Finite Group Databasebeta

← #36711878

Subgroups — #36711878

Order 22784.

504 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26032600C211yesmaximal, characteristic
2#26022539C211yesmaximal, characteristic, complemented
3#26022452C211yesmaximal, characteristic, complemented
4#26022447C211yesmaximal, characteristic, complemented
5#26022443C211yesmaximal, characteristic, complemented
6#26022432C211yesmaximal, characteristic, complemented
7#26022406C211yesmaximal, characteristic, complemented
8#26022404C211yesmaximal, characteristic, complemented
9#26022367C211yesmaximal, characteristic, complemented
10#26022350C211yesmaximal, characteristic, complemented
11#26022114C211yesmaximal, characteristic, complemented
12#26022078C211yesmaximal, characteristic, complemented
13#26022041C211yesmaximal, characteristic, complemented
14C2×#19151423C211yesmaximal, characteristic, complemented
15C89×#3336C211yesmaximal, characteristic, complemented
16#19152252—21no—
17C2×#15667955C2211yescharacteristic
18C2×#15667952C2211yescharacteristic
19C22×#13083549C2211yescharacteristic
20#19152003C2211yescharacteristic
21#19151912—21no—
22#19151912—21no—
23#19151879—21no—
24#19151834C2211yescharacteristic
25#19151834C2211yescharacteristic
26C4×D712C2211yescharacteristic
27C2×#15667946C2211yescharacteristic
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#552×C89—21no—
49#499×C89C2211yescharacteristic
50#485×C89—21no—
51#483×C89—21no—
52#479×C89—21no—
53#467×C89C2211yescharacteristic
54#467×C89C2211yescharacteristic
55C4×Q16×C89C2211yescharacteristic
56C8×D4×C89C2211yescharacteristic
57#427×C89C2211yescharacteristic
58C2×#107×C89C2211yescharacteristic
59C2×#13083573—41no—
60C2×D712C2311yescharacteristic
61#15668012—21no—
62#15667996—41no—
63#15667996—41no—
64#15667992—41no—
65#15667986—21no—
66#15667986—21no—
67C4×D356—21no—
68C2×C4⋊Dic89—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#15667946—21no—
79#15667946—21no—
80#15667946C2311yescharacteristic
81#15667946C2311yescharacteristic
82#15667946C2311yescharacteristic
83#15667934—21no—
84#15667934—21no—
85#15667934—21no—
86C2×Q16×C89C2311yescharacteristic
87C22×C8×C89C2311yescharacteristic
88C22×C8×C89C2311yescharacteristic
89#126×C89—41no—
90#125×C89—21no—
91#125×C89—41no—
92#123×C89—41no—
93C22⋊Q8×C89—21no—
94C22⋊Q8×C89—21no—
95C4×Q8×C89—21no—
96C4×D4×C89D411yescharacteristic
97#117×C89—21no—
98#117×C89—41no—
99C2×C4⋊C4×C89—21no—
100#107×C89—21no—
101#107×C89C2311yescharacteristic
102#107×C89C2311yescharacteristic
103#107×C89C2311yescharacteristic
104#106×C89—21no—
105#106×C89—21no—
106C4⋊C8×C89D411yescharacteristic
107#103×C89—21no—
108#103×C89—21no—
109#103×C89—21no—
110C22⋊C8×C89D411yescharacteristic
111C22⋊C8×C89D411yescharacteristic
112C4×C8×C89C2311yescharacteristic
113#13083573—84no—
114C2×Dic178—41no—
115C2×D356—21no—
116C2×C22⋊D89—41no—
117C2×C22⋊D89—41no—
118C22×Dic89—41no—
119C22×Dic89—41no—
120C2×C4×D89—41no—
121D712—21no—
122D712—21no—
123C4⋊Dic89—21no—
124C4⋊Dic89—21no—
125C4⋊Dic89—21no—
126C4⋊Dic89—41no—
127C4⋊Dic89—41no—
128C4⋊Dic89—41no—
129#13083551—41no—
130#13083551—41no—
131#13083551—41no—
132#13083549—21no—
133#13083549—21no—
134#13083549—21no—
135#13083549#4112yescomplemented
136C2×Q8×C89—21no—
137C2×D4×C89D811yescharacteristic
138C22×C4×C89—41no—
139C22×C4×C89C2×D411yescharacteristic
140C22×C4×C89C2×D411yescharacteristic
141Q16×C89—21no—
142Q16×C89—21no—
143C2×C8×C89—21no—
144C2×C8×C89—21no—
145C2×C8×C89—21no—
146C2×C8×C89—21no—
147C2×C8×C89C2×D411yescharacteristic
148C2×C8×C89C2×D411yescharacteristic
149C2×C8×C89C2×D411yescharacteristic
150C2×C8×C89C2411yescharacteristic
151C4⋊C4×C89—21no—
152C4⋊C4×C89—21no—
153C4⋊C4×C89—21no—
154C4⋊C4×C89—41no—
155C4⋊C4×C89—41no—
156C4⋊C4×C89—41no—
157C4⋊C4×C89—41no—
158C4⋊C4×C89—41no—
159C4⋊C4×C89—41no—
160C4⋊C4×C89D811yescharacteristic
161C22⋊C4×C89—41no—
162C22⋊C4×C89—41no—
163C22⋊C4×C89—41no—
164C22⋊C4×C89D811yescharacteristic
165C22⋊C4×C89D811yescharacteristic
166C42×C89—41no—
167C42×C89—41no—
168C42×C89C2×D411yescharacteristic
169C23×C89C2×D811yescharacteristic
170C23×C89C2×D811yescharacteristic
171C22×D89—41no—
172Q8×C89—41no—
173Q8×C89—41no—
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—82no—
184C2×C4×C89C2×D811yescharacteristic
185C2×C4×C89C2×D811yescharacteristic
186C2×C4×C89C2×D811yescharacteristic
187C2×C4×C89C2×D811yescharacteristic
188C2×C4×C89C22×D411yescharacteristic
189C22⋊D89—82no—
190C22⋊D89—82no—
191C2×Dic89—41no—
192C2×Dic89—41no—
193C2×Dic89—41no—
194C2×Dic89—82no—
195C2×Dic89—82no—
196D356—41no—
197D356—41no—
198C4×D89—82no—
199Dic178—82no—
200C8×C89—21no—
201C8×C89—21no—
202C8×C89—21no—
203C8×C89C2×#4111yescharacteristic
204C8×C892−1+411yescharacteristic
205C22×C89—21no—
206C22×C89—21no—
207C22×C89—21no—
208C22×C89—21no—
209C22×C89—21no—
210C22×C89#50912yes—
211C22×C89C22×D811yescharacteristic
212C2×D89—81no—
213C2×D89—81no—
214C2×D89—81no—
215C4×C89—21no—
216C4×C89—21no—
217C4×C89—21no—
218C4×C89—21no—
219C4×C89—81no—
220C4×C89—81no—
221C4×C89—81no—
222C4×C89—81no—
223C4×C89#54711yescharacteristic
224C4×C89#57811yescharacteristic
225Dic89—81no—
226Dic89—81no—
227Dic89—81no—
228#33910—891nomaximal, Sylow
229C2×C89—22no—
230C2×C89—41no—
231C2×C89#333411yescharacteristic
232C2×C89C2×#50911yescharacteristic
233C2×C89#341711yescharacteristic
234D89—161no—
235#3336D8911yescharacteristic, complemented
236#3334—891no—
237#2255—891no—
238#2236—891no—
239#2235—891no—
240#2234—891no—
241#2232—891no—
242#2228—891no—
243#2227—891no—
244#2220—891no—
245#2217—891no—
246#2175—891no—
247#2168—891no—
248#2159—891no—
249C2×#366—891no—
250C89#3391011yescharacteristic, complemented, Sylow
251#552—21no—
252#547—1781no—
253C2×SD32—891no—
254C2×M32—891no—
255C22×C16—891no—
256#500—891no—
257#499C2×D8911yescharacteristic
258#485—21no—
259#483—21no—
260#480—1781no—
261#480—1781no—
262#479—21no—
263#474—1781no—
264#467C2×D8911yescharacteristic
265#467C2×D8911yescharacteristic
266#466—891no—
267#466—891no—
268C4×Q16C2×D8911yescharacteristic
269C4×D8—891no—
270C8×D4C2×D8911yescharacteristic
271#427C2×D8911yescharacteristic
272C2×#107—891no—
273C2×#107—891no—
274C2×#107C2×D8911yescharacteristic
275#367—891no—
276#367—891no—
277#367—891no—
278#366—891no—
279#366—891no—
280#366—891no—
281#366—891no—
282#366—891no—
283#366—891no—
284C4⋊C16—891no—
285#358—891no—
286#358—891no—
287#358—891no—
288#357—891no—
289#357—891no—
290#357—891no—
291C22⋊C16—891no—
292C22⋊C16—891no—
293#346—891no—
294C2×#41—3561no—
295C2×Q16C22×D8911yescharacteristic
296C2×D8—891no—
297C22×C8C22×D8911yescharacteristic
298C22×C8C22×D8911yescharacteristic
299C4⋊Q8—1781no—
300#126—41no—
301#125—21no—
302#125—41no—
303#123—41no—
304#123—3561no—
305#123—3561no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—3561no—
309#121—1781no—
310#121—1781no—
311C4×Q8—21no—
312C4×D4—1781no—
313C4×D4C22⋊D8911yescharacteristic
314#117—21no—
315#117—41no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—1781no—
318C2×C4⋊C4—1781no—
319SD32—1782no—
320M32—1782no—
321C2×C16—891no—
322C2×C16—891no—
323C2×C16—891no—
324C2×C16—891no—
325C2×C16—1782no—
326#107—21no—
327#107—891no—
328#107—891no—
329#107—891no—
330#107—1781no—
331#107—1781no—
332#107—1781no—
333#107C22×D8911yescharacteristic
334#107C22×D8911yescharacteristic
335#107C22×D8911yescharacteristic
336#106—21no—
337#106—21no—
338C4⋊C8C22⋊D8911yescharacteristic
339#103—21no—
340#103—21no—
341#103—21no—
342#102—1781no—
343#102—1781no—
344#102—1781no—
345C22⋊C8C22⋊D8911yescharacteristic
346C22⋊C8C22⋊D8911yescharacteristic
347C4×C8C22×D8911yescharacteristic
348#41—7124no—
349C2×Q8—21no—
350C2×Q8—3561no—
351C2×D4—3561no—
352C2×D4—3561no—
353C2×D4—1781no—
354C2×D4#1308355911yescharacteristic
355C22×C4—41no—
356C22×C4—3561no—
357C22×C4—3561no—
358C22×C4—3561no—
359C22×C4C2×C22⋊D8911yescharacteristic
360C22×C4C2×C22⋊D8911yescharacteristic
361Q16—21no—
362Q16—21no—
363D8—1781no—
364D8—1781no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8C2×C22⋊D8911yescharacteristic
370C2×C8C2×C22⋊D8911yescharacteristic
371C2×C8C2×C22⋊D8911yescharacteristic
372C2×C8C23×D8911yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—3561no—
383C4⋊C4—3561no—
384C4⋊C4—3561no—
385C4⋊C4—1781no—
386C4⋊C4—1781no—
387C4⋊C4—1781no—
388C4⋊C4#1308355911yescharacteristic
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—3561no—
393C22⋊C4—3561no—
394C22⋊C4—3561no—
395C22⋊C4#1308355911yescharacteristic
396C22⋊C4#1308355911yescharacteristic
397C42—41no—
398C42—41no—
399C42C2×C22⋊D8911yescharacteristic
400C16—892no—
401C16—1781no—
402C16—1781no—
403C16—1781no—
404C23—3561no—
405C23C2×#1308355911yescharacteristic
406C23C2×#1308355911yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—7122no—
410D4—22no—
411D4—3561no—
412D4—3561no—
413D4—7122no—
414D4—7122no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—82no—
424C2×C4—3561no—
425C2×C4—3561no—
426C2×C4—3561no—
427C2×C4—7122no—
428C2×C4—7122no—
429C2×C4—7122no—
430C2×C4C2×#1308355911yescharacteristic
431C2×C4C2×#1308355911yescharacteristic
432C2×C4C2×#1308355911yescharacteristic
433C2×C4C2×#1308355911yescharacteristic
434C2×C4C22×C22⋊D8911yescharacteristic
435C8—21no—
436C8—21no—
437C8—21no—
438C8#41×D8911yescharacteristic
439C8#1566805911yescharacteristic
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—7121no—
446C22—7121no—
447C22—7121no—
448C22#1915203312yes—
449C22C22×#1308355911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—7121no—
459C4—7121no—
460C4—7121no—
461C4#1915224711yescharacteristic
462C4#1915241311yescharacteristic
463C2—22no—
464C2—41no—
465C2—14241no—
466C2#2603259511yescharacteristic
467C2C2×#1915203311yescharacteristic
468C2#2603340311yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3336C2
socleC22×C89C22×D8
prime radical#26032600C2
hypercenter#3336D89
hypocenterC89#33910
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #26022539C2semidirect
3. #26022452C2semidirect
4. #26022447C2semidirect
5. #26022443C2semidirect
6. #26022432C2semidirect
7. #26022406C2semidirect
8. #26022404C2semidirect
9. #26022367C2semidirect
10. #26022350C2semidirect
11. #26022114C2semidirect
12. #26022078C2semidirect
13. #26022041C2semidirect
14. C2×#19151423C2semidirect
15. C89×#3336C2semidirect
135. #13083549#41semidirect
235. #3336D89semidirect
250. C89#33910semidirect

Maximal subgroups

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

subgroupindex
#260326002
#260225392
#260224522
#260224472
#260224432
#260224322
#260224062
#260224042
#260223672
#260223502
#260221142
#260220782
#260220412
C2×#191514232
C89×#33362
#3391089