Finite Group Databasebeta

← C31×#18288 #21475047

Subgroups — C31×#18288 #21475047

Order 7936.

1034 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic, complemented
2C31×#3187C211yesmaximal, characteristic, complemented
3C31×#3186C211yesmaximal, characteristic
4C31×#2618C211yesmaximal, characteristic, complemented
5C31×#1609C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1520C211yesmaximal, characteristic, complemented
8C31×#1519C211yesmaximal, characteristic, complemented
9C2×C31×#324C211yesmaximal, characteristic, complemented
10C31×#539—22no—
11C31×#539—22no—
12C31×#539—22no—
13C31×#539C2214yes—
14C2×C31×#126—21no—
15C2×C31×#125—21no—
16C2×C31×#123—21no—
17C2×C31×#123—21no—
18C2×C31×#123—21no—
19C2×C31×C22⋊Q8—21no—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C31×#121C2211yescharacteristic
23C2×C4×D4×C31—21no—
24C22×C4⋊C4×C31C2211yescharacteristic
25C31×#483C2212yescomplemented
26C31×#481—21no—
27C31×#480—21no—
28C31×#480C2212yes—
29C31×#466—22no—
30C31×#465—22no—
31C31×#450C2212yescomplemented
32C31×#448—21no—
33C31×#447—21no—
34C31×#447C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C31×#398C2212yescomplemented
38C31×#396—22no—
39C31×#394C2212yescomplemented
40C31×#386—21no—
41C31×#380C2211yescharacteristic
42C31×#328C2214yescomplemented
43C31×#327C2214yes—
44C31×#324C2214yescomplemented
45C22×Q8×C31D411yescharacteristic
46C22×D4×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31D411yescharacteristic
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31—21no—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×SD16×C31—22no—
55C2×D8×C31—22no—
56C22×C8×C31—21no—
57C22×C8×C31—21no—
58C31×#126—24no—
59C31×#126—24no—
60C31×#125—24no—
61C31×#125—24no—
62C31×#123—24no—
63C31×#123—24no—
64C31×#123—24no—
65C31×#123—24no—
66C31×#123—44no—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8D412yes—
72C31×C22⋊Q8C2312yes—
73C31×#121—21no—
74C31×#121—21no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121—21no—
78C31×#121—21no—
79C31×#121D412yes—
80C31×#121C2312yes—
81C4×D4×C31—24no—
82C4×D4×C31—24no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31D412yes—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31D411yescharacteristic
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C4⋊C4×C31C2312yes—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—22no—
107C2×C42×C31—21no—
108C31×#107—22no—
109C31×#106—22no—
110C31×#103—22no—
111C31×#102—22no—
112C31×#102—22no—
113C31×#102—22no—
114C31×C22⋊C8—21no—
115C31×C22⋊C8—21no—
116C31×C22⋊C8C2312yes—
117C31×C22⋊C8C2312yes—
118C31×#95—22no—
119C31×#95C2312yes—
120C24×C31—21no—
121C24×C31—21no—
122C24×C31C2×D411yescharacteristic
123C2×Q8×C31—21no—
124C2×Q8×C31—81no—
125C2×Q8×C31C2×D412yes—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—81no—
138C2×D4×C31—81no—
139C2×D4×C31C2×D412yes—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—81no—
153C22×C4×C31—81no—
154C22×C4×C31—82no—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2411yescharacteristic
163SD16×C31—82no—
164D8×C31—82no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—81no—
168C2×C8×C31—81no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—44no—
183C4⋊C4×C31—81no—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—44no—
194C22⋊C4×C31—42no—
195C22⋊C4×C31—41no—
196C22⋊C4×C31—41no—
197C22⋊C4×C31—82no—
198C42×C31—24no—
199#18288C3111yesmaximal, characteristic, direct product factor, Sylow
200C23×C31—22no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31#12011yescharacteristic
212C23×C31#12011yescharacteristic
213C23×C31#13512yes—
214C23×C31AGL(1,ℤ/8)12yes—
215C23×C31C22×D411yescharacteristic
216Q8×C31—42no—
217Q8×C31—81no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—42no—
223D4×C31—44no—
224D4×C31—81no—
225D4×C31—82no—
226D4×C31—82no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—81no—
254C2×C4×C31—84no—
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C22×D411yescharacteristic
259C8×C31—42no—
260C8×C31—42no—
261C2×#539C2×C3111yescharacteristic, complemented
262#3187C2×C3111yescharacteristic, complemented
263#3186C2×C3111yescharacteristic
264#2618C2×C3111yescharacteristic, complemented
265#1609C2×C3114yescomplemented
266#1597C2×C3114yescomplemented
267#1520C2×C3111yescharacteristic, complemented
268#1519C2×C3111yescharacteristic, complemented
269C2×#324C2×C3111yescharacteristic, complemented
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—21no—
273C22×C31—21no—
274C22×C31—21no—
275C22×C31—21no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—84no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31#44912yes—
288C22×C31#44912yes—
289C22×C31C2×#12011yescharacteristic
290C22×C31C2×#13511yescharacteristic
291C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—42no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—42no—
300C4×C31—42no—
301C4×C31—82no—
302C4×C31—81no—
303#539—22no—
304#539—22no—
305#539—22no—
306#539C22×C3114yes—
307C2×#126—21no—
308C2×#125—21no—
309C2×#123—21no—
310C2×#123—21no—
311C2×#123—21no—
312C2×C22⋊Q8—21no—
313C2×C22⋊Q8C22×C3111yescharacteristic
314C2×#121C22×C3111yescharacteristic
315C2×#121C22×C3111yescharacteristic
316C2×C4×D4—21no—
317C22×C4⋊C4C22×C3111yescharacteristic
318#483C22×C3112yescomplemented
319#481—21no—
320#480—21no—
321#480C22×C3112yes—
322#466—22no—
323#465—22no—
324#450C22×C3112yescomplemented
325#448—21no—
326#447—21no—
327#447C22×C3112yes—
328C2×C22⋊C8C22×C3111yescharacteristic
329C2×C22⋊C8C22×C3111yescharacteristic
330#398C22×C3112yescomplemented
331#396—22no—
332#394C22×C3112yescomplemented
333#386—21no—
334#380C22×C3111yescharacteristic
335#328C22×C3114yescomplemented
336#327C22×C3114yes—
337#324C22×C3114yescomplemented
338C2×C31—21no—
339C2×C31—21no—
340C2×C31—22no—
341C2×C31—22no—
342C2×C31—82no—
343C2×C31—161no—
344C2×C31C2×#44911yescharacteristic
345C2×C31#302011yescharacteristic
346C2×C31#302611yescharacteristic
347C22×Q8D4×C3111yescharacteristic
348C22×D4—21no—
349C22×D4—21no—
350C22×D4D4×C3111yescharacteristic
351C22×D4C23×C3111yescharacteristic
352C23×C4—21no—
353C23×C4D4×C3111yescharacteristic
354C23×C4D4×C3111yescharacteristic
355C23×C4C23×C3111yescharacteristic
356C2×SD16—22no—
357C2×D8—22no—
358C22×C8—21no—
359C22×C8—21no—
360#126—24no—
361#126—24no—
362#125—24no—
363#125—24no—
364#123—24no—
365#123—24no—
366#123—24no—
367#123—24no—
368#123—44no—
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8D4×C3112yes—
374C22⋊Q8C23×C3112yes—
375#121—21no—
376#121—21no—
377#121—21no—
378#121—21no—
379#121—21no—
380#121—21no—
381#121D4×C3112yes—
382#121C23×C3112yes—
383C4×D4—24no—
384C4×D4—24no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4D4×C3112yes—
396C2×C4⋊C4D4×C3111yescharacteristic
397C2×C4⋊C4D4×C3111yescharacteristic
398C2×C4⋊C4C23×C3111yescharacteristic
399C2×C4⋊C4C23×C3112yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C42—21no—
410#107—22no—
411#106—22no—
412#103—22no—
413#102—22no—
414#102—22no—
415#102—22no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8C23×C3112yes—
419C22⋊C8C23×C3112yes—
420#95—22no—
421#95C23×C3112yes—
422C31#1828811yescharacteristic, direct product factor, Sylow
423C24—21no—
424C24—21no—
425C24C2×D4×C3111yescharacteristic
426C2×Q8—21no—
427C2×Q8—81no—
428C2×Q8C2×D4×C3112yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4C2×D4×C3112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4C2×D4×C3112yes—
459C22×C4C2×D4×C3112yes—
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C24×C3111yescharacteristic
466SD16—82no—
467D8—82no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—81no—
471C2×C8—81no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4C2×D4×C3112yes—
488C4⋊C4C2×D4×C3112yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—82no—
501C42—24no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C31×#12011yescharacteristic
514C23C31×#12011yescharacteristic
515C23C31×#13512yes—
516C23C31×AGL(1,ℤ/8)12yes—
517C23C22×D4×C3111yescharacteristic
518Q8—42no—
519Q8—81no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—42no—
525D4—42no—
526D4—81no—
527D4—82no—
528D4—82no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—81no—
556C2×C4—84no—
557C2×C4C31×#12011yescharacteristic
558C2×C4C31×#12011yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C2×C4C22×D4×C3111yescharacteristic
561C8—42no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C31×#44912yes—
581C22C31×#44912yes—
582C22C2×C31×#12011yescharacteristic
583C22C2×C31×#13511yescharacteristic
584C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—81no—
596C2—21no—
597C2—22no—
598C2—21no—
599C2—22no—
600C2—82no—
601C2—161no—
602C2C2×C31×#44911yescharacteristic
603C2C31×#302011yescharacteristic
604C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3186C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#539C2semidirect
2. C31×#3187C2semidirect
4. C31×#2618C2semidirect
5. C31×#1609C2semidirect
6. C31×#1597C2semidirect
7. C31×#1520C2semidirect
8. C31×#1519C2semidirect
9. C2×C31×#324C2semidirect
25. C31×#483C22semidirect
31. C31×#450C22semidirect
37. C31×#398C22semidirect
39. C31×#394C22semidirect
42. C31×#328C22semidirect
44. C31×#324C22semidirect
199. #18288C31direct (N × H)
261. C2×#539C2×C31semidirect
262. #3187C2×C31semidirect
264. #2618C2×C31semidirect
265. #1609C2×C31semidirect
266. #1597C2×C31semidirect
267. #1520C2×C31semidirect
268. #1519C2×C31semidirect
269. C2×#324C2×C31semidirect
318. #483C22×C31semidirect
324. #450C22×C31semidirect
330. #398C22×C31semidirect
332. #394C22×C31semidirect
335. #328C22×C31semidirect
337. #324C22×C31semidirect
422. C31#18288direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C31×#31872
C31×#31862
C31×#26182
C31×#16092
C31×#15972
C31×#15202
C31×#15192
C2×C31×#3242
#1828831