Finite Group Databasebeta

← #36553838

Subgroups — #36553838

Order 22784.

1462 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
1C2×#19151572C212yesmaximal
2#26022870C212yesmaximal, complemented
3#26020240C218yesmaximal, complemented
4#26019985C212yesmaximal, complemented
5C2×#520×C89C211yesmaximal, characteristic
6C2×#15667971C2214yes—
7#19151572—28no—
8C2×#15667927C2218yes—
9#19151497—28no—
10#19151458C2218yes—
11#19151365C2218yes—
12C22×#41×C89C2211yescharacteristic
13#520×C89—28no—
14C2×C4×Q8×C89C412yescomplemented
15C2×C4×D4×C89C412yescomplemented
16C2×C4×D4×C89C2214yes—
17C2×#117×C89C412yescomplemented
18C2×#117×C89C2211yescharacteristic
19C22×C4⋊C4×C89C412yescomplemented
20C22×C4⋊C4×C89C2211yescharacteristic
21#15667971—28no—
22#15667971—28no—
23C2×C4⋊Dic89—24no—
24C2×C4⋊Dic89—28no—
25C2×C22⋊Dic89C2318yes—
26C2×C4×Dic89—24no—
27#15667927—216no—
28#15667919—28no—
29C2×#41×C89—24no—
30C2×#41×C89—24no—
31C2×#41×C89D414yes—
32C2×#41×C89Q814yes—
33C22×Q8×C89C2×C411yescharacteristic
34C22×D4×C89C2×C411yescharacteristic
35C22×D4×C89C2312yes—
36C23×C4×C89C2×C412yes—
37C23×C4×C89C2×C414yes—
38C23×C4×C89C2312yes—
39C23×C4×C89C2311yescharacteristic
40C4×Q8×C89—28no—
41C4×D4×C89—216no—
42C4×D4×C89—28no—
43#117×C89—24no—
44#117×C89—28no—
45C2×C4⋊C4×C89—28no—
46C2×C4⋊C4×C89C2×C411yescharacteristic
47C2×C4⋊C4×C89C2×C414yes—
48C2×C4⋊C4×C89C2×C411yescharacteristic
49C2×C4⋊C4×C89C2×C412yes—
50C2×C4⋊C4×C89C2×C412yes—
51C2×C4⋊C4×C89D418yes—
52C2×C22⋊C4×C89C2×C414yes—
53C2×C22⋊C4×C89C2312yes—
54C2×C42×C89C2×C414yes—
55C2×C42×C89C2×C412yes—
56C22×Dic89—28no—
57C4⋊Dic89—48no—
58C4⋊Dic89—88no—
59C22⋊Dic89—28no—
60C22⋊Dic89—28no—
61C4×Dic89—48no—
62C24×C89C22×C412yes—
63C24×C89C2411yescharacteristic
64#41×C89—216no—
65#41×C89—216no—
66C2×Q8×C89—22no—
67C2×Q8×C89—22no—
68C2×Q8×C89C4⋊C414yescomplemented
69C2×D4×C89—24no—
70C2×D4×C89—24no—
71C2×D4×C89—22no—
72C2×D4×C89—22no—
73C2×D4×C89C4⋊C414yescomplemented
74C2×D4×C89C2×D414yes—
75C2×D4×C89C2×Q814yes—
76C22×C4×C89—24no—
77C22×C4×C89—24no—
78C22×C4×C89—28no—
79C22×C4×C89—82no—
80C22×C4×C89—84no—
81C22×C4×C89C4⋊C418yescomplemented
82C22×C4×C89C22×C412yes—
83C22×C4×C89C22×C412yes—
84C22×C4×C89C22×C412yes—
85C22×C4×C89C22×C414yes—
86C22×C4×C89C22×C411yescharacteristic
87C22×C4×C89C22×C411yescharacteristic
88C22×C4×C89C2×D418yes—
89C22×C4×C89C2×D414yes—
90C22×C4×C89C2×D412yes—
91C22×C4×C89C2×Q812yes—
92C22×C4×C89#4114yes—
93C4⋊C4×C89—22no—
94C4⋊C4×C89—28no—
95C4⋊C4×C89—22no—
96C4⋊C4×C89C22⋊C418yes—
97C4⋊C4×C89C22⋊C418yes—
98C22⋊C4×C89—24no—
99C22⋊C4×C89—28no—
100C42×C89—28no—
101C42×C89—24no—
102C23×C89—22no—
103C23×C89—22no—
104C23×C89—41no—
105C23×C89—41no—
106C23×C89—42no—
107C23×C89—42no—
108C23×C89C2×C4⋊C414yes—
109C23×C89C23×C411yescharacteristic
110C23×C89C22×D412yes—
111C23×C89C22×D411yescharacteristic
112C23×C89C22×Q811yescharacteristic
113C23×C89C2×#4112yes—
114Q8×C89—28no—
115D4×C89—28no—
116D4×C89—28no—
117D4×C89—28no—
118C2×C4×C89—22no—
119C2×C4×C89—22no—
120C2×C4×C89—216no—
121C2×C4×C89—22no—
122C2×C4×C89—22no—
123C2×C4×C89—44no—
124C2×C4×C89—48no—
125C2×C4×C89—42no—
126C2×C4×C89—42no—
127C2×C4×C89—88no—
128C2×C4×C89C2×C22⋊C418yes—
129C2×C4×C89C2×C22⋊C412yes—
130C2×C4×C89C2×C22⋊C412yes—
131C2×C4×C89C2×C4⋊C414yes—
132C2×C4×C89C2×C4⋊C412yes—
133C2×C4×C89C2×C4⋊C412yes—
134C2×C4×C89#11712yes—
135C2×C4×C89#11712yes—
136C2×C4×C89#12118yes—
137C2×C4×C89C22⋊Q818yes—
138C2×Dic89—48no—
139C2×Dic89—48no—
140C2×Dic89—48no—
141C22×C89—22no—
142C22×C89—21no—
143C22×C89—22no—
144C22×C89—21no—
145C22×C89—24no—
146C22×C89—41no—
147C22×C89—41no—
148C22×C89—42no—
149C22×C89—44no—
150C22×C89—44no—
151C22×C89—42no—
152C22×C89—42no—
153C22×C89C22×C22⋊C411yescharacteristic
154C22×C89C22×C4⋊C411yescharacteristic
155C22×C89C2×#11711yescharacteristic
156C22×C89C2×#12112yes—
157C22×C89C2×C22⋊Q812yes—
158C4×C89—28no—
159C4×C89—24no—
160C4×C89—42no—
161C4×C89—42no—
162C4×C89#38014yes—
163C4×C89#38014yes—
164Dic89—88no—
165#20634—891nomaximal, Sylow
166C2×C89—22no—
167C2×C89—21no—
168C2×C89—21no—
169C2×C89—22no—
170C2×C89—44no—
171C2×C89C2×#38011yescharacteristic
172C2×C89#288311yescharacteristic
173C2×C89#288411yescharacteristic
174C2×#520D8911yescharacteristic
175C2×#410—892no—
176#2316—892no—
177#1799—898no—
178#1740—892no—
179C89#2063411yescharacteristic, complemented, Sylow
180C22×#41C2×D8911yescharacteristic
181#520—28no—
182C2×C4×Q8Dic8912yescomplemented
183C2×C4×D4Dic8912yescomplemented
184C2×C4×D4C2×D8914yes—
185C2×#117—894no—
186C2×#117Dic8912yescomplemented
187C2×#117C2×D8911yescharacteristic
188C22×C4⋊C4Dic8912yescomplemented
189C22×C4⋊C4C2×D8911yescharacteristic
190#410—1788no—
191C2×#99—898no—
192#389—1788no—
193#380—898no—
194#342—898no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41Dic17814yes—
198C2×#41D35614yes—
199C22×Q8C2×Dic8911yescharacteristic
200C22×D4C2×Dic8911yescharacteristic
201C22×D4C22×D8912yes—
202C23×C4C2×Dic8912yes—
203C23×C4C2×Dic8914yes—
204C23×C4C22×D8912yes—
205C23×C4C22×D8911yescharacteristic
206C4×Q8—28no—
207C4×D4—216no—
208C4×D4—28no—
209#117—24no—
210#117—28no—
211#117—1788no—
212#117—1788no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—1784no—
215C2×C4⋊C4—1788no—
216C2×C4⋊C4C2×Dic8911yescharacteristic
217C2×C4⋊C4C2×Dic8914yes—
218C2×C4⋊C4C2×Dic8911yescharacteristic
219C2×C4⋊C4C2×Dic8912yes—
220C2×C4⋊C4C2×Dic8912yes—
221C2×C4⋊C4C22⋊D8918yes—
222C2×C22⋊C4—898no—
223C2×C22⋊C4C2×Dic8914yes—
224C2×C22⋊C4C22×D8912yes—
225C2×C42—1784no—
226C2×C42C2×Dic8914yes—
227C2×C42C2×Dic8912yes—
228#99—17816no—
229#95—1788no—
230C24C22×Dic8912yes—
231C24C23×D8911yescharacteristic
232#41—216no—
233#41—216no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8C4⋊Dic8914yescomplemented
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4C4⋊Dic8914yescomplemented
242C2×D4C2×D35614yes—
243C2×D4C2×Dic17814yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—1788no—
250C22×C4C4⋊Dic8918yescomplemented
251C22×C4C22×Dic8912yes—
252C22×C4C22×Dic8912yes—
253C22×C4C22×Dic8912yes—
254C22×C4C22×Dic8914yes—
255C22×C4C22×Dic8911yescharacteristic
256C22×C4C22×Dic8911yescharacteristic
257C22×C4C2×C22⋊D8918yes—
258C22×C4C2×C22⋊D8914yes—
259C22×C4C2×D35612yes—
260C22×C4C2×Dic17812yes—
261C22×C4#1308357414yes—
262C4⋊C4—22no—
263C4⋊C4—28no—
264C4⋊C4—22no—
265C4⋊C4—3568no—
266C4⋊C4—7128no—
267C4⋊C4C22⋊Dic8918yes—
268C4⋊C4C22⋊Dic8918yes—
269C22⋊C4—24no—
270C22⋊C4—28no—
271C22⋊C4—1788no—
272C22⋊C4—1788no—
273C42—28no—
274C42—24no—
275C42—3568no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23C2×C4⋊Dic8914yes—
283C23C23×Dic8911yescharacteristic
284C23C22×C22⋊D8912yes—
285C23C22×D35611yescharacteristic
286C23C22×Dic17811yescharacteristic
287C23C2×#1308357412yes—
288Q8—28no—
289D4—28no—
290D4—28no—
291D4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—216no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—88no—
302C2×C4—3568no—
303C2×C4—3568no—
304C2×C4—3568no—
305C2×C4C2×C22⋊Dic8918yes—
306C2×C4C2×C22⋊Dic8912yes—
307C2×C4C2×C22⋊Dic8912yes—
308C2×C4C2×C4⋊Dic8914yes—
309C2×C4C2×C4⋊Dic8912yes—
310C2×C4C2×C4⋊Dic8912yes—
311C2×C4#1566797112yes—
312C2×C4#1566797112yes—
313C2×C4#1566798618yes—
314C2×C4#1566799118yes—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—24no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22C22×C22⋊Dic8911yescharacteristic
328C22C22×C4⋊Dic8911yescharacteristic
329C22C2×#1566797111yescharacteristic
330C22C2×#1566798612yes—
331C22C2×#1566799112yes—
332C4—28no—
333C4—24no—
334C4—42no—
335C4—42no—
336C4—7128no—
337C4#1915145814yes—
338C4#1915145814yes—
339C2—22no—
340C2—21no—
341C2—21no—
342C2—22no—
343C2—44no—
344C2C2×#1915145811yescharacteristic
345C2#2602871311yescharacteristic
346C2#2602871611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic89
commutatorC23×C89C23×C4
FrattiniC24C23×D89
FittingC2×#520×C89C2
socleC22×C89C22×C22⋊C4
prime radicalC22×#41×C89C22
cosocleC24×C89C24
hypercenterC2×#520D89
hypocenterC89#20634
LCSG ▷ C23×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. #26022870C2semidirect
3. #26020240C2semidirect
4. #26019985C2semidirect
14. C2×C4×Q8×C89C4semidirect
15. C2×C4×D4×C89C4semidirect
17. C2×#117×C89C4semidirect
19. C22×C4⋊C4×C89C4semidirect
68. C2×Q8×C89C4⋊C4semidirect
73. C2×D4×C89C4⋊C4semidirect
81. C22×C4×C89C4⋊C4semidirect
179. C89#20634semidirect
182. C2×C4×Q8Dic89semidirect
183. C2×C4×D4Dic89semidirect
186. C2×#117Dic89semidirect
188. C22×C4⋊C4Dic89semidirect
236. C2×Q8C4⋊Dic89semidirect
241. C2×D4C4⋊Dic89semidirect
250. C22×C4C4⋊Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191515722
#260228702
#260202402
#260199852
C2×#520×C892
#2063489