Finite Group Databasebeta

← C89×#32871 #36410719

Subgroups — C89×#32871 #36410719

Order 22784.

492 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2103C211yesmaximal, characteristic, complemented
2C89×#2103C211yesmaximal, characteristic, complemented
3C89×#2098C211yesmaximal, characteristic, complemented
4C89×#2082C211yesmaximal, characteristic, complemented
5C89×#2081C211yesmaximal, characteristic, complemented
6C89×#2070C211yesmaximal, characteristic, complemented
7C89×#2069C211yesmaximal, characteristic, complemented
8C89×#2049C211yesmaximal, characteristic, complemented
9C89×#2048C211yesmaximal, characteristic
10C89×#2046C211yesmaximal, characteristic, complemented
11C89×#2046C211yesmaximal, characteristic, complemented
12C89×#2019C211yesmaximal, characteristic
13C89×#1903C211yesmaximal, characteristic
14C89×#1902C211yesmaximal, characteristic, complemented
15C89×#1858C211yesmaximal, characteristic, complemented
16C2×#137×C89C2211yescharacteristic
17C2×AGL(1,ℤ/8)×C89C2211yescharacteristic
18C22×SD16×C89C2211yescharacteristic
19#568×C89C2211yescharacteristic
20#430×C89C2211yescharacteristic
21#430×C89C2211yescharacteristic
22#430×C89C2211yescharacteristic
23#430×C89C2211yescharacteristic
24#430×C89C2211yescharacteristic
25#424×C89C2211yescharacteristic
26#424×C89C2211yescharacteristic
27#421×C89C2211yescharacteristic
28#421×C89C2211yescharacteristic
29#419×C89C2211yescharacteristic
30#419×C89C2211yescharacteristic
31#418×C89C2211yescharacteristic
32#418×C89C2211yescharacteristic
33#413×C89C2211yescharacteristic
34C2×#101×C89C2211yescharacteristic
35C2×#101×C89C2211yescharacteristic
36C2×#101×C89C2211yescharacteristic
37C2×#100×C89C2211yescharacteristic
38C2×#100×C89C2211yescharacteristic
39C2×#100×C89C2211yescharacteristic
40#408×C89C2211yescharacteristic
41#344×C89C2211yescharacteristic
42#344×C89C2211yescharacteristic
43#344×C89C2211yescharacteristic
44#344×C89C2211yescharacteristic
45#341×C89C2211yescharacteristic
46#341×C89C2211yescharacteristic
47#341×C89C2211yescharacteristic
48#338×C89C2211yescharacteristic
49#338×C89C2211yescharacteristic
50H(3,ℤ/4)×C89C2211yescharacteristic
51C2×#41×C89C2311yescharacteristic
52C22×Q8×C89C2311yescharacteristic
53C22×D4×C89C2311yescharacteristic
54#137×C89—41no—
55#137×C89—41no—
56AGL(1,ℤ/8)×C89—41no—
57AGL(1,ℤ/8)×C89—41no—
58C2×Q16×C89—21no—
59C2×SD16×C89—21no—
60C2×SD16×C89—21no—
61C2×SD16×C89—41no—
62C2×SD16×C89—41no—
63C2×SD16×C89—41no—
64C2×D8×C89—21no—
65#131×C89—41no—
66#131×C89—41no—
67C2×M16×C89—41no—
68C2×M16×C89C2311yescharacteristic
69C2×M16×C89C2311yescharacteristic
70C2×M16×C89C2311yescharacteristic
71C2×M16×C89C2311yescharacteristic
72C2×M16×C89C2311yescharacteristic
73C2×M16×C89C2311yescharacteristic
74C2×M16×C89C2311yescharacteristic
75C2×M16×C89C2311yescharacteristic
76C2×M16×C89C2311yescharacteristic
77C22×C8×C89C2311yescharacteristic
78#117×C89C2311yescharacteristic
79#117×C89C2311yescharacteristic
80#108×C89—41no—
81#108×C89—41no—
82#108×C89—41no—
83#108×C89—41no—
84#108×C89—41no—
85C4⋊C8×C89—41no—
86C4⋊C8×C89—41no—
87#104×C89—41no—
88#104×C89—41no—
89#103×C89—41no—
90#103×C89—41no—
91#102×C89—41no—
92#102×C89—41no—
93#101×C89—21no—
94#101×C89—41no—
95#101×C89—41no—
96#101×C89—41no—
97#100×C89—21no—
98#100×C89—41no—
99#100×C89—41no—
100#100×C89—41no—
101C22⋊C8×C89—41no—
102C24×C89—21no—
103#41×C89—21no—
104#41×C89—21no—
105#41×C89—21no—
106#41×C89—21no—
107C2×Q8×C89—21no—
108C2×Q8×C89—21no—
109C2×Q8×C89—21no—
110C2×Q8×C89—21no—
111C2×Q8×C89—21no—
112C2×Q8×C89—21no—
113C2×Q8×C89#4111yescharacteristic
114C2×D4×C89—21no—
115C2×D4×C89—21no—
116C2×D4×C89—21no—
117C2×D4×C89—21no—
118C2×D4×C89—21no—
119C2×D4×C89—21no—
120C2×D4×C89—21no—
121C2×D4×C89#4111yescharacteristic
122C22×C4×C89—21no—
123C22×C4×C89—21no—
124C22×C4×C89C2411yescharacteristic
125Q16×C89—42no—
126SD16×C89—42no—
127SD16×C89—42no—
128SD16×C89—44no—
129D8×C89—42no—
130M16×C89—21no—
131M16×C89—21no—
132M16×C89—21no—
133M16×C89—21no—
134M16×C89—21no—
135M16×C89—21no—
136M16×C89—21no—
137M16×C89—21no—
138M16×C89—21no—
139M16×C89—21no—
140M16×C89—21no—
141M16×C89—21no—
142M16×C89—21no—
143M16×C89—21no—
144M16×C89—21no—
145M16×C89—21no—
146M16×C89—21no—
147M16×C89—21no—
148M16×C89—42no—
149C2×C8×C89—21no—
150C2×C8×C89—21no—
151C2×C8×C89—21no—
152C2×C8×C89—21no—
153C2×C8×C89—21no—
154C2×C8×C89—21no—
155C2×C8×C89—21no—
156C2×C8×C89—21no—
157C2×C8×C89—21no—
158C2×C8×C89—21no—
159C2×C8×C89—21no—
160C2×C8×C89—21no—
161C2×C8×C89—42no—
162C4⋊C4×C89—21no—
163C4⋊C4×C89—21no—
164C22⋊C4×C89—21no—
165C22⋊C4×C89—21no—
166C42×C89—21no—
167C42×C89—21no—
168C23×C89—21no—
169C23×C89—41no—
170C23×C89—41no—
171C23×C89—41no—
172C23×C89—161no—
173C23×C892−1+411yescharacteristic
174Q8×C89—21no—
175Q8×C89—21no—
176Q8×C89—41no—
177Q8×C89—41no—
178Q8×C89—41no—
179Q8×C89—41no—
180D4×C89—21no—
181D4×C89—21no—
182D4×C89—41no—
183D4×C89—41no—
184D4×C89—41no—
185D4×C89—41no—
186D4×C89—41no—
187D4×C89—41no—
188C2×C4×C89—21no—
189C2×C4×C89—41no—
190C2×C4×C89—41no—
191C2×C4×C89—41no—
192C2×C4×C89—41no—
193C2×C4×C89—41no—
194C2×C4×C89—41no—
195C2×C4×C89—41no—
196C2×C4×C89C2×#4111yescharacteristic
197C2×C4×C89H(5,𝔽2)11yescharacteristic
198C2×C4×C89H(5,𝔽2)11yescharacteristic
199C2×C4×C89H(5,𝔽2)11yescharacteristic
200C2×C4×C89H(5,𝔽2)11yescharacteristic
201C2×C4×C89H(5,𝔽2)11yescharacteristic
202C8×C89—41no—
203C8×C89—41no—
204C8×C89—41no—
205C8×C89—41no—
206C8×C89—41no—
207C8×C89—41no—
208C8×C89—41no—
209C8×C89—41no—
210C8×C89—41no—
211C8×C89—41no—
212C22×C89—41no—
213C22×C89—41no—
214C22×C89—81no—
215C22×C89—81no—
216C22×C89—161no—
217C22×C89—161no—
218C22×C89—161no—
219C22×C89#53911yescharacteristic
220C22×C89#56211yescharacteristic
221C22×C89#56211yescharacteristic
222C4×C89—41no—
223C4×C89—81no—
224C4×C89—81no—
225C4×C89—81no—
226C4×C89#53811yescharacteristic
227C4×C89#53811yescharacteristic
228C4×C89#56011yescharacteristic
229C4×C89#56011yescharacteristic
230#32871C8911yesmaximal, characteristic, direct product factor, Sylow
231C2×C89—21no—
232C2×C89—21no—
233C2×C89—21no—
234C2×C89—81no—
235C2×C89—161no—
236C2×C89#282611yescharacteristic
237#2103C2×C8911yescharacteristic, complemented
238#2103C2×C8911yescharacteristic, complemented
239#2098C2×C8911yescharacteristic, complemented
240#2082C2×C8911yescharacteristic, complemented
241#2081C2×C8911yescharacteristic, complemented
242#2070C2×C8911yescharacteristic, complemented
243#2069C2×C8911yescharacteristic, complemented
244#2049C2×C8911yescharacteristic, complemented
245#2048C2×C8911yescharacteristic
246#2046C2×C8911yescharacteristic, complemented
247#2046C2×C8911yescharacteristic, complemented
248#2019C2×C8911yescharacteristic
249#1903C2×C8911yescharacteristic
250#1902C2×C8911yescharacteristic, complemented
251#1858C2×C8911yescharacteristic, complemented
252C89#3287111yescharacteristic, direct product factor, Sylow
253C2×#137C22×C8911yescharacteristic
254C2×AGL(1,ℤ/8)C22×C8911yescharacteristic
255C22×SD16C22×C8911yescharacteristic
256#568C22×C8911yescharacteristic
257#430C22×C8911yescharacteristic
258#430C22×C8911yescharacteristic
259#430C22×C8911yescharacteristic
260#430C22×C8911yescharacteristic
261#430C22×C8911yescharacteristic
262#424C22×C8911yescharacteristic
263#424C22×C8911yescharacteristic
264#421C22×C8911yescharacteristic
265#421C22×C8911yescharacteristic
266#419C22×C8911yescharacteristic
267#419C22×C8911yescharacteristic
268#418C22×C8911yescharacteristic
269#418C22×C8911yescharacteristic
270#413C22×C8911yescharacteristic
271C2×#101C22×C8911yescharacteristic
272C2×#101C22×C8911yescharacteristic
273C2×#101C22×C8911yescharacteristic
274C2×#100C22×C8911yescharacteristic
275C2×#100C22×C8911yescharacteristic
276C2×#100C22×C8911yescharacteristic
277#408C22×C8911yescharacteristic
278#344C22×C8911yescharacteristic
279#344C22×C8911yescharacteristic
280#344C22×C8911yescharacteristic
281#344C22×C8911yescharacteristic
282#341C22×C8911yescharacteristic
283#341C22×C8911yescharacteristic
284#341C22×C8911yescharacteristic
285#338C22×C8911yescharacteristic
286#338C22×C8911yescharacteristic
287H(3,ℤ/4)C22×C8911yescharacteristic
288C2×#41C23×C8911yescharacteristic
289C22×Q8C23×C8911yescharacteristic
290C22×D4C23×C8911yescharacteristic
291#137—41no—
292#137—41no—
293AGL(1,ℤ/8)—41no—
294AGL(1,ℤ/8)—41no—
295C2×Q16—21no—
296C2×SD16—21no—
297C2×SD16—21no—
298C2×SD16—41no—
299C2×SD16—41no—
300C2×SD16—41no—
301C2×D8—21no—
302#131—41no—
303#131—41no—
304C2×M16—41no—
305C2×M16C23×C8911yescharacteristic
306C2×M16C23×C8911yescharacteristic
307C2×M16C23×C8911yescharacteristic
308C2×M16C23×C8911yescharacteristic
309C2×M16C23×C8911yescharacteristic
310C2×M16C23×C8911yescharacteristic
311C2×M16C23×C8911yescharacteristic
312C2×M16C23×C8911yescharacteristic
313C2×M16C23×C8911yescharacteristic
314C22×C8C23×C8911yescharacteristic
315#117C23×C8911yescharacteristic
316#117C23×C8911yescharacteristic
317#108—41no—
318#108—41no—
319#108—41no—
320#108—41no—
321#108—41no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324#104—41no—
325#104—41no—
326#103—41no—
327#103—41no—
328#102—41no—
329#102—41no—
330#101—21no—
331#101—41no—
332#101—41no—
333#101—41no—
334#100—21no—
335#100—41no—
336#100—41no—
337#100—41no—
338C22⋊C8—41no—
339C24—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8#41×C8911yescharacteristic
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4#41×C8911yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4C24×C8911yescharacteristic
362Q16—42no—
363SD16—42no—
364SD16—42no—
365SD16—44no—
366D8—42no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—42no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—42no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C42—21no—
404C42—21no—
405C23—21no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—161no—
410C232−1+4×C8911yescharacteristic
411Q8—21no—
412Q8—21no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4C2×#41×C8911yescharacteristic
434C2×C4H(5,𝔽2)×C8911yescharacteristic
435C2×C4H(5,𝔽2)×C8911yescharacteristic
436C2×C4H(5,𝔽2)×C8911yescharacteristic
437C2×C4H(5,𝔽2)×C8911yescharacteristic
438C2×C4H(5,𝔽2)×C8911yescharacteristic
439C8—41no—
440C8—41no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—41no—
447C8—41no—
448C8—41no—
449C22—41no—
450C22—41no—
451C22—81no—
452C22—81no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22#539×C8911yescharacteristic
457C22#562×C8911yescharacteristic
458C22#562×C8911yescharacteristic
459C4—41no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4#538×C8911yescharacteristic
464C4#538×C8911yescharacteristic
465C4#560×C8911yescharacteristic
466C4#560×C8911yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—81no—
471C2—161no—
472C2C89×#282611yescharacteristic
subgroupG/H
center, socleC2×C89#2826
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC2×AGL(1,ℤ/8)×C89C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C89 ◁ C22×C4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#2103C2semidirect
2. C89×#2103C2semidirect
3. C89×#2098C2semidirect
4. C89×#2082C2semidirect
5. C89×#2081C2semidirect
6. C89×#2070C2semidirect
7. C89×#2069C2semidirect
8. C89×#2049C2semidirect
10. C89×#2046C2semidirect
11. C89×#2046C2semidirect
14. C89×#1902C2semidirect
15. C89×#1858C2semidirect
230. #32871C89direct (N × H)
237. #2103C2×C89semidirect
238. #2103C2×C89semidirect
239. #2098C2×C89semidirect
240. #2082C2×C89semidirect
241. #2081C2×C89semidirect
242. #2070C2×C89semidirect
243. #2069C2×C89semidirect
244. #2049C2×C89semidirect
246. #2046C2×C89semidirect
247. #2046C2×C89semidirect
250. #1902C2×C89semidirect
251. #1858C2×C89semidirect
252. C89#32871direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#21032
C89×#20982
C89×#20822
C89×#20812
C89×#20702
C89×#20692
C89×#20492
C89×#20482
C89×#20462
C89×#20192
C89×#19032
C89×#19022
C89×#18582
#3287189