Finite Group Databasebeta

← C31×#16698 #21473457

Subgroups — C31×#16698 #21473457

Order 7936.

1362 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#381C211yesmaximal, characteristic
2C2×C31×#380C211yesmaximal, characteristic
3C31×#1445C211yesmaximal, characteristic, complemented
4C31×#1445C211yesmaximal, characteristic, complemented
5C31×#1445C212yesmaximal, complemented
6C31×#1444C211yesmaximal, characteristic, complemented
7C31×#1443C212yesmaximal, complemented
8C31×#1442C211yesmaximal, characteristic, complemented
9C31×#1440C212yesmaximal
10C31×#1439C212yesmaximal
11C31×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C31×#381—24no—
21C31×#380—24no—
22C2×C31×#95C2211yescharacteristic
23C2×C31×#95C2211yescharacteristic
24C2×C31×#95C2212yes—
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2311yescharacteristic
38C23×C4×C31—24no—
39C23×C4×C31—22no—
40C23×C4×C31—22no—
41C23×C4×C31—42no—
42C23×C4×C31D414yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2312yes—
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—42no—
60C2×C4⋊C4×C31—42no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
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—
71C31×#95—24no—
72C31×#95—22no—
73C31×#95—22no—
74C31×#95—22no—
75C31×#95—22no—
76C31×#95—22no—
77C31×#95—24no—
78C31×#95—22no—
79C31×#95—42no—
80C31×#95—42no—
81C31×#95—42no—
82C31×#95—42no—
83C31×#95—42no—
84C31×#95—42no—
85C31×#95—42no—
86C24×C31—22no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—21no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—21no—
95C24×C31C2×D412yes—
96C24×C31#4114yes—
97C24×C31C2411yescharacteristic
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—28no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—28no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
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—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
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—21no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—44no—
154C22×C4×C31—48no—
155C22×C4×C31—48no—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31#4114yes—
159C22×C4×C31#4112yes—
160C22×C4×C31#4112yes—
161C4⋊C4×C31—44no—
162C4⋊C4×C31—48no—
163C4⋊C4×C31—44no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—48no—
167C22⋊C4×C31—44no—
168#16698C3111yesmaximal, characteristic, direct product factor, Sylow
169C23×C31—22no—
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—24no—
176C23×C31—24no—
177C23×C31—24no—
178C23×C31—24no—
179C23×C31—24no—
180C23×C31—24no—
181C23×C31—42no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—42no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31#12314yes—
192C23×C31C22×D411yescharacteristic
193C23×C31C2×#4112yes—
194C23×C31C2×#4111yescharacteristic
195C23×C31C2×#4111yescharacteristic
196C23×C31C2×#4112yes—
197C23×C31H(5,𝔽2)12yes—
198C23×C31H(5,𝔽2)11yescharacteristic
199C23×C31H(5,𝔽2)11yescharacteristic
200C23×C31H(5,𝔽2)11yescharacteristic
201C23×C312−1+412yes—
202C23×C312−1+411yescharacteristic
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—416no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
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—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#12314yes—
262C2×#381C2×C3111yescharacteristic
263C2×#380C2×C3111yescharacteristic
264#1445C2×C3111yescharacteristic, complemented
265#1445C2×C3111yescharacteristic, complemented
266#1445C2×C3112yescomplemented
267#1444C2×C3111yescharacteristic, complemented
268#1443C2×C3112yescomplemented
269#1442C2×C3111yescharacteristic, complemented
270#1440C2×C3112yes—
271#1439C2×C3112yes—
272C22⋊#95C2×C3111yescharacteristic
273C22×C31—24no—
274C22×C31—24no—
275C22×C31—24no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—44no—
279C22×C31—44no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31C2×#12311yescharacteristic
285C22×C31C2×#12311yescharacteristic
286C22×C31#53211yescharacteristic
287C22×C31#53411yescharacteristic
288C22×C31#53511yescharacteristic
289C22×C31#53611yescharacteristic
290C22×C31#53711yescharacteristic
291C22×C31#53812yes—
292C22×C31#53912yes—
293C22×C31#53912yes—
294C22×C31#54011yescharacteristic
295C22×C31#54011yescharacteristic
296C22×C31#54012yes—
297C22×C31#54211yescharacteristic
298C22×C31#54211yescharacteristic
299C22×C31#55112yes—
300C22×C31#55112yes—
301C22×C31#55112yes—
302C22×C31#55212yes—
303C22×C31#55212yes—
304C22×C31#55612yes—
305C22×C31#56212yes—
306C22×C31#56312yes—
307C4×C31—24no—
308C4×C31—42no—
309C4×C31—44no—
310C4×C31—81no—
311C4×C31—82no—
312C4×C31—81no—
313C4×C31—82no—
314C4×C31—82no—
315C4×C31—81no—
316C4×C31—81no—
317C24×C4C22×C3111yescharacteristic
318C22×C4⋊C4C22×C3111yescharacteristic
319C22×C4⋊C4C22×C3112yes—
320C22×C4⋊C4C22×C3111yescharacteristic
321C22×C22⋊C4C22×C3111yescharacteristic
322C22×C22⋊C4C22×C3112yes—
323C22×C22⋊C4C22×C3111yescharacteristic
324C22×C22⋊C4C22×C3112yes—
325#381—24no—
326#380—24no—
327C2×#95C22×C3111yescharacteristic
328C2×#95C22×C3111yescharacteristic
329C2×#95C22×C3112yes—
330C2×#95C22×C3111yescharacteristic
331C2×#95C22×C3111yescharacteristic
332C2×#95C22×C3112yes—
333C2×#95C22×C3111yescharacteristic
334C2×#95C22×C3111yescharacteristic
335C2×#95C22×C3112yes—
336C2×#95C22×C3112yes—
337C2×#95C22×C3112yes—
338C2×#95C22×C3112yes—
339C2×#95C22×C3112yes—
340C2×#95C22×C3112yes—
341C2×#95C22×C3112yes—
342C2×C31—44no—
343C2×C31#248311yescharacteristic
344C2×C31#249811yescharacteristic
345C2×C31#249911yescharacteristic
346C2×C31#262311yescharacteristic
347C2×C31#262411yescharacteristic
348C2×C31#266111yescharacteristic
349C2×C31#273812yes—
350C2×C31#274012yes—
351C2×C31#274312yes—
352C2×C31#274512yes—
353C2×C31#281411yescharacteristic
354C25C23×C3111yescharacteristic
355C23×C4—24no—
356C23×C4—22no—
357C23×C4—22no—
358C23×C4—42no—
359C23×C4D4×C3114yes—
360C23×C4C23×C3111yescharacteristic
361C23×C4C23×C3112yes—
362C23×C4C23×C3112yes—
363C23×C4C23×C3111yescharacteristic
364C23×C4C23×C3111yescharacteristic
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3112yes—
367C23×C4C23×C3111yescharacteristic
368C23×C4C23×C3112yes—
369C23×C4C23×C3111yescharacteristic
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—42no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—42no—
397#95—42no—
398#95—42no—
399#95—42no—
400#95—42no—
401#95—42no—
402#95—42no—
403C31#1669811yescharacteristic, direct product factor, Sylow
404C24—22no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24C2×D4×C3112yes—
414C24#41×C3114yes—
415C24C24×C3111yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
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—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
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—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3112yes—
476C22×C4#41×C3114yes—
477C22×C4#41×C3112yes—
478C22×C4#41×C3112yes—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C31×#12314yes—
509C23C22×D4×C3111yescharacteristic
510C23C2×#41×C3112yes—
511C23C2×#41×C3111yescharacteristic
512C23C2×#41×C3111yescharacteristic
513C23C2×#41×C3112yes—
514C23C31×H(5,𝔽2)12yes—
515C23C31×H(5,𝔽2)11yescharacteristic
516C23C31×H(5,𝔽2)11yescharacteristic
517C23C31×H(5,𝔽2)11yescharacteristic
518C23C31×2−1+412yes—
519C23C31×2−1+411yescharacteristic
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
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—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4C31×#12314yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22C2×C31×#12311yescharacteristic
591C22C2×C31×#12311yescharacteristic
592C22C31×#53211yescharacteristic
593C22C31×#53411yescharacteristic
594C22C31×#53511yescharacteristic
595C22C31×#53611yescharacteristic
596C22C31×#53711yescharacteristic
597C22C31×#53812yes—
598C22C31×#53912yes—
599C22C31×#53912yes—
600C22C31×#54011yescharacteristic
601C22C31×#54011yescharacteristic
602C22C31×#54012yes—
603C22C31×#54211yescharacteristic
604C22C31×#54211yescharacteristic
605C22C31×#55112yes—
606C22C31×#55112yes—
607C22C31×#55112yes—
608C22C31×#55212yes—
609C22C31×#55212yes—
610C22C31×#55612yes—
611C22C31×#56212yes—
612C22C31×#56312yes—
613C4—24no—
614C4—42no—
615C4—44no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—82no—
621C4—81no—
622C4—81no—
623C2—44no—
624C2C31×#248311yescharacteristic
625C2C31×#249811yescharacteristic
626C2C31×#249911yescharacteristic
627C2C31×#262311yescharacteristic
628C2C31×#262411yescharacteristic
629C2C31×#266111yescharacteristic
630C2C31×#273812yes—
631C2C31×#274012yes—
632C2C31×#274312yes—
633C2C31×#274512yes—
634C2C31×#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. C31×#1445C2semidirect
4. C31×#1445C2semidirect
5. C31×#1445C2semidirect
6. C31×#1444C2semidirect
7. C31×#1443C2semidirect
8. C31×#1442C2semidirect
168. #16698C31direct (N × H)
264. #1445C2×C31semidirect
265. #1445C2×C31semidirect
266. #1445C2×C31semidirect
267. #1444C2×C31semidirect
268. #1443C2×C31semidirect
269. #1442C2×C31semidirect
403. C31#16698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3812
C2×C31×#3802
C31×#14452
C31×#14442
C31×#14432
C31×#14422
C31×#14402
C31×#14392
C31×C22⋊#952
#1669831