Finite Group Databasebeta

← #36711917

Subgroups — #36711917

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26032655C211yesmaximal, characteristic
2#26022535C211yesmaximal, characteristic, complemented
3#26022457C211yesmaximal, characteristic, complemented
4#26022452C211yesmaximal, characteristic, complemented
5#26022437C211yesmaximal, characteristic, complemented
6#26022432C211yesmaximal, characteristic, complemented
7#26022416C211yesmaximal, characteristic, complemented
8#26022386C211yesmaximal, characteristic, complemented
9#26022367C211yesmaximal, characteristic, complemented
10#26022350C211yesmaximal, characteristic, complemented
11#26022114C211yesmaximal, characteristic, complemented
12#26022078C211yesmaximal, characteristic, complemented
13#26022038C211yesmaximal, characteristic, complemented
14C2×#19151423C211yesmaximal, characteristic, complemented
15C89×#3338C211yesmaximal, characteristic, complemented
16#19152294—21no—
17C2×#15667955C2211yescharacteristic
18C2×#15667952C2211yescharacteristic
19C22×#13083549C2211yescharacteristic
20#19152000C2211yescharacteristic
21#19151923—21no—
22#19151917—21no—
23#19151899—21no—
24#19151834C2211yescharacteristic
25#19151834C2211yescharacteristic
26C4×D712C2211yescharacteristic
27#19151636C2211yescharacteristic
28C2×#15667946C2211yescharacteristic
29#19151425C2211yescharacteristic
30#19151425C2211yescharacteristic
31#19151425C2211yescharacteristic
32#19151423C2211yescharacteristic
33#19151423C2211yescharacteristic
34#19151423C2211yescharacteristic
35#19151423C2211yescharacteristic
36#19151423C2211yescharacteristic
37#19151423C2211yescharacteristic
38C4⋊#13083549C2211yescharacteristic
39#19151402C2211yescharacteristic
40#19151402C2211yescharacteristic
41#19151402C2211yescharacteristic
42#19151398C2211yescharacteristic
43#19151398C2211yescharacteristic
44#19151398C2211yescharacteristic
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×D712C2311yescharacteristic
60#15668007—41no—
61#15668004—21no—
62#15667994—41no—
63#15667986—21no—
64#15667986—21no—
65#15667984—41no—
66C4×D356—21no—
67#15667972—41no—
68#15667971—21no—
69C2×C4⋊Dic89—21no—
70#15667955—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#15667934—21no—
84#15667934—21no—
85#15667934—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×D356—21no—
114C2×C22⋊D89—41no—
115C2×C22⋊D89—41no—
116C22×Dic89—41no—
117C2×C4×D89—41no—
118D712—21no—
119D712—21no—
120C4⋊Dic89—21no—
121C4⋊Dic89—21no—
122C4⋊Dic89—21no—
123C4⋊Dic89—41no—
124#13083553—41no—
125#13083553—41no—
126C22⋊Dic89—41no—
127#13083551—41no—
128#13083551—41no—
129#13083551—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
170C22×D89—41no—
171Q8×C89—41no—
172Q8×C89—41no—
173Q8×C89—161no—
174D4×C89—22no—
175C2×C4×C89—21no—
176C2×C4×C89—21no—
177C2×C4×C89—21no—
178C2×C4×C89—21no—
179C2×C4×C89—41no—
180C2×C4×C89—41no—
181C2×C4×C89—41no—
182C2×C4×C89—41no—
183C2×C4×C89—161no—
184C2×C4×C89—161no—
185C2×C4×C89C2×D811yescharacteristic
186C2×C4×C89C2×D811yescharacteristic
187C2×C4×C89C2×D811yescharacteristic
188C2×C4×C89C2×D811yescharacteristic
189C2×C4×C89C22×D411yescharacteristic
190C22⋊D89—82no—
191C22⋊D89—161no—
192C2×Dic89—41no—
193C2×Dic89—41no—
194C2×Dic89—41no—
195C2×Dic89—161no—
196D356—41no—
197D356—41no—
198C4×D89—161no—
199C8×C89—21no—
200C8×C89—21no—
201C8×C89—21no—
202C8×C89C2×#4111yescharacteristic
203C8×C892−1+411yescharacteristic
204C22×C89—21no—
205C22×C89—21no—
206C22×C89—21no—
207C22×C89—21no—
208C22×C89—21no—
209C22×C89#51012yes—
210C22×C89C22×D811yescharacteristic
211C2×D89—81no—
212C2×D89—81no—
213C2×D89—81no—
214C4×C89—21no—
215C4×C89—21no—
216C4×C89—21no—
217C4×C89—21no—
218C4×C89—81no—
219C4×C89—81no—
220C4×C89—81no—
221C4×C89—81no—
222C4×C89#54711yescharacteristic
223C4×C89#57811yescharacteristic
224Dic89—81no—
225Dic89—81no—
226Dic89—81no—
227#33913—891nomaximal, Sylow
228C2×C89—22no—
229C2×C89—41no—
230C2×C89#333411yescharacteristic
231C2×C89C2×#51011yescharacteristic
232C2×C89#341511yescharacteristic
233D89—161no—
234#3339—891no—
235#3338D8911yescharacteristic, complemented
236#2255—891no—
237#2237—891no—
238#2236—891no—
239#2233—891no—
240#2232—891no—
241#2229—891no—
242#2224—891no—
243#2220—891no—
244#2217—891no—
245#2175—891no—
246#2168—891no—
247#2159—891no—
248C2×#366—891no—
249C89#3391311yescharacteristic, complemented, Sylow
250#554—21no—
251#553—1781no—
252C2×SD32—891no—
253C2×M32—891no—
254C22×C16—891no—
255#500C2×D8911yescharacteristic
256#499—891no—
257#484—21no—
258#484—21no—
259#482—1781no—
260#481—1781no—
261#478—1781no—
262#477—21no—
263#467C2×D8911yescharacteristic
264#467C2×D8911yescharacteristic
265#466—891no—
266#466—891no—
267C4×Q16C2×D8911yescharacteristic
268C4×D8—891no—
269C8×D4C2×D8911yescharacteristic
270#427—891no—
271C2×#107—891no—
272C2×#107C2×D8911yescharacteristic
273C2×#107C2×D8911yescharacteristic
274#367—891no—
275#367—891no—
276#367—891no—
277#366—891no—
278#366—891no—
279#366—891no—
280#366—891no—
281#366—891no—
282#366—891no—
283C4⋊C16—891no—
284#358—891no—
285#358—891no—
286#358—891no—
287#357—891no—
288#357—891no—
289#357—891no—
290C22⋊C16—891no—
291C22⋊C16—891no—
292#346—891no—
293C2×Q16C22×D8911yescharacteristic
294C2×D8—891no—
295C22×C8C22×D8911yescharacteristic
296C22×C8C22×D8911yescharacteristic
297C4⋊Q8—21no—
298#126—3561no—
299#125—41no—
300#125—1781no—
301#123—3561no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—41no—
305C22⋊Q8—41no—
306#121—3561no—
307#121—1781no—
308#121—1781no—
309C4×Q8—21no—
310C4×D4—41no—
311C4×D4—3561no—
312C4×D4—1781no—
313C4×D4C22⋊D8911yescharacteristic
314#117—1781no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—1781no—
318SD32—1782no—
319M32—1782no—
320C2×C16—891no—
321C2×C16—891no—
322C2×C16—891no—
323C2×C16—891no—
324C2×C16—1782no—
325#107—21no—
326#107—21no—
327#107—21no—
328#107—891no—
329#107—891no—
330#107—891no—
331#107—1781no—
332#107C22×D8911yescharacteristic
333#107C22×D8911yescharacteristic
334#107C22×D8911yescharacteristic
335#106—1781no—
336#106—1781no—
337C4⋊C8C22⋊D8911yescharacteristic
338#103—21no—
339#103—21no—
340#103—21no—
341#102—1781no—
342#102—1781no—
343#102—1781no—
344C22⋊C8C22⋊D8911yescharacteristic
345C22⋊C8C22⋊D8911yescharacteristic
346C4×C8C22×D8911yescharacteristic
347C2×Q8—21no—
348C2×Q8—41no—
349C2×D4—3561no—
350C2×D4—3561no—
351C2×D4—1781no—
352C2×D4#1308355911yescharacteristic
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—3561no—
356C22×C4—3561no—
357C22×C4C2×C22⋊D8911yescharacteristic
358C22×C4C2×C22⋊D8911yescharacteristic
359Q16—21no—
360Q16—21no—
361D8—1781no—
362D8—1781no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8C2×C22⋊D8911yescharacteristic
368C2×C8C2×C22⋊D8911yescharacteristic
369C2×C8C2×C22⋊D8911yescharacteristic
370C2×C8C23×D8911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—3561no—
381C4⋊C4—3561no—
382C4⋊C4—3561no—
383C4⋊C4—1781no—
384C4⋊C4—1781no—
385C4⋊C4—1781no—
386C4⋊C4#1308355911yescharacteristic
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—3561no—
390C22⋊C4—3561no—
391C22⋊C4—3561no—
392C22⋊C4—3561no—
393C22⋊C4#1308355911yescharacteristic
394C22⋊C4#1308355911yescharacteristic
395C42—41no—
396C42—3561no—
397C42C2×C22⋊D8911yescharacteristic
398C16—892no—
399C16—1781no—
400C16—1781no—
401C16—1781no—
402C23—3561no—
403C23C2×#1308355911yescharacteristic
404C23C2×#1308355911yescharacteristic
405Q8—41no—
406Q8—41no—
407Q8—161no—
408D4—22no—
409D4—3561no—
410D4—3561no—
411D4—14241no—
412D4—7122no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—161no—
422C2×C4—161no—
423C2×C4—3561no—
424C2×C4—3561no—
425C2×C4—3561no—
426C2×C4—14241no—
427C2×C4—14241no—
428C2×C4C2×#1308355911yescharacteristic
429C2×C4C2×#1308355911yescharacteristic
430C2×C4C2×#1308355911yescharacteristic
431C2×C4C2×#1308355911yescharacteristic
432C2×C4C22×C22⋊D8911yescharacteristic
433C8—21no—
434C8—21no—
435C8—21no—
436C8#41×D8911yescharacteristic
437C8#1566805911yescharacteristic
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—7121no—
444C22—7121no—
445C22—7121no—
446C22#1915203912yes—
447C22C22×#1308355911yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—7121no—
457C4—7121no—
458C4—7121no—
459C4#1915224711yescharacteristic
460C4#1915241311yescharacteristic
461C2—22no—
462C2—41no—
463C2—14241no—
464C2#2603259511yescharacteristic
465C2C2×#1915203911yescharacteristic
466C2#2603339111yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3338C2
socleC22×C89C22×D8
prime radical#26032655C2
hypercenter#3338D89
hypocenterC89#33913
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. #26022452C2semidirect
5. #26022437C2semidirect
6. #26022432C2semidirect
7. #26022416C2semidirect
8. #26022386C2semidirect
9. #26022367C2semidirect
10. #26022350C2semidirect
11. #26022114C2semidirect
12. #26022078C2semidirect
13. #26022038C2semidirect
14. C2×#19151423C2semidirect
15. C89×#3338C2semidirect
235. #3338D89semidirect
249. C89#33913semidirect

Maximal subgroups

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

subgroupindex
#260326552
#260225352
#260224572
#260224522
#260224372
#260224322
#260224162
#260223862
#260223672
#260223502
#260221142
#260220782
#260220382
C2×#191514232
C89×#33382
#3391389