Finite Group Databasebeta

← C89×#60330 #36438178

Subgroups — C89×#60330 #36438178

Order 22784.

1710 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3461C211yesmaximal, characteristic
2C89×#2993C214yesmaximal, complemented
3C89×#2992C214yesmaximal, complemented
4C89×#2989C212yesmaximal, complemented
5C89×#2985C214yesmaximal, complemented
6C89×#2983C211yesmaximal, characteristic, complemented
7C89×#2966C214yesmaximal, complemented
8C89×#2966C214yesmaximal, complemented
9C8×#41×C89C212yesmaximal, complemented
10C89×#2960C214yesmaximal, complemented
11C2×C8×Q8×C89C211yesmaximal, characteristic, complemented
12#544×C89C411yescharacteristic
13#542×C89C414yes—
14#541×C89C414yes—
15#539×C89C414yes—
16#537×C89C411yescharacteristic
17#536×C89C412yes—
18#533×C89C2214yes—
19#529×C89C2214yes—
20#529×C89C2214yes—
21C4×#41×C89C2212yes—
22C2×C4×Q8×C89C2211yescharacteristic
23#446×C89C2214yes—
24#446×C89C2214yes—
25#446×C89C2218yes—
26#446×C89C2214yes—
27#446×C89C2214yes—
28C8×Q8×C89C2214yes—
29C8×Q8×C89C2214yes—
30C8×Q8×C89C2212yes—
31C8×Q8×C89C2214yes—
32C8×Q8×C89C2212yes—
33#436×C89C2214yes—
34#436×C89C2214yes—
35#435×C89C2218yes—
36#435×C89C2218yes—
37C8×D4×C89C2212yes—
38C8×D4×C89C2214yes—
39C8×D4×C89C2214yes—
40C8×D4×C89C2212yes—
41C8×D4×C89C2214yes—
42#433×C89C2218yes—
43#433×C89C2218yes—
44#432×C89C2214yes—
45#432×C89C2214yes—
46#431×C89C2212yes—
47#431×C89C2214yes—
48#424×C89C2218yes—
49#423×C89C2214yes—
50C2×C4⋊C8×C89C2211yescharacteristic
51C2×C4⋊C8×C89C2212yes—
52#405×C89C2218yes—
53C2×C4×C8×C89C2211yescharacteristic
54C2×C4×C8×C89C2212yes—
55C2×C4×C8×C89C2214yes—
56C2×#41×C89C2×C412yes—
57C22×Q8×C89C2×C411yescharacteristic
58C2×M16×C89C2318yes—
59C22×C8×C89C2311yescharacteristic
60C22×C8×C89C2314yes—
61C22×C8×C89C2311yescharacteristic
62C22×C8×C89C2312yes—
63C22×C8×C89C2314yes—
64C4⋊Q8×C89C2×C414yes—
65C4⋊Q8×C89C2×C414yes—
66#126×C89C2×C418yes—
67#126×C89C2×C418yes—
68#125×C89C2×C414yes—
69#125×C89C2×C414yes—
70#125×C89C2×C418yes—
71#124×C89C2×C414yes—
72#124×C89C2×C414yes—
73#123×C89C2×C418yes—
74#123×C89C2×C418yes—
75C22⋊Q8×C89C2×C414yes—
76C22⋊Q8×C89C2×C414yes—
77C22⋊Q8×C89C2×C414yes—
78C22⋊Q8×C89C2×C414yes—
79C22⋊Q8×C89C2×C418yes—
80#121×C89C2×C414yes—
81#121×C89C2×C414yes—
82C4×Q8×C89C2×C414yes—
83C4×Q8×C89C2×C412yes—
84C4×Q8×C89C2314yes—
85C4×Q8×C89C2312yes—
86C4×Q8×C89C2314yes—
87C4×D4×C89C2×C414yes—
88C4×D4×C89C2×C412yes—
89C4×D4×C89C2314yes—
90C4×D4×C89C2312yes—
91C4×D4×C89C2314yes—
92#117×C89C2×C412yes—
93#117×C89C2×C414yes—
94#117×C89C2318yes—
95C2×C4⋊C4×C89C2×C411yescharacteristic
96C2×C4⋊C4×C89C2×C412yes—
97C2×C42×C89C2311yescharacteristic
98C2×C42×C89C2312yes—
99C2×C42×C89C2314yes—
100C4⋊C8×C89C2314yes—
101C4⋊C8×C89C2318yes—
102C4⋊C8×C89C2312yes—
103C4⋊C8×C89C2318yes—
104C4⋊C8×C89C2314yes—
105C4⋊C8×C89C2314yes—
106C4⋊C8×C89C2312yes—
107C4⋊C8×C89C2314yes—
108C4⋊C8×C89C2314yes—
109C22⋊C8×C89C2314yes—
110C22⋊C8×C89C2314yes—
111C22⋊C8×C89C2314yes—
112C22⋊C8×C89C2318yes—
113#97×C89C2318yes—
114#97×C89C2318yes—
115C4×C8×C89C2312yes—
116C4×C8×C89C2314yes—
117C4×C8×C89C2314yes—
118C4×C8×C89C2314yes—
119C4×C8×C89C2312yes—
120C4×C8×C89C2314yes—
121C4×C8×C89C2314yes—
122#41×C89—28no—
123C2×Q8×C89—24no—
124C2×Q8×C89C22×C414yes—
125C2×Q8×C89C22×C412yes—
126C2×Q8×C89C22×C414yes—
127C2×D4×C89C22×C414yes—
128C2×D4×C89C22×C412yes—
129C2×D4×C89C22×C414yes—
130C22×C4×C89C22×C412yes—
131C22×C4×C89C22×C414yes—
132C22×C4×C89C22×C411yescharacteristic
133C22×C4×C89C22×C412yes—
134C22×C4×C89C22×C414yes—
135C22×C4×C89C22×C411yescharacteristic
136C22×C4×C89C2411yescharacteristic
137C22×C4×C89C2414yes—
138M16×C89—48no—
139C2×C8×C89—22no—
140C2×C8×C89—28no—
141C2×C8×C89—41no—
142C2×C8×C89—44no—
143C2×C8×C89—42no—
144C2×C8×C89C2411yescharacteristic
145C2×C8×C89C2412yes—
146C2×C8×C89C2414yes—
147C2×C8×C89C2414yes—
148C2×C8×C89C2414yes—
149C2×C8×C89C2411yescharacteristic
150C4⋊C4×C89C22×C414yes—
151C4⋊C4×C89C22×C418yes—
152C4⋊C4×C89C22×C414yes—
153C4⋊C4×C89C22×C412yes—
154C4⋊C4×C89C22×C412yes—
155C4⋊C4×C89C22×C414yes—
156C4⋊C4×C89C22×C418yes—
157C4⋊C4×C89C22×C418yes—
158C4⋊C4×C89C22×C414yes—
159C4⋊C4×C89C22×C414yes—
160C4⋊C4×C89C22×C418yes—
161C22⋊C4×C89C22×C414yes—
162C22⋊C4×C89C22×C414yes—
163C22⋊C4×C89C22×C414yes—
164C22⋊C4×C89C22×C418yes—
165C22⋊C4×C89C22×C418yes—
166C22⋊C4×C89C22×C418yes—
167C42×C89C22×C412yes—
168C42×C89C22×C414yes—
169C42×C89C22×C414yes—
170C42×C89C22×C414yes—
171C42×C89C2412yes—
172C42×C89C2414yes—
173C42×C89C2414yes—
174C23×C89C23×C414yes—
175C23×C89C23×C411yescharacteristic
176Q8×C89—28no—
177Q8×C89—24no—
178Q8×C89—44no—
179D4×C89—28no—
180D4×C89—24no—
181D4×C89—44no—
182C2×C4×C89—24no—
183C2×C4×C89—28no—
184C2×C4×C89—21no—
185C2×C4×C89—24no—
186C2×C4×C89—21no—
187C2×C4×C89—24no—
188C2×C4×C89—41no—
189C2×C4×C89—44no—
190C2×C4×C89—42no—
191C2×C4×C89#13114yes—
192C2×C4×C89C23×C411yescharacteristic
193C2×C4×C89C23×C412yes—
194C2×C4×C89C23×C414yes—
195C2×C4×C89C23×C414yes—
196C2×C4×C89C23×C414yes—
197C2×C4×C89C23×C414yes—
198C2×C4×C89C23×C414yes—
199C2×C4×C89C23×C412yes—
200C2×C4×C89C2511yescharacteristic
201C8×C89—21no—
202C8×C89—24no—
203C8×C89—22no—
204C8×C89—24no—
205C8×C89—24no—
206C8×C892−1+412yes—
207C22×C89—21no—
208C22×C89—21no—
209C22×C89—24no—
210C22×C89—24no—
211C22×C89—24no—
212C22×C89C2×#13112yes—
213C22×C89C24×C411yescharacteristic
214C4×C89—21no—
215C4×C89—24no—
216C4×C89—24no—
217C4×C89—24no—
218C4×C89—22no—
219C4×C89—24no—
220C4×C89—24no—
221C4×C89C2×#13114yes—
222C4×C89C2×2−1+411yescharacteristic
223C4×C89#58511yescharacteristic
224#60330C8911yesmaximal, characteristic, direct product factor, Sylow
225C2×C89—22no—
226C2×C89—44no—
227C2×C89C4×2−1+411yescharacteristic
228C2×C89C22×#13111yescharacteristic
229C2×C89#357311yescharacteristic
230#3461C2×C8911yescharacteristic
231#2993C2×C8914yescomplemented
232#2992C2×C8914yescomplemented
233#2989C2×C8912yescomplemented
234#2985C2×C8914yescomplemented
235#2983C2×C8911yescharacteristic, complemented
236#2966C2×C8914yescomplemented
237#2966C2×C8914yescomplemented
238C8×#41C2×C8912yescomplemented
239#2960C2×C8914yescomplemented
240C2×C8×Q8C2×C8911yescharacteristic, complemented
241C89#6033011yescharacteristic, direct product factor, Sylow
242#544C4×C8911yescharacteristic
243#542C4×C8914yes—
244#541C4×C8914yes—
245#539C4×C8914yes—
246#537C4×C8911yescharacteristic
247#536C4×C8912yes—
248#533C22×C8914yes—
249#529C22×C8914yes—
250#529C22×C8914yes—
251C4×#41C22×C8912yes—
252C2×C4×Q8C22×C8911yescharacteristic
253#446C22×C8914yes—
254#446C22×C8914yes—
255#446C22×C8918yes—
256#446C22×C8914yes—
257#446C22×C8914yes—
258C8×Q8C22×C8914yes—
259C8×Q8C22×C8914yes—
260C8×Q8C22×C8912yes—
261C8×Q8C22×C8914yes—
262C8×Q8C22×C8912yes—
263#436C22×C8914yes—
264#436C22×C8914yes—
265#435C22×C8918yes—
266#435C22×C8918yes—
267C8×D4C22×C8912yes—
268C8×D4C22×C8914yes—
269C8×D4C22×C8914yes—
270C8×D4C22×C8912yes—
271C8×D4C22×C8914yes—
272#433C22×C8918yes—
273#433C22×C8918yes—
274#432C22×C8914yes—
275#432C22×C8914yes—
276#431C22×C8912yes—
277#431C22×C8914yes—
278#424C22×C8918yes—
279#423C22×C8914yes—
280C2×C4⋊C8C22×C8911yescharacteristic
281C2×C4⋊C8C22×C8912yes—
282#405C22×C8918yes—
283C2×C4×C8C22×C8911yescharacteristic
284C2×C4×C8C22×C8912yes—
285C2×C4×C8C22×C8914yes—
286C2×#41C2×C4×C8912yes—
287C22×Q8C2×C4×C8911yescharacteristic
288C2×M16C23×C8918yes—
289C22×C8C23×C8911yescharacteristic
290C22×C8C23×C8914yes—
291C22×C8C23×C8911yescharacteristic
292C22×C8C23×C8912yes—
293C22×C8C23×C8914yes—
294C4⋊Q8C2×C4×C8914yes—
295C4⋊Q8C2×C4×C8914yes—
296#126C2×C4×C8918yes—
297#126C2×C4×C8918yes—
298#125C2×C4×C8914yes—
299#125C2×C4×C8914yes—
300#125C2×C4×C8918yes—
301#124C2×C4×C8914yes—
302#124C2×C4×C8914yes—
303#123C2×C4×C8918yes—
304#123C2×C4×C8918yes—
305C22⋊Q8C2×C4×C8914yes—
306C22⋊Q8C2×C4×C8914yes—
307C22⋊Q8C2×C4×C8914yes—
308C22⋊Q8C2×C4×C8914yes—
309C22⋊Q8C2×C4×C8918yes—
310#121C2×C4×C8914yes—
311#121C2×C4×C8914yes—
312C4×Q8C2×C4×C8914yes—
313C4×Q8C2×C4×C8912yes—
314C4×Q8C23×C8914yes—
315C4×Q8C23×C8912yes—
316C4×Q8C23×C8914yes—
317C4×D4C2×C4×C8914yes—
318C4×D4C2×C4×C8912yes—
319C4×D4C23×C8914yes—
320C4×D4C23×C8912yes—
321C4×D4C23×C8914yes—
322#117C2×C4×C8912yes—
323#117C2×C4×C8914yes—
324#117C23×C8918yes—
325C2×C4⋊C4C2×C4×C8911yescharacteristic
326C2×C4⋊C4C2×C4×C8912yes—
327C2×C42C23×C8911yescharacteristic
328C2×C42C23×C8912yes—
329C2×C42C23×C8914yes—
330C4⋊C8C23×C8914yes—
331C4⋊C8C23×C8918yes—
332C4⋊C8C23×C8912yes—
333C4⋊C8C23×C8918yes—
334C4⋊C8C23×C8914yes—
335C4⋊C8C23×C8914yes—
336C4⋊C8C23×C8912yes—
337C4⋊C8C23×C8914yes—
338C4⋊C8C23×C8914yes—
339C22⋊C8C23×C8914yes—
340C22⋊C8C23×C8914yes—
341C22⋊C8C23×C8914yes—
342C22⋊C8C23×C8918yes—
343#97C23×C8918yes—
344#97C23×C8918yes—
345C4×C8C23×C8912yes—
346C4×C8C23×C8914yes—
347C4×C8C23×C8914yes—
348C4×C8C23×C8912yes—
349C4×C8C23×C8914yes—
350C4×C8C23×C8914yes—
351C4×C8C23×C8914yes—
352#41—28no—
353C2×Q8—24no—
354C2×Q8C22×C4×C8914yes—
355C2×Q8C22×C4×C8912yes—
356C2×Q8C22×C4×C8914yes—
357C2×D4C22×C4×C8914yes—
358C2×D4C22×C4×C8912yes—
359C2×D4C22×C4×C8914yes—
360C22×C4C22×C4×C8912yes—
361C22×C4C22×C4×C8914yes—
362C22×C4C22×C4×C8911yescharacteristic
363C22×C4C22×C4×C8912yes—
364C22×C4C22×C4×C8914yes—
365C22×C4C22×C4×C8911yescharacteristic
366C22×C4C24×C8911yescharacteristic
367C22×C4C24×C8914yes—
368M16—48no—
369C2×C8—22no—
370C2×C8—28no—
371C2×C8—41no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8C24×C8911yescharacteristic
375C2×C8C24×C8912yes—
376C2×C8C24×C8914yes—
377C2×C8C24×C8914yes—
378C2×C8C24×C8914yes—
379C2×C8C24×C8911yescharacteristic
380C4⋊C4C22×C4×C8914yes—
381C4⋊C4C22×C4×C8918yes—
382C4⋊C4C22×C4×C8914yes—
383C4⋊C4C22×C4×C8912yes—
384C4⋊C4C22×C4×C8912yes—
385C4⋊C4C22×C4×C8914yes—
386C4⋊C4C22×C4×C8918yes—
387C4⋊C4C22×C4×C8918yes—
388C4⋊C4C22×C4×C8914yes—
389C4⋊C4C22×C4×C8914yes—
390C4⋊C4C22×C4×C8918yes—
391C22⋊C4C22×C4×C8914yes—
392C22⋊C4C22×C4×C8914yes—
393C22⋊C4C22×C4×C8914yes—
394C22⋊C4C22×C4×C8918yes—
395C22⋊C4C22×C4×C8918yes—
396C22⋊C4C22×C4×C8918yes—
397C42C22×C4×C8912yes—
398C42C22×C4×C8914yes—
399C42C22×C4×C8914yes—
400C42C22×C4×C8914yes—
401C42C24×C8912yes—
402C42C24×C8914yes—
403C42C24×C8914yes—
404C23C23×C4×C8914yes—
405C23C23×C4×C8911yescharacteristic
406Q8—28no—
407Q8—24no—
408Q8—44no—
409D4—28no—
410D4—24no—
411D4—44no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4#131×C8914yes—
422C2×C4C23×C4×C8911yescharacteristic
423C2×C4C23×C4×C8912yes—
424C2×C4C23×C4×C8914yes—
425C2×C4C23×C4×C8914yes—
426C2×C4C23×C4×C8914yes—
427C2×C4C23×C4×C8914yes—
428C2×C4C23×C4×C8914yes—
429C2×C4C23×C4×C8912yes—
430C2×C4C25×C8911yescharacteristic
431C8—21no—
432C8—24no—
433C8—22no—
434C8—24no—
435C8—24no—
436C82−1+4×C8912yes—
437C22—21no—
438C22—21no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22C2×#131×C8912yes—
443C22C24×C4×C8911yescharacteristic
444C4—21no—
445C4—24no—
446C4—24no—
447C4—24no—
448C4—22no—
449C4—24no—
450C4—24no—
451C4C2×#131×C8914yes—
452C4C2×2−1+4×C8911yescharacteristic
453C4#585×C8911yescharacteristic
454C2—22no—
455C2—44no—
456C2C4×2−1+4×C8911yescharacteristic
457C2C22×#131×C8911yescharacteristic
458C2C89×#357311yescharacteristic
subgroupG/H
centerC2×C8×C89C24
commutatorC22C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC22×C89C24×C4
prime radical#537×C89C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2993C2semidirect
3. C89×#2992C2semidirect
4. C89×#2989C2semidirect
5. C89×#2985C2semidirect
6. C89×#2983C2semidirect
7. C89×#2966C2semidirect
8. C89×#2966C2semidirect
9. C8×#41×C89C2semidirect
10. C89×#2960C2semidirect
11. C2×C8×Q8×C89C2semidirect
224. #60330C89direct (N × H)
231. #2993C2×C89semidirect
232. #2992C2×C89semidirect
233. #2989C2×C89semidirect
234. #2985C2×C89semidirect
235. #2983C2×C89semidirect
236. #2966C2×C89semidirect
237. #2966C2×C89semidirect
238. C8×#41C2×C89semidirect
239. #2960C2×C89semidirect
240. C2×C8×Q8C2×C89semidirect
241. C89#60330direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#34612
C89×#29932
C89×#29922
C89×#29892
C89×#29852
C89×#29832
C89×#29662
C8×#41×C892
C89×#29602
C2×C8×Q8×C892
#6033089