Finite Group Databasebeta

← C31×#16886 #21473645

Subgroups — C31×#16886 #21473645

Order 7936.

1514 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#390C211yesmaximal, characteristic
2C2×C31×#387C211yesmaximal, characteristic, complemented
3C31×#1445C211yesmaximal, characteristic, complemented
4C31×#1445C211yesmaximal, characteristic, complemented
5C31×#1440C212yesmaximal, complemented
6C31×#1440C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1439C212yesmaximal, complemented
9C31×#1438C211yesmaximal, characteristic
10C31×#1438C211yesmaximal, characteristic
11C31×#1434C211yesmaximal, characteristic, complemented
12C23×D4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C42×C31C2211yescharacteristic
23C31×#390—24no—
24C31×#387—24no—
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2312yes—
38C22×D4×C31—24no—
39C22×D4×C31—22no—
40C22×D4×C31—22no—
41C22×D4×C31—22no—
42C22×D4×C31—42no—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2312yes—
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—42no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31—42no—
76C2×C22⋊C4×C31—42no—
77C2×C22⋊C4×C31—42no—
78C2×C22⋊C4×C31—42no—
79C2×C42×C31—42no—
80C2×C42×C31D414yes—
81C31×#95—22no—
82C31×#95—22no—
83C31×#95—22no—
84C31×#95—22no—
85C31×#95—22no—
86C31×#95—22no—
87C31×#95—42no—
88C31×#95—42no—
89C31×#95—42no—
90C31×#95—42no—
91C31×#95—42no—
92C31×#95—42no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31—22no—
107C24×C31#4114yes—
108C24×C31C2411yescharacteristic
109C2×D4×C31—28no—
110C2×D4×C31—48no—
111C2×D4×C31—48no—
112C2×D4×C31—44no—
113C2×D4×C31—44no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
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—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
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—21no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31#4112yes—
167C22×C4×C31#4112yes—
168C22×C4×C31#4112yes—
169C22×C4×C31#4112yes—
170C4⋊C4×C31—28no—
171C4⋊C4×C31—44no—
172C4⋊C4×C31—44no—
173C22⋊C4×C31—48no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—48no—
179C42×C31—44no—
180#16886C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
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×C31C22×D411yescharacteristic
217C23×C31C2×#4111yescharacteristic
218C23×C31C2×#4112yes—
219C23×C31C2×#4111yescharacteristic
220C23×C31C2×#4111yescharacteristic
221C23×C31C2×#4111yescharacteristic
222C23×C31H(5,𝔽2)12yes—
223C23×C31H(5,𝔽2)12yes—
224C23×C31H(5,𝔽2)12yes—
225C23×C31H(5,𝔽2)11yescharacteristic
226C23×C312−1+411yescharacteristic
227D4×C31—416no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
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#12414yes—
284C2×#390C2×C3111yescharacteristic
285C2×#387C2×C3111yescharacteristic, complemented
286#1445C2×C3111yescharacteristic, complemented
287#1445C2×C3111yescharacteristic, complemented
288#1440C2×C3112yescomplemented
289#1440C2×C3112yescomplemented
290#1439C2×C3112yescomplemented
291#1439C2×C3112yescomplemented
292#1438C2×C3111yescharacteristic
293#1438C2×C3111yescharacteristic
294#1434C2×C3111yescharacteristic, complemented
295C22×C31—44no—
296C22×C31—44no—
297C22×C31—44no—
298C22×C31—44no—
299C22×C31—44no—
300C22×C31—44no—
301C22×C31—44no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31#52511yescharacteristic
311C22×C31#52511yescharacteristic
312C22×C31C2×#12411yescharacteristic
313C22×C31#53411yescharacteristic
314C22×C31#53511yescharacteristic
315C22×C31#53511yescharacteristic
316C22×C31#53711yescharacteristic
317C22×C31#53812yes—
318C22×C31#53812yes—
319C22×C31#53912yes—
320C22×C31#54012yes—
321C22×C31#54012yes—
322C22×C31#54012yes—
323C22×C31#55112yes—
324C22×C31#55112yes—
325C22×C31#55112yes—
326C22×C31#55312yes—
327C22×C31#55611yescharacteristic
328C22×C31#55611yescharacteristic
329C22×C31#55911yescharacteristic
330C22×C31#55911yescharacteristic
331C22×C31#56012yes—
332C22×C31#56212yes—
333C4×C31—24no—
334C4×C31—42no—
335C4×C31—42no—
336C4×C31—42no—
337C4×C31—42no—
338C4×C31—81no—
339C4×C31—82no—
340C4×C31—82no—
341C4×C31—82no—
342C4×C31—81no—
343C23×D4C22×C3111yescharacteristic
344C22×C4⋊C4C22×C3111yescharacteristic
345C22×C4⋊C4C22×C3111yescharacteristic
346C22×C4⋊C4C22×C3111yescharacteristic
347C22×C22⋊C4C22×C3112yes—
348C22×C22⋊C4C22×C3112yes—
349C22×C22⋊C4C22×C3112yes—
350C22×C22⋊C4C22×C3112yes—
351C22×C22⋊C4C22×C3112yes—
352C22×C22⋊C4C22×C3112yes—
353C22×C42C22×C3111yescharacteristic
354#390—24no—
355#387—24no—
356C2×#95C22×C3111yescharacteristic
357C2×#95C22×C3111yescharacteristic
358C2×#95C22×C3111yescharacteristic
359C2×#95C22×C3111yescharacteristic
360C2×#95C22×C3111yescharacteristic
361C2×#95C22×C3111yescharacteristic
362C2×#95C22×C3112yes—
363C2×#95C22×C3112yes—
364C2×#95C22×C3112yes—
365C2×#95C22×C3112yes—
366C2×#95C22×C3112yes—
367C2×#95C22×C3112yes—
368C2×C31—84no—
369C2×C31#256111yescharacteristic
370C2×C31#259211yescharacteristic
371C2×C31#259811yescharacteristic
372C2×C31#263511yescharacteristic
373C2×C31#263611yescharacteristic
374C2×C31#264111yescharacteristic
375C2×C31#274012yes—
376C2×C31#275212yes—
377C2×C31#278612yes—
378C2×C31#279712yes—
379C2×C31#281311yescharacteristic
380C25C23×C3112yes—
381C22×D4—24no—
382C22×D4—22no—
383C22×D4—22no—
384C22×D4—22no—
385C22×D4—42no—
386C23×C4C23×C3111yescharacteristic
387C23×C4C23×C3112yes—
388C23×C4C23×C3111yescharacteristic
389C23×C4C23×C3112yes—
390C23×C4C23×C3112yes—
391C23×C4C23×C3111yescharacteristic
392C23×C4C23×C3111yescharacteristic
393C23×C4C23×C3111yescharacteristic
394C23×C4C23×C3111yescharacteristic
395C23×C4C23×C3111yescharacteristic
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—42no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—42no—
419C2×C22⋊C4—42no—
420C2×C22⋊C4—42no—
421C2×C22⋊C4—42no—
422C2×C42—42no—
423C2×C42D4×C3114yes—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—42no—
431#95—42no—
432#95—42no—
433#95—42no—
434#95—42no—
435#95—42no—
436C31#1688611yescharacteristic, direct product factor, Sylow
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24#41×C3114yes—
452C24C24×C3111yescharacteristic
453C2×D4—28no—
454C2×D4—48no—
455C2×D4—48no—
456C2×D4—44no—
457C2×D4—44no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
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—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4C2×D4×C3112yes—
508C22×C4C2×D4×C3112yes—
509C22×C4C2×D4×C3112yes—
510C22×C4#41×C3112yes—
511C22×C4#41×C3112yes—
512C22×C4#41×C3112yes—
513C22×C4#41×C3112yes—
514C4⋊C4—28no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C42—44no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C22×D4×C3111yescharacteristic
560C23C2×#41×C3111yescharacteristic
561C23C2×#41×C3112yes—
562C23C2×#41×C3111yescharacteristic
563C23C2×#41×C3111yescharacteristic
564C23C2×#41×C3111yescharacteristic
565C23C31×H(5,𝔽2)12yes—
566C23C31×H(5,𝔽2)12yes—
567C23C31×H(5,𝔽2)12yes—
568C23C31×H(5,𝔽2)11yescharacteristic
569C23C31×2−1+411yescharacteristic
570D4—416no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
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×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4C31×#12414yes—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C31×#52511yescharacteristic
643C22C31×#52511yescharacteristic
644C22C2×C31×#12411yescharacteristic
645C22C31×#53411yescharacteristic
646C22C31×#53511yescharacteristic
647C22C31×#53511yescharacteristic
648C22C31×#53711yescharacteristic
649C22C31×#53812yes—
650C22C31×#53812yes—
651C22C31×#53912yes—
652C22C31×#54012yes—
653C22C31×#54012yes—
654C22C31×#54012yes—
655C22C31×#55112yes—
656C22C31×#55112yes—
657C22C31×#55112yes—
658C22C31×#55312yes—
659C22C31×#55611yescharacteristic
660C22C31×#55611yescharacteristic
661C22C31×#55911yescharacteristic
662C22C31×#55911yescharacteristic
663C22C31×#56012yes—
664C22C31×#56212yes—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—81no—
675C2—84no—
676C2C31×#256111yescharacteristic
677C2C31×#259211yescharacteristic
678C2C31×#259811yescharacteristic
679C2C31×#263511yescharacteristic
680C2C31×#263611yescharacteristic
681C2C31×#264111yescharacteristic
682C2C31×#274012yes—
683C2C31×#275212yes—
684C2C31×#278612yes—
685C2C31×#279712yes—
686C2C31×#281311yescharacteristic
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×#387C2semidirect
3. C31×#1445C2semidirect
4. C31×#1445C2semidirect
5. C31×#1440C2semidirect
6. C31×#1440C2semidirect
7. C31×#1439C2semidirect
8. C31×#1439C2semidirect
11. C31×#1434C2semidirect
180. #16886C31direct (N × H)
285. C2×#387C2×C31semidirect
286. #1445C2×C31semidirect
287. #1445C2×C31semidirect
288. #1440C2×C31semidirect
289. #1440C2×C31semidirect
290. #1439C2×C31semidirect
291. #1439C2×C31semidirect
294. #1434C2×C31semidirect
436. C31#16886direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3902
C2×C31×#3872
C31×#14452
C31×#14402
C31×#14392
C31×#14382
C31×#14342
#1688631