Finite Group Databasebeta

← #36481084

Subgroups — #36481084

Order 22784.

1934 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668000C211yesmaximal, characteristic
2C2×C4×C4⋊Dic89C211yesmaximal, characteristic, complemented
3#26018522C212yesmaximal, complemented
4#26018498C214yesmaximal, complemented
5#26018492C212yesmaximal
6#26018488C214yesmaximal, complemented
7C2×#383×C89C211yesmaximal, characteristic, complemented
8C23×Dic178C2211yescharacteristic
9C23×D356C2211yescharacteristic
10C2×#15668000—24no—
11C2×#15668000—24no—
12C2×#15668000C418yescomplemented
13C22×C4⋊Dic89C2212yes—
14C22×#13083551C2212yes—
15C22×#13083551C2212yes—
16C22×#13083551C2218yes—
17C22×C4×Dic89C2212yes—
18C4×C4⋊Dic89—24no—
19C2×#15667919C2212yes—
20C2×#15667919C2218yes—
21C22×C42×C89C2211yescharacteristic
22#383×C89—24no—
23C2×#95×C89C2212yes—
24C2×#95×C89C2212yes—
25C2×#95×C89C2212yes—
26C24×D89C2312yes—
27C22×Dic178—22no—
28C22×Dic178—22no—
29C22×Dic178—44no—
30C22×Dic178C2×C414yes—
31C22×D356—22no—
32C22×D356—22no—
33C22×D356—44no—
34C22×D356C2×C414yes—
35C23×Dic89C2312yes—
36C23×Dic89C2314yes—
37#15668000—232no—
38C2×C4⋊Dic89—28no—
39C2×C4⋊Dic89—24no—
40C2×#13083551—216no—
41C2×#13083551—24no—
42C2×#13083551—24no—
43C2×#13083551—24no—
44C2×#13083551—24no—
45C2×#13083551—48no—
46C2×#13083551—48no—
47C2×#13083551C2×C418yes—
48C2×#13083551C2×C418yes—
49C2×C4×Dic89—24no—
50C2×C4×Dic89—44no—
51#15667919—24no—
52#15667919—48no—
53C23×C4×C89C2314yes—
54C23×C4×C89C2311yescharacteristic
55C23×C4×C89C2311yescharacteristic
56C23×C4×C89C2311yescharacteristic
57C2×C42×C89C2×C414yes—
58C2×C42×C89D414yes—
59C2×C42×C89D414yes—
60#95×C89—24no—
61#95×C89—24no—
62#95×C89—24no—
63C23×D89—22no—
64C23×D89—22no—
65C23×D89—28no—
66C23×D89—28no—
67C23×D89—24no—
68C23×D89—24no—
69C23×D89C22×C414yes—
70C2×Dic178—28no—
71C2×Dic178—48no—
72C2×Dic178—48no—
73C2×Dic178—48no—
74C2×D356—28no—
75C2×D356—48no—
76C2×D356—48no—
77C2×D356—48no—
78C22×Dic89—22no—
79C22×Dic89—22no—
80C22×Dic89—24no—
81C22×Dic89—24no—
82C22×Dic89—28no—
83C22×Dic89—28no—
84C22×Dic89—28no—
85C22×Dic89C22×C414yes—
86C22×Dic89#4118yes—
87C4⋊Dic89—216no—
88#13083551—216no—
89#13083551—216no—
90#13083551—432no—
91C4×Dic89—48no—
92C24×C89C2411yescharacteristic
93C22×C4×C89—24no—
94C22×C4×C89—24no—
95C22×C4×C89—22no—
96C22×C4×C89—22no—
97C22×C4×C89—24no—
98C22×C4×C89—28no—
99C22×C4×C89—28no—
100C22×C4×C89—24no—
101C22×C4×C89—24no—
102C22×C4×C89C22×C412yes—
103C22×C4×C89C22×C412yes—
104C22×C4×C89C22×C412yes—
105C22×C4×C89C2×D412yes—
106C22×C4×C89C2×D412yes—
107C22×C4×C89C2×D412yes—
108C22×C4×C89C2×D412yes—
109C22×C4×C89C2×D412yes—
110C22×C4×C89C2×D412yes—
111C42×C89C22⋊C4116yescomplemented
112C23×C89C23×C411yescharacteristic
113C23×C89C22×D411yescharacteristic
114C23×C89C22×D411yescharacteristic
115C23×C89C2×#4114yes—
116C23×C89H(5,𝔽2)14yes—
117C23×C89H(5,𝔽2)12yes—
118C23×C892−1+412yes—
119C22×D89—24no—
120C22×D89—28no—
121C22×D89—28no—
122C22×D89—28no—
123C22×D89—44no—
124C22×D89—44no—
125C22×D89—44no—
126C22×D89—44no—
127C22×D89—42no—
128C22×D89—42no—
129C22×D89—44no—
130C22×D89—48no—
131C22×D89—48no—
132C22×D89—416no—
133C2×C4×C89—24no—
134C2×C4×C89—24no—
135C2×C4×C89—24no—
136C2×C4×C89—24no—
137C2×C4×C89—24no—
138C2×C4×C89—22no—
139C2×C4×C89—24no—
140C2×C4×C89—24no—
141C2×C4×C89—22no—
142C2×C4×C89—24no—
143C2×C4×C89—44no—
144C2×C4×C89—44no—
145C2×C4×C89—44no—
146C2×C4×C89—48no—
147C2×C4×C89—48no—
148C2×C4×C89C2×C22⋊C414yes—
149C2×C4×C89C2×C22⋊C414yes—
150C2×C4×C89C2×C22⋊C414yes—
151C2×Dic89—24no—
152C2×Dic89—28no—
153C2×Dic89—28no—
154C2×Dic89—216no—
155C2×Dic89—44no—
156C2×Dic89—44no—
157C2×Dic89—44no—
158C2×Dic89—44no—
159C2×Dic89—48no—
160D356—416no—
161Dic178—416no—
162C22×C89C22×C22⋊C411yescharacteristic
163C22×C89C4×#4112yes—
164C22×C89#51812yes—
165C22×C89#52012yes—
166C22×C89#52512yes—
167C22×C89#53212yes—
168C22×C89#53512yes—
169C22×C89#53512yes—
170C22×C89#53511yescharacteristic
171C22×C89#53611yescharacteristic
172C22×C89#53712yes—
173C22×C89#53818yes—
174C22×C89#54218yes—
175C2×D89—44no—
176C2×D89—44no—
177C2×D89—416no—
178C2×D89—44no—
179C2×D89—84no—
180C2×D89—84no—
181C2×D89—88no—
182C4×C89—24no—
183C4×C89—24no—
184C4×C89—24no—
185C4×C89—84no—
186Dic89—44no—
187Dic89—48no—
188#14288—891nomaximal, Sylow
189C2×C89#229711yescharacteristic
190C2×C89#230911yescharacteristic
191C2×C89#231111yescharacteristic
192C2×C89#233714yes—
193C2×C89#263112yes—
194C2×C89#263212yes—
195C2×C89#266214yes—
196D89—84no—
197C22×#124—891no—
198C2×#383D8911yescharacteristic, complemented
199C2×C4×C4⋊C4—891no—
200#1446—892no—
201#1439—894no—
202#1438—892no—
203#1437—894no—
204C89#1428811yescharacteristic, complemented, Sylow
205C23×Q8—891no—
206C23×D4—891no—
207C2×#124—898no—
208C2×#124—1784no—
209C2×#124—1784no—
210C22×C4⋊C4—892no—
211C22×C22⋊C4—892no—
212C22×C22⋊C4—892no—
213C22×C22⋊C4—898no—
214C22×C42—892no—
215C22×C42C2×D8911yescharacteristic
216#383—24no—
217C4×C4⋊C4—1784no—
218C2×#95—898no—
219C2×#95—892no—
220C2×#95C2×D8912yes—
221C2×#95C2×D8912yes—
222C2×#95C2×D8912yes—
223C25—892no—
224C22×Q8—894no—
225C22×Q8—3564no—
226C22×Q8—1782no—
227C22×Q8—1782no—
228C22×D4—894no—
229C22×D4—3564no—
230C22×D4—1782no—
231C22×D4—1782no—
232C23×C4—894no—
233C23×C4—892no—
234C23×C4C22×D8914yes—
235C23×C4C22×D8911yescharacteristic
236C23×C4C22×D8911yescharacteristic
237C23×C4C22×D8911yescharacteristic
238#124—17832no—
239C2×C4⋊C4—1788no—
240C2×C4⋊C4—1784no—
241C2×C22⋊C4—898no—
242C2×C22⋊C4—898no—
243C2×C22⋊C4—3568no—
244C2×C22⋊C4—3568no—
245C2×C22⋊C4—17816no—
246C2×C22⋊C4—1784no—
247C2×C22⋊C4—1784no—
248C2×C22⋊C4—1784no—
249C2×C22⋊C4—1784no—
250C2×C42—3564no—
251C2×C42—1784no—
252C2×C42C4×D8914yes—
253C2×C42D35614yes—
254C2×C42C22⋊D8914yes—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—3568no—
259#95—1784no—
260C24—894no—
261C24—1782no—
262C24—1782no—
263C24—1788no—
264C24—1784no—
265C24—1788no—
266C24—1784no—
267C24C23×D8911yescharacteristic
268C2×Q8—3568no—
269C2×Q8—3568no—
270C2×Q8—3568no—
271C2×Q8—1788no—
272C2×D4—3568no—
273C2×D4—3568no—
274C2×D4—3568no—
275C2×D4—1788no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—894no—
286C22×C4—898no—
287C22×C4—1782no—
288C22×C4—1784no—
289C22×C4—1782no—
290C22×C4—1784no—
291C22×C4—1788no—
292C22×C4—1788no—
293C22×C4—1788no—
294C22×C4C2×C4×D8912yes—
295C22×C4C2×C4×D8912yes—
296C22×C4C2×C4×D8912yes—
297C22×C4C2×C22⋊D8912yes—
298C22×C4C2×C22⋊D8912yes—
299C22×C4C2×C22⋊D8912yes—
300C22×C4C2×D35612yes—
301C22×C4C2×D35612yes—
302C22×C4C2×D35612yes—
303C4⋊C4—17816no—
304C22⋊C4—35632no—
305C22⋊C4—17816no—
306C22⋊C4—17816no—
307C42—3568no—
308C42#13083551116yescomplemented
309C23—3564no—
310C23—3564no—
311C23—3564no—
312C23—3562no—
313C23—3562no—
314C23—3564no—
315C23—3568no—
316C23—3568no—
317C23—35616no—
318C23—3564no—
319C23—1784no—
320C23—1788no—
321C23—1788no—
322C23—1788no—
323C23C22×C4×D8911yescharacteristic
324C23C22×C22⋊D8911yescharacteristic
325C23C22×D35611yescharacteristic
326C23#41×D8914yes—
327C23#1566805814yes—
328C23#1566805812yes—
329C23#1566805912yes—
330Q8—35616no—
331D4—35616no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—3564no—
348C2×C4—3564no—
349C2×C4—3564no—
350C2×C4—3564no—
351C2×C4—3568no—
352C2×C4—17816no—
353C2×C4—1784no—
354C2×C4—1788no—
355C2×C4—1788no—
356C2×C4C2×#1308355114yes—
357C2×C4C2×#1308355114yes—
358C2×C4C2×#1308355114yes—
359C22—3564no—
360C22—35616no—
361C22—3564no—
362C22—3564no—
363C22—7124no—
364C22—7124no—
365C22—7128no—
366C22C22×#1308355111yescharacteristic
367C22#1915206612yes—
368C22#1915207212yes—
369C22#1915207812yes—
370C22#1915211012yes—
371C22#1915214412yes—
372C22#1915216212yes—
373C22#1915216711yescharacteristic
374C22#1915216712yes—
375C22#1915217411yescharacteristic
376C22#1915217612yes—
377C22#1915218718yes—
378C22#1915221418yes—
379C4—24no—
380C4—24no—
381C4—24no—
382C4—84no—
383C4—3568no—
384C4—3564no—
385C2—7124no—
386C2#2602274011yescharacteristic
387C2#2602283211yescharacteristic
388C2#2602284011yescharacteristic
389C2#2602306214yes—
390C2#2602610712yes—
391C2#2602611512yes—
392C2#2602639214yes—
subgroupG/H
center, FrattiniC24C23×D89
commutatorC23×C89C23×C4
FittingC2×#383×C89C2
socle, cosocleC24×C89C24
prime radicalC23×D356C22
hypercenterC2×#383D89
hypocenterC89#14288
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic89C2semidirect
3. #26018522C2semidirect
4. #26018498C2semidirect
6. #26018488C2semidirect
7. C2×#383×C89C2semidirect
12. C2×#15668000C4semidirect
111. C42×C89C22⋊C4semidirect
198. C2×#383D89semidirect
204. C89#14288semidirect
308. C42#13083551semidirect

Maximal subgroups

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

subgroupindex
C22×#156680002
C2×C4×C4⋊Dic892
#260185222
#260184982
#260184922
#260184882
C2×#383×C892
#1428889