Finite Group Databasebeta

← C89×#34485 #36412333

Subgroups — C89×#34485 #36412333

Order 22784.

2340 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C89C211yesmaximal, characteristic
2C89×#2379C214yesmaximal, complemented
3C89×#2369C211yesmaximal, characteristic, complemented
4C89×#2361C214yesmaximal, complemented
5D4×C4⋊C4×C89C212yesmaximal, complemented
6C89×#2346C214yesmaximal, complemented
7C89×#2326C214yesmaximal, complemented
8C89×#2324C214yesmaximal, complemented
9C89×#2317C214yesmaximal, complemented
10C4×C4⋊Q8×C89C211yesmaximal, characteristic, complemented
11C89×#2293C212yesmaximal, complemented
12#555×C89C4116yescomplemented
13C2×C4⋊Q8×C89C2211yescharacteristic
14C2×#124×C89C2214yes—
15C2×#121×C89C2214yes—
16C2×C4×Q8×C89C2212yes—
17C2×C4×D4×C89C2214yes—
18C2×C4×D4×C89C2214yes—
19C2×C4×D4×C89C2212yes—
20C2×#117×C89C2214yes—
21C2×#117×C89C2214yes—
22C22×C4⋊C4×C89C2214yes—
23#391×C89C2212yes—
24#391×C89C2214yes—
25#390×C89C2214yes—
26#390×C89C2212yes—
27#390×C89C2212yes—
28#389×C89C2214yes—
29#389×C89C2212yes—
30#389×C89C2214yes—
31#389×C89C2214yes—
32#389×C89C2212yes—
33#388×C89C2218yes—
34#388×C89C2214yes—
35#388×C89C2218yes—
36#388×C89C2214yes—
37#386×C89C2218yes—
38#385×C89C2218yes—
39#385×C89C2218yes—
40#385×C89C2218yes—
41#384×C89C2211yescharacteristic
42#384×C89C2214yes—
43#382×C89C2218yes—
44#380×C89C2214yes—
45C4×C4⋊C4×C89C2214yes—
46C4×C4⋊C4×C89C2214yes—
47C4×C4⋊C4×C89C2212yes—
48C4×C22⋊C4×C89C2214yes—
49#376×C89C2214yes—
50C43×C89C2211yescharacteristic
51C22×Q8×C89C2312yes—
52C22×D4×C89C2312yes—
53C22×D4×C89C2314yes—
54C23×C4×C89C2314yes—
55C23×C4×C89C2318yes—
56C4⋊Q8×C89C2×C418yescomplemented
57#124×C89C2×C4132yescomplemented
58#121×C89C2×C4132yescomplemented
59C4×Q8×C89—44no—
60C4×D4×C89—48no—
61C4×D4×C89—48no—
62C4×D4×C89C2×C4116yescomplemented
63#117×C89—48no—
64#117×C89C2×C4132yescomplemented
65C2×C4⋊C4×C89—48no—
66C2×C4⋊C4×C89C2312yes—
67C2×C4⋊C4×C89C2314yes—
68C2×C4⋊C4×C89C2312yes—
69C2×C4⋊C4×C89C2314yes—
70C2×C4⋊C4×C89C2318yes—
71C2×C4⋊C4×C89C2314yes—
72C2×C4⋊C4×C89C2312yes—
73C2×C4⋊C4×C89C2314yes—
74C2×C4⋊C4×C89C2314yes—
75C2×C4⋊C4×C89C2314yes—
76C2×C4⋊C4×C89C2312yes—
77C2×C22⋊C4×C89C2314yes—
78C2×C22⋊C4×C89C2314yes—
79C2×C22⋊C4×C89C2318yes—
80C2×C22⋊C4×C89C2314yes—
81C2×C22⋊C4×C89C2314yes—
82C2×C22⋊C4×C89C2318yes—
83C2×C22⋊C4×C89C2314yes—
84C2×C42×C89C2312yes—
85C2×C42×C89C2311yescharacteristic
86C2×C42×C89C2312yes—
87C2×C42×C89C2314yes—
88C2×C42×C89C2314yes—
89C2×C42×C89C2311yescharacteristic
90C2×C42×C89C2314yes—
91C2×C42×C89C2311yescharacteristic
92C2×C42×C89C2314yes—
93#95×C89C2314yes—
94#95×C89C2318yes—
95#95×C89C2314yes—
96#95×C89C2318yes—
97#95×C89C2318yes—
98#95×C89C2314yes—
99C24×C89C2414yes—
100C2×Q8×C89—42no—
101C2×Q8×C89—42no—
102C2×Q8×C89C22×C418yes—
103C2×D4×C89—24no—
104C2×D4×C89—24no—
105C2×D4×C89—44no—
106C2×D4×C89—44no—
107C2×D4×C89C22×C418yes—
108C2×D4×C89C22×C4116yes—
109C22×C4×C89—28no—
110C22×C4×C89—28no—
111C22×C4×C89—48no—
112C22×C4×C89—48no—
113C22×C4×C89—48no—
114C22×C4×C89C22×C4116yes—
115C22×C4×C89C2414yes—
116C22×C4×C89C2412yes—
117C22×C4×C89C2414yes—
118C22×C4×C89C2414yes—
119C22×C4×C89C2411yescharacteristic
120C22×C4×C89C2411yescharacteristic
121C22×C4×C89C2414yes—
122C22×C4×C89C2414yes—
123C22×C4×C89C2411yescharacteristic
124C22×C4×C89C2412yes—
125C4⋊C4×C89—24no—
126C4⋊C4×C89—28no—
127C4⋊C4×C89—24no—
128C4⋊C4×C89—48no—
129C4⋊C4×C89—44no—
130C4⋊C4×C89—44no—
131C4⋊C4×C89—44no—
132C4⋊C4×C89—44no—
133C4⋊C4×C89—42no—
134C4⋊C4×C89C22×C418yes—
135C4⋊C4×C89C22×C4116yes—
136C22⋊C4×C89—28no—
137C22⋊C4×C89—44no—
138C22⋊C4×C89—44no—
139C22⋊C4×C89—44no—
140C22⋊C4×C89—48no—
141C22⋊C4×C89C22×C4132yes—
142C22⋊C4×C89C22×C4116yes—
143C42×C89—24no—
144C42×C89—28no—
145C42×C89—28no—
146C42×C89—22no—
147C42×C89—24no—
148C42×C89—44no—
149C42×C89—41no—
150C42×C89C22×C4116yes—
151C42×C89C22×C414yes—
152C23×C89—24no—
153C23×C89—24no—
154C23×C89—24no—
155C23×C89—24no—
156C23×C89—24no—
157C23×C89—24no—
158C23×C89C23×C418yes—
159C23×C89C2511yescharacteristic
160Q8×C89—48no—
161D4×C89—216no—
162D4×C89—416no—
163C2×C4×C89—24no—
164C2×C4×C89—21no—
165C2×C4×C89—24no—
166C2×C4×C89—22no—
167C2×C4×C89—24no—
168C2×C4×C89—24no—
169C2×C4×C89—21no—
170C2×C4×C89—232no—
171C2×C4×C89—24no—
172C2×C4×C89—22no—
173C2×C4×C89—22no—
174C2×C4×C89—24no—
175C2×C4×C89—21no—
176C2×C4×C89—24no—
177C2×C4×C89—24no—
178C2×C4×C89—28no—
179C2×C4×C89—24no—
180C2×C4×C89—22no—
181C2×C4×C89—24no—
182C2×C4×C89—24no—
183C2×C4×C89—22no—
184C2×C4×C89—816no—
185C2×C4×C89C23×C418yes—
186C2×C4×C89C23×C412yes—
187C2×C4×C89C23×C418yes—
188C2×C4×C89C23×C414yes—
189C2×C4×C89H(5,𝔽2)12yes—
190C2×C4×C89H(5,𝔽2)14yes—
191C2×C4×C892−1+414yes—
192C22×C89—28no—
193C22×C89—28no—
194C22×C89—28no—
195C22×C89—44no—
196C22×C89—44no—
197C22×C89—44no—
198C22×C89—44no—
199C22×C89C24×C411yescharacteristic
200C22×C89C2×H(5,𝔽2)12yes—
201C22×C89C2×H(5,𝔽2)11yescharacteristic
202C22×C89C2×2−1+412yes—
203C22×C89#58511yescharacteristic
204C4×C89—28no—
205C4×C89—28no—
206C4×C89—22no—
207C4×C89—22no—
208C4×C89—44no—
209C4×C89—42no—
210C4×C89—41no—
211C4×C89—44no—
212C4×C89—44no—
213C4×C89#52018yes—
214#34485C8911yesmaximal, characteristic, direct product factor, Sylow
215C2×C89—48no—
216C2×C89C2×#52012yes—
217C2×C89C4×H(5,𝔽2)11yescharacteristic
218C2×C89#354411yescharacteristic
219C2×C89#354712yes—
220C2×C89#354911yescharacteristic
221C2×#555C2×C8911yescharacteristic
222#2379C2×C8914yescomplemented
223#2369C2×C8911yescharacteristic, complemented
224#2361C2×C8914yescomplemented
225D4×C4⋊C4C2×C8912yescomplemented
226#2346C2×C8914yescomplemented
227#2326C2×C8914yescomplemented
228#2324C2×C8914yescomplemented
229#2317C2×C8914yescomplemented
230C4×C4⋊Q8C2×C8911yescharacteristic, complemented
231#2293C2×C8912yescomplemented
232C89#3448511yescharacteristic, direct product factor, Sylow
233#555C4×C89116yescomplemented
234C2×C4⋊Q8C22×C8911yescharacteristic
235C2×#124C22×C8914yes—
236C2×#121C22×C8914yes—
237C2×C4×Q8C22×C8912yes—
238C2×C4×D4C22×C8914yes—
239C2×C4×D4C22×C8914yes—
240C2×C4×D4C22×C8912yes—
241C2×#117C22×C8914yes—
242C2×#117C22×C8914yes—
243C22×C4⋊C4C22×C8914yes—
244#391C22×C8912yes—
245#391C22×C8914yes—
246#390C22×C8914yes—
247#390C22×C8912yes—
248#390C22×C8912yes—
249#389C22×C8914yes—
250#389C22×C8912yes—
251#389C22×C8914yes—
252#389C22×C8914yes—
253#389C22×C8912yes—
254#388C22×C8918yes—
255#388C22×C8914yes—
256#388C22×C8918yes—
257#388C22×C8914yes—
258#386C22×C8918yes—
259#385C22×C8918yes—
260#385C22×C8918yes—
261#385C22×C8918yes—
262#384C22×C8911yescharacteristic
263#384C22×C8914yes—
264#382C22×C8918yes—
265#380C22×C8914yes—
266C4×C4⋊C4C22×C8914yes—
267C4×C4⋊C4C22×C8914yes—
268C4×C4⋊C4C22×C8912yes—
269C4×C22⋊C4C22×C8914yes—
270#376C22×C8914yes—
271C43C22×C8911yescharacteristic
272C22×Q8C23×C8912yes—
273C22×D4C23×C8912yes—
274C22×D4C23×C8914yes—
275C23×C4C23×C8914yes—
276C23×C4C23×C8918yes—
277C4⋊Q8C2×C4×C8918yescomplemented
278#124C2×C4×C89132yescomplemented
279#121C2×C4×C89132yescomplemented
280C4×Q8—44no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4C2×C4×C89116yescomplemented
284#117—48no—
285#117C2×C4×C89132yescomplemented
286C2×C4⋊C4—48no—
287C2×C4⋊C4C23×C8912yes—
288C2×C4⋊C4C23×C8914yes—
289C2×C4⋊C4C23×C8912yes—
290C2×C4⋊C4C23×C8918yes—
291C2×C4⋊C4C23×C8914yes—
292C2×C4⋊C4C23×C8914yes—
293C2×C4⋊C4C23×C8914yes—
294C2×C4⋊C4C23×C8912yes—
295C2×C4⋊C4C23×C8914yes—
296C2×C4⋊C4C23×C8912yes—
297C2×C4⋊C4C23×C8914yes—
298C2×C22⋊C4C23×C8914yes—
299C2×C22⋊C4C23×C8914yes—
300C2×C22⋊C4C23×C8914yes—
301C2×C22⋊C4C23×C8918yes—
302C2×C22⋊C4C23×C8914yes—
303C2×C22⋊C4C23×C8918yes—
304C2×C22⋊C4C23×C8914yes—
305C2×C42C23×C8912yes—
306C2×C42C23×C8914yes—
307C2×C42C23×C8914yes—
308C2×C42C23×C8911yescharacteristic
309C2×C42C23×C8912yes—
310C2×C42C23×C8911yescharacteristic
311C2×C42C23×C8914yes—
312C2×C42C23×C8914yes—
313C2×C42C23×C8911yescharacteristic
314#95C23×C8914yes—
315#95C23×C8918yes—
316#95C23×C8914yes—
317#95C23×C8918yes—
318#95C23×C8918yes—
319#95C23×C8914yes—
320C24C24×C8914yes—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8C22×C4×C8918yes—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4C22×C4×C8918yes—
329C2×D4C22×C4×C89116yes—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4C22×C4×C89116yes—
336C22×C4C24×C8914yes—
337C22×C4C24×C8912yes—
338C22×C4C24×C8914yes—
339C22×C4C24×C8914yes—
340C22×C4C24×C8911yescharacteristic
341C22×C4C24×C8911yescharacteristic
342C22×C4C24×C8914yes—
343C22×C4C24×C8914yes—
344C22×C4C24×C8911yescharacteristic
345C22×C4C24×C8912yes—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4C22×C4×C8918yes—
356C4⋊C4C22×C4×C89116yes—
357C22⋊C4—28no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4C22×C4×C89132yes—
363C22⋊C4C22×C4×C89116yes—
364C42—24no—
365C42—28no—
366C42—28no—
367C42—22no—
368C42—24no—
369C42—44no—
370C42—41no—
371C42C22×C4×C89116yes—
372C42C22×C4×C8914yes—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23C23×C4×C8918yes—
380C23C25×C8911yescharacteristic
381Q8—48no—
382D4—216no—
383D4—416no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—232no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—816no—
406C2×C4C23×C4×C8918yes—
407C2×C4C23×C4×C8912yes—
408C2×C4C23×C4×C8918yes—
409C2×C4C23×C4×C8914yes—
410C2×C4H(5,𝔽2)×C8912yes—
411C2×C4H(5,𝔽2)×C8914yes—
412C2×C42−1+4×C8914yes—
413C22—28no—
414C22—28no—
415C22—28no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22C24×C4×C8911yescharacteristic
421C22C2×H(5,𝔽2)×C8912yes—
422C22C2×H(5,𝔽2)×C8911yescharacteristic
423C22C2×2−1+4×C8912yes—
424C22#585×C8911yescharacteristic
425C4—28no—
426C4—28no—
427C4—22no—
428C4—22no—
429C4—44no—
430C4—42no—
431C4—44no—
432C4—44no—
433C4—41no—
434C4#520×C8918yes—
435C2—48no—
436C2C2×#520×C8912yes—
437C2C4×H(5,𝔽2)×C8911yescharacteristic
438C2C89×#354411yescharacteristic
439C2C89×#354712yes—
440C2C89×#354911yescharacteristic
subgroupG/H
center, socleC23×C89C25
commutatorC22C24×C4×C89
Frattini, cosocleC23C25×C89
prime radicalC2×#555×C89C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2379C2semidirect
3. C89×#2369C2semidirect
4. C89×#2361C2semidirect
5. D4×C4⋊C4×C89C2semidirect
6. C89×#2346C2semidirect
7. C89×#2326C2semidirect
8. C89×#2324C2semidirect
9. C89×#2317C2semidirect
10. C4×C4⋊Q8×C89C2semidirect
11. C89×#2293C2semidirect
12. #555×C89C4semidirect
56. C4⋊Q8×C89C2×C4semidirect
57. #124×C89C2×C4semidirect
58. #121×C89C2×C4semidirect
62. C4×D4×C89C2×C4semidirect
64. #117×C89C2×C4semidirect
214. #34485C89direct (N × H)
222. #2379C2×C89semidirect
223. #2369C2×C89semidirect
224. #2361C2×C89semidirect
225. D4×C4⋊C4C2×C89semidirect
226. #2346C2×C89semidirect
227. #2326C2×C89semidirect
228. #2324C2×C89semidirect
229. #2317C2×C89semidirect
230. C4×C4⋊Q8C2×C89semidirect
231. #2293C2×C89semidirect
232. C89#34485direct (N × H)
233. #555C4×C89semidirect
277. C4⋊Q8C2×C4×C89semidirect
278. #124C2×C4×C89semidirect
279. #121C2×C4×C89semidirect
283. C4×D4C2×C4×C89semidirect
285. #117C2×C4×C89semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C892
C89×#23792
C89×#23692
C89×#23612
D4×C4⋊C4×C892
C89×#23462
C89×#23262
C89×#23242
C89×#23172
C4×C4⋊Q8×C892
C89×#22932
#3448589