Finite Group Databasebeta

← #36448728

Subgroups — #36448728

Order 22784.

438 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
1#26020272C211yesmaximal, characteristic
2#26019026C211yesmaximal, characteristic, complemented
3#26019010C211yesmaximal, characteristic, complemented
4#26018634C211yesmaximal, characteristic
5#26018153C211yesmaximal, characteristic, complemented
6#26018091C211yesmaximal, characteristic, complemented
7C89×#2906C211yesmaximal, characteristic
8#19151651C2211yescharacteristic
9#19151622C2211yescharacteristic
10C2×#15667939—21no—
11C2×C4×#10623494C2211yescharacteristic
12C4×C4⋊Dic89C2211yescharacteristic
13#19151357—21no—
14#19151350—22no—
15#19151350—21no—
16#19151350—21no—
17Q8⋊#10623494C411yescharacteristic, complemented
18Q8⋊#10623494C411yescharacteristic, complemented
19D4⋊#10623494C411yescharacteristic, complemented
20D4⋊#10623494C411yescharacteristic, complemented
21C2×#131×C89—21no—
22C4×#41×C89C2211yescharacteristic
23#424×C89—21no—
24#424×C89—22no—
25#423×C89—21no—
26#423×C89C2211yescharacteristic
27C2×C4⋊C8×C89—21no—
28C2×C4⋊C8×C89C2211yescharacteristic
29C2×#13083558—21no—
30C22×#10623494—21no—
31C2×C4⋊Dic89—21no—
32C2×C4×Dic89—21no—
33C4⋊#10623494—21no—
34C4⋊#10623494—21no—
35C4⋊#10623494C2×C411yescharacteristic
36C4⋊#10623494C2×C411yescharacteristic
37#15667939—42no—
38C22⋊#10623494—21no—
39C4×#10623494—21no—
40C4×#10623494—21no—
41C4×#10623494C2×C411yescharacteristic
42C4×#10623494C2×C411yescharacteristic
43#15667919—21no—
44C2×#41×C89D411yescharacteristic
45#131×C89—42no—
46#131×C89—42no—
47C2×M16×C89—22no—
48C2×M16×C89—21no—
49C22×C8×C89—22no—
50C22×C8×C89—21no—
51C4×Q8×C89C2×C411yescharacteristic
52C4×Q8×C89Q811yescharacteristic
53C4×D4×C89—21no—
54C4×D4×C89—21no—
55C4×D4×C89C2×C411yescharacteristic
56C4×D4×C89Q811yescharacteristic
57#117×C89—21no—
58#117×C89D411yescharacteristic
59C2×C42×C89—21no—
60C2×C42×C89C2311yescharacteristic
61C4⋊C8×C89—22no—
62C4⋊C8×C89—22no—
63C4⋊C8×C89—21no—
64C4⋊C8×C89—21no—
65C4⋊C8×C89D411yescharacteristic
66C4⋊C8×C89D411yescharacteristic
67C4⋊C8×C89D412yes—
68C22×Dic89—41no—
69#13083558—41no—
70#13083558—41no—
71C2×#10623494—21no—
72C2×#10623494—21no—
73C2×#10623494—21no—
74C2×#10623494—21no—
75C2×#10623494—42no—
76C2×#10623494—41no—
77C4⋊Dic89—22no—
78C4⋊Dic89—41no—
79C4×Dic89—41no—
80C4×Dic89—41no—
81#41×C89—21no—
82#41×C89—21no—
83#41×C89D812yes—
84#41×C89Q1612yes—
85C2×Q8×C89C4⋊C411yescharacteristic
86C2×D4×C89—21no—
87C2×D4×C89C4⋊C411yescharacteristic
88C22×C4×C89—21no—
89C22×C4×C89—21no—
90C22×C4×C89—21no—
91C22×C4×C89C2×D411yescharacteristic
92C22×C4×C89#4111yescharacteristic
93C22×C4×C89#4111yescharacteristic
94M16×C89—44no—
95M16×C89—41no—
96M16×C89—41no—
97C2×C8×C89—21no—
98C2×C8×C89—22no—
99C2×C8×C89—21no—
100C2×C8×C89—44no—
101C2×C8×C89—41no—
102C2×C8×C89—41no—
103C4⋊C4×C89—21no—
104C4⋊C4×C89—21no—
105C4⋊C4×C89C4⋊C411yescharacteristic
106C4⋊C4×C89C4⋊C411yescharacteristic
107C22⋊C4×C89—21no—
108C22⋊C4×C89—21no—
109C22⋊C4×C89—21no—
110C42×C89—21no—
111C42×C89—21no—
112C42×C89—21no—
113C42×C89C22×C411yescharacteristic
114C42×C89C2×D411yescharacteristic
115C42×C89C2×D411yescharacteristic
116C42×C89C2×Q811yescharacteristic
117C23×C89—21no—
118C23×C89#12311yescharacteristic
119Q8×C89—21no—
120Q8×C89#10712yes—
121D4×C89—22no—
122D4×C89—22no—
123D4×C89—21no—
124D4×C89#10712yes—
125C2×C4×C89—21no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
128C2×C4×C89—22no—
129C2×C4×C89—22no—
130C2×C4×C89—21no—
131C2×C4×C89—21no—
132C2×C4×C89—41no—
133C2×C4×C89—41no—
134C2×C4×C89—41no—
135C2×C4×C89—81no—
136C2×C4×C89C2×C4⋊C411yescharacteristic
137C2×C4×C89C4×D411yescharacteristic
138C2×C4×C89C4×D411yescharacteristic
139C2×C4×C89#12011yescharacteristic
140C2×C4×C89C22⋊Q811yescharacteristic
141C2×C4×C89C22⋊Q811yescharacteristic
142C2×C4×C89C2×D811yescharacteristic
143C2×C4×C89C2×Q1611yescharacteristic
144C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C89#13711yescharacteristic
146C2×Dic89—42no—
147C2×Dic89—81no—
148C2×Dic89—81no—
149C8×C89—41no—
150C8×C89—42no—
151C8×C89—41no—
152#10623494—22no—
153#10623494—41no—
154#10623494—41no—
155#10623494—41no—
156#10623494—81no—
157C22×C89—21no—
158C22×C89—21no—
159C22×C89—22no—
160C22×C89—41no—
161C22×C89—41no—
162C22×C89#38511yescharacteristic
163C22×C89#48011yescharacteristic
164C22×C89#48411yescharacteristic
165C4×C89—22no—
166C4×C89—21no—
167C4×C89—21no—
168C4×C89—21no—
169C4×C89—41no—
170C4×C89—41no—
171C4×C89C2×#10711yescharacteristic
172C4×C89#42811yescharacteristic
173C4×C89#44311yescharacteristic
174C4×C89#44411yescharacteristic
175C4×C89#44711yescharacteristic
176C4×C89#45111yescharacteristic
177Dic89—82no—
178#8902—891nomaximal, Sylow
179C2×C89—21no—
180C2×C89—21no—
181C2×C89—42no—
182C2×C89#185211yescharacteristic
183C2×C89#186111yescharacteristic
184C2×C89#186611yescharacteristic
185#2906D8911yescharacteristic
186#1806—891no—
187#1571—891no—
188#1567—891no—
189#1476—891no—
190#1303—891no—
191#1279—891no—
192C89#890211yescharacteristic, complemented, Sylow
193C2×#131—21no—
194C4×#41C2×D8911yescharacteristic
195#431—891no—
196#424—21no—
197#424—22no—
198#423—21no—
199#423—891no—
200#423C2×D8911yescharacteristic
201C2×C4⋊C8—21no—
202C2×C4⋊C8C2×D8911yescharacteristic
203C2×#104—1781no—
204C2×C4×C8—891no—
205C4×C4⋊C4—891no—
206#339—1781no—
207#335—1782no—
208#335—1781no—
209#335—1781no—
210Q8⋊C8—891no—
211Q8⋊C8—891no—
212D4⋊C8—891no—
213D4⋊C8—891no—
214C2×#41C22⋊D8911yescharacteristic
215#131—42no—
216#131—42no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—1781no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—1781no—
223C4×Q8Dic17811yescharacteristic
224C4×Q8C4×D8911yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic17811yescharacteristic
228C4×D4C4×D8911yescharacteristic
229#117—21no—
230#117C22⋊D8911yescharacteristic
231C2×C4⋊C4—1781no—
232C2×C42—21no—
233C2×C42—1781no—
234C2×C42C22×D8911yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—891no—
240C4⋊C8—891no—
241C4⋊C8—1781no—
242C4⋊C8—1781no—
243C4⋊C8C22⋊D8911yescharacteristic
244C4⋊C8C22⋊D8911yescharacteristic
245C4⋊C8C22⋊D8912yes—
246#104—3562no—
247C22⋊C8—1781no—
248C4×C8—891no—
249C4×C8—891no—
250C4×C8—1781no—
251C4×C8—1781no—
252#95—1781no—
253#41—21no—
254#41—21no—
255#41#1308355912yes—
256#41#1308356412yes—
257C2×Q8#1308355311yescharacteristic
258C2×D4—21no—
259C2×D4#1308355311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—3561no—
264C22×C4C2×C22⋊D8911yescharacteristic
265C22×C4#1308357411yescharacteristic
266C22×C4#1308357411yescharacteristic
267M16—44no—
268M16—41no—
269M16—41no—
270M16—3561no—
271M16—3561no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—44no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—3562no—
279C2×C8—3561no—
280C2×C8—1781no—
281C2×C8—1781no—
282C2×C8—1781no—
283C2×C8—1781no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—3561no—
287C4⋊C4—1782no—
288C4⋊C4#1308355311yescharacteristic
289C4⋊C4#1308355311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—3561no—
297C42—3561no—
298C42C2×C4×D8911yescharacteristic
299C42C2×C22⋊D8911yescharacteristic
300C42C2×C22⋊D8911yescharacteristic
301C42C2×Dic17811yescharacteristic
302C23—21no—
303C23#1566799511yescharacteristic
304Q8—21no—
305Q8#1566794512yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1566794512yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—3562no—
322C2×C4—7121no—
323C2×C4—7121no—
324C2×C4C2×#1308355311yescharacteristic
325C2×C4C4×C22⋊D8911yescharacteristic
326C2×C4C4×C22⋊D8911yescharacteristic
327C2×C4#1566798111yescharacteristic
328C2×C4#1566799111yescharacteristic
329C2×C4#1566799111yescharacteristic
330C2×C4C2×#1308355911yescharacteristic
331C2×C4C2×#1308356411yescharacteristic
332C2×C4#1566803811yescharacteristic
333C2×C4#1566804311yescharacteristic
334C8—41no—
335C8—42no—
336C8—41no—
337C8—3561no—
338C8—3561no—
339C8—3561no—
340C8—1782no—
341C8—7121no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1915147111yescharacteristic
348C22#1915191111yescharacteristic
349C22#1915193311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—7122no—
357C4C2×#1566794511yescharacteristic
358C4#1915163811yescharacteristic
359C4#1915171311yescharacteristic
360C4#1915171811yescharacteristic
361C4#1915173211yescharacteristic
362C4#1915175411yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2602044011yescharacteristic
367C2#2602049211yescharacteristic
368C2#2602052711yescharacteristic
subgroupG/H
centerC2×C4#15667981
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#2906C2
socleC22×C89#385
prime radicalC2×#41×C89D4
cosocleC2×C42×C89C23
hypercenter#2906D89
hypocenterC89#8902
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #26019026C2semidirect
3. #26019010C2semidirect
5. #26018153C2semidirect
6. #26018091C2semidirect
17. Q8⋊#10623494C4semidirect
18. Q8⋊#10623494C4semidirect
19. D4⋊#10623494C4semidirect
20. D4⋊#10623494C4semidirect
192. C89#8902semidirect

Maximal subgroups

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

subgroupindex
#260202722
#260190262
#260190102
#260186342
#260181532
#260180912
C89×#29062
#890289