Finite Group Databasebeta

← C31×#15217 #21471976

Subgroups — C31×#15217 #21471976

Order 7936.

2054 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#124C211yesmaximal, characteristic
2C2×C31×#391C211yesmaximal, characteristic, complemented
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C2×C31×#390C211yesmaximal, characteristic
5C2×C31×#390C211yesmaximal, characteristic
6C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
7C31×#1440C214yesmaximal, complemented
8C31×#1439C214yesmaximal, complemented
9C31×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Q8×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C2×C31×#124—24no—
13C2×C31×#124—24no—
14C2×C31×#124—24no—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2214yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2214yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C42×C31C2211yescharacteristic
23C22×C42×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#391—24no—
26C31×#391—24no—
27C31×#390—24no—
28C31×#390—24no—
29C4×C4⋊C4×C31—24no—
30C2×C31×#95C2212yes—
31C2×C31×#95C2214yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2214yes—
35C25×C31C2312yes—
36C22×Q8×C31—22no—
37C22×Q8×C31—22no—
38C22×Q8×C31—22no—
39C22×Q8×C31—22no—
40C22×Q8×C31—22no—
41C22×Q8×C31—22no—
42C22×Q8×C31D414yes—
43C22×D4×C31—22no—
44C22×D4×C31—22no—
45C22×D4×C31—22no—
46C22×D4×C31—22no—
47C22×D4×C31—22no—
48C22×D4×C31—22no—
49C22×D4×C31D414yes—
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2312yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2311yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C31×#124—232no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—22no—
68C2×C22⋊C4×C31—28no—
69C2×C22⋊C4×C31—28no—
70C2×C22⋊C4×C31—28no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—28no—
75C2×C22⋊C4×C31—28no—
76C2×C22⋊C4×C31—44no—
77C2×C22⋊C4×C31—44no—
78C2×C42×C31—22no—
79C2×C42×C31—22no—
80C2×C42×C31—22no—
81C2×C42×C31—22no—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—22no—
85C2×C42×C31D414yes—
86C2×C42×C31D414yes—
87C31×#95—24no—
88C31×#95—28no—
89C31×#95—24no—
90C31×#95—24no—
91C31×#95—44no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—24no—
101C24×C31—24no—
102C24×C31C2×D414yes—
103C24×C31C2411yescharacteristic
104C2×Q8×C31—28no—
105C2×Q8×C31—28no—
106C2×Q8×C31—28no—
107C2×Q8×C31—44no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C2×Q8×C31—44no—
111C2×D4×C31—28no—
112C2×D4×C31—28no—
113C2×D4×C31—28no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—44no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D414yes—
155C22×C4×C31C2×D414yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31#4114yes—
162C22×C4×C31#4114yes—
163C22×C4×C31#4112yes—
164C22×C4×C31#4112yes—
165C22×C4×C31#4112yes—
166C4⋊C4×C31—216no—
167C4⋊C4×C31—44no—
168C4⋊C4×C31—44no—
169C22⋊C4×C31—232no—
170C22⋊C4×C31—416no—
171C22⋊C4×C31—48no—
172C22⋊C4×C31—48no—
173C42×C31—28no—
174C42×C31—28no—
175C42×C31—44no—
176#15217C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—24no—
178C23×C31—24no—
179C23×C31—24no—
180C23×C31—28no—
181C23×C31—28no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—44no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—44no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—44no—
193C23×C31—44no—
194C23×C31—44no—
195C23×C31—44no—
196C23×C31—44no—
197C23×C31—44no—
198C23×C31—44no—
199C23×C31—44no—
200C23×C31C22×D411yescharacteristic
201C23×C31C22×D411yescharacteristic
202C23×C31C22×D411yescharacteristic
203C23×C31C22×D411yescharacteristic
204C23×C31C2×#4111yescharacteristic
205C23×C31C2×#4111yescharacteristic
206C23×C31C2×#4111yescharacteristic
207C23×C31C2×#4112yes—
208C23×C31C2×#4112yes—
209C23×C31H(5,𝔽2)12yes—
210C23×C312−1+412yes—
211Q8×C31—416no—
212D4×C31—416no—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
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—44no—
250C2×C4×C31—44no—
251C2×C4×C31—44no—
252C2×C4×C31—44no—
253C2×C4×C31#12014yes—
254C2×C4×C31#12114yes—
255C2×C4×C31#12114yes—
256C2×C4×C31#12114yes—
257C2×C4×C31#12314yes—
258C2×C4×C31#12314yes—
259C2×C4×C31#12414yes—
260C2×C4×C31#12414yescomplemented
261C22×#124C2×C3111yescharacteristic
262C2×#391C2×C3111yescharacteristic, complemented
263C2×#391C2×C3111yescharacteristic, complemented
264C2×#390C2×C3111yescharacteristic
265C2×#390C2×C3111yescharacteristic
266C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
267#1440C2×C3114yescomplemented
268#1439C2×C3114yescomplemented
269C4⋊#95C2×C3111yescharacteristic, complemented
270C22×C31—44no—
271C22×C31—48no—
272C22×C31—48no—
273C22×C31—44no—
274C22×C31—44no—
275C22×C31—84no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—84no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31C2×#12011yescharacteristic
282C22×C31C2×#12111yescharacteristic
283C22×C31C2×#12111yescharacteristic
284C22×C31C2×#12111yescharacteristic
285C22×C31C2×#12311yescharacteristic
286C22×C31C2×#12311yescharacteristic
287C22×C31#52512yes—
288C22×C31C2×#12411yescharacteristic
289C22×C31C2×#12411yescharacteristic
290C22×C31#52912yes—
291C22×C31#53211yescharacteristic
292C22×C31#53511yescharacteristic
293C22×C31#53611yescharacteristic
294C22×C31#53712yes—
295C22×C31#54212yes—
296C22×C31#54612yes—
297C22×C31#54712yes—
298C22×C31#54812yes—
299C22×C31#55012yes—
300C22×C31#55612yes—
301C22×C31#55612yes—
302C22×C31#55912yes—
303C22×C31#55912yes—
304C4×C31—24no—
305C4×C31—42no—
306C4×C31—44no—
307C4×C31—44no—
308C4×C31—44no—
309C4×C31—82no—
310C4×C31—82no—
311C4×C31#39418yes—
312C23×Q8C22×C3111yescharacteristic
313C23×D4C22×C3111yescharacteristic
314C2×#124—24no—
315C2×#124—24no—
316C2×#124—24no—
317C22×C4⋊C4C22×C3111yescharacteristic
318C22×C4⋊C4C22×C3111yescharacteristic
319C22×C4⋊C4C22×C3112yes—
320C22×C22⋊C4C22×C3114yes—
321C22×C22⋊C4C22×C3112yes—
322C22×C22⋊C4C22×C3114yes—
323C22×C22⋊C4C22×C3112yes—
324C22×C42C22×C3111yescharacteristic
325C22×C42C22×C3111yescharacteristic
326C22×C42C22×C3111yescharacteristic
327#391—24no—
328#391—24no—
329#390—24no—
330#390—24no—
331C4×C4⋊C4—24no—
332C2×#95C22×C3112yes—
333C2×#95C22×C3114yes—
334C2×#95C22×C3112yes—
335C2×#95C22×C3112yes—
336C2×#95C22×C3114yes—
337C2×C31—84no—
338C2×C31C2×#39411yescharacteristic
339C2×C31#240712yes—
340C2×C31#244712yes—
341C2×C31#244812yes—
342C2×C31#254411yescharacteristic
343C2×C31#254511yescharacteristic
344C2×C31#254711yescharacteristic
345C2×C31#254811yescharacteristic
346C2×C31#259212yes—
347C2×C31#262011yescharacteristic
348C2×C31#285011yescharacteristic
349C25C23×C3112yes—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8D4×C3114yes—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4D4×C3114yes—
364C23×C4C23×C3112yes—
365C23×C4C23×C3112yes—
366C23×C4C23×C3112yes—
367C23×C4C23×C3112yes—
368C23×C4C23×C3111yescharacteristic
369C23×C4C23×C3112yes—
370C23×C4C23×C3111yescharacteristic
371C23×C4C23×C3111yescharacteristic
372#124—232no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—28no—
384C2×C22⋊C4—28no—
385C2×C22⋊C4—28no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—28no—
389C2×C22⋊C4—28no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—44no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42D4×C3114yes—
400C2×C42D4×C3114yes—
401#95—24no—
402#95—28no—
403#95—24no—
404#95—24no—
405#95—44no—
406C31#1521711yescharacteristic, direct product factor, Sylow
407C24—22no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24C2×D4×C3114yes—
418C24C24×C3111yescharacteristic
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4C2×D4×C3112yes—
469C22×C4C2×D4×C3114yes—
470C22×C4C2×D4×C3114yes—
471C22×C4C2×D4×C3112yes—
472C22×C4C2×D4×C3112yes—
473C22×C4C2×D4×C3112yes—
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3112yes—
476C22×C4#41×C3114yes—
477C22×C4#41×C3114yes—
478C22×C4#41×C3112yes—
479C22×C4#41×C3112yes—
480C22×C4#41×C3112yes—
481C4⋊C4—216no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C22⋊C4—232no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C42—28no—
489C42—28no—
490C42—44no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C22×D4×C3111yescharacteristic
515C23C22×D4×C3111yescharacteristic
516C23C22×D4×C3111yescharacteristic
517C23C22×D4×C3111yescharacteristic
518C23C2×#41×C3111yescharacteristic
519C23C2×#41×C3111yescharacteristic
520C23C2×#41×C3111yescharacteristic
521C23C2×#41×C3112yes—
522C23C2×#41×C3112yes—
523C23C31×H(5,𝔽2)12yes—
524C23C31×2−1+412yes—
525Q8—416no—
526D4—416no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
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—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4C31×#12014yes—
568C2×C4C31×#12114yes—
569C2×C4C31×#12114yes—
570C2×C4C31×#12114yes—
571C2×C4C31×#12314yes—
572C2×C4C31×#12314yes—
573C2×C4C31×#12414yes—
574C2×C4C31×#12414yescomplemented
575C22—44no—
576C22—48no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—84no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22C2×C31×#12011yescharacteristic
587C22C2×C31×#12111yescharacteristic
588C22C2×C31×#12111yescharacteristic
589C22C2×C31×#12111yescharacteristic
590C22C2×C31×#12311yescharacteristic
591C22C2×C31×#12311yescharacteristic
592C22C31×#52512yes—
593C22C2×C31×#12411yescharacteristic
594C22C2×C31×#12411yescharacteristic
595C22C31×#52912yes—
596C22C31×#53211yescharacteristic
597C22C31×#53511yescharacteristic
598C22C31×#53611yescharacteristic
599C22C31×#53712yes—
600C22C31×#54212yes—
601C22C31×#54612yes—
602C22C31×#54712yes—
603C22C31×#54812yes—
604C22C31×#55012yes—
605C22C31×#55612yes—
606C22C31×#55612yes—
607C22C31×#55912yes—
608C22C31×#55912yes—
609C4—24no—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4C31×#39418yes—
617C2—84no—
618C2C2×C31×#39411yescharacteristic
619C2C31×#240712yes—
620C2C31×#244712yes—
621C2C31×#244812yes—
622C2C31×#254411yescharacteristic
623C2C31×#254511yescharacteristic
624C2C31×#254711yescharacteristic
625C2C31×#254811yescharacteristic
626C2C31×#259212yes—
627C2C31×#262011yescharacteristic
628C2C31×#285011yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#391C2semidirect
3. C2×C31×#391C2semidirect
6. C2×C4×C4⋊C4×C31C2semidirect
7. C31×#1440C2semidirect
8. C31×#1439C2semidirect
9. C31×C4⋊#95C2semidirect
176. #15217C31direct (N × H)
260. C2×C4×C31#124semidirect
262. C2×#391C2×C31semidirect
263. C2×#391C2×C31semidirect
266. C2×C4×C4⋊C4C2×C31semidirect
267. #1440C2×C31semidirect
268. #1439C2×C31semidirect
269. C4⋊#95C2×C31semidirect
406. C31#15217direct (N × H)
574. C2×C4C31×#124semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1242
C2×C31×#3912
C2×C31×#3902
C2×C4×C4⋊C4×C312
C31×#14402
C31×#14392
C31×C4⋊#952
#1521731