Finite Group Databasebeta

← C31×#15784 #21472543

Subgroups — C31×#15784 #21472543

Order 7936.

1998 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#124C211yesmaximal, characteristic
2C2×C31×#397C212yesmaximal, complemented
3C2×C31×#386C211yesmaximal, characteristic
4C31×#1446C212yesmaximal, complemented
5C31×#1440C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1438C211yesmaximal, characteristic
8C31×#1436C212yesmaximal, complemented
9C31×C22⋊#95C212yesmaximal, complemented
10C23×Q8×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C24×C4×C31C2211yescharacteristic
13C2×C31×#124—24no—
14C2×C31×#124—28no—
15C22×C4⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2211yescharacteristic
23C22×C42×C31C2211yescharacteristic
24C31×#397—28no—
25C31×#386—24no—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C25×C31C2311yescharacteristic
36C25×C31C2311yescharacteristic
37C22×Q8×C31—24no—
38C22×Q8×C31—22no—
39C22×Q8×C31—24no—
40C22×Q8×C31—41no—
41C22×Q8×C31—41no—
42C22×D4×C31—24no—
43C22×D4×C31—24no—
44C22×D4×C31—41no—
45C22×D4×C31—41no—
46C22×D4×C31D414yes—
47C23×C4×C31—22no—
48C23×C4×C31—24no—
49C23×C4×C31—41no—
50C23×C4×C31—41no—
51C23×C4×C31—42no—
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2312yes—
57C23×C4×C31C2312yes—
58C23×C4×C31C2311yescharacteristic
59C31×#124—232no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—42no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—41no—
76C2×C22⋊C4×C31—42no—
77C2×C22⋊C4×C31—42no—
78C2×C22⋊C4×C31—41no—
79C2×C22⋊C4×C31—42no—
80C2×C22⋊C4×C31D418yes—
81C2×C42×C31—24no—
82C2×C42×C31—22no—
83C31×#95—24no—
84C31×#95—24no—
85C31×#95—24no—
86C31×#95—24no—
87C31×#95—24no—
88C31×#95—24no—
89C31×#95—42no—
90C31×#95—42no—
91C31×#95—42no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—21no—
97C24×C31—21no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—21no—
105C24×C31—21no—
106C24×C31—22no—
107C24×C31C2×D412yes—
108C24×C31C2×D414yes—
109C24×C31C2×D412yes—
110C24×C31C2411yescharacteristic
111C2×Q8×C31—28no—
112C2×Q8×C31—48no—
113C2×Q8×C31—48no—
114C2×Q8×C31—44no—
115C2×Q8×C31—44no—
116C2×D4×C31—28no—
117C2×D4×C31—28no—
118C2×D4×C31—44no—
119C2×D4×C31—48no—
120C2×D4×C31—48no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
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—22no—
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—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—44no—
163C22×C4×C31—48no—
164C22×C4×C31—48no—
165C22×C4×C31—44no—
166C22×C4×C31—44no—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D414yes—
169C22×C4×C31C2×D412yes—
170C22×C4×C31#4114yes—
171C22×C4×C31#4114yes—
172C22×C4×C31#4114yes—
173C4⋊C4×C31—48no—
174C22⋊C4×C31—216no—
175C22⋊C4×C31—216no—
176C22⋊C4×C31—44no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—48no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C42×C31—28no—
182#15784C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—24no—
185C23×C31—22no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—24no—
192C23×C31—24no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—24no—
197C23×C31—24no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31#12014yes—
225C23×C31C22×D411yescharacteristic
226C23×C31C22×D412yes—
227C23×C31C2×#4112yes—
228C23×C31C2×#4112yes—
229C23×C31C2×#4112yes—
230C23×C31H(5,𝔽2)11yescharacteristic
231C23×C31H(5,𝔽2)11yescharacteristic
232C23×C31H(5,𝔽2)12yes—
233C23×C312−1+412yes—
234Q8×C31—416no—
235D4×C31—416no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—24no—
245C2×C4×C31—24no—
246C2×C4×C31—24no—
247C2×C4×C31—24no—
248C2×C4×C31—24no—
249C2×C4×C31—24no—
250C2×C4×C31—24no—
251C2×C4×C31—24no—
252C2×C4×C31—416no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—42no—
283C2×C4×C31#12314yes—
284C22×#124C2×C3111yescharacteristic
285C2×#397C2×C3112yescomplemented
286C2×#386C2×C3111yescharacteristic
287#1446C2×C3112yescomplemented
288#1440C2×C3112yescomplemented
289#1439C2×C3112yescomplemented
290#1438C2×C3111yescharacteristic
291#1436C2×C3112yescomplemented
292C22⋊#95C2×C3112yescomplemented
293C22×C31—24no—
294C22×C31—28no—
295C22×C31—42no—
296C22×C31—44no—
297C22×C31—44no—
298C22×C31—44no—
299C22×C31—42no—
300C22×C31—44no—
301C22×C31—44no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—44no—
305C22×C31—44no—
306C22×C31—42no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31—82no—
311C22×C31C2×#12011yescharacteristic
312C22×C31C2×#12311yescharacteristic
313C22×C31#52912yes—
314C22×C31#53212yes—
315C22×C31#53411yescharacteristic
316C22×C31#53712yes—
317C22×C31#53812yes—
318C22×C31#54212yes—
319C22×C31#54212yes—
320C22×C31#54612yes—
321C22×C31#54612yes—
322C22×C31#54612yes—
323C22×C31#54612yes—
324C22×C31#54712yes—
325C22×C31#54712yes—
326C22×C31#55111yescharacteristic
327C22×C31#55111yescharacteristic
328C22×C31#55612yes—
329C22×C31#55912yes—
330C22×C31H(3,𝔽4)11yescharacteristic
331C22×C31#56311yescharacteristic
332C4×C31—24no—
333C4×C31—44no—
334C4×C31—44no—
335C4×C31—44no—
336C4×C31—44no—
337C4×C31—82no—
338C4×C31—82no—
339C23×Q8C22×C3111yescharacteristic
340C23×D4C22×C3111yescharacteristic
341C24×C4C22×C3111yescharacteristic
342C2×#124—24no—
343C2×#124—28no—
344C22×C4⋊C4C22×C3112yes—
345C22×C22⋊C4C22×C3111yescharacteristic
346C22×C22⋊C4C22×C3112yes—
347C22×C22⋊C4C22×C3112yes—
348C22×C22⋊C4C22×C3111yescharacteristic
349C22×C22⋊C4C22×C3112yes—
350C22×C22⋊C4C22×C3112yes—
351C22×C22⋊C4C22×C3111yescharacteristic
352C22×C42C22×C3111yescharacteristic
353#397—28no—
354#386—24no—
355C2×#95C22×C3112yes—
356C2×#95C22×C3112yes—
357C2×#95C22×C3112yes—
358C2×#95C22×C3112yes—
359C2×#95C22×C3112yes—
360C2×#95C22×C3112yes—
361C2×#95C22×C3112yes—
362C2×#95C22×C3112yes—
363C2×#95C22×C3112yes—
364C2×C31—44no—
365C2×C31—82no—
366C2×C31#240511yescharacteristic
367C2×C31#243012yes—
368C2×C31#244712yes—
369C2×C31#252112yes—
370C2×C31#255712yes—
371C2×C31#268411yescharacteristic
372C2×C31#271212yes—
373C2×C31#271512yes—
374C2×C31#271711yescharacteristic
375C25C23×C3111yescharacteristic
376C25C23×C3111yescharacteristic
377C22×Q8—24no—
378C22×Q8—22no—
379C22×Q8—24no—
380C22×Q8—41no—
381C22×Q8—41no—
382C22×D4—24no—
383C22×D4—24no—
384C22×D4—41no—
385C22×D4—41no—
386C22×D4D4×C3114yes—
387C23×C4—22no—
388C23×C4—24no—
389C23×C4—41no—
390C23×C4—41no—
391C23×C4—42no—
392C23×C4C23×C3112yes—
393C23×C4C23×C3112yes—
394C23×C4C23×C3112yes—
395C23×C4C23×C3112yes—
396C23×C4C23×C3112yes—
397C23×C4C23×C3112yes—
398C23×C4C23×C3111yescharacteristic
399#124—232no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—42no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—41no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—41no—
419C2×C22⋊C4—42no—
420C2×C22⋊C4D4×C3118yes—
421C2×C42—24no—
422C2×C42—22no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—42no—
430#95—42no—
431#95—42no—
432C31#1578411yescharacteristic, direct product factor, Sylow
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—21no—
438C24—21no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—21no—
446C24—21no—
447C24—22no—
448C24C2×D4×C3112yes—
449C24C2×D4×C3114yes—
450C24C2×D4×C3112yes—
451C24C24×C3111yescharacteristic
452C2×Q8—28no—
453C2×Q8—48no—
454C2×Q8—48no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×D4—28no—
458C2×D4—28no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—48no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—44no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4C2×D4×C3114yes—
509C22×C4C2×D4×C3114yes—
510C22×C4C2×D4×C3112yes—
511C22×C4#41×C3114yes—
512C22×C4#41×C3114yes—
513C22×C4#41×C3114yes—
514C4⋊C4—48no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C42—28no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23C31×#12014yes—
565C23C22×D4×C3111yescharacteristic
566C23C22×D4×C3112yes—
567C23C2×#41×C3112yes—
568C23C2×#41×C3112yes—
569C23C2×#41×C3112yes—
570C23C31×H(5,𝔽2)11yescharacteristic
571C23C31×H(5,𝔽2)11yescharacteristic
572C23C31×H(5,𝔽2)12yes—
573C23C31×2−1+412yes—
574Q8—416no—
575D4—416no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4C31×#12314yes—
624C22—24no—
625C22—28no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C2×C31×#12011yescharacteristic
643C22C2×C31×#12311yescharacteristic
644C22C31×#52912yes—
645C22C31×#53212yes—
646C22C31×#53411yescharacteristic
647C22C31×#53712yes—
648C22C31×#53812yes—
649C22C31×#54212yes—
650C22C31×#54212yes—
651C22C31×#54612yes—
652C22C31×#54612yes—
653C22C31×#54612yes—
654C22C31×#54612yes—
655C22C31×#54712yes—
656C22C31×#54712yes—
657C22C31×#55111yescharacteristic
658C22C31×#55111yescharacteristic
659C22C31×#55612yes—
660C22C31×#55912yes—
661C22C31×H(3,𝔽4)11yescharacteristic
662C22C31×#56311yescharacteristic
663C4—24no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—82no—
669C4—82no—
670C2—44no—
671C2—82no—
672C2C31×#240511yescharacteristic
673C2C31×#243012yes—
674C2C31×#244712yes—
675C2C31×#252112yes—
676C2C31×#255712yes—
677C2C31×#268411yescharacteristic
678C2C31×#271212yes—
679C2C31×#271512yes—
680C2C31×#271711yescharacteristic
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×#397C2semidirect
4. C31×#1446C2semidirect
5. C31×#1440C2semidirect
6. C31×#1439C2semidirect
8. C31×#1436C2semidirect
9. C31×C22⋊#95C2semidirect
182. #15784C31direct (N × H)
285. C2×#397C2×C31semidirect
287. #1446C2×C31semidirect
288. #1440C2×C31semidirect
289. #1439C2×C31semidirect
291. #1436C2×C31semidirect
292. C22⋊#95C2×C31semidirect
432. C31#15784direct (N × H)

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×#3972
C2×C31×#3862
C31×#14462
C31×#14402
C31×#14392
C31×#14382
C31×#14362
C31×C22⋊#952
#1578431