Finite Group Databasebeta

← C31×#33557 #21490316

Subgroups — C31×#33557 #21490316

Order 7936.

1258 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3543C211yesmaximal, characteristic, complemented
2C31×#3026C212yesmaximal, complemented
3C2×C31×#453C211yesmaximal, characteristic, complemented
4C31×#2200C214yesmaximal, complemented
5C31×#2198C212yesmaximal, complemented
6C31×#2196C212yesmaximal, complemented
7C2×C31×#353C211yesmaximal, characteristic, complemented
8C31×#2121C212yesmaximal, complemented
9C2×C31×H(5,𝔽2)C2211yescharacteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11C31×#560—21no—
12C31×#560C2212yescomplemented
13C31×#559C2211yescharacteristic, complemented
14C31×#550—21no—
15C31×#546—22no—
16D42×C31—22no—
17D42×C31C2211yescharacteristic, complemented
18C31×#540—22no—
19C31×#535—22no—
20C31×#534—21no—
21C31×#534—22no—
22C31×#534C2212yes—
23C2×C31×#127C2211yescharacteristic
24C31×#529—21no—
25C31×#458—22no—
26C31×#458C2214yescomplemented
27C31×B(4,𝔽2)—22no—
28C31×B(4,𝔽2)C2214yescomplemented
29C31×#453—21no—
30C31×#453—21no—
31C31×#453C2212yes—
32C31×#453C2212yescomplemented
33C2×C31×#104—21no—
34C2×C31×#100C2211yescharacteristic, complemented
35C2×C31×#99—22no—
36C2×C31×#99C2212yescomplemented
37C31×#353C2212yescomplemented
38C31×#353C2212yescomplemented
39C31×#352C2214yescomplemented
40C31×AΣL(1,𝔽16)C2214yescomplemented
41C31×H(5,𝔽2)—22no—
42C31×H(5,𝔽2)—21no—
43C31×H(5,𝔽2)—42no—
44C31×H(5,𝔽2)C2312yes—
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C2×#41×C31—41no—
48C22×D4×C31—22no—
49C22×D4×C31—21no—
50C22×D4×C31—22no—
51C22×D4×C31—22no—
52C22×D4×C31—41no—
53C22×D4×C31D412yescomplemented
54C22×D4×C31D411yescharacteristic, complemented
55C22×D4×C31C2311yescharacteristic
56C31×AGL(1,ℤ/8)—22no—
57C31×AGL(1,ℤ/8)—22no—
58C31×AGL(1,ℤ/8)—42no—
59C2×SD16×C31—41no—
60C2×D8×C31—41no—
61C2×M16×C31—21no—
62C31×#127—22no—
63C31×#127—22no—
64C31×#127D411yescharacteristic, complemented
65C31×#127D411yescharacteristic, complemented
66C31×#127C2311yescharacteristic
67C31×#127C2311yescharacteristic
68C31×#126—22no—
69C31×#125—21no—
70C31×#124—21no—
71C31×#124—22no—
72C31×#123—21no—
73C31×#123—22no—
74C31×#123—21no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—24no—
78C31×#123—42no—
79C31×#123D412yescomplemented
80C31×#123C2312yes—
81C31×C22⋊Q8—21no—
82C31×#121—21no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—44no—
94C31×#120—21no—
95C31×#120—22no—
96C31×#120—22no—
97C31×#120—22no—
98C31×#120—22no—
99C31×#120—21no—
100C31×#120—24no—
101C31×#120—42no—
102C31×#120D412yescomplemented
103C31×#120C2312yes—
104C4×Q8×C31—21no—
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—22no—
108C4×D4×C31—21no—
109C31×#117—22no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31D412yescomplemented
113C2×C42×C31D411yescharacteristic, complemented
114C31×#104—22no—
115C31×#104—22no—
116C31×#100—21no—
117C31×#100C2312yes—
118C31×#99—22no—
119C31×#99—24no—
120C31×#99—24no—
121C31×#99C2314yes—
122C24×C31—22no—
123C24×C31—21no—
124C24×C31—21no—
125C24×C31C2×D412yes—
126C24×C31C2×D412yes—
127#41×C31—22no—
128#41×C31—22no—
129#41×C31—42no—
130#41×C31—41no—
131#41×C31—44no—
132#41×C31—42no—
133#41×C31—41no—
134#41×C31—84no—
135C2×Q8×C31—21no—
136C2×Q8×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—24no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—24no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—41no—
154C2×D4×C31—41no—
155C2×D4×C31—41no—
156C2×D4×C31—41no—
157C2×D4×C31—42no—
158C2×D4×C31—42no—
159C2×D4×C31—42no—
160C2×D4×C31—42no—
161C2×D4×C31—44no—
162C2×D4×C31—42no—
163C2×D4×C31—41no—
164C2×D4×C31—42no—
165C2×D4×C31—42no—
166C2×D4×C31—44no—
167C2×D4×C31—84no—
168C2×D4×C31—84no—
169C2×D4×C31—81no—
170C2×D4×C31—81no—
171C2×D4×C31C2×D412yes—
172C2×D4×C31C2×D411yescharacteristic
173C2×D4×C31C2×D411yescharacteristic
174C2×D4×C31C2×D411yescharacteristic
175C2×D4×C31C2411yescharacteristic
176C22×C4×C31—22no—
177C22×C4×C31—22no—
178C22×C4×C31—21no—
179C22×C4×C31—22no—
180C22×C4×C31—21no—
181C22×C4×C31—22no—
182C22×C4×C31—41no—
183C22×C4×C31—42no—
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D412yes—
186SD16×C31—42no—
187SD16×C31—42no—
188D8×C31—42no—
189D8×C31—42no—
190M16×C31—22no—
191M16×C31—41no—
192C2×C8×C31—41no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—44no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—21no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—24no—
212C22⋊C4×C31—44no—
213C22⋊C4×C31—42no—
214C22⋊C4×C31—44no—
215C22⋊C4×C31C2×D412yes—
216C22⋊C4×C31C2×D412yes—
217C42×C31—22no—
218C42×C31—21no—
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221#33557C3111yesmaximal, characteristic, direct product factor, Sylow
222C23×C31—22no—
223C23×C31—22no—
224C23×C31—24no—
225C23×C31—22no—
226C23×C31—21no—
227C23×C31—21no—
228C23×C31—42no—
229C23×C31—42no—
230C23×C31—41no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—82no—
234C23×C31—82no—
235C23×C31—82no—
236C23×C31—82no—
237C23×C31—81no—
238C23×C31—162no—
239C23×C31#12012yes—
240C23×C31#12011yescharacteristic
241C23×C31C22×D412yes—
242Q8×C31—22no—
243Q8×C31—41no—
244Q8×C31—42no—
245D4×C31—22no—
246D4×C31—41no—
247D4×C31—42no—
248D4×C31—41no—
249D4×C31—41no—
250D4×C31—41no—
251D4×C31—41no—
252D4×C31—44no—
253D4×C31—42no—
254D4×C31—44no—
255D4×C31—44no—
256D4×C31—44no—
257D4×C31—42no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—81no—
262D4×C31—82no—
263D4×C31—82no—
264D4×C31—82no—
265D4×C31—82no—
266D4×C31—82no—
267D4×C31—82no—
268D4×C31—82no—
269D4×C31—82no—
270D4×C31—164no—
271C2×C4×C31—21no—
272C2×C4×C31—21no—
273C2×C4×C31—21no—
274C2×C4×C31—24no—
275C2×C4×C31—21no—
276C2×C4×C31—22no—
277C2×C4×C31—22no—
278C2×C4×C31—22no—
279C2×C4×C31—42no—
280C2×C4×C31—44no—
281C2×C4×C31—42no—
282C2×C4×C31—42no—
283C2×C4×C31—42no—
284C2×C4×C31—41no—
285C2×C4×C31—42no—
286C2×C4×C31—44no—
287C2×C4×C31—81no—
288C2×C4×C31—82no—
289C2×C4×C31—82no—
290C2×C4×C31—162no—
291C2×C4×C31#12011yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C8×C31—42no—
294#3543C2×C3111yescharacteristic, complemented
295#3026C2×C3112yescomplemented
296C2×#453C2×C3111yescharacteristic, complemented
297#2200C2×C3114yescomplemented
298#2198C2×C3112yescomplemented
299#2196C2×C3112yescomplemented
300C2×#353C2×C3111yescharacteristic, complemented
301#2121C2×C3112yescomplemented
302C22×C31—22no—
303C22×C31—22no—
304C22×C31—41no—
305C22×C31—42no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—41no—
315C22×C31—82no—
316C22×C31—82no—
317C22×C31—82no—
318C22×C31—84no—
319C22×C31—84no—
320C22×C31—161no—
321C22×C31—161no—
322C22×C31—162no—
323C22×C31—162no—
324C22×C31B(4,𝔽2)11yescharacteristic
325C22×C31B(4,𝔽2)11yescharacteristic
326C22×C31C2×#12011yescharacteristic
327C4×C31—22no—
328C4×C31—21no—
329C4×C31—21no—
330C4×C31—42no—
331C4×C31—41no—
332C4×C31—42no—
333C4×C31—42no—
334C4×C31—42no—
335C4×C31—84no—
336C2×H(5,𝔽2)C22×C3111yescharacteristic, complemented
337C2×AGL(1,ℤ/8)—21no—
338#560—21no—
339#560C22×C3112yescomplemented
340#559C22×C3111yescharacteristic, complemented
341#550—21no—
342#546—22no—
343D42—22no—
344D42C22×C3111yescharacteristic, complemented
345#540—22no—
346#535—22no—
347#534—21no—
348#534—22no—
349#534C22×C3112yes—
350C2×#127C22×C3111yescharacteristic
351#529—21no—
352#458—22no—
353#458C22×C3114yescomplemented
354B(4,𝔽2)—22no—
355B(4,𝔽2)C22×C3114yescomplemented
356#453—21no—
357#453—21no—
358#453C22×C3112yes—
359#453C22×C3112yescomplemented
360C2×#104—21no—
361C2×#100C22×C3111yescharacteristic, complemented
362C2×#99—22no—
363C2×#99C22×C3112yescomplemented
364#353C22×C3112yescomplemented
365#353C22×C3112yescomplemented
366#352C22×C3114yescomplemented
367AΣL(1,𝔽16)C22×C3114yescomplemented
368C2×C31—21no—
369C2×C31—21no—
370C2×C31—21no—
371C2×C31—42no—
372C2×C31—42no—
373C2×C31—82no—
374C2×C31—81no—
375C2×C31—81no—
376C2×C31—82no—
377C2×C31C2×B(4,𝔽2)11yescharacteristic
378H(5,𝔽2)—22no—
379H(5,𝔽2)—21no—
380H(5,𝔽2)—42no—
381H(5,𝔽2)C23×C3112yes—
382C2×#41—22no—
383C2×#41—21no—
384C2×#41—41no—
385C22×D4—22no—
386C22×D4—21no—
387C22×D4—22no—
388C22×D4—22no—
389C22×D4—41no—
390C22×D4D4×C3112yescomplemented
391C22×D4D4×C3111yescharacteristic, complemented
392C22×D4C23×C3111yescharacteristic
393AGL(1,ℤ/8)—22no—
394AGL(1,ℤ/8)—22no—
395AGL(1,ℤ/8)—42no—
396C2×SD16—41no—
397C2×D8—41no—
398C2×M16—21no—
399#127—22no—
400#127—22no—
401#127D4×C3111yescharacteristic, complemented
402#127D4×C3111yescharacteristic, complemented
403#127C23×C3111yescharacteristic
404#127C23×C3111yescharacteristic
405#126—22no—
406#125—21no—
407#124—21no—
408#124—22no—
409#123—21no—
410#123—22no—
411#123—21no—
412#123—22no—
413#123—22no—
414#123—24no—
415#123—42no—
416#123D4×C3112yescomplemented
417#123C23×C3112yes—
418C22⋊Q8—21no—
419#121—21no—
420#121—22no—
421#121—22no—
422#121—22no—
423#121—22no—
424#121—22no—
425#121—22no—
426#121—22no—
427#121—22no—
428#121—22no—
429#121—22no—
430#121—44no—
431#120—21no—
432#120—22no—
433#120—22no—
434#120—22no—
435#120—22no—
436#120—21no—
437#120—24no—
438#120—42no—
439#120D4×C3112yescomplemented
440#120C23×C3112yes—
441C4×Q8—21no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—21no—
446#117—22no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4D4×C3112yescomplemented
450C2×C42D4×C3111yescharacteristic, complemented
451#104—22no—
452#104—22no—
453#100—21no—
454#100C23×C3112yes—
455#99—22no—
456#99—24no—
457#99—24no—
458#99C23×C3114yes—
459C31#3355711yescharacteristic, direct product factor, Sylow
460C24—22no—
461C24—21no—
462C24—21no—
463C24C2×D4×C3112yes—
464C24C2×D4×C3112yes—
465#41—22no—
466#41—22no—
467#41—42no—
468#41—41no—
469#41—44no—
470#41—42no—
471#41—41no—
472#41—84no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—41no—
492C2×D4—41no—
493C2×D4—41no—
494C2×D4—41no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—44no—
500C2×D4—41no—
501C2×D4—42no—
502C2×D4—42no—
503C2×D4—42no—
504C2×D4—44no—
505C2×D4—84no—
506C2×D4—84no—
507C2×D4—81no—
508C2×D4—81no—
509C2×D4C2×D4×C3112yes—
510C2×D4C2×D4×C3111yescharacteristic
511C2×D4C2×D4×C3111yescharacteristic
512C2×D4C2×D4×C3111yescharacteristic
513C2×D4C24×C3111yescharacteristic
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—41no—
521C22×C4—42no—
522C22×C4C2×D4×C3111yescharacteristic
523C22×C4C2×D4×C3112yes—
524SD16—42no—
525SD16—42no—
526D8—42no—
527D8—42no—
528M16—22no—
529M16—41no—
530C2×C8—41no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—24no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4C2×D4×C3112yes—
554C22⋊C4C2×D4×C3112yes—
555C42—22no—
556C42—21no—
557C42C2×D4×C3111yescharacteristic
558C42C2×D4×C3111yescharacteristic
559C23—22no—
560C23—22no—
561C23—24no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—81no—
575C23—162no—
576C23C31×#12012yes—
577C23C31×#12011yescharacteristic
578C23C22×D4×C3112yes—
579Q8—22no—
580Q8—41no—
581Q8—42no—
582D4—22no—
583D4—41no—
584D4—42no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—44no—
590D4—42no—
591D4—44no—
592D4—44no—
593D4—44no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—81no—
599D4—82no—
600D4—82no—
601D4—82no—
602D4—82no—
603D4—82no—
604D4—82no—
605D4—82no—
606D4—82no—
607D4—164no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—24no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—162no—
628C2×C4C31×#12011yescharacteristic
629C2×C4C22×D4×C3111yescharacteristic
630C8—42no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—161no—
650C22—161no—
651C22—162no—
652C22—162no—
653C22C31×B(4,𝔽2)11yescharacteristic
654C22C31×B(4,𝔽2)11yescharacteristic
655C22C2×C31×#12011yescharacteristic
656C4—22no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—84no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—42no—
669C2—42no—
670C2—82no—
671C2—81no—
672C2—81no—
673C2—82no—
674C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C31
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3543C2semidirect
2. C31×#3026C2semidirect
3. C2×C31×#453C2semidirect
4. C31×#2200C2semidirect
5. C31×#2198C2semidirect
6. C31×#2196C2semidirect
7. C2×C31×#353C2semidirect
8. C31×#2121C2semidirect
9. C2×C31×H(5,𝔽2)C22semidirect
12. C31×#560C22semidirect
13. C31×#559C22semidirect
17. D42×C31C22semidirect
26. C31×#458C22semidirect
28. C31×B(4,𝔽2)C22semidirect
32. C31×#453C22semidirect
34. C2×C31×#100C22semidirect
36. C2×C31×#99C22semidirect
37. C31×#353C22semidirect
38. C31×#353C22semidirect
39. C31×#352C22semidirect
40. C31×AΣL(1,𝔽16)C22semidirect
53. C22×D4×C31D4semidirect
54. C22×D4×C31D4semidirect
64. C31×#127D4semidirect
65. C31×#127D4semidirect
79. C31×#123D4semidirect
102. C31×#120D4semidirect
112. C2×C22⋊C4×C31D4semidirect
113. C2×C42×C31D4semidirect
221. #33557C31direct (N × H)
294. #3543C2×C31semidirect
295. #3026C2×C31semidirect
296. C2×#453C2×C31semidirect
297. #2200C2×C31semidirect
298. #2198C2×C31semidirect
299. #2196C2×C31semidirect
300. C2×#353C2×C31semidirect
301. #2121C2×C31semidirect
336. C2×H(5,𝔽2)C22×C31semidirect
339. #560C22×C31semidirect
340. #559C22×C31semidirect
344. D42C22×C31semidirect
353. #458C22×C31semidirect
355. B(4,𝔽2)C22×C31semidirect
359. #453C22×C31semidirect
361. C2×#100C22×C31semidirect
363. C2×#99C22×C31semidirect
364. #353C22×C31semidirect
365. #353C22×C31semidirect
366. #352C22×C31semidirect
367. AΣL(1,𝔽16)C22×C31semidirect
390. C22×D4D4×C31semidirect
391. C22×D4D4×C31semidirect
401. #127D4×C31semidirect
402. #127D4×C31semidirect
416. #123D4×C31semidirect
439. #120D4×C31semidirect
449. C2×C22⋊C4D4×C31semidirect
450. C2×C42D4×C31semidirect
459. C31#33557direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35432
C31×#30262
C2×C31×#4532
C31×#22002
C31×#21982
C31×#21962
C2×C31×#3532
C31×#21212
#3355731