Finite Group Databasebeta

← C31×#16805 #21473564

Subgroups — C31×#16805 #21473564

Order 7936.

1402 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#397C211yesmaximal, characteristic
2C2×C31×#393C211yesmaximal, characteristic
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C2×C31×#388C211yesmaximal, characteristic
5C31×#1446C211yesmaximal, characteristic, complemented
6C31×#1445C211yesmaximal, characteristic, complemented
7C31×#1443C212yesmaximal, complemented
8C31×#1441C212yesmaximal, complemented
9C31×#1440C212yesmaximal
10C31×#1439C212yesmaximal
11C31×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C31×#397—24no—
23C31×#393—24no—
24C31×#391—24no—
25C31×#388—24no—
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2211yescharacteristic
34C2×C31×#95C2211yescharacteristic
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C25×C31C2311yescharacteristic
40C22×Q8×C31—22no—
41C22×Q8×C31—22no—
42C22×Q8×C31—24no—
43C22×Q8×C31—22no—
44C22×Q8×C31—42no—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—42no—
62C2×C4⋊C4×C31—42no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31D414yes—
76C2×C42×C31—42no—
77C2×C42×C31D414yes—
78C31×#95—22no—
79C31×#95—24no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—24no—
83C31×#95—24no—
84C31×#95—24no—
85C31×#95—22no—
86C31×#95—24no—
87C31×#95—42no—
88C31×#95—42no—
89C31×#95—42no—
90C31×#95—42no—
91C24×C31—21no—
92C24×C31—21no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—21no—
96C24×C31—21no—
97C24×C31—22no—
98C24×C31—21no—
99C24×C31—21no—
100C24×C31—22no—
101C24×C31C2×D412yes—
102C24×C31C2411yescharacteristic
103C2×Q8×C31—28no—
104C2×Q8×C31—48no—
105C2×Q8×C31—48no—
106C2×Q8×C31—44no—
107C2×Q8×C31—44no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—21no—
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—21no—
143C22×C4×C31—21no—
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—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31#4114yes—
164C22×C4×C31#4112yes—
165C22×C4×C31#4114yes—
166C4⋊C4×C31—48no—
167C4⋊C4×C31—44no—
168C4⋊C4×C31—44no—
169C22⋊C4×C31—28no—
170C22⋊C4×C31—48no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—48no—
173C22⋊C4×C31—44no—
174C42×C31—44no—
175#16805C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—24no—
180C23×C31—24no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31C22×D411yescharacteristic
200C23×C31C22×D411yescharacteristic
201C23×C31C2×#4111yescharacteristic
202C23×C31C2×#4112yes—
203C23×C31C2×#4112yes—
204C23×C31H(5,𝔽2)12yes—
205C23×C31H(5,𝔽2)11yescharacteristic
206C23×C31H(5,𝔽2)11yescharacteristic
207C23×C312−1+411yescharacteristic
208C23×C312−1+411yescharacteristic
209C23×C312−1+412yes—
210Q8×C31—416no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
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—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
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#12114yes—
268C2×C4×C31#12414yes—
269C2×#397C2×C3111yescharacteristic
270C2×#393C2×C3111yescharacteristic
271C2×#391C2×C3111yescharacteristic, complemented
272C2×#388C2×C3111yescharacteristic
273#1446C2×C3111yescharacteristic, complemented
274#1445C2×C3111yescharacteristic, complemented
275#1443C2×C3112yescomplemented
276#1441C2×C3112yescomplemented
277#1440C2×C3112yes—
278#1439C2×C3112yes—
279C4⋊#95C2×C3111yescharacteristic, complemented
280C22×C31—42no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—82no—
289C22×C31—82no—
290C22×C31—81no—
291C22×C31C2×#12111yescharacteristic
292C22×C31#52511yescharacteristic
293C22×C31C2×#12411yescharacteristic
294C22×C31#53411yescharacteristic
295C22×C31#53511yescharacteristic
296C22×C31#53611yescharacteristic
297C22×C31#53611yescharacteristic
298C22×C31#53711yescharacteristic
299C22×C31#53711yescharacteristic
300C22×C31#53812yes—
301C22×C31#53912yes—
302C22×C31#54112yes—
303C22×C31#54211yescharacteristic
304C22×C31#54211yescharacteristic
305C22×C31#54212yes—
306C22×C31#54612yes—
307C22×C31#54712yes—
308C22×C31#55112yes—
309C22×C31#55612yes—
310C22×C31#55612yes—
311C22×C31#55912yes—
312C22×C31#56212yes—
313C22×C31#56312yes—
314C4×C31—24no—
315C4×C31—24no—
316C4×C31—44no—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—81no—
320C4×C31—82no—
321C4×C31—81no—
322C4×C31—82no—
323C4×C31—82no—
324C23×Q8C22×C3111yescharacteristic
325C22×C4⋊C4C22×C3111yescharacteristic
326C22×C4⋊C4C22×C3111yescharacteristic
327C22×C4⋊C4C22×C3112yes—
328C22×C22⋊C4C22×C3111yescharacteristic
329C22×C22⋊C4C22×C3112yes—
330C22×C22⋊C4C22×C3111yescharacteristic
331C22×C22⋊C4C22×C3112yes—
332C22×C22⋊C4C22×C3111yescharacteristic
333C22×C42C22×C3111yescharacteristic
334#397—24no—
335#393—24no—
336#391—24no—
337#388—24no—
338C2×#95C22×C3111yescharacteristic
339C2×#95C22×C3112yes—
340C2×#95C22×C3111yescharacteristic
341C2×#95C22×C3112yes—
342C2×#95C22×C3112yes—
343C2×#95C22×C3112yes—
344C2×#95C22×C3112yes—
345C2×#95C22×C3111yescharacteristic
346C2×#95C22×C3111yescharacteristic
347C2×#95C22×C3112yes—
348C2×#95C22×C3112yes—
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×C31—82no—
352C2×C31#253411yescharacteristic
353C2×C31#254711yescharacteristic
354C2×C31#259211yescharacteristic
355C2×C31#262011yescharacteristic
356C2×C31#263711yescharacteristic
357C2×C31#263911yescharacteristic
358C2×C31#268912yes—
359C2×C31#271012yes—
360C2×C31#275912yes—
361C2×C31#279812yes—
362C2×C31#281411yescharacteristic
363C25C23×C3111yescharacteristic
364C22×Q8—22no—
365C22×Q8—22no—
366C22×Q8—24no—
367C22×Q8—22no—
368C22×Q8—42no—
369C23×C4C23×C3112yes—
370C23×C4C23×C3112yes—
371C23×C4C23×C3112yes—
372C23×C4C23×C3111yescharacteristic
373C23×C4C23×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3111yescharacteristic
376C23×C4C23×C3112yes—
377C23×C4C23×C3111yescharacteristic
378C23×C4C23×C3111yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—42no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4D4×C3114yes—
400C2×C42—42no—
401C2×C42D4×C3114yes—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—42no—
412#95—42no—
413#95—42no—
414#95—42no—
415C31#1680511yescharacteristic, direct product factor, Sylow
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—21no—
421C24—21no—
422C24—22no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24C2×D4×C3112yes—
427C24C24×C3111yescharacteristic
428C2×Q8—28no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
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—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
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—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3112yes—
485C22×C4C2×D4×C3112yes—
486C22×C4C2×D4×C3112yes—
487C22×C4C2×D4×C3112yes—
488C22×C4#41×C3114yes—
489C22×C4#41×C3112yes—
490C22×C4#41×C3114yes—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C42—44no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C22×D4×C3111yescharacteristic
524C23C22×D4×C3111yescharacteristic
525C23C2×#41×C3111yescharacteristic
526C23C2×#41×C3112yes—
527C23C2×#41×C3112yes—
528C23C31×H(5,𝔽2)12yes—
529C23C31×H(5,𝔽2)11yescharacteristic
530C23C31×H(5,𝔽2)11yescharacteristic
531C23C31×2−1+411yescharacteristic
532C23C31×2−1+411yescharacteristic
533C23C31×2−1+412yes—
534Q8—416no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4C31×#12114yes—
592C2×C4C31×#12414yes—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22C2×C31×#12111yescharacteristic
605C22C31×#52511yescharacteristic
606C22C2×C31×#12411yescharacteristic
607C22C31×#53411yescharacteristic
608C22C31×#53511yescharacteristic
609C22C31×#53611yescharacteristic
610C22C31×#53611yescharacteristic
611C22C31×#53711yescharacteristic
612C22C31×#53711yescharacteristic
613C22C31×#53812yes—
614C22C31×#53912yes—
615C22C31×#54112yes—
616C22C31×#54211yescharacteristic
617C22C31×#54211yescharacteristic
618C22C31×#54212yes—
619C22C31×#54612yes—
620C22C31×#54712yes—
621C22C31×#55112yes—
622C22C31×#55612yes—
623C22C31×#55612yes—
624C22C31×#55912yes—
625C22C31×#56212yes—
626C22C31×#56312yes—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4—82no—
637C2—82no—
638C2C31×#253411yescharacteristic
639C2C31×#254711yescharacteristic
640C2C31×#259211yescharacteristic
641C2C31×#262011yescharacteristic
642C2C31×#263711yescharacteristic
643C2C31×#263911yescharacteristic
644C2C31×#268912yes—
645C2C31×#271012yes—
646C2C31×#275912yes—
647C2C31×#279812yes—
648C2C31×#281411yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#391C2semidirect
5. C31×#1446C2semidirect
6. C31×#1445C2semidirect
7. C31×#1443C2semidirect
8. C31×#1441C2semidirect
11. C31×C4⋊#95C2semidirect
175. #16805C31direct (N × H)
271. C2×#391C2×C31semidirect
273. #1446C2×C31semidirect
274. #1445C2×C31semidirect
275. #1443C2×C31semidirect
276. #1441C2×C31semidirect
279. C4⋊#95C2×C31semidirect
415. C31#16805direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3972
C2×C31×#3932
C2×C31×#3912
C2×C31×#3882
C31×#14462
C31×#14452
C31×#14432
C31×#14412
C31×#14402
C31×#14392
C31×C4⋊#952
#1680531