C4×C89C2×D411yescharacteristic
105C22×C4×C89C2×D411yescharacteristic
106C22×C4×C89#4111yescharacteristic
107M16×C89—41no—
108M16×C89—41no—
109C2×C8×C89—41no—
110C4⋊C4×C89—21no—
111C4⋊C4×C89—21no—
112C4⋊C4×C89—22no—
113C4⋊C4×C89—41no—
114C4⋊C4×C89C4⋊C411yescharacteristic
115C4⋊C4×C89C4⋊C411yescharacteristic
116C22⋊C4×C89—21no—
117C22⋊C4×C89—21no—
118C22⋊C4×C89—21no—
119C42×C89—21no—
120C42×C89—21no—
121C42×C89—21no—
122C42×C89—41no—
123C42×C89—41no—
124C42×C89C22×C411yescharacteristic
125C42×C89C2×D411yescharacteristic
126C42×C89C2×Q811yescharacteristic
127C42×C89#4111yescharacteristic
128C23×C89—21no—
129C23×C89#12111yescharacteristic
130Q8×C89—21no—
131Q8×C89#10712yescomplemented
132D4×C89—22no—
133D4×C89—22no—
134D4×C89—21no—
135D4×C89#10712yescomplemented
136C2×C4×C89—21no—
137C2×C4×C89—21no—
138C2×C4×C89—21no—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—21no—
142C2×C4×C89—21no—
143C2×C4×C89—41no—
144C2×C4×C89—42no—
145C2×C4×C89—41no—
146C2×C4×C89—41no—
147C2×C4×C89—42no—
148C2×C4×C89—81no—
149C2×C4×C89—81no—
150C2×C4×C89C2×C22⋊C411yescharacteristic
151C2×C4×C89C2×C4⋊C411yescharacteristic
152C2×C4×C89#11711yescharacteristic
153C2×C4×C89#12111yescharacteristic
154C2×C4×C89C22⋊Q811yescharacteristic
155C2×C4×C89C22⋊Q811yescharacteristic
156C2×C4×C89C2×D811yescharacteristic
157C2×C4×C89C2×Q1611yescharacteristic
158C2×C4×C89#13511yescharacteristic
159C2×C4×C89#13511yescharacteristic
160C2×Dic89—42no—
161C2×Dic89—81no—
162C2×Dic89—81no—
163C8×C89—81no—
164#10623494—21no—
165#10623494—21no—
166#10623494—21no—
167#10623494—41no—
168#10623494—41no—
169#10623494—81no—
170#10623494#10412yescomplemented
171C22×C89—21no—
172C22×C89—21no—
173C22×C89—22no—
174C22×C89—41no—
175C22×C89—41no—
176C22×C89#38011yescharacteristic
177C22×C89#46611yescharacteristic
178C22×C89#46711yescharacteristic
179C4×C89—21no—
180C4×C89—22no—
181C4×C89—21no—
182C4×C89—21no—
183C4×C89—41no—
184C4×C89—41no—
185C4×C89—82no—
186C4×C89C2×#10411yescharacteristic
187C4×C89#42111yescharacteristic
188C4×C89C2×#10711yescharacteristic
189C4×C89#42711yescharacteristic
190C4×C89#46611yescharacteristic
191C4×C89#46711yescharacteristic
192Dic89—82no—
193#8667—891nomaximal, Sylow
194C2×C89—21no—
195C2×C89—21no—
196C2×C89—42no—
197C2×C89#180611yescharacteristic
198C2×C89#180811yescharacteristic
199C2×C89#181411yescharacteristic
200C8×#41—891no—
201#1806—891no—
202#1806D8911yescharacteristic
203C4×#107—891no—
204#1567—891no—
205#1303—891no—
206#1303—891no—
207C89#866711yescharacteristic, complemented, Sylow
208C4×#41C2×D8911yescharacteristic
209C8×Q8—891no—
210C8×Q8—891no—
211C8×D4—891no—
212C8×D4—891no—
213C8×D4—1781no—
214C8×D4—1781no—
215#431—891no—
216#431—1781no—
217C2×#107—1781no—
218#423—891no—
219#423C2×D8911yescharacteristic
220C2×#104—21no—
221C2×#104—1781no—
222C2×C4×C8—891no—
223C2×C4×C8—1781no—
224C4×C4⋊C4—891no—
225C4×C4⋊C4C2×D8911yescharacteristic
226#340—1781no—
227#339—21no—
228#339—1781no—
229#335—1781no—
230#335—1781no—
231C2×#41C22⋊D8911yescharacteristic
232C2×M16—21no—
233C2×M16—1781no—
234C22×C8—891no—
235C22×C8—891no—
236C22×C8—1781no—
237C22×C8—1781no—
238C4×Q8Dic17811yescharacteristic
239C4×Q8C4×D8911yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4Dic17811yescharacteristic
243C4×D4C4×D8911yescharacteristic
244#117—21no—
245#117C22⋊D8911yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1781no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—1781no—
251C2×C42C22×D8911yescharacteristic
252#107—1784no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—891no—
256C4⋊C8—891no—
257C4⋊C8—1781no—
258C4⋊C8—1781no—
259C4⋊C8—1781no—
260C4⋊C8—1781no—
261#104—42no—
262#104—3562no—
263C22⋊C8—1781no—
264C22⋊C8—1781no—
265C22⋊C8—1781no—
266C4×C8—891no—
267C4×C8—891no—
268C4×C8—891no—
269C4×C8—891no—
270C4×C8—1781no—
271C4×C8—1781no—
272C4×C8—1781no—
273#95—21no—
274#95—1781no—
275#41—21no—
276#41—21no—
277#41#1308355912yes—
278#41#1308356412yes—
279C2×Q8#1308355311yescharacteristic
280C2×D4—21no—
281C2×D4#1308355311yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—3561no—
287C22×C4D4×D8911yescharacteristic
288C22×C4C2×C22⋊D8911yescharacteristic
289C22×C4#1308357311yescharacteristic
290M16—41no—
291M16—41no—
292M16—3561no—
293M16—3561no—
294C2×C8—41no—
295C2×C8—891no—
296C2×C8—891no—
297C2×C8—891no—
298C2×C8—891no—
299C2×C8—3562no—
300C2×C8—3562no—
301C2×C8—3561no—
302C2×C8—3561no—
303C2×C8—1782no—
304C2×C8—1781no—
305C2×C8—1781no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—3561no—
311C4⋊C4—1782no—
312C4⋊C4#1308355311yescharacteristic
313C4⋊C4#1308355311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—3561no—
323C42—3561no—
324C42C2×C4×D8911yescharacteristic
325C42D4×D8911yescharacteristic
326C42C2×Dic17811yescharacteristic
327C42#1308357311yescharacteristic
328C23—21no—
329C23#1566798411yescharacteristic
330Q8—21no—
331Q8#1566794512yescomplemented
332D4—22no—
333D4—22no—
334D4—21no—
335D4#1566794512yescomplemented
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—3562no—
351C2×C4—7121no—
352C2×C4—7121no—
353C2×C4C22⋊C4×D8911yescharacteristic
354C2×C4C2×#1308355311yescharacteristic
355C2×C4#1566796811yescharacteristic
356C2×C4#1566798411yescharacteristic
357C2×C4#1566798911yescharacteristic
358C2×C4#1566798911yescharacteristic
359C2×C4C2×#1308355911yescharacteristic
360C2×C4C2×#1308356411yescharacteristic
361C2×C4#1566803211yescharacteristic
362C2×C4#1566803211yescharacteristic
363C8—81no—
364C8—892no—
365C8—3561no—
366C8—3561no—
367C8—1781no—
368C8—1781no—
369C8—1781no—
370C8—7121no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22#1915145611yescharacteristic
377C22#1915183211yescharacteristic
378C22#1915183711yescharacteristic
379C4—21no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—7122no—
387C4#104×D8911yescharacteristic
388C4#1915161311yescharacteristic
389C4C2×#1566794511yescharacteristic
390C4#1915163411yescharacteristic
391C4#1915183211yescharacteristic
392C4#1915183711yescharacteristic
393C2—21no—
394C2—21no—
395C2—42no—
396C2#2602027011yescharacteristic
397C2#2602027911yescharacteristic
398C2#2602030411yescharacteristic
subgroupG/H
centerC2×C4#15667984
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#1806C2
socleC22×C89#380
prime radicalC2×#41×C89D4
cosocleC2×C42×C89C23
hypercenter#1806D89
hypocenterC89#8667
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. C4×#15667945C2semidirect
4. #26019010C2semidirect
5. #26018155C2semidirect
6. #26018153C2semidirect
8. Q8×#10623494C4semidirect
9. Q8×#10623494C4semidirect
12. D4×#10623494C4semidirect
13. D4×#10623494C4semidirect
131. Q8×C89#107semidirect
135. D4×C89#107semidirect
170. #10623494#104semidirect
207. C89#8667semidirect
331. Q8#15667945semidirect
335. D4#15667945semidirect

Maximal subgroups

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

subgroupindex
#41×#106234942
#260202722
C4×#156679452
#260190102
#260181552
#260181532
C89×#18062
#866789