Finite Group Databasebeta

← #36572741

Subgroups — #36572741

Order 22784.

914 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151705C211yesmaximal, characteristic
2C2×Q16×Dic89C211yesmaximal, characteristic
3Q8×#13083553C212yesmaximal
4#26020990C211yesmaximal, characteristic
5#26020988C211yesmaximal, characteristic
6#26020520C212yesmaximal
7#26020518C212yesmaximal
8#26020407C211yesmaximal, characteristic
9#26020255C212yesmaximal
10#26020120C211yesmaximal, characteristic
11C2×#441×C89C211yesmaximal, characteristic
12C2×Q8×Dic89C2212yes—
13C2×Q8×Dic89C2212yes—
14#19151705—24no—
15Q16×Dic89C418yescomplemented
16C2×#15667946C2211yescharacteristic
17C2×#15667944C2211yescharacteristic
18C2×#15667943C2212yes—
19C2×#15667941C2211yescharacteristic
20C2×#15667936C2212yes—
21C2×#15667936C2212yes—
22C2×#15667922C2211yescharacteristic
23C2×C8×Dic89C2211yescharacteristic
24#19151496—22no—
25#19151496C2212yes—
26#19151494—22no—
27#19151494C2212yes—
28#19151467—22no—
29#19151467C2212yes—
30C4×#13083553—22no—
31C4×#13083553C2211yescharacteristic
32#19151361C2212yes—
33#19151360C2212yes—
34#19151353C2211yescharacteristic
35#19151352C2211yescharacteristic
36C22×Q16×C89C2211yescharacteristic
37C2×C4×Q8×C89C2212yes—
38#441×C89—24no—
39C2×#106×C89C2211yescharacteristic
40C2×#103×C89C2212yes—
41C2×#97×C89C2211yescharacteristic
42C22×#10623494C2311yescharacteristic
43C22×#10623494C2311yescharacteristic
44Q8×Dic89—24no—
45Q8×Dic89—24no—
46Q8×Dic89—42no—
47Q8×Dic89C2×C418yes—
48C2×C4⋊Dic89—22no—
49C2×C4⋊Dic89—22no—
50C2×C4⋊Dic89C2312yes—
51C2×C4⋊Dic89C2312yes—
52C2×#13083553—22no—
53C2×#13083553—22no—
54C2×#13083553—22no—
55C2×#13083553—22no—
56C2×#13083553D412yes—
57C2×C4×Dic89—22no—
58C2×C4×Dic89—22no—
59C2×C4×Dic89C2311yescharacteristic
60C2×C4×Dic89C2311yescharacteristic
61#15667946C2×C414yes—
62#15667944—22no—
63#15667943—42no—
64#15667941—22no—
65#15667936—24no—
66#15667936C2×C418yes—
67#15667922—22no—
68C8×Dic89C2×C414yes—
69#15667919—22no—
70#15667919—22no—
71#15667919D412yes—
72C22×Q8×C89C2312yes—
73C2×Q16×C89C2×C414yes—
74C2×Q16×C89D414yes—
75C2×Q16×C89Q814yes—
76C22×C8×C89C2311yescharacteristic
77C22×C8×C89C2311yescharacteristic
78C4×Q8×C89—24no—
79C4×Q8×C89—42no—
80C2×C4⋊C4×C89—22no—
81C2×C4⋊C4×C89C2312yes—
82C2×C42×C89—22no—
83C2×C42×C89C2311yescharacteristic
84#106×C89—22no—
85#103×C89—24no—
86#97×C89—22no—
87C22×Dic89—22no—
88C22×Dic89—22no—
89C22×Dic89C2×D412yes—
90C22×Dic89C2×D411yescharacteristic
91C22×Dic89C2×D411yescharacteristic
92C2×#10623494—21no—
93C2×#10623494—21no—
94C2×#10623494—21no—
95C2×#10623494—21no—
96C2×#10623494—21no—
97C2×#10623494C22×C412yes—
98C4⋊Dic89—22no—
99C4⋊Dic89—28no—
100C4⋊Dic89—22no—
101C4⋊Dic89—22no—
102C4⋊Dic89—44no—
103C4⋊Dic89C22×C414yes—
104#13083553—24no—
105#13083553—44no—
106#13083553—82no—
107#13083553—82no—
108#13083553—82no—
109C4×Dic89—21no—
110C4×Dic89—21no—
111C4×Dic89—28no—
112C4×Dic89—44no—
113C4×Dic89C22×C412yes—
114C4×Dic89#4112yes—
115C2×Q8×C89—22no—
116C2×Q8×C89—22no—
117C2×Q8×C89—22no—
118C2×Q8×C89C22×C414yes—
119C2×Q8×C89C2×D414yes—
120C2×Q8×C89C2×Q814yes—
121C22×C4×C89—22no—
122C22×C4×C89—22no—
123C22×C4×C89C2×D412yes—
124C22×C4×C89C2411yescharacteristic
125Q16×C89C4⋊C4116yescomplemented
126C2×C8×C89—21no—
127C2×C8×C89—21no—
128C2×C8×C89—21no—
129C2×C8×C89C22×C412yes—
130C2×C8×C89C2×D412yes—
131C2×C8×C89C2×Q812yes—
132C4⋊C4×C89—22no—
133C4⋊C4×C89—22no—
134C4⋊C4×C89—44no—
135C42×C89—21no—
136C42×C89—21no—
137C42×C89—44no—
138C23×C89C22×D411yescharacteristic
139Q8×C89—24no—
140Q8×C89—24no—
141Q8×C89—24no—
142Q8×C89C2×C4⋊C418yes—
143C2×C4×C89—24no—
144C2×C4×C89—24no—
145C2×C4×C89—24no—
146C2×C4×C89—22no—
147C2×C4×C89—22no—
148C2×C4×C89—22no—
149C2×C4×C89—42no—
150C2×C4×C89—42no—
151C2×C4×C89—42no—
152C2×C4×C89C23×C411yescharacteristic
153C2×C4×C89C22×D411yescharacteristic
154C2×C4×C89C22×Q811yescharacteristic
155C2×C4×C89C2×#4111yescharacteristic
156C2×C4×C89H(5,𝔽2)11yescharacteristic
157C2×C4×C892−1+411yescharacteristic
158C2×Dic89—22no—
159C2×Dic89—24no—
160C2×Dic89—22no—
161C2×Dic89—22no—
162C2×Dic89—22no—
163C2×Dic89—42no—
164C2×Dic89—42no—
165C2×Dic89—42no—
166C2×Dic89—42no—
167C2×Dic89—42no—
168C2×Dic89C4×D412yes—
169C2×Dic89C4×D412yes—
170C2×Dic89#13714yes—
171C8×C89—41no—
172C8×C89C2×C4⋊C414yes—
173#10623494—22no—
174#10623494—41no—
175C22×C89C2×C4×D411yescharacteristic
176C22×C89D4211yescharacteristic
177C22×C89D4×Q811yescharacteristic
178C22×C89C2×#13712yes—
179C22×C89#57712yes—
180C4×C89—28no—
181C4×C89—42no—
182C4×C89—82no—
183C4×C89C22×C4⋊C411yescharacteristic
184C4×C89#52011yescharacteristic
185C4×C89#54711yescharacteristic
186C4×C89#55411yescharacteristic
187Dic89—22no—
188Dic89—44no—
189Dic89—42no—
190Dic89—82no—
191Dic89#44114yescomplemented
192#22674—891nomaximal, Sylow
193C2×C89D4×C4⋊C411yescharacteristic
194C2×C89C2×#44111yescharacteristic
195C2×C89#295211yescharacteristic
196C2×C89#327812yes—
197C2×C89#338712yes—
198C2×#441—891no—
199C2×#441D8911yescharacteristic
200C2×C4×Q16—891no—
201Q8×C4⋊C4—892no—
202#1943—891no—
203#1943—891no—
204#1865—892no—
205#1865—892no—
206#1844—891no—
207#1803—892no—
208#1770—891no—
209C89#2267411yescharacteristic, complemented, Sylow
210C22×Q16C2×D8911yescharacteristic
211C2×C4×Q8—892no—
212C2×C4×Q8—892no—
213C2×C4×Q8C2×D8912yes—
214#441—24no—
215#441—1784no—
216C4×Q16—898no—
217C2×#107—891no—
218C2×#106—892no—
219C2×#106—891no—
220C2×#106C2×D8911yescharacteristic
221C2×C4⋊C8—891no—
222C2×#103—892no—
223C2×#103—892no—
224C2×#103C2×D8912yes—
225C2×#97—891no—
226C2×#97C2×D8911yescharacteristic
227C2×C4×C8—891no—
228#389—892no—
229#389—892no—
230#389—1782no—
231#389—1782no—
232#384—892no—
233#384—1782no—
234C4×C4⋊C4—891no—
235C4×C4⋊C4—1782no—
236#340—892no—
237#340—892no—
238#336—891no—
239#336—891no—
240C22×Q8C22×D8912yes—
241C2×Q16Dic17814yes—
242C2×Q16C4×D8914yes—
243C2×Q16C22⋊D8914yes—
244C22×C8—891no—
245C22×C8—891no—
246C22×C8C22×D8911yescharacteristic
247C22×C8C22×D8911yescharacteristic
248C4×Q8—24no—
249C4×Q8—42no—
250C4×Q8—898no—
251C4×Q8—3562no—
252C4×Q8—1784no—
253C4×Q8—1784no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—892no—
256C2×C4⋊C4—892no—
257C2×C4⋊C4—892no—
258C2×C4⋊C4—1782no—
259C2×C4⋊C4—1782no—
260C2×C4⋊C4—1782no—
261C2×C4⋊C4—1782no—
262C2×C4⋊C4—1782no—
263C2×C4⋊C4—1782no—
264C2×C4⋊C4C22×D8912yes—
265C2×C42—22no—
266C2×C42—891no—
267C2×C42—891no—
268C2×C42—1782no—
269C2×C42—1782no—
270C2×C42C22×D8911yescharacteristic
271#107—894no—
272#106—22no—
273#106—3562no—
274#106—1782no—
275C4⋊C8—1782no—
276#103—24no—
277#103—898no—
278#103—1784no—
279#97—22no—
280#97—1782no—
281C4×C8—894no—
282#95—892no—
283#95—1782no—
284#95—1782no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8C2×C4×D8914yes—
289C2×Q8C2×C22⋊D8914yes—
290C2×Q8C2×Dic17814yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—891no—
294C22×C4—891no—
295C22×C4—892no—
296C22×C4—1782no—
297C22×C4—1782no—
298C22×C4D4×D8912yes—
299C22×C4C23×D8911yescharacteristic
300Q16#13083553116yescomplemented
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—892no—
305C2×C8—1781no—
306C2×C8—1781no—
307C2×C8—1781no—
308C2×C8—1781no—
309C2×C8—1781no—
310C2×C8C2×C4×D8912yes—
311C2×C8C2×C22⋊D8912yes—
312C2×C8C2×Dic17812yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—44no—
316C4⋊C4—894no—
317C4⋊C4—3564no—
318C4⋊C4—3564no—
319C4⋊C4—1784no—
320C4⋊C4—1788no—
321C4⋊C4—1782no—
322C4⋊C4—1782no—
323C4⋊C4—1782no—
324C4⋊C4—7122no—
325C4⋊C4—7122no—
326C4⋊C4—7122no—
327C42—21no—
328C42—21no—
329C42—44no—
330C42—892no—
331C42—892no—
332C42—3564no—
333C42—1788no—
334C42—1781no—
335C42—1781no—
336C23C2×D4×D8911yescharacteristic
337Q8—24no—
338Q8—24no—
339Q8—24no—
340Q8C2×#1308355318yes—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—894no—
351C2×C4—892no—
352C2×C4—892no—
353C2×C4—3562no—
354C2×C4—3562no—
355C2×C4—3562no—
356C2×C4—3562no—
357C2×C4—3562no—
358C2×C4—1782no—
359C2×C4—1782no—
360C2×C4—1782no—
361C2×C4—1784no—
362C2×C4—1782no—
363C2×C4C22×C4×D8911yescharacteristic
364C2×C4C22×C22⋊D8911yescharacteristic
365C2×C4C22×Dic17811yescharacteristic
366C2×C4#41×D8911yescharacteristic
367C2×C4#1566805711yescharacteristic
368C2×C4#1566805911yescharacteristic
369C8—41no—
370C8—3561no—
371C8—1782no—
372C8C2×#1308355314yes—
373C22C4×D4×D8911yescharacteristic
374C22D4×C22⋊D8911yescharacteristic
375C22D4×Dic17811yescharacteristic
376C22#137×D8912yes—
377C22#1915240712yes—
378C4—28no—
379C4—42no—
380C4—82no—
381C4—894no—
382C4—3562no—
383C4—3564no—
384C4—1782no—
385C4—7122no—
386C4C22×#1308355311yescharacteristic
387C4#1915207711yescharacteristic
388C4#1915224711yescharacteristic
389C4#1915230711yescharacteristic
390C2D4×#1308355311yescharacteristic
391C2#441×D8911yescharacteristic
392C2#2602913211yescharacteristic
393C2#2603191212yes—
394C2#2603318512yes—
subgroupG/H
centerC23C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC2×#441×C89C2
socle, prime radicalC23×C89C22×D4
cosocleC22×C4×C89C24
hypercenterC2×#441D89
hypocenterC89#22674
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
15. Q16×Dic89C4semidirect
125. Q16×C89C4⋊C4semidirect
191. Dic89#441semidirect
209. C89#22674semidirect
300. Q16#13083553semidirect

Maximal subgroups

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

subgroupindex
C2×#191517052
C2×Q16×Dic892
Q8×#130835532
#260209902
#260209882
#260205202
#260205182
#260204072
#260202552
#260201202
C2×#441×C892
#2267489