Finite Group Databasebeta

← #36471367

Subgroups — #36471367

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15667946C211yesmaximal, characteristic, complemented
2C2×#19151399C212yesmaximal, complemented
3C2×C22⋊#13083549C211yesmaximal, characteristic, complemented
4#26018343C212yesmaximal, complemented
5C2×#466×C89C211yesmaximal, characteristic
6C22×C4⋊Dic89—21no—
7C22×#13083549C2212yes—
8C2×#15667946—24no—
9C2×#15667946—21no—
10C2×#15667946C2212yes—
11#19151399—24no—
12C22⋊#13083549—22no—
13C22×D8×C89C2211yescharacteristic
14C23×C8×C89C2211yescharacteristic
15C2×#121×C89—21no—
16#466×C89—22no—
17#466×C89C414yescomplemented
18C2×#107×C89C2211yescharacteristic
19C2×#102×C89—21no—
20C23×Dic89—41no—
21C2×C4⋊Dic89—24no—
22C2×C4⋊Dic89—22no—
23C2×C4⋊Dic89—42no—
24C2×C4⋊Dic89—41no—
25C2×#13083549—22no—
26C2×#13083549—22no—
27C2×#13083549—22no—
28#15667946—24no—
29#15667946—22no—
30#15667946D414yescomplemented
31C22×D4×C89—21no—
32C22×D4×C89—41no—
33C23×C4×C89D411yescharacteristic
34C2×D8×C89—22no—
35C2×D8×C89—21no—
36C2×D8×C89—22no—
37C2×D8×C89C2×C412yescomplemented
38C22×C8×C89—22no—
39C22×C8×C89—22no—
40C22×C8×C89C2×C412yes—
41C22×C8×C89D411yescharacteristic
42C22×C8×C89D412yes—
43C22×C8×C89C2311yescharacteristic
44#121×C89—24no—
45#121×C89—42no—
46C2×C4⋊C4×C89—21no—
47C2×C22⋊C4×C89—41no—
48#107×C89—21no—
49#107×C89C2×C412yescomplemented
50#102×C89—24no—
51C22×Dic89—44no—
52C22×Dic89—42no—
53C22×Dic89—82no—
54C22×Dic89—82no—
55C4⋊Dic89—24no—
56C4⋊Dic89—44no—
57C4⋊Dic89—42no—
58#13083549—42no—
59C24×C89—41no—
60C24×C89D811yescharacteristic
61C2×D4×C89—22no—
62C2×D4×C89—44no—
63C2×D4×C89—41no—
64C2×D4×C89—41no—
65C2×D4×C89—41no—
66C2×D4×C89—41no—
67C2×D4×C89—41no—
68C2×D4×C89—82no—
69C2×D4×C89—82no—
70C22×C4×C89—21no—
71C22×C4×C89—21no—
72C22×C4×C89—21no—
73C22×C4×C89—21no—
74C22×C4×C89—41no—
75C22×C4×C89C22⋊C412yes—
76C22×C4×C89D811yescharacteristic
77C22×C4×C89SD1612yes—
78C22×C4×C89C2×D411yescharacteristic
79D8×C89—24no—
80D8×C89—42no—
81C2×C8×C89—21no—
82C2×C8×C89—21no—
83C2×C8×C89—24no—
84C2×C8×C89—24no—
85C2×C8×C89C22⋊C412yes—
86C2×C8×C89C22×C411yescharacteristic
87C2×C8×C89C2×D412yes—
88C2×C8×C89C2×D411yescharacteristic
89C2×C8×C89#4112yes—
90C4⋊C4×C89—22no—
91C4⋊C4×C89—41no—
92C22⋊C4×C89—44no—
93C23×C89—21no—
94C23×C89—21no—
95C23×C89—21no—
96C23×C89—21no—
97C23×C89—42no—
98C23×C89—81no—
99C23×C89—81no—
100C23×C89—81no—
101C23×C89—81no—
102C23×C89—81no—
103C23×C89—81no—
104C23×C89#10212yes—
105C23×C89D1612yes—
106C23×C89SD3212yes—
107C23×C89C2×D811yescharacteristic
108D4×C89—42no—
109D4×C89—42no—
110D4×C89—88no—
111D4×C89—81no—
112D4×C89—81no—
113C2×C4×C89—21no—
114C2×C4×C89—21no—
115C2×C4×C89—22no—
116C2×C4×C89—22no—
117C2×C4×C89—22no—
118C2×C4×C89—22no—
119C2×C4×C89—42no—
120C2×C4×C89—81no—
121C2×C4×C89—81no—
122C2×C4×C89#10212yes—
123C2×C4×C89C2×C22⋊C411yescharacteristic
124C2×C4×C89#12011yescharacteristic
125C2×C4×C89#12311yescharacteristic
126C2×C4×C89C2×SD1611yescharacteristic
127C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C89#13711yescharacteristic
129C2×Dic89—44no—
130C2×Dic89—88no—
131C2×Dic89—82no—
132C2×Dic89—82no—
133C8×C89—22no—
134C8×C89C4×D412yes—
135C8×C89#12112yes—
136C22×C89—22no—
137C22×C89—22no—
138C22×C89—22no—
139C22×C89—22no—
140C22×C89—22no—
141C22×C89—22no—
142C22×C89—82no—
143C22×C89—82no—
144C22×C89—82no—
145C22×C89—161no—
146C22×C89—161no—
147C22×C89—161no—
148C22×C89—161no—
149C22×C89#35714yescomplemented
150C22×C89C2×#10211yescharacteristic
151C22×C89#44711yescharacteristic
152C22×C89#48011yescharacteristic
153C22×C89C2×D1611yescharacteristic
154C22×C89C2×SD3211yescharacteristic
155C22×C89#50911yescharacteristic
156C22×C89#51011yescharacteristic
157C4×C89—22no—
158C4×C89—82no—
159C4×C89#38611yescharacteristic
160C4×C89#41711yescharacteristic
161C4×C89#44811yescharacteristic
162C4×C89#48111yescharacteristic
163Dic89—84no—
164#12991—891nomaximal, Sylow
165C2×C89—24no—
166C2×C89—162no—
167C2×C89#187411yescharacteristic
168C2×C89C2×#35711yescharacteristic
169C2×C89#213911yescharacteristic
170C2×C89#218411yescharacteristic
171C2×C89#218511yescharacteristic
172C2×C89#223211yescharacteristic
173C2×C89#223311yescharacteristic
174C2×#466D8911yescharacteristic
175C22×#107—891no—
176C2×#357—892no—
177C2×C22⋊C16—891no—
178#1380—892no—
179C89#1299111yescharacteristic, complemented, Sylow
180C22×D8C2×D8911yescharacteristic
181C23×C8C2×D8911yescharacteristic
182C2×#121—21no—
183C22×C4⋊C4—1781no—
184C22×C16—892no—
185#466—22no—
186#466Dic8914yescomplemented
187C2×#107—892no—
188C2×#107—1784no—
189C2×#107—1781no—
190C2×#107C2×D8911yescharacteristic
191C2×#102—21no—
192#357—1784no—
193C22⋊C16—1782no—
194C22×D4—21no—
195C22×D4—41no—
196C23×C4—3561no—
197C23×C4C22⋊D8911yescharacteristic
198C2×D8—22no—
199C2×D8—21no—
200C2×D8—22no—
201C2×D8C2×Dic8912yescomplemented
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×Dic8912yes—
205C22×C8C22⋊D8911yescharacteristic
206C22×C8C22⋊D8912yes—
207C22×C8C22×D8911yescharacteristic
208#121—24no—
209#121—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—3562no—
212C2×C4⋊C4—3561no—
213C2×C4⋊C4—1784no—
214C2×C4⋊C4—1782no—
215C2×C22⋊C4—41no—
216C2×C16—1782no—
217C2×C16—1782no—
218C2×C16—1782no—
219#107—21no—
220#107—894no—
221#107—1784no—
222#107—1782no—
223#107C2×Dic8912yescomplemented
224#102—24no—
225C24—41no—
226C24#1308355911yescharacteristic
227C2×D4—22no—
228C2×D4—44no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—3564no—
242C22×C4—3562no—
243C22×C4—7122no—
244C22×C4—7122no—
245C22×C4C22⋊Dic8912yes—
246C22×C4#1308355911yescharacteristic
247C22×C4#1308356212yes—
248C22×C4C2×C22⋊D8911yescharacteristic
249D8—24no—
250D8—42no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8C22⋊Dic8912yes—
256C2×C8C22×Dic8911yescharacteristic
257C2×C8D4×D8912yes—
258C2×C8C2×C22⋊D8911yescharacteristic
259C2×C8#1308357312yes—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—3564no—
263C4⋊C4—3562no—
264C4⋊C4—1784no—
265C22⋊C4—44no—
266C16—3562no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23#1566793312yes—
279C23#1566795312yes—
280C23#1566795612yes—
281C23C2×#1308355911yescharacteristic
282D4—42no—
283D4—42no—
284D4—88no—
285D4—81no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—3564no—
297C2×C4—7128no—
298C2×C4—7122no—
299C2×C4—7122no—
300C2×C4#1566793312yes—
301C2×C4C2×C22⋊Dic8911yescharacteristic
302C2×C4#1566798011yescharacteristic
303C2×C4#1566799311yescharacteristic
304C2×C4C2×#1308356211yescharacteristic
305C2×C4#1566803711yescharacteristic
306C2×C4#1566804211yescharacteristic
307C8—22no—
308C8D4×Dic8912yes—
309C8#1566798412yes—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—161no—
320C22—161no—
321C22—161no—
322C22—161no—
323C22#1915139914yescomplemented
324C22C2×#1566793311yescharacteristic
325C22#1915173111yescharacteristic
326C22#1915190911yescharacteristic
327C22C2×#1566795311yescharacteristic
328C22C2×#1566795611yescharacteristic
329C22#1915203311yescharacteristic
330C22#1915203811yescharacteristic
331C4—22no—
332C4—82no—
333C4—7124no—
334C4#1915147811yescharacteristic
335C4#1915159911yescharacteristic
336C4#1915173611yescharacteristic
337C4#1915191411yescharacteristic
338C2—24no—
339C2—162no—
340C2#2602057011yescharacteristic
341C2C2×#1915139911yescharacteristic
342C2#2602196411yescharacteristic
343C2#2602216311yescharacteristic
344C2#2602216811yescharacteristic
345C2#2602243011yescharacteristic
346C2#2602243511yescharacteristic
subgroupG/H
centerC23C2×#13083559
commutatorC2×C8×C89C22×C4
FrattiniC22×C8C22×D89
Fitting, prime radicalC2×#466×C89C2
socleC23×C89C2×D8
cosocleC22×C8×C89C23
hypercenterC2×#466D89
hypocenterC89#12991
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#15667946C2semidirect
2. C2×#19151399C2semidirect
3. C2×C22⋊#13083549C2semidirect
4. #26018343C2semidirect
17. #466×C89C4semidirect
30. #15667946D4semidirect
37. C2×D8×C89C2×C4semidirect
49. #107×C89C2×C4semidirect
149. C22×C89#357semidirect
179. C89#12991semidirect
186. #466Dic89semidirect
201. C2×D8C2×Dic89semidirect
223. #107C2×Dic89semidirect
323. C22#19151399semidirect

Maximal subgroups

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

subgroupindex
C22×#156679462
C2×#191513992
C2×C22⋊#130835492
#260183432
C2×#466×C892
#1299189