Finite Group Databasebeta

← C89×#9317 #36387165

Subgroups — C89×#9317 #36387165

Order 22784.

496 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
1C89×#3079C211yesmaximal, characteristic
2C89×#2988C211yesmaximal, characteristic
3C89×#1981C211yesmaximal, characteristic, complemented
4C89×#1806C211yesmaximal, characteristic
5C89×#1549C211yesmaximal, characteristic, complemented
6C89×#1303C211yesmaximal, characteristic, complemented
7C89×#1292C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C89C2211yescharacteristic
9C4×#41×C89C2211yescharacteristic
10#477×C89C412yescomplemented
11#476×C89C412yescomplemented
12#475×C89—21no—
13#474×C89—21no—
14C8×Q8×C89—21no—
15C8×D4×C89—22no—
16C8×D4×C89—21no—
17#431×C89—21no—
18#431×C89C2211yescharacteristic
19#428×C89—21no—
20#423×C89C2211yescharacteristic
21#423×C89C2211yescharacteristic
22C2×C4⋊C8×C89—21no—
23C2×C4⋊C8×C89C2211yescharacteristic
24C2×#104×C89—21no—
25#417×C89—21no—
26C2×#103×C89—22no—
27C4×C4⋊C4×C89C2211yescharacteristic
28#339×C89—21no—
29#332×C89—21no—
30#332×C89—21no—
31C2×#41×C89D411yescharacteristic
32C22×Q8×C89—21no—
33C2×M16×C89—21no—
34C2×M16×C89—21no—
35C22×C8×C89—22no—
36C22×C8×C89D412yes—
37C4⋊Q8×C89—21no—
38C4⋊Q8×C89—21no—
39C4⋊Q8×C89—21no—
40C4⋊Q8×C89C2×C412yes—
41C4×Q8×C89C2×C411yescharacteristic
42C4×Q8×C89D411yescharacteristic
43C4×D4×C89—21no—
44C4×D4×C89—21no—
45C4×D4×C89C2×C411yescharacteristic
46C4×D4×C89D411yescharacteristic
47#117×C89—21no—
48#117×C89D411yescharacteristic
49C2×C4⋊C4×C89—21no—
50C2×C4⋊C4×C89—21no—
51C2×C4⋊C4×C89—21no—
52C2×C42×C89—21no—
53C2×C42×C89—21no—
54C2×C42×C89C2311yescharacteristic
55#107×C89—22no—
56#106×C89—22no—
57C4⋊C8×C89—22no—
58C4⋊C8×C89—21no—
59C4⋊C8×C89—21no—
60C4⋊C8×C89—21no—
61C4⋊C8×C89—21no—
62C4⋊C8×C89—21no—
63C4⋊C8×C89C2×C412yes—
64C4⋊C8×C89D411yescharacteristic
65C4⋊C8×C89D411yescharacteristic
66#104×C89—42no—
67#103×C89—22no—
68#103×C89—44no—
69#102×C89—22no—
70C22⋊C8×C89—22no—
71C22⋊C8×C89—21no—
72C4×C8×C89—22no—
73C4×C8×C89—21no—
74#95×C89—21no—
75#41×C89—21no—
76#41×C89—21no—
77#41×C89SD1612yes—
78#41×C89Q1612yes—
79C2×Q8×C89—22no—
80C2×Q8×C89—42no—
81C2×Q8×C89—42no—
82C2×Q8×C89—41no—
83C2×Q8×C89C22⋊C411yescharacteristic
84C2×D4×C89—21no—
85C2×D4×C89C22⋊C411yescharacteristic
86C22×C4×C89—21no—
87C22×C4×C89—21no—
88C22×C4×C89—21no—
89C22×C4×C89—41no—
90C22×C4×C89—41no—
91C22×C4×C89C2×D411yescharacteristic
92C22×C4×C89C2×D411yescharacteristic
93C22×C4×C89#4111yescharacteristic
94M16×C89—41no—
95M16×C89—41no—
96M16×C89—41no—
97M16×C89—41no—
98C2×C8×C89—22no—
99C2×C8×C89—21no—
100C2×C8×C89—22no—
101C2×C8×C89—21no—
102C2×C8×C89—42no—
103C2×C8×C89—41no—
104C2×C8×C89—82no—
105C4⋊C4×C89—21no—
106C4⋊C4×C89—21no—
107C4⋊C4×C89—22no—
108C4⋊C4×C89—22no—
109C4⋊C4×C89—22no—
110C4⋊C4×C89—41no—
111C4⋊C4×C89—41no—
112C4⋊C4×C89—41no—
113C4⋊C4×C89C22⋊C411yescharacteristic
114C4⋊C4×C89C22⋊C411yescharacteristic
115C22⋊C4×C89—21no—
116C22⋊C4×C89—21no—
117C22⋊C4×C89—21no—
118C42×C89—21no—
119C42×C89—21no—
120C42×C89—21no—
121C42×C89—41no—
122C42×C89—41no—
123C42×C89C22×C411yescharacteristic
124C42×C89C2×D411yescharacteristic
125C42×C89C2×D411yescharacteristic
126C42×C89#4111yescharacteristic
127C23×C89—21no—
128C23×C89#12111yescharacteristic
129Q8×C89—21no—
130Q8×C89—44no—
131Q8×C89—81no—
132Q8×C89—81no—
133Q8×C89#10312yescomplemented
134D4×C89—22no—
135D4×C89—22no—
136D4×C89—21no—
137D4×C89#10312yescomplemented
138C2×C4×C89—21no—
139C2×C4×C89—21no—
140C2×C4×C89—21no—
141C2×C4×C89—22no—
142C2×C4×C89—22no—
143C2×C4×C89—21no—
144C2×C4×C89—21no—
145C2×C4×C89—41no—
146C2×C4×C89—41no—
147C2×C4×C89—41no—
148C2×C4×C89—42no—
149C2×C4×C89—42no—
150C2×C4×C89—81no—
151C2×C4×C89—81no—
152C2×C4×C89—82no—
153C2×C4×C89—81no—
154C2×C4×C89C2×C22⋊C411yescharacteristic
155C2×C4×C89C4×D411yescharacteristic
156C2×C4×C89C4×D411yescharacteristic
157C2×C4×C89#12011yescharacteristic
158C2×C4×C89#12111yescharacteristic
159C2×C4×C89#12311yescharacteristic
160C2×C4×C89C2×SD1611yescharacteristic
161C2×C4×C89C2×Q1611yescharacteristic
162C2×C4×C89#13511yescharacteristic
163C2×C4×C89#13511yescharacteristic
164C8×C89—42no—
165C8×C89—42no—
166C8×C89—41no—
167C8×C89—41no—
168C8×C89—81no—
169C22×C89—21no—
170C22×C89—21no—
171C22×C89—22no—
172C22×C89—41no—
173C22×C89—41no—
174C22×C89#38611yescharacteristic
175C22×C89#46511yescharacteristic
176C22×C89#46711yescharacteristic
177C4×C89—21no—
178C4×C89—22no—
179C4×C89—21no—
180C4×C89—21no—
181C4×C89—41no—
182C4×C89—41no—
183C4×C89—82no—
184C4×C89—82no—
185C4×C89C2×#10311yescharacteristic
186C4×C89#41611yescharacteristic
187C4×C89#45311yescharacteristic
188C4×C89#45411yescharacteristic
189C4×C89#46511yescharacteristic
190C4×C89#46711yescharacteristic
191#9317C8911yesmaximal, characteristic, direct product factor, Sylow
192C2×C89—21no—
193C2×C89—21no—
194C2×C89—42no—
195C2×C89#188311yescharacteristic
196C2×C89#189211yescharacteristic
197C2×C89#190211yescharacteristic
198#3079C2×C8911yescharacteristic
199#2988C2×C8911yescharacteristic
200#1981C2×C8911yescharacteristic, complemented
201#1806C2×C8911yescharacteristic
202#1549C2×C8911yescharacteristic, complemented
203#1303C2×C8911yescharacteristic, complemented
204#1292C2×C8911yescharacteristic, complemented
205C89#931711yescharacteristic, direct product factor, Sylow
206C2×C4⋊Q8C22×C8911yescharacteristic
207C4×#41C22×C8911yescharacteristic
208#477C4×C8912yescomplemented
209#476C4×C8912yescomplemented
210#475—21no—
211#474—21no—
212C8×Q8—21no—
213C8×D4—22no—
214C8×D4—21no—
215#431—21no—
216#431C22×C8911yescharacteristic
217#428—21no—
218#423C22×C8911yescharacteristic
219#423C22×C8911yescharacteristic
220C2×C4⋊C8—21no—
221C2×C4⋊C8C22×C8911yescharacteristic
222C2×#104—21no—
223#417—21no—
224C2×#103—22no—
225C4×C4⋊C4C22×C8911yescharacteristic
226#339—21no—
227#332—21no—
228#332—21no—
229C2×#41D4×C8911yescharacteristic
230C22×Q8—21no—
231C2×M16—21no—
232C2×M16—21no—
233C22×C8—22no—
234C22×C8D4×C8912yes—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C2×C4×C8912yes—
239C4×Q8C2×C4×C8911yescharacteristic
240C4×Q8D4×C8911yescharacteristic
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C2×C4×C8911yescharacteristic
244C4×D4D4×C8911yescharacteristic
245#117—21no—
246#117D4×C8911yescharacteristic
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C23×C8911yescharacteristic
253#107—22no—
254#106—22no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C2×C4×C8912yes—
262C4⋊C8D4×C8911yescharacteristic
263C4⋊C8D4×C8911yescharacteristic
264#104—42no—
265#103—22no—
266#103—44no—
267#102—22no—
268C22⋊C8—22no—
269C22⋊C8—21no—
270C4×C8—22no—
271C4×C8—21no—
272#95—21no—
273#41—21no—
274#41—21no—
275#41SD16×C8912yes—
276#41Q16×C8912yes—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C8911yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊C4×C8911yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×D4×C8911yescharacteristic
290C22×C4C2×D4×C8911yescharacteristic
291C22×C4#41×C8911yescharacteristic
292M16—41no—
293M16—41no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—82no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊C4×C8911yescharacteristic
312C4⋊C4C22⋊C4×C8911yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—41no—
321C42C22×C4×C8911yescharacteristic
322C42C2×D4×C8911yescharacteristic
323C42C2×D4×C8911yescharacteristic
324C42#41×C8911yescharacteristic
325C23—21no—
326C23#121×C8911yescharacteristic
327Q8—21no—
328Q8—44no—
329Q8—81no—
330Q8—81no—
331Q8#103×C8912yescomplemented
332D4—22no—
333D4—22no—
334D4—21no—
335D4#103×C8912yescomplemented
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C8911yescharacteristic
353C2×C4C4×D4×C8911yescharacteristic
354C2×C4C4×D4×C8911yescharacteristic
355C2×C4#120×C8911yescharacteristic
356C2×C4#121×C8911yescharacteristic
357C2×C4#123×C8911yescharacteristic
358C2×C4C2×SD16×C8911yescharacteristic
359C2×C4C2×Q16×C8911yescharacteristic
360C2×C4#135×C8911yescharacteristic
361C2×C4#135×C8911yescharacteristic
362C8—42no—
363C8—42no—
364C8—41no—
365C8—41no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22#386×C8911yescharacteristic
373C22#465×C8911yescharacteristic
374C22#467×C8911yescharacteristic
375C4—21no—
376C4—22no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4—82no—
383C4C2×#103×C8911yescharacteristic
384C4#416×C8911yescharacteristic
385C4#453×C8911yescharacteristic
386C4#454×C8911yescharacteristic
387C4#465×C8911yescharacteristic
388C4#467×C8911yescharacteristic
389C2—21no—
390C2—21no—
391C2—42no—
392C2C89×#188311yescharacteristic
393C2C89×#189211yescharacteristic
394C2C89×#190211yescharacteristic
subgroupG/H
center, socleC22×C89#386
commutatorC42C22×C4×C89
Frattini, cosocleC2×C42C23×C89
prime radicalC2×#41×C89D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
3. C89×#1981C2semidirect
5. C89×#1549C2semidirect
6. C89×#1303C2semidirect
7. C89×#1292C2semidirect
10. #477×C89C4semidirect
11. #476×C89C4semidirect
133. Q8×C89#103semidirect
137. D4×C89#103semidirect
191. #9317C89direct (N × H)
200. #1981C2×C89semidirect
202. #1549C2×C89semidirect
203. #1303C2×C89semidirect
204. #1292C2×C89semidirect
205. C89#9317direct (N × H)
208. #477C4×C89semidirect
209. #476C4×C89semidirect
331. Q8#103×C89semidirect
335. D4#103×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#30792
C89×#29882
C89×#19812
C89×#18062
C89×#15492
C89×#13032
C89×#12922
#931789