Finite Group Databasebeta

← C89×#62153 #36440001

Subgroups — C89×#62153 #36440001

Order 22784.

1696 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3518C211yesmaximal, characteristic
2Q8×#41×C89C211yesmaximal, characteristic, complemented
3Q8×Q16×C89C211yesmaximal, characteristic, complemented
4Q8×SD16×C89C211yesmaximal, characteristic, complemented
5C89×#3363C211yesmaximal, characteristic, complemented
6C89×#3359C211yesmaximal, characteristic, complemented
7C89×#3357C213yesmaximal, complemented
8C89×#3356C213yesmaximal, complemented
9C89×#3329C213yesmaximal, complemented
10C89×#3328C213yesmaximal, complemented
11C89×#3252C213yesmaximal, complemented
12C89×#3231C213yesmaximal, complemented
13C89×#3108C213yesmaximal, complemented
14C89×#3078C213yesmaximal, complemented
15C89×#2991C211yesmaximal, characteristic, complemented
16Q82×C89C2211yescharacteristic
17Q82×C89C2211yescharacteristic
18#557×C89—21no—
19#557×C89C2211yescharacteristic
20#556×C89—23no—
21#556×C89C2213yes—
22#555×C89C2213yes—
23#554×C89—23no—
24#554×C89C2213yes—
25D4×Q8×C89C2213yes—
26D4×Q8×C89C2211yescharacteristic
27#548×C89—21no—
28#542×C89—23no—
29#533×C89—23no—
30#533×C89C2213yes—
31#533×C89C2213yes—
32#529×C89C2213yes—
33#519×C89—21no—
34C4×#41×C89—21no—
35C4×#41×C89C2211yescharacteristic
36C2×C4×Q8×C89C2213yes—
37#500×C89C2213yes—
38#498×C89C2213yes—
39#487×C89C2213yes—
40#487×C89C2213yes—
41#485×C89C2216yes—
42#485×C89C2216yes—
43#483×C89C2213yes—
44#480×C89C2213yes—
45#479×C89C2213yes—
46#479×C89C2216yes—
47#479×C89C2213yes—
48#477×C89C2213yes—
49#477×C89C2213yes—
50#477×C89C2213yes—
51#476×C89C2213yes—
52#475×C89C2213yes—
53#475×C89C2213yes—
54#470×C89C2216yes—
55#468×C89C2216yes—
56#464×C89C2211yescharacteristic
57#464×C89C2213yes—
58#462×C89C2213yes—
59#461×C89C2213yes—
60#451×C89C2213yes—
61#448×C89C2213yes—
62C8×Q8×C89C2211yescharacteristic
63C8×Q8×C89C2211yescharacteristic
64C4×Q16×C89C2213yes—
65C4×Q16×C89C2213yes—
66C4×SD16×C89C2213yes—
67C4×SD16×C89C2213yes—
68#436×C89C2213yes—
69#436×C89C2213yes—
70#431×C89C2213yes—
71#428×C89C2216yes—
72#423×C89C2213yes—
73#419×C89C2216yes—
74C2×#41×C89D411yescharacteristic
75C22×Q8×C89—21no—
76C22×Q8×C89C2313yes—
77C2×Q16×C89C2311yescharacteristic
78C2×Q16×C89C2313yes—
79C2×SD16×C89C2311yescharacteristic
80C2×SD16×C89C2313yes—
81C2×M16×C89C2316yes—
82C4⋊Q8×C89—23no—
83C4⋊Q8×C89—23no—
84C4⋊Q8×C89—23no—
85C4⋊Q8×C89C2313yes—
86C4⋊Q8×C89C2313yes—
87C4⋊Q8×C89C2313yes—
88C4⋊Q8×C89C2313yes—
89#126×C89—26no—
90#126×C89—23no—
91#125×C89—23no—
92#125×C89—23no—
93#125×C89—23no—
94#125×C89C2313yes—
95#124×C89—23no—
96#124×C89—26no—
97#124×C89C2311yescharacteristic
98#123×C89—23no—
99C22⋊Q8×C89—23no—
100C22⋊Q8×C89—23no—
101C22⋊Q8×C89—23no—
102C22⋊Q8×C89—23no—
103C22⋊Q8×C89—26no—
104C22⋊Q8×C89—26no—
105C22⋊Q8×C89C2313yes—
106C22⋊Q8×C89C2316yes—
107#121×C89—23no—
108#121×C89C2313yes—
109C4×Q8×C89—21no—
110C4×Q8×C89—21no—
111C4×Q8×C89—21no—
112C4×Q8×C89—23no—
113C4×Q8×C89—23no—
114C4×Q8×C89—23no—
115C4×Q8×C89—23no—
116C4×Q8×C89—23no—
117C4×Q8×C89—23no—
118C4×Q8×C89D411yescharacteristic
119C4×Q8×C89C2313yes—
120C4×Q8×C89C2313yes—
121C4×Q8×C89C2313yes—
122C4×Q8×C89C2311yescharacteristic
123C4×Q8×C89C2313yes—
124C4×Q8×C89C2311yescharacteristic
125C4×D4×C89—23no—
126C4×D4×C89—23no—
127C4×D4×C89—23no—
128C4×D4×C89D413yes—
129C4×D4×C89C2313yes—
130C4×D4×C89C2313yes—
131#117×C89—23no—
132#117×C89—23no—
133#117×C89—23no—
134#117×C89D413yes—
135#117×C89C2316yes—
136C2×C4⋊C4×C89—23no—
137C2×C4⋊C4×C89C2313yes—
138C2×C42×C89—23no—
139C2×C42×C89C2313yes—
140#107×C89C2313yes—
141#107×C89C2316yes—
142#107×C89C2313yes—
143#106×C89C2313yes—
144#106×C89C2316yes—
145#106×C89C2313yes—
146C4⋊C8×C89C2313yes—
147C4⋊C8×C89C2313yes—
148C4⋊C8×C89C2313yes—
149C4⋊C8×C89C2311yescharacteristic
150#103×C89C2313yes—
151#103×C89C2313yes—
152#103×C89C2316yes—
153#103×C89C2316yes—
154#103×C89C2313yes—
155#103×C89C2311yescharacteristic
156#103×C89C2313yes—
157#103×C89C2313yes—
158#102×C89C2311yescharacteristic
159#102×C89C2313yes—
160C22⋊C8×C89C2313yes—
161C22⋊C8×C89C2313yes—
162C4×C8×C89C2313yes—
163C4×C8×C89C2313yes—
164#41×C89—24no—
165C2×Q8×C89—23no—
166C2×Q8×C89—23no—
167C2×Q8×C89—23no—
168C2×Q8×C89—23no—
169C2×Q8×C89—26no—
170C2×Q8×C89—43no—
171C2×Q8×C89—44no—
172C2×Q8×C89C2×D411yescharacteristic
173C2×Q8×C89C2413yes—
174C2×Q8×C89C2411yescharacteristic
175C2×Q8×C89C2411yescharacteristic
176C2×D4×C89—23no—
177C2×D4×C89C2×D413yes—
178C2×D4×C89C2411yescharacteristic
179C22×C4×C89—23no—
180C22×C4×C89—23no—
181C22×C4×C89—23no—
182C22×C4×C89C2×D413yes—
183C22×C4×C89C2×D413yes—
184C22×C4×C89C2413yes—
185Q16×C89—22no—
186Q16×C89—43no—
187SD16×C89—22no—
188SD16×C89—43no—
189M16×C89—46no—
190C2×C8×C89C2411yescharacteristic
191C2×C8×C89C2411yescharacteristic
192C2×C8×C89C2413yes—
193C2×C8×C89C2413yes—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—23no—
196C4⋊C4×C89—23no—
197C4⋊C4×C89—23no—
198C4⋊C4×C89—23no—
199C4⋊C4×C89—23no—
200C4⋊C4×C89—23no—
201C4⋊C4×C89—23no—
202C4⋊C4×C89—23no—
203C4⋊C4×C89—23no—
204C4⋊C4×C89—23no—
205C4⋊C4×C89—23no—
206C4⋊C4×C89—23no—
207C4⋊C4×C89—26no—
208C4⋊C4×C89—23no—
209C4⋊C4×C89C2×D413yes—
210C4⋊C4×C89C2×D413yes—
211C4⋊C4×C89C2411yescharacteristic
212C4⋊C4×C89C2413yes—
213C4⋊C4×C89C2413yes—
214C4⋊C4×C89C2413yes—
215C22⋊C4×C89—26no—
216C22⋊C4×C89—26no—
217C22⋊C4×C89—23no—
218C22⋊C4×C89—23no—
219C22⋊C4×C89—21no—
220C22⋊C4×C89—21no—
221C22⋊C4×C89—23no—
222C22⋊C4×C89C2×D416yes—
223C42×C89—21no—
224C42×C89—23no—
225C42×C89—23no—
226C42×C89—23no—
227C42×C89—26no—
228C42×C89—23no—
229C42×C89C2×D413yes—
230C42×C89C2×D413yes—
231C42×C89C2413yes—
232C42×C89C2411yescharacteristic
233C23×C89—21no—
234C23×C89C22×D413yes—
235Q8×C89—21no—
236Q8×C89—21no—
237Q8×C89—23no—
238Q8×C89—23no—
239Q8×C89—21no—
240Q8×C89—46no—
241Q8×C89—46no—
242Q8×C89—83no—
243Q8×C89—83no—
244Q8×C89#13714yes—
245Q8×C892−1+412yes—
246D4×C89—26no—
247D4×C89—21no—
248D4×C89—21no—
249D4×C89—83no—
250C2×C4×C89—21no—
251C2×C4×C89—26no—
252C2×C4×C89—23no—
253C2×C4×C89—23no—
254C2×C4×C89—21no—
255C2×C4×C89—21no—
256C2×C4×C89—23no—
257C2×C4×C89—23no—
258C2×C4×C89—23no—
259C2×C4×C89—23no—
260C2×C4×C89—43no—
261C2×C4×C89—46no—
262C2×C4×C89—46no—
263C2×C4×C89—83no—
264C2×C4×C89C22×D411yescharacteristic
265C2×C4×C89C22×D413yes—
266C2×C4×C89C22×D411yescharacteristic
267C2×C4×C89C22×D413yes—
268C2×C4×C89C22×D413yes—
269C2×C4×C89C2511yescharacteristic
270C8×C89—21no—
271C8×C89—21no—
272C8×C89—23no—
273C8×C89—23no—
274C22×C89—23no—
275C22×C89—23no—
276C22×C89—23no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89C23×D411yescharacteristic
281C4×C89—21no—
282C4×C89—23no—
283C4×C89—23no—
284C4×C89—22no—
285C4×C89—21no—
286C4×C89—41no—
287C4×C89—43no—
288C4×C89—43no—
289C4×C89—43no—
290C4×C89—41no—
291C4×C89—43no—
292C4×C89C2×#13716yes—
293C4×C89C2×2−1+411yescharacteristic
294C4×C89#58511yescharacteristic
295#62153C8911yesmaximal, characteristic, direct product factor, Sylow
296C2×C89—43no—
297C2×C89—81no—
298C2×C89#348911yescharacteristic
299C2×C89C22×#13711yescharacteristic
300C2×C89#358411yescharacteristic
301#3518C2×C8911yescharacteristic
302Q8×#41C2×C8911yescharacteristic, complemented
303Q8×Q16C2×C8911yescharacteristic, complemented
304Q8×SD16C2×C8911yescharacteristic, complemented
305#3363C2×C8911yescharacteristic, complemented
306#3359C2×C8911yescharacteristic, complemented
307#3357C2×C8913yescomplemented
308#3356C2×C8913yescomplemented
309#3329C2×C8913yescomplemented
310#3328C2×C8913yescomplemented
311#3252C2×C8913yescomplemented
312#3231C2×C8913yescomplemented
313#3108C2×C8913yescomplemented
314#3078C2×C8913yescomplemented
315#2991C2×C8911yescharacteristic, complemented
316C89#6215311yescharacteristic, direct product factor, Sylow
317Q82C22×C8911yescharacteristic
318Q82C22×C8911yescharacteristic
319#557—21no—
320#557C22×C8911yescharacteristic
321#556—23no—
322#556C22×C8913yes—
323#555C22×C8913yes—
324#554—23no—
325#554C22×C8913yes—
326D4×Q8C22×C8913yes—
327D4×Q8C22×C8911yescharacteristic
328#548—21no—
329#542—23no—
330#533—23no—
331#533C22×C8913yes—
332#533C22×C8913yes—
333#529C22×C8913yes—
334#519—21no—
335C4×#41—21no—
336C4×#41C22×C8911yescharacteristic
337C2×C4×Q8C22×C8913yes—
338#500C22×C8913yes—
339#498C22×C8913yes—
340#487C22×C8913yes—
341#487C22×C8913yes—
342#485C22×C8916yes—
343#485C22×C8916yes—
344#483C22×C8913yes—
345#480C22×C8913yes—
346#479C22×C8913yes—
347#479C22×C8916yes—
348#479C22×C8913yes—
349#477C22×C8913yes—
350#477C22×C8913yes—
351#477C22×C8913yes—
352#476C22×C8913yes—
353#475C22×C8913yes—
354#475C22×C8913yes—
355#470C22×C8916yes—
356#468C22×C8916yes—
357#464C22×C8911yescharacteristic
358#464C22×C8913yes—
359#462C22×C8913yes—
360#461C22×C8913yes—
361#451C22×C8913yes—
362#448C22×C8913yes—
363C8×Q8C22×C8911yescharacteristic
364C8×Q8C22×C8911yescharacteristic
365C4×Q16C22×C8913yes—
366C4×Q16C22×C8913yes—
367C4×SD16C22×C8913yes—
368C4×SD16C22×C8913yes—
369#436C22×C8913yes—
370#436C22×C8913yes—
371#431C22×C8913yes—
372#428C22×C8916yes—
373#423C22×C8913yes—
374#419C22×C8916yes—
375C2×#41D4×C8911yescharacteristic
376C22×Q8—21no—
377C22×Q8C23×C8913yes—
378C2×Q16C23×C8911yescharacteristic
379C2×Q16C23×C8913yes—
380C2×SD16C23×C8911yescharacteristic
381C2×SD16C23×C8913yes—
382C2×M16C23×C8916yes—
383C4⋊Q8—23no—
384C4⋊Q8—23no—
385C4⋊Q8—23no—
386C4⋊Q8C23×C8913yes—
387C4⋊Q8C23×C8913yes—
388C4⋊Q8C23×C8913yes—
389C4⋊Q8C23×C8913yes—
390#126—26no—
391#126—23no—
392#125—23no—
393#125—23no—
394#125—23no—
395#125C23×C8913yes—
396#124—23no—
397#124—26no—
398#124C23×C8911yescharacteristic
399#123—23no—
400C22⋊Q8—23no—
401C22⋊Q8—23no—
402C22⋊Q8—23no—
403C22⋊Q8—23no—
404C22⋊Q8—26no—
405C22⋊Q8—26no—
406C22⋊Q8C23×C8913yes—
407C22⋊Q8C23×C8916yes—
408#121—23no—
409#121C23×C8913yes—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—23no—
414C4×Q8—23no—
415C4×Q8—23no—
416C4×Q8—23no—
417C4×Q8—23no—
418C4×Q8—23no—
419C4×Q8D4×C8911yescharacteristic
420C4×Q8C23×C8911yescharacteristic
421C4×Q8C23×C8913yes—
422C4×Q8C23×C8913yes—
423C4×Q8C23×C8913yes—
424C4×Q8C23×C8913yes—
425C4×Q8C23×C8911yescharacteristic
426C4×D4—23no—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4D4×C8913yes—
430C4×D4C23×C8913yes—
431C4×D4C23×C8913yes—
432#117—23no—
433#117—23no—
434#117—23no—
435#117D4×C8913yes—
436#117C23×C8916yes—
437C2×C4⋊C4—23no—
438C2×C4⋊C4C23×C8913yes—
439C2×C42—23no—
440C2×C42C23×C8913yes—
441#107C23×C8913yes—
442#107C23×C8916yes—
443#107C23×C8913yes—
444#106C23×C8913yes—
445#106C23×C8916yes—
446#106C23×C8913yes—
447C4⋊C8C23×C8913yes—
448C4⋊C8C23×C8913yes—
449C4⋊C8C23×C8913yes—
450C4⋊C8C23×C8911yescharacteristic
451#103C23×C8913yes—
452#103C23×C8913yes—
453#103C23×C8916yes—
454#103C23×C8916yes—
455#103C23×C8913yes—
456#103C23×C8911yescharacteristic
457#103C23×C8913yes—
458#103C23×C8913yes—
459#102C23×C8911yescharacteristic
460#102C23×C8913yes—
461C22⋊C8C23×C8913yes—
462C22⋊C8C23×C8913yes—
463C4×C8C23×C8913yes—
464C4×C8C23×C8913yes—
465#41—24no—
466C2×Q8—23no—
467C2×Q8—23no—
468C2×Q8—23no—
469C2×Q8—23no—
470C2×Q8—26no—
471C2×Q8—43no—
472C2×Q8—44no—
473C2×Q8C2×D4×C8911yescharacteristic
474C2×Q8C24×C8913yes—
475C2×Q8C24×C8911yescharacteristic
476C2×Q8C24×C8911yescharacteristic
477C2×D4—23no—
478C2×D4C2×D4×C8913yes—
479C2×D4C24×C8911yescharacteristic
480C22×C4—23no—
481C22×C4—23no—
482C22×C4—23no—
483C22×C4C2×D4×C8913yes—
484C22×C4C2×D4×C8913yes—
485C22×C4C24×C8913yes—
486Q16—22no—
487Q16—43no—
488SD16—22no—
489SD16—43no—
490M16—46no—
491C2×C8C24×C8911yescharacteristic
492C2×C8C24×C8911yescharacteristic
493C2×C8C24×C8913yes—
494C2×C8C24×C8913yes—
495C4⋊C4—21no—
496C4⋊C4—23no—
497C4⋊C4—23no—
498C4⋊C4—23no—
499C4⋊C4—23no—
500C4⋊C4—23no—
501C4⋊C4—23no—
502C4⋊C4—23no—
503C4⋊C4—23no—
504C4⋊C4—23no—
505C4⋊C4—23no—
506C4⋊C4—23no—
507C4⋊C4—23no—
508C4⋊C4—26no—
509C4⋊C4—23no—
510C4⋊C4C2×D4×C8913yes—
511C4⋊C4C2×D4×C8913yes—
512C4⋊C4C24×C8911yescharacteristic
513C4⋊C4C24×C8913yes—
514C4⋊C4C24×C8913yes—
515C4⋊C4C24×C8913yes—
516C22⋊C4—26no—
517C22⋊C4—26no—
518C22⋊C4—23no—
519C22⋊C4—23no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—23no—
523C22⋊C4C2×D4×C8916yes—
524C42—21no—
525C42—23no—
526C42—23no—
527C42—23no—
528C42—26no—
529C42—23no—
530C42C2×D4×C8913yes—
531C42C2×D4×C8913yes—
532C42C24×C8913yes—
533C42C24×C8911yescharacteristic
534C23—21no—
535C23C22×D4×C8913yes—
536Q8—21no—
537Q8—21no—
538Q8—23no—
539Q8—23no—
540Q8—21no—
541Q8—46no—
542Q8—46no—
543Q8—83no—
544Q8—83no—
545Q8#137×C8914yes—
546Q82−1+4×C8912yes—
547D4—26no—
548D4—21no—
549D4—21no—
550D4—83no—
551C2×C4—21no—
552C2×C4—26no—
553C2×C4—23no—
554C2×C4—23no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—23no—
558C2×C4—23no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—43no—
562C2×C4—46no—
563C2×C4—46no—
564C2×C4—83no—
565C2×C4C22×D4×C8911yescharacteristic
566C2×C4C22×D4×C8913yes—
567C2×C4C22×D4×C8911yescharacteristic
568C2×C4C22×D4×C8913yes—
569C2×C4C22×D4×C8913yes—
570C2×C4C25×C8911yescharacteristic
571C8—21no—
572C8—21no—
573C8—23no—
574C8—23no—
575C22—23no—
576C22—23no—
577C22—23no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C23×D4×C8911yescharacteristic
582C4—21no—
583C4—21no—
584C4—23no—
585C4—23no—
586C4—22no—
587C4—43no—
588C4—41no—
589C4—43no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4C2×#137×C8916yes—
594C4C2×2−1+4×C8911yescharacteristic
595C4#585×C8911yescharacteristic
596C2—43no—
597C2—81no—
598C2C89×#348911yescharacteristic
599C2C22×#137×C8911yescharacteristic
600C2C89×#358411yescharacteristic
subgroupG/H
center, socleC22×C89C23×D4
commutator, Frattini, cosocleC2×C4C25×C89
prime radicalC89×#3518C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×#41×C89C2semidirect
3. Q8×Q16×C89C2semidirect
4. Q8×SD16×C89C2semidirect
5. C89×#3363C2semidirect
6. C89×#3359C2semidirect
7. C89×#3357C2semidirect
8. C89×#3356C2semidirect
9. C89×#3329C2semidirect
10. C89×#3328C2semidirect
11. C89×#3252C2semidirect
12. C89×#3231C2semidirect
13. C89×#3108C2semidirect
14. C89×#3078C2semidirect
15. C89×#2991C2semidirect
295. #62153C89direct (N × H)
302. Q8×#41C2×C89semidirect
303. Q8×Q16C2×C89semidirect
304. Q8×SD16C2×C89semidirect
305. #3363C2×C89semidirect
306. #3359C2×C89semidirect
307. #3357C2×C89semidirect
308. #3356C2×C89semidirect
309. #3329C2×C89semidirect
310. #3328C2×C89semidirect
311. #3252C2×C89semidirect
312. #3231C2×C89semidirect
313. #3108C2×C89semidirect
314. #3078C2×C89semidirect
315. #2991C2×C89semidirect
316. C89#62153direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#35182
Q8×#41×C892
Q8×Q16×C892
Q8×SD16×C892
C89×#33632
C89×#33592
C89×#33572
C89×#33562
C89×#33292
C89×#33282
C89×#32522
C89×#32312
C89×#31082
C89×#30782
C89×#29912
#6215389