Finite Group Databasebeta

← C89×#60305 #36438153

Subgroups — C89×#60305 #36438153

Order 22784.

2004 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3481C211yesmaximal, characteristic
2C89×#2994C214yesmaximal, complemented
3C89×#2990C212yesmaximal, complemented
4C89×#2979C211yesmaximal, characteristic, complemented
5C89×#2977C214yesmaximal, complemented
6C89×#2974C214yesmaximal, complemented
7C89×#2966C214yesmaximal, complemented
8C8×#41×C89C212yesmaximal, complemented
9C89×#2931C211yesmaximal, characteristic, complemented
10C89×#2929C214yesmaximal, complemented
11C2×C8×D4×C89C214yesmaximal, complemented
12C2×#131×C89C2212yescomplemented
13C23×C8×C89C2214yescomplemented
14#559×C89C411yescharacteristic
15#554×C89C412yes—
16#553×C89C414yes—
17#551×C89C414yes—
18#546×C89C414yes—
19D42×C89C411yescharacteristic
20#532×C89C2211yescharacteristic
21#529×C89C2214yes—
22#525×C89C2214yes—
23C4×#41×C89C2212yes—
24C2×C4×D4×C89C2214yes—
25#446×C89C2214yescomplemented
26C8×Q8×C89C2212yescomplemented
27C8×Q8×C89C2212yescomplemented
28#436×C89C2214yescomplemented
29#436×C89C2214yescomplemented
30#435×C89C2218yescomplemented
31#435×C89C2218yescomplemented
32#435×C89C2218yescomplemented
33C8×D4×C89C2212yes—
34C8×D4×C89C2214yes—
35C8×D4×C89C2212yes—
36C8×D4×C89C2218yescomplemented
37C8×D4×C89C2214yescomplemented
38C8×D4×C89C2218yescomplemented
39C8×D4×C89C2214yescomplemented
40C8×D4×C89C2214yescomplemented
41C8×D4×C89C2214yescomplemented
42#433×C89C2218yescomplemented
43#432×C89C2214yescomplemented
44#431×C89C2214yescomplemented
45#431×C89C2212yescomplemented
46#424×C89C2214yescomplemented
47#423×C89C2211yescharacteristic, complemented
48C2×C4⋊C8×C89C2214yescomplemented
49#408×C89C2214yescomplemented
50#408×C89C2212yescomplemented
51#407×C89C2214yescomplemented
52C2×C22⋊C8×C89C2218yescomplemented
53#405×C89C2214yescomplemented
54C2×C4×C8×C89C2214yescomplemented
55C2×C4×C8×C89C2211yescharacteristic, complemented
56C2×#41×C89C2×C412yes—
57C2×#41×C89C2312yes—
58C22×D4×C89C2×C414yes—
59C23×C4×C89C2314yes—
60#131×C89—44no—
61C2×M16×C89C2314yes—
62C2×M16×C89C2314yes—
63C2×M16×C89C2312yes—
64C22×C8×C89—48no—
65C22×C8×C89C2314yes—
66C22×C8×C89C2311yescharacteristic
67C22×C8×C89C2314yes—
68C22×C8×C89C2314yes—
69C22×C8×C89C2311yescharacteristic
70C22×C8×C89C2318yes—
71C22×C8×C89C2314yes—
72C4⋊Q8×C89C2×C411yescharacteristic
73#127×C89C2×C411yescharacteristic
74#126×C89C2×C418yes—
75#125×C89C2×C414yes—
76#124×C89C2×C414yes—
77#124×C89C2×C412yes—
78#123×C89C2×C414yes—
79#123×C89C2×C414yes—
80#123×C89C2×C418yes—
81C22⋊Q8×C89C2×C418yes—
82C22⋊Q8×C89C2×C414yes—
83#121×C89C2×C418yes—
84#121×C89C2×C414yes—
85#121×C89C2×C414yes—
86#120×C89C2×C414yes—
87#120×C89C2×C414yes—
88C4×Q8×C89C2×C412yes—
89C4×Q8×C89C2312yes—
90C4×D4×C89C2×C414yes—
91C4×D4×C89C2×C418yes—
92C4×D4×C89C2×C414yes—
93C4×D4×C89C2×C412yes—
94C4×D4×C89C2×C414yes—
95C4×D4×C89C2312yes—
96C4×D4×C89C2318yes—
97C4×D4×C89C2314yes—
98C4×D4×C89C2314yes—
99#117×C89C2×C414yes—
100#117×C89C2×C412yes—
101#117×C89C2314yes—
102C2×C4⋊C4×C89C2×C414yes—
103C2×C22⋊C4×C89C2×C418yes—
104C2×C42×C89C2314yes—
105C2×C42×C89C2311yescharacteristic
106C4⋊C8×C89C2312yes—
107C4⋊C8×C89C2314yes—
108C4⋊C8×C89C2318yes—
109C4⋊C8×C89C2314yes—
110C4⋊C8×C89C2312yes—
111C4⋊C8×C89C2314yes—
112C22⋊C8×C89C2314yes—
113C22⋊C8×C89C2318yes—
114C22⋊C8×C89C2318yes—
115C22⋊C8×C89C2318yes—
116C22⋊C8×C89C2314yes—
117C22⋊C8×C89C2314yes—
118#97×C89C2318yes—
119C4×C8×C89C2314yes—
120C4×C8×C89C2312yes—
121C4×C8×C89C2314yes—
122C4×C8×C89C2312yes—
123C4×C8×C89C2314yes—
124C24×C89C22×C414yes—
125#41×C89—28no—
126#41×C89—42no—
127#41×C89—42no—
128C2×Q8×C89C22×C412yes—
129C2×Q8×C89C22×C412yes—
130C2×D4×C89—216no—
131C2×D4×C89C22×C414yes—
132C2×D4×C89C22×C418yes—
133C2×D4×C89C22×C414yes—
134C2×D4×C89C22×C412yes—
135C2×D4×C89C22×C414yes—
136C2×D4×C89C22×C412yes—
137C22×C4×C89—44no—
138C22×C4×C89—44no—
139C22×C4×C89C22×C414yes—
140C22×C4×C89C22×C414yes—
141C22×C4×C89C22×C412yes—
142C22×C4×C89C22×C418yes—
143C22×C4×C89C22×C414yes—
144C22×C4×C89C2414yes—
145C22×C4×C89C2411yescharacteristic
146C22×C4×C89C2414yes—
147M16×C89—44no—
148M16×C89—42no—
149M16×C89—44no—
150C2×C8×C89—28no—
151C2×C8×C89—28no—
152C2×C8×C89—22no—
153C2×C8×C89—44no—
154C2×C8×C89—48no—
155C2×C8×C89—41no—
156C2×C8×C89—44no—
157C2×C8×C89C2412yes—
158C2×C8×C89C2414yes—
159C2×C8×C89C2414yes—
160C2×C8×C89C2411yescharacteristic
161C2×C8×C89C2414yes—
162C2×C8×C89C2411yescharacteristic
163C4⋊C4×C89C22×C412yes—
164C4⋊C4×C89C22×C414yes—
165C4⋊C4×C89C22×C418yes—
166C4⋊C4×C89C22×C412yes—
167C4⋊C4×C89C22×C414yes—
168C4⋊C4×C89C22×C414yes—
169C4⋊C4×C89C22×C418yes—
170C22⋊C4×C89C22×C414yes—
171C22⋊C4×C89C22×C418yes—
172C22⋊C4×C89C22×C418yes—
173C22⋊C4×C89C22×C418yes—
174C22⋊C4×C89C22×C414yes—
175C22⋊C4×C89C22×C414yes—
176C22⋊C4×C89C22×C414yes—
177C22⋊C4×C89C22×C414yes—
178C42×C89C22×C411yescharacteristic
179C42×C89C22×C414yes—
180C42×C89C22×C414yes—
181C42×C89C22×C411yescharacteristic
182C42×C89C2412yes—
183C42×C89C2414yes—
184C23×C89—216no—
185C23×C89—44no—
186C23×C89C23×C414yes—
187C23×C89C23×C411yescharacteristic
188C23×C89C23×C414yes—
189Q8×C89—24no—
190Q8×C89—42no—
191D4×C89—216no—
192D4×C89—28no—
193D4×C89—28no—
194D4×C89—44no—
195D4×C89—42no—
196D4×C89#13118yescomplemented
197C2×C4×C89—28no—
198C2×C4×C89—28no—
199C2×C4×C89—24no—
200C2×C4×C89—24no—
201C2×C4×C89—24no—
202C2×C4×C89—21no—
203C2×C4×C89—24no—
204C2×C4×C89—21no—
205C2×C4×C89—24no—
206C2×C4×C89—44no—
207C2×C4×C89—48no—
208C2×C4×C89C23×C412yes—
209C2×C4×C89C23×C414yes—
210C2×C4×C89C23×C411yescharacteristic
211C2×C4×C89C23×C414yes—
212C2×C4×C89C23×C414yes—
213C2×C4×C89C23×C414yes—
214C2×C4×C89C23×C412yes—
215C2×C4×C89C2511yescharacteristic
216C8×C89—24no—
217C8×C89—24no—
218C8×C89—21no—
219C8×C89—22no—
220C8×C89—24no—
221C8×C89H(5,𝔽2)12yes—
222C22×C89—24no—
223C22×C89—24no—
224C22×C89—24no—
225C22×C89—21no—
226C22×C89—22no—
227C22×C89—28no—
228C22×C89—416no—
229C22×C89C2×#13118yes—
230C22×C89C24×C411yescharacteristic
231C4×C89—24no—
232C4×C89—24no—
233C4×C89—21no—
234C4×C89—22no—
235C4×C89—24no—
236C4×C89—24no—
237C4×C89C2×#13114yes—
238C4×C89C2×H(5,𝔽2)11yescharacteristic
239C4×C89#58511yescharacteristic
240#60305C8911yesmaximal, characteristic, direct product factor, Sylow
241C2×C89—28no—
242C2×C89—41no—
243C2×C89—44no—
244C2×C89C4×H(5,𝔽2)11yescharacteristic
245C2×C89C22×#13112yes—
246#3481C2×C8911yescharacteristic
247#2994C2×C8914yescomplemented
248#2990C2×C8912yescomplemented
249#2979C2×C8911yescharacteristic, complemented
250#2977C2×C8914yescomplemented
251#2974C2×C8914yescomplemented
252#2966C2×C8914yescomplemented
253C8×#41C2×C8912yescomplemented
254#2931C2×C8911yescharacteristic, complemented
255#2929C2×C8914yescomplemented
256C2×C8×D4C2×C8914yescomplemented
257C89#6030511yescharacteristic, direct product factor, Sylow
258C2×#131C22×C8912yescomplemented
259C23×C8C22×C8914yescomplemented
260#559C4×C8911yescharacteristic
261#554C4×C8912yes—
262#553C4×C8914yes—
263#551C4×C8914yes—
264#546C4×C8914yes—
265D42C4×C8911yescharacteristic
266#532C22×C8911yescharacteristic
267#529C22×C8914yes—
268#525C22×C8914yes—
269C4×#41C22×C8912yes—
270C2×C4×D4C22×C8914yes—
271#446C22×C8914yescomplemented
272C8×Q8C22×C8912yescomplemented
273C8×Q8C22×C8912yescomplemented
274#436C22×C8914yescomplemented
275#436C22×C8914yescomplemented
276#435C22×C8918yescomplemented
277#435C22×C8918yescomplemented
278#435C22×C8918yescomplemented
279C8×D4C22×C8912yes—
280C8×D4C22×C8914yes—
281C8×D4C22×C8912yes—
282C8×D4C22×C8918yescomplemented
283C8×D4C22×C8914yescomplemented
284C8×D4C22×C8918yescomplemented
285C8×D4C22×C8914yescomplemented
286C8×D4C22×C8914yescomplemented
287C8×D4C22×C8914yescomplemented
288#433C22×C8918yescomplemented
289#432C22×C8914yescomplemented
290#431C22×C8914yescomplemented
291#431C22×C8912yescomplemented
292#424C22×C8914yescomplemented
293#423C22×C8911yescharacteristic, complemented
294C2×C4⋊C8C22×C8914yescomplemented
295#408C22×C8914yescomplemented
296#408C22×C8912yescomplemented
297#407C22×C8914yescomplemented
298C2×C22⋊C8C22×C8918yescomplemented
299#405C22×C8914yescomplemented
300C2×C4×C8C22×C8914yescomplemented
301C2×C4×C8C22×C8911yescharacteristic, complemented
302C2×#41C2×C4×C8912yes—
303C2×#41C23×C8912yes—
304C22×D4C2×C4×C8914yes—
305C23×C4C23×C8914yes—
306#131—44no—
307C2×M16C23×C8914yes—
308C2×M16C23×C8912yes—
309C2×M16C23×C8914yes—
310C22×C8—48no—
311C22×C8C23×C8914yes—
312C22×C8C23×C8914yes—
313C22×C8C23×C8911yescharacteristic
314C22×C8C23×C8918yes—
315C22×C8C23×C8914yes—
316C22×C8C23×C8914yes—
317C22×C8C23×C8911yescharacteristic
318C4⋊Q8C2×C4×C8911yescharacteristic
319#127C2×C4×C8911yescharacteristic
320#126C2×C4×C8918yes—
321#125C2×C4×C8914yes—
322#124C2×C4×C8914yes—
323#124C2×C4×C8912yes—
324#123C2×C4×C8914yes—
325#123C2×C4×C8914yes—
326#123C2×C4×C8918yes—
327C22⋊Q8C2×C4×C8918yes—
328C22⋊Q8C2×C4×C8914yes—
329#121C2×C4×C8918yes—
330#121C2×C4×C8914yes—
331#121C2×C4×C8914yes—
332#120C2×C4×C8914yes—
333#120C2×C4×C8914yes—
334C4×Q8C2×C4×C8912yes—
335C4×Q8C23×C8912yes—
336C4×D4C2×C4×C8914yes—
337C4×D4C2×C4×C8918yes—
338C4×D4C2×C4×C8914yes—
339C4×D4C2×C4×C8912yes—
340C4×D4C2×C4×C8914yes—
341C4×D4C23×C8912yes—
342C4×D4C23×C8918yes—
343C4×D4C23×C8914yes—
344C4×D4C23×C8914yes—
345#117C2×C4×C8914yes—
346#117C2×C4×C8912yes—
347#117C23×C8914yes—
348C2×C4⋊C4C2×C4×C8914yes—
349C2×C22⋊C4C2×C4×C8918yes—
350C2×C42C23×C8914yes—
351C2×C42C23×C8911yescharacteristic
352C4⋊C8C23×C8912yes—
353C4⋊C8C23×C8914yes—
354C4⋊C8C23×C8918yes—
355C4⋊C8C23×C8914yes—
356C4⋊C8C23×C8912yes—
357C4⋊C8C23×C8914yes—
358C22⋊C8C23×C8914yes—
359C22⋊C8C23×C8918yes—
360C22⋊C8C23×C8918yes—
361C22⋊C8C23×C8918yes—
362C22⋊C8C23×C8914yes—
363C22⋊C8C23×C8914yes—
364#97C23×C8918yes—
365C4×C8C23×C8912yes—
366C4×C8C23×C8914yes—
367C4×C8C23×C8914yes—
368C4×C8C23×C8912yes—
369C4×C8C23×C8914yes—
370C24C22×C4×C8914yes—
371#41—28no—
372#41—42no—
373#41—42no—
374C2×Q8C22×C4×C8912yes—
375C2×Q8C22×C4×C8912yes—
376C2×D4—216no—
377C2×D4C22×C4×C8914yes—
378C2×D4C22×C4×C8918yes—
379C2×D4C22×C4×C8914yes—
380C2×D4C22×C4×C8912yes—
381C2×D4C22×C4×C8914yes—
382C2×D4C22×C4×C8912yes—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4C22×C4×C8914yes—
386C22×C4C22×C4×C8914yes—
387C22×C4C22×C4×C8912yes—
388C22×C4C22×C4×C8918yes—
389C22×C4C22×C4×C8914yes—
390C22×C4C24×C8914yes—
391C22×C4C24×C8911yescharacteristic
392C22×C4C24×C8914yes—
393M16—42no—
394M16—44no—
395M16—44no—
396C2×C8—28no—
397C2×C8—28no—
398C2×C8—22no—
399C2×C8—44no—
400C2×C8—48no—
401C2×C8—41no—
402C2×C8—44no—
403C2×C8C24×C8912yes—
404C2×C8C24×C8914yes—
405C2×C8C24×C8914yes—
406C2×C8C24×C8911yescharacteristic
407C2×C8C24×C8914yes—
408C2×C8C24×C8911yescharacteristic
409C4⋊C4C22×C4×C8912yes—
410C4⋊C4C22×C4×C8914yes—
411C4⋊C4C22×C4×C8918yes—
412C4⋊C4C22×C4×C8912yes—
413C4⋊C4C22×C4×C8914yes—
414C4⋊C4C22×C4×C8914yes—
415C4⋊C4C22×C4×C8918yes—
416C22⋊C4C22×C4×C8914yes—
417C22⋊C4C22×C4×C8918yes—
418C22⋊C4C22×C4×C8918yes—
419C22⋊C4C22×C4×C8918yes—
420C22⋊C4C22×C4×C8914yes—
421C22⋊C4C22×C4×C8914yes—
422C22⋊C4C22×C4×C8914yes—
423C22⋊C4C22×C4×C8914yes—
424C42C22×C4×C8911yescharacteristic
425C42C22×C4×C8914yes—
426C42C22×C4×C8914yes—
427C42C22×C4×C8911yescharacteristic
428C42C24×C8912yes—
429C42C24×C8914yes—
430C23—216no—
431C23—44no—
432C23C23×C4×C8914yes—
433C23C23×C4×C8911yescharacteristic
434C23C23×C4×C8914yes—
435Q8—24no—
436Q8—42no—
437D4—216no—
438D4—28no—
439D4—28no—
440D4—44no—
441D4—42no—
442D4#131×C8918yescomplemented
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—44no—
453C2×C4—48no—
454C2×C4C23×C4×C8912yes—
455C2×C4C23×C4×C8914yes—
456C2×C4C23×C4×C8911yescharacteristic
457C2×C4C23×C4×C8914yes—
458C2×C4C23×C4×C8914yes—
459C2×C4C23×C4×C8914yes—
460C2×C4C23×C4×C8912yes—
461C2×C4C25×C8911yescharacteristic
462C8—24no—
463C8—24no—
464C8—21no—
465C8—22no—
466C8—24no—
467C8H(5,𝔽2)×C8912yes—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—21no—
472C22—22no—
473C22—28no—
474C22—416no—
475C22C2×#131×C8918yes—
476C22C24×C4×C8911yescharacteristic
477C4—24no—
478C4—24no—
479C4—21no—
480C4—22no—
481C4—24no—
482C4—24no—
483C4C2×#131×C8914yes—
484C4C2×H(5,𝔽2)×C8911yescharacteristic
485C4#585×C8911yescharacteristic
486C2—28no—
487C2—41no—
488C2—44no—
489C2C4×H(5,𝔽2)×C8911yescharacteristic
490C2C22×#131×C8912yes—
subgroupG/H
centerC2×C8×C89C24
commutatorC22C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC22×C89C24×C4
prime radicalC89×#3481C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2994C2semidirect
3. C89×#2990C2semidirect
4. C89×#2979C2semidirect
5. C89×#2977C2semidirect
6. C89×#2974C2semidirect
7. C89×#2966C2semidirect
8. C8×#41×C89C2semidirect
9. C89×#2931C2semidirect
10. C89×#2929C2semidirect
11. C2×C8×D4×C89C2semidirect
12. C2×#131×C89C22semidirect
13. C23×C8×C89C22semidirect
25. #446×C89C22semidirect
26. C8×Q8×C89C22semidirect
27. C8×Q8×C89C22semidirect
28. #436×C89C22semidirect
29. #436×C89C22semidirect
30. #435×C89C22semidirect
31. #435×C89C22semidirect
32. #435×C89C22semidirect
36. C8×D4×C89C22semidirect
37. C8×D4×C89C22semidirect
38. C8×D4×C89C22semidirect
39. C8×D4×C89C22semidirect
40. C8×D4×C89C22semidirect
41. C8×D4×C89C22semidirect
42. #433×C89C22semidirect
43. #432×C89C22semidirect
44. #431×C89C22semidirect
45. #431×C89C22semidirect
46. #424×C89C22semidirect
47. #423×C89C22semidirect
48. C2×C4⋊C8×C89C22semidirect
49. #408×C89C22semidirect
50. #408×C89C22semidirect
51. #407×C89C22semidirect
52. C2×C22⋊C8×C89C22semidirect
53. #405×C89C22semidirect
54. C2×C4×C8×C89C22semidirect
55. C2×C4×C8×C89C22semidirect
196. D4×C89#131semidirect
240. #60305C89direct (N × H)
247. #2994C2×C89semidirect
248. #2990C2×C89semidirect
249. #2979C2×C89semidirect
250. #2977C2×C89semidirect
251. #2974C2×C89semidirect
252. #2966C2×C89semidirect
253. C8×#41C2×C89semidirect
254. #2931C2×C89semidirect
255. #2929C2×C89semidirect
256. C2×C8×D4C2×C89semidirect
257. C89#60305direct (N × H)
258. C2×#131C22×C89semidirect
259. C23×C8C22×C89semidirect
271. #446C22×C89semidirect
272. C8×Q8C22×C89semidirect
273. C8×Q8C22×C89semidirect
274. #436C22×C89semidirect
275. #436C22×C89semidirect
276. #435C22×C89semidirect
277. #435C22×C89semidirect
278. #435C22×C89semidirect
282. C8×D4C22×C89semidirect
283. C8×D4C22×C89semidirect
284. C8×D4C22×C89semidirect
285. C8×D4C22×C89semidirect
286. C8×D4C22×C89semidirect
287. C8×D4C22×C89semidirect
288. #433C22×C89semidirect
289. #432C22×C89semidirect
290. #431C22×C89semidirect
291. #431C22×C89semidirect
292. #424C22×C89semidirect
293. #423C22×C89semidirect
294. C2×C4⋊C8C22×C89semidirect
295. #408C22×C89semidirect
296. #408C22×C89semidirect
297. #407C22×C89semidirect
298. C2×C22⋊C8C22×C89semidirect
299. #405C22×C89semidirect
300. C2×C4×C8C22×C89semidirect
301. C2×C4×C8C22×C89semidirect
442. D4#131×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#34812
C89×#29942
C89×#29902
C89×#29792
C89×#29772
C89×#29742
C89×#29662
C8×#41×C892
C89×#29312
C89×#29292
C2×C8×D4×C892
#6030589