Finite Group Databasebeta

← #36711887

Subgroups — #36711887

Order 22784.

472 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26032621C211yesmaximal, characteristic
2#26022535C211yesmaximal, characteristic, complemented
3#26022457C211yesmaximal, characteristic, complemented
4#26022454C211yesmaximal, characteristic, complemented
5#26022449C211yesmaximal, characteristic, complemented
6#26022447C211yesmaximal, characteristic, complemented
7#26022416C211yesmaximal, characteristic, complemented
8#26022391C211yesmaximal, characteristic, complemented
9#26022358C211yesmaximal, characteristic, complemented
10#26022341C211yesmaximal, characteristic, complemented
11#26022121C211yesmaximal, characteristic, complemented
12#26022078C211yesmaximal, characteristic
13#26022038C211yesmaximal, characteristic, complemented
14C2×#19151425C211yesmaximal, characteristic, complemented
15C89×#3338C211yesmaximal, characteristic
16#19152288—21no—
17C2×#15667958C2211yescharacteristic
18C2×#15667952C2211yescharacteristic
19C22×#13083549C2211yescharacteristic
20#19152000C2211yescharacteristic
21#19151940—21no—
22#19151929—21no—
23#19151906—21no—
24#19151839C2211yescharacteristic
25#19151839C2211yescharacteristic
26C4×Dic356C2211yescharacteristic
27#19151636C2211yescharacteristic
28C2×#15667946C2211yescharacteristic
29#19151425C2211yescharacteristic
30#19151425C2211yescharacteristic
31#19151425C2211yescharacteristic
32#19151425C2211yescharacteristic
33#19151425C2211yescharacteristic
34#19151425C2211yescharacteristic
35#19151423C2211yescharacteristic
36#19151423C2211yescharacteristic
37#19151423C2211yescharacteristic
38C4⋊#13083549C2211yescharacteristic
39#19151402C2211yescharacteristic
40#19151402C2211yescharacteristic
41#19151402C2211yescharacteristic
42#19151401C2211yescharacteristic
43#19151401C2211yescharacteristic
44#19151401C2211yescharacteristic
45C22⋊#13083549C2211yescharacteristic
46C22⋊#13083549C2211yescharacteristic
47#19151374C2211yescharacteristic
48#554×C89—21no—
49#500×C89C2211yescharacteristic
50#484×C89—21no—
51#484×C89—21no—
52#477×C89—21no—
53#467×C89C2211yescharacteristic
54#467×C89C2211yescharacteristic
55C4×Q16×C89C2211yescharacteristic
56C8×D4×C89C2211yescharacteristic
57C2×#107×C89C2211yescharacteristic
58C2×#107×C89C2211yescharacteristic
59C2×Dic356C2311yescharacteristic
60#15668005—41no—
61#15668004—21no—
62#15668002—41no—
63#15667993—41no—
64#15667991—21no—
65#15667991—21no—
66C4×Dic178—21no—
67#15667971—21no—
68#15667968—41no—
69C2×C4⋊Dic89—21no—
70#15667958—22no—
71#15667952—22no—
72C2×#13083549—22no—
73C2×#13083549C2311yescharacteristic
74C2×#13083549C2311yescharacteristic
75C2×#13083549C2311yescharacteristic
76C2×#13083549C2311yescharacteristic
77#15667946—21no—
78#15667946C2311yescharacteristic
79#15667946C2311yescharacteristic
80#15667946C2311yescharacteristic
81#15667944—21no—
82#15667944—21no—
83#15667937—21no—
84#15667937—21no—
85#15667937—21no—
86C2×Q16×C89C2311yescharacteristic
87C22×C8×C89C2311yescharacteristic
88C22×C8×C89C2311yescharacteristic
89C4⋊Q8×C89—21no—
90#125×C89—41no—
91C22⋊Q8×C89—21no—
92C22⋊Q8×C89—21no—
93C22⋊Q8×C89—41no—
94C22⋊Q8×C89—41no—
95C4×Q8×C89—21no—
96C4×D4×C89—41no—
97C4×D4×C89D411yescharacteristic
98C2×C4⋊C4×C89—21no—
99C2×C4⋊C4×C89—21no—
100#107×C89—21no—
101#107×C89—21no—
102#107×C89—21no—
103#107×C89C2311yescharacteristic
104#107×C89C2311yescharacteristic
105#107×C89C2311yescharacteristic
106C4⋊C8×C89D411yescharacteristic
107#103×C89—21no—
108#103×C89—21no—
109#103×C89—21no—
110C22⋊C8×C89D411yescharacteristic
111C22⋊C8×C89D411yescharacteristic
112C4×C8×C89C2311yescharacteristic
113C2×Dic178—21no—
114C22×Dic89—41no—
115Dic356—21no—
116Dic356—21no—
117C4⋊Dic89—21no—
118C4⋊Dic89—21no—
119C4⋊Dic89—21no—
120C4⋊Dic89—41no—
121#13083553—41no—
122#13083553—41no—
123#13083553—41no—
124#13083553—41no—
125#13083553—41no—
126C22⋊Dic89—41no—
127C22⋊Dic89—41no—
128C22⋊Dic89—41no—
129C4×Dic89—41no—
130C4×Dic89—41no—
131#13083549—21no—
132#13083549—21no—
133#13083549—21no—
134#13083549#4112yes—
135C2×Q8×C89—21no—
136C2×Q8×C89—41no—
137C2×D4×C89D811yescharacteristic
138C22×C4×C89—41no—
139C22×C4×C89—41no—
140C22×C4×C89C2×D411yescharacteristic
141C22×C4×C89C2×D411yescharacteristic
142Q16×C89—21no—
143Q16×C89—21no—
144C2×C8×C89—21no—
145C2×C8×C89—21no—
146C2×C8×C89—21no—
147C2×C8×C89—21no—
148C2×C8×C89C2×D411yescharacteristic
149C2×C8×C89C2×D411yescharacteristic
150C2×C8×C89C2×D411yescharacteristic
151C2×C8×C89C2411yescharacteristic
152C4⋊C4×C89—21no—
153C4⋊C4×C89—21no—
154C4⋊C4×C89—21no—
155C4⋊C4×C89—41no—
156C4⋊C4×C89—41no—
157C4⋊C4×C89—41no—
158C4⋊C4×C89—41no—
159C4⋊C4×C89—41no—
160C4⋊C4×C89—41no—
161C4⋊C4×C89D811yescharacteristic
162C22⋊C4×C89—41no—
163C22⋊C4×C89—41no—
164C22⋊C4×C89D811yescharacteristic
165C22⋊C4×C89D811yescharacteristic
166C42×C89—41no—
167C42×C89C2×D411yescharacteristic
168C23×C89C2×D811yescharacteristic
169C23×C89C2×D811yescharacteristic
170Q8×C89—41no—
171Q8×C89—41no—
172Q8×C89—161no—
173D4×C89—22no—
174C2×C4×C89—21no—
175C2×C4×C89—21no—
176C2×C4×C89—21no—
177C2×C4×C89—21no—
178C2×C4×C89—41no—
179C2×C4×C89—41no—
180C2×C4×C89—41no—
181C2×C4×C89—41no—
182C2×C4×C89—161no—
183C2×C4×C89—161no—
184C2×C4×C89C2×D811yescharacteristic
185C2×C4×C89C2×D811yescharacteristic
186C2×C4×C89C2×D811yescharacteristic
187C2×C4×C89C2×D811yescharacteristic
188C2×C4×C89C22×D411yescharacteristic
189C2×Dic89—41no—
190C2×Dic89—41no—
191C2×Dic89—41no—
192C2×Dic89—41no—
193C2×Dic89—82no—
194Dic178—41no—
195Dic178—41no—
196C8×C89—21no—
197C8×C89—21no—
198C8×C89—21no—
199C8×C89C2×#4111yescharacteristic
200C8×C892−1+411yescharacteristic
201C22×C89—21no—
202C22×C89—21no—
203C22×C89—21no—
204C22×C89—21no—
205C22×C89—21no—
206C22×C89#51012yes—
207C22×C89C22×D811yescharacteristic
208C4×C89—21no—
209C4×C89—21no—
210C4×C89—21no—
211C4×C89—21no—
212C4×C89—81no—
213C4×C89—81no—
214C4×C89—81no—
215C4×C89—81no—
216C4×C89#54711yescharacteristic
217C4×C89#57811yescharacteristic
218Dic89—81no—
219Dic89—81no—
220Dic89—81no—
221Dic89—81no—
222#33911—891nomaximal, Sylow
223C2×C89—22no—
224C2×C89—41no—
225C2×C89#333411yescharacteristic
226C2×C89C2×#51011yescharacteristic
227C2×C89#341611yescharacteristic
228#3338D8911yescharacteristic
229#3336—891no—
230#2255—891no—
231#2237—891no—
232#2236—891no—
233#2235—891no—
234#2235—891no—
235#2229—891no—
236#2225—891no—
237#2219—891no—
238#2216—891no—
239#2176—891no—
240#2168—891no—
241#2159—891no—
242C2×#367—891no—
243C89#3391111yescharacteristic, complemented, Sylow
244#554—21no—
245#552—1781no—
246C2×Q32—891no—
247C2×M32—891no—
248C22×C16—891no—
249#500C2×D8911yescharacteristic
250#499—891no—
251#485—1781no—
252#484—21no—
253#484—21no—
254#483—1781no—
255#479—1781no—
256#477—21no—
257#467—891no—
258#467—891no—
259#467C2×D8911yescharacteristic
260#467C2×D8911yescharacteristic
261C4×Q16—891no—
262C4×Q16C2×D8911yescharacteristic
263C8×D4C2×D8911yescharacteristic
264#427—891no—
265C2×#107—891no—
266C2×#107C2×D8911yescharacteristic
267C2×#107C2×D8911yescharacteristic
268#367—891no—
269#367—891no—
270#367—891no—
271#367—891no—
272#367—891no—
273#367—891no—
274#366—891no—
275#366—891no—
276#366—891no—
277C4⋊C16—891no—
278#358—891no—
279#358—891no—
280#358—891no—
281#358—891no—
282#358—891no—
283#358—891no—
284C22⋊C16—891no—
285C22⋊C16—891no—
286#346—891no—
287C2×Q16—891no—
288C2×Q16C22×D8911yescharacteristic
289C22×C8C22×D8911yescharacteristic
290C22×C8C22×D8911yescharacteristic
291C4⋊Q8—21no—
292#126—3561no—
293#125—41no—
294#125—3561no—
295#125—1781no—
296#123—3561no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—41no—
300C22⋊Q8—41no—
301C22⋊Q8—1781no—
302C22⋊Q8—1781no—
303C4×Q8—21no—
304C4×Q8—1781no—
305C4×D4—41no—
306C4×D4C22⋊D8911yescharacteristic
307#117—3561no—
308#117—1781no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—1781no—
312Q32—1782no—
313M32—1782no—
314C2×C16—891no—
315C2×C16—891no—
316C2×C16—891no—
317C2×C16—891no—
318C2×C16—1782no—
319#107—21no—
320#107—21no—
321#107—21no—
322#107—891no—
323#107—891no—
324#107—891no—
325#107—1781no—
326#107C22×D8911yescharacteristic
327#107C22×D8911yescharacteristic
328#107C22×D8911yescharacteristic
329#106—1781no—
330#106—1781no—
331C4⋊C8C22⋊D8911yescharacteristic
332#103—21no—
333#103—21no—
334#103—21no—
335#103—1781no—
336#103—1781no—
337#103—1781no—
338C22⋊C8C22⋊D8911yescharacteristic
339C22⋊C8C22⋊D8911yescharacteristic
340C4×C8C22×D8911yescharacteristic
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—1781no—
344C2×D4#1308355911yescharacteristic
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—3561no—
348C22×C4C2×C22⋊D8911yescharacteristic
349C22×C4C2×C22⋊D8911yescharacteristic
350Q16—21no—
351Q16—21no—
352Q16—1781no—
353Q16—1781no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×C22⋊D8911yescharacteristic
359C2×C8C2×C22⋊D8911yescharacteristic
360C2×C8C2×C22⋊D8911yescharacteristic
361C2×C8C23×D8911yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—3561no—
372C4⋊C4—3561no—
373C4⋊C4—3561no—
374C4⋊C4—3561no—
375C4⋊C4—3561no—
376C4⋊C4—3561no—
377C4⋊C4—1781no—
378C4⋊C4—1781no—
379C4⋊C4—1781no—
380C4⋊C4#1308355911yescharacteristic
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—3561no—
384C22⋊C4—3561no—
385C22⋊C4—3561no—
386C22⋊C4#1308355911yescharacteristic
387C22⋊C4#1308355911yescharacteristic
388C42—41no—
389C42—3561no—
390C42—3561no—
391C42C2×C22⋊D8911yescharacteristic
392C16—892no—
393C16—1781no—
394C16—1781no—
395C16—1781no—
396C23C2×#1308355911yescharacteristic
397C23C2×#1308355911yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—161no—
401Q8—3561no—
402Q8—3561no—
403D4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—161no—
413C2×C4—161no—
414C2×C4—3561no—
415C2×C4—3561no—
416C2×C4—3561no—
417C2×C4—3561no—
418C2×C4—7122no—
419C2×C4C2×#1308355911yescharacteristic
420C2×C4C2×#1308355911yescharacteristic
421C2×C4C2×#1308355911yescharacteristic
422C2×C4C2×#1308355911yescharacteristic
423C2×C4C22×C22⋊D8911yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8#41×D8911yescharacteristic
428C8#1566805911yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22#1915203912yes—
435C22C22×#1308355911yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—7121no—
445C4—7121no—
446C4—7121no—
447C4—7121no—
448C4#1915224711yescharacteristic
449C4#1915241311yescharacteristic
450C2—22no—
451C2—41no—
452C2#2603259511yescharacteristic
453C2C2×#1915203911yescharacteristic
454C2#2603339911yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3338C2
socleC22×C89C22×D8
prime radicalC2×D4×C89D8
hypercenter#3338D89
hypocenterC89#33911
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #26022535C2semidirect
3. #26022457C2semidirect
4. #26022454C2semidirect
5. #26022449C2semidirect
6. #26022447C2semidirect
7. #26022416C2semidirect
8. #26022391C2semidirect
9. #26022358C2semidirect
10. #26022341C2semidirect
11. #26022121C2semidirect
13. #26022038C2semidirect
14. C2×#19151425C2semidirect
243. C89#33911semidirect

Maximal subgroups

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

subgroupindex
#260326212
#260225352
#260224572
#260224542
#260224492
#260224472
#260224162
#260223912
#260223582
#260223412
#260221212
#260220782
#260220382
C2×#191514252
C89×#33382
#3391189