Finite Group Databasebeta

← C31×#15941 #21472700

Subgroups — C31×#15941 #21472700

Order 7936.

1538 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#394C211yesmaximal, characteristic
2C2×C31×#382C211yesmaximal, characteristic, complemented
3C31×#1443C212yesmaximal, complemented
4C31×#1440C212yesmaximal, complemented
5C31×#1439C212yesmaximal, complemented
6C31×#1438C212yesmaximal
7C31×#1437C212yesmaximal, complemented
8C31×#1437C212yesmaximal, complemented
9C31×#1436C211yesmaximal, characteristic, complemented
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C22⋊C4×C31C2212yes—
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C42×C31C2211yescharacteristic
20C22×C42×C31C2211yescharacteristic
21C31×#394—24no—
22C31×#382—24no—
23C2×C31×#95C2211yescharacteristic
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C25×C31C2312yes—
34C22×D4×C31—24no—
35C22×D4×C31—22no—
36C22×D4×C31—42no—
37C22×D4×C31—41no—
38C22×D4×C31—41no—
39C23×C4×C31C2312yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C2×C4⋊C4×C31—22no—
48C2×C4⋊C4×C31—41no—
49C2×C4⋊C4×C31—41no—
50C2×C4⋊C4×C31—41no—
51C2×C4⋊C4×C31—41no—
52C2×C4⋊C4×C31D414yes—
53C2×C22⋊C4×C31—24no—
54C2×C22⋊C4×C31—24no—
55C2×C22⋊C4×C31—24no—
56C2×C22⋊C4×C31—24no—
57C2×C22⋊C4×C31—24no—
58C2×C22⋊C4×C31—24no—
59C2×C22⋊C4×C31—24no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—42no—
64C2×C22⋊C4×C31—42no—
65C2×C22⋊C4×C31—42no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—42no—
71C2×C42×C31—22no—
72C2×C42×C31—24no—
73C2×C42×C31—42no—
74C2×C42×C31—41no—
75C31×#95—24no—
76C31×#95—24no—
77C31×#95—22no—
78C31×#95—24no—
79C31×#95—24no—
80C31×#95—42no—
81C31×#95—42no—
82C31×#95—42no—
83C31×#95—41no—
84C31×#95—42no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31#4114yes—
100C24×C31C2411yescharacteristic
101C2×D4×C31—48no—
102C2×D4×C31—44no—
103C2×D4×C31—44no—
104C2×D4×C31—48no—
105C2×D4×C31—44no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
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—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
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×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31#4114yes—
154C22×C4×C31#4114yes—
155C22×C4×C31#4114yes—
156C4⋊C4×C31—44no—
157C4⋊C4×C31—44no—
158C22⋊C4×C31—48no—
159C22⋊C4×C31—48no—
160C22⋊C4×C31—48no—
161C22⋊C4×C31—48no—
162C22⋊C4×C31—48no—
163C22⋊C4×C31—48no—
164C42×C31—28no—
165C42×C31—44no—
166#15941C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—24no—
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
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×C31C22×D411yescharacteristic
203C23×C31C2×#4112yes—
204C23×C31C2×#4112yes—
205C23×C31C2×#4112yes—
206C23×C31C2×#4112yes—
207C23×C31H(5,𝔽2)11yescharacteristic
208C23×C31H(5,𝔽2)11yescharacteristic
209C23×C31H(5,𝔽2)12yes—
210C23×C31H(5,𝔽2)12yes—
211D4×C31—416no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
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#12414yes—
262C2×#394C2×C3111yescharacteristic
263C2×#382C2×C3111yescharacteristic, complemented
264#1443C2×C3112yescomplemented
265#1440C2×C3112yescomplemented
266#1439C2×C3112yescomplemented
267#1438C2×C3112yes—
268#1437C2×C3112yescomplemented
269#1437C2×C3112yescomplemented
270#1436C2×C3111yescharacteristic, complemented
271C22×C31—44no—
272C22×C31—44no—
273C22×C31—44no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—44no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—82no—
286C22×C31#52511yescharacteristic
287C22×C31C2×#12411yescharacteristic
288C22×C31#52912yes—
289C22×C31#52912yes—
290C22×C31#53411yescharacteristic
291C22×C31#53812yes—
292C22×C31#53812yes—
293C22×C31#54012yes—
294C22×C31#54012yes—
295C22×C31#54612yes—
296C22×C31#55012yes—
297C22×C31#55111yescharacteristic
298C22×C31#55112yes—
299C22×C31#55112yes—
300C22×C31#55312yes—
301C22×C31#55312yes—
302C22×C31#55312yes—
303C22×C31#55312yes—
304C22×C31#55511yescharacteristic
305C22×C31H(3,𝔽4)11yescharacteristic
306C22×C31H(3,𝔽4)11yescharacteristic
307C4×C31—24no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—44no—
311C4×C31—44no—
312C4×C31—82no—
313C4×C31—82no—
314C4×C31—82no—
315C23×D4C22×C3111yescharacteristic
316C22×C4⋊C4C22×C3111yescharacteristic
317C22×C4⋊C4C22×C3111yescharacteristic
318C22×C22⋊C4C22×C3112yes—
319C22×C22⋊C4C22×C3112yes—
320C22×C22⋊C4C22×C3112yes—
321C22×C22⋊C4C22×C3112yes—
322C22×C22⋊C4C22×C3112yes—
323C22×C22⋊C4C22×C3112yes—
324C22×C42C22×C3111yescharacteristic
325C22×C42C22×C3111yescharacteristic
326#394—24no—
327#382—24no—
328C2×#95C22×C3111yescharacteristic
329C2×#95C22×C3112yes—
330C2×#95C22×C3112yes—
331C2×#95C22×C3112yes—
332C2×#95C22×C3112yes—
333C2×#95C22×C3112yes—
334C2×#95C22×C3111yescharacteristic
335C2×#95C22×C3112yes—
336C2×#95C22×C3112yes—
337C2×#95C22×C3112yes—
338C2×C31—84no—
339C2×C31#247112yes—
340C2×C31#249012yes—
341C2×C31#251812yes—
342C2×C31#257612yes—
343C2×C31#259111yescharacteristic
344C2×C31#268411yescharacteristic
345C2×C31#273311yescharacteristic
346C2×C31#275312yes—
347C2×C31#275312yes—
348C25C23×C3112yes—
349C22×D4—24no—
350C22×D4—22no—
351C22×D4—42no—
352C22×D4—41no—
353C22×D4—41no—
354C23×C4C23×C3112yes—
355C23×C4C23×C3112yes—
356C23×C4C23×C3112yes—
357C23×C4C23×C3111yescharacteristic
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360C23×C4C23×C3112yes—
361C23×C4C23×C3111yescharacteristic
362C2×C4⋊C4—22no—
363C2×C4⋊C4—41no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4D4×C3114yes—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—42no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C42—22no—
387C2×C42—24no—
388C2×C42—42no—
389C2×C42—41no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—42no—
396#95—42no—
397#95—42no—
398#95—41no—
399#95—42no—
400C31#1594111yescharacteristic, direct product factor, Sylow
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24#41×C3114yes—
416C24C24×C3111yescharacteristic
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—44no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
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—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4C2×D4×C3112yes—
467C22×C4C2×D4×C3112yes—
468C22×C4C2×D4×C3112yes—
469C22×C4#41×C3114yes—
470C22×C4#41×C3114yes—
471C22×C4#41×C3114yes—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—28no—
481C42—44no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C22×D4×C3111yescharacteristic
518C23C2×#41×C3112yes—
519C23C2×#41×C3112yes—
520C23C2×#41×C3112yes—
521C23C2×#41×C3112yes—
522C23C31×H(5,𝔽2)11yescharacteristic
523C23C31×H(5,𝔽2)11yescharacteristic
524C23C31×H(5,𝔽2)12yes—
525C23C31×H(5,𝔽2)12yes—
526D4—416no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
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—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×C4C31×#12414yes—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22C31×#52511yescharacteristic
593C22C2×C31×#12411yescharacteristic
594C22C31×#52912yes—
595C22C31×#52912yes—
596C22C31×#53411yescharacteristic
597C22C31×#53812yes—
598C22C31×#53812yes—
599C22C31×#54012yes—
600C22C31×#54012yes—
601C22C31×#54612yes—
602C22C31×#55012yes—
603C22C31×#55111yescharacteristic
604C22C31×#55112yes—
605C22C31×#55112yes—
606C22C31×#55312yes—
607C22C31×#55312yes—
608C22C31×#55312yes—
609C22C31×#55312yes—
610C22C31×#55511yescharacteristic
611C22C31×H(3,𝔽4)11yescharacteristic
612C22C31×H(3,𝔽4)11yescharacteristic
613C4—24no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—82no—
621C2—84no—
622C2C31×#247112yes—
623C2C31×#249012yes—
624C2C31×#251812yes—
625C2C31×#257612yes—
626C2C31×#259111yescharacteristic
627C2C31×#268411yescharacteristic
628C2C31×#273311yescharacteristic
629C2C31×#275312yes—
630C2C31×#275312yes—
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×#382C2semidirect
3. C31×#1443C2semidirect
4. C31×#1440C2semidirect
5. C31×#1439C2semidirect
7. C31×#1437C2semidirect
8. C31×#1437C2semidirect
9. C31×#1436C2semidirect
166. #15941C31direct (N × H)
263. C2×#382C2×C31semidirect
264. #1443C2×C31semidirect
265. #1440C2×C31semidirect
266. #1439C2×C31semidirect
268. #1437C2×C31semidirect
269. #1437C2×C31semidirect
270. #1436C2×C31semidirect
400. C31#15941direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3942
C2×C31×#3822
C31×#14432
C31×#14402
C31×#14392
C31×#14382
C31×#14372
C31×#14362
#1594131