Finite Group Databasebeta

← C31×#13877 #21470636

Subgroups — C31×#13877 #21470636

Order 7936.

4046 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
1C23×C22⋊C4×C31C211yesmaximal, characteristic
2C23×C22⋊C4×C31C211yesmaximal, characteristic
3C2×C31×#385C212yesmaximal, complemented
4C2×C31×#380C212yesmaximal, complemented
5C2×C31×#379C211yesmaximal, characteristic
6C31×#1439C214yesmaximal, complemented
7C31×C22⋊#95C212yesmaximal, complemented
8C31×C22⋊#95C212yesmaximal, complemented
9C26×C31C2211yescharacteristic
10C24×C4×C31C2212yes—
11C24×C4×C31C2212yes—
12C22×C4⋊C4×C31C2214yescomplemented
13C22×C22⋊C4×C31—24no—
14C22×C22⋊C4×C31—28no—
15C22×C22⋊C4×C31—28no—
16C22×C22⋊C4×C31C418yescomplemented
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2212yes—
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2212yes—
25C22×C22⋊C4×C31C2211yescharacteristic
26C31×#385—28no—
27C31×#380—28no—
28C31×#379—24no—
29C2×C31×#95C2214yescomplemented
30C2×C31×#95C2214yescomplemented
31C2×C31×#95C2214yescomplemented
32C25×C31—24no—
33C25×C31—24no—
34C25×C31—24no—
35C25×C31—24no—
36C25×C31—44no—
37C25×C31C2×C414yes—
38C25×C31D414yes—
39C25×C31Q814yes—
40C25×C31C2311yescharacteristic
41C25×C31C2311yescharacteristic
42C25×C31C2311yescharacteristic
43C23×C4×C31—24no—
44C23×C4×C31—24no—
45C23×C4×C31—24no—
46C23×C4×C31—28no—
47C23×C4×C31—28no—
48C23×C4×C31—24no—
49C23×C4×C31—24no—
50C23×C4×C31—44no—
51C23×C4×C31—44no—
52C23×C4×C31C2×C418yescomplemented
53C23×C4×C31C2312yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2314yes—
57C23×C4×C31C2312yes—
58C2×C4⋊C4×C31—28no—
59C2×C4⋊C4×C31—28no—
60C2×C4⋊C4×C31—44no—
61C2×C22⋊C4×C31—232no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—28no—
67C2×C22⋊C4×C31—28no—
68C2×C22⋊C4×C31—232no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—44no—
78C2×C22⋊C4×C31C2×C414yes—
79C2×C22⋊C4×C31C2×C414yes—
80C2×C22⋊C4×C31C2×C418yescomplemented
81C2×C22⋊C4×C31D418yes—
82C2×C22⋊C4×C31D418yes—
83C31×#95—28no—
84C31×#95—28no—
85C31×#95—44no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—24no—
91C24×C31—24no—
92C24×C31—28no—
93C24×C31—28no—
94C24×C31—216no—
95C24×C31—28no—
96C24×C31—216no—
97C24×C31—216no—
98C24×C31—216no—
99C24×C31—28no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—24no—
105C24×C31—48no—
106C24×C31—48no—
107C24×C31—48no—
108C24×C31—416no—
109C24×C31—416no—
110C24×C31—416no—
111C24×C31—416no—
112C24×C31C4⋊C4116yescomplemented
113C24×C31C22×C412yes—
114C24×C31C22×C412yes—
115C24×C31C22×C412yes—
116C24×C31C2×D412yes—
117C24×C31C2×D414yes—
118C24×C31C2×D412yes—
119C24×C31C2×D412yes—
120C24×C31C2×D414yes—
121C24×C31C2×Q812yes—
122C24×C31C2×Q812yes—
123C24×C31C2×Q812yes—
124C24×C31#4114yes—
125C24×C31#4114yes—
126C24×C31C2411yescharacteristic
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—24no—
130C22×C4×C31—216no—
131C22×C4×C31—216no—
132C22×C4×C31—22no—
133C22×C4×C31—24no—
134C22×C4×C31—24no—
135C22×C4×C31—24no—
136C22×C4×C31—24no—
137C22×C4×C31—22no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31—216no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—48no—
154C22×C4×C31—416no—
155C22×C4×C31—416no—
156C22×C4×C31—416no—
157C22×C4×C31—416no—
158C22×C4×C31—48no—
159C22×C4×C31—48no—
160C22×C4×C31C22×C414yes—
161C22×C4×C31C22×C414yes—
162C22×C4×C31C2×D418yes—
163C22×C4×C31C2×D418yes—
164C4⋊C4×C31—416no—
165C22⋊C4×C31—216no—
166C22⋊C4×C31—28no—
167C22⋊C4×C31—216no—
168C22⋊C4×C31—216no—
169C22⋊C4×C31—48no—
170C22⋊C4×C31—44no—
171C22⋊C4×C31—48no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31C22⋊C4116yescomplemented
174#13877C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—24no—
178C23×C31—24no—
179C23×C31—24no—
180C23×C31—22no—
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—22no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—28no—
190C23×C31—28no—
191C23×C31—28no—
192C23×C31—28no—
193C23×C31—28no—
194C23×C31—28no—
195C23×C31—28no—
196C23×C31—28no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—24no—
200C23×C31—24no—
201C23×C31—22no—
202C23×C31—24no—
203C23×C31—24no—
204C23×C31—24no—
205C23×C31—22no—
206C23×C31—24no—
207C23×C31—24no—
208C23×C31—232no—
209C23×C31—232no—
210C23×C31—416no—
211C23×C31—432no—
212C23×C31—432no—
213C23×C31—432no—
214C23×C31—432no—
215C23×C31—464no—
216C23×C31—44no—
217C23×C31—44no—
218C23×C31—44no—
219C23×C31—44no—
220C23×C31C2×C22⋊C414yes—
221C23×C31C2×C4⋊C414yes—
222C23×C31C2×C4⋊C414yes—
223C23×C31C2×C4⋊C414yes—
224C23×C31#11714yes—
225C23×C31C4×D418yes—
226C23×C31#12014yes—
227C23×C31#12118yes—
228C23×C31C22⋊Q818yes—
229C23×C31C22⋊Q818yes—
230C23×C31#12314yes—
231C23×C31C23×C411yescharacteristic
232C23×C31C22×D412yes—
233C23×C31C22×D411yescharacteristic
234C23×C31C22×D412yes—
235C23×C31C22×Q811yescharacteristic
236C23×C31C2×#4112yes—
237C23×C31C2×#4112yes—
238C23×C31H(5,𝔽2)14yes—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—28no—
242C2×C4×C31—28no—
243C2×C4×C31—24no—
244C2×C4×C31—24no—
245C2×C4×C31—24no—
246C2×C4×C31—28no—
247C2×C4×C31—28no—
248C2×C4×C31—28no—
249C2×C4×C31—432no—
250C2×C4×C31—432no—
251C2×C4×C31—44no—
252C2×C4×C31—44no—
253C2×C4×C31—44no—
254C2×C4×C31—44no—
255C2×C4×C31—44no—
256C2×C4×C31—44no—
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—44no—
264C2×C4×C31—44no—
265C2×C4×C31—44no—
266C2×C4×C31—44no—
267C2×C4×C31C2×C22⋊C418yescomplemented
268C23×C22⋊C4C2×C3111yescharacteristic
269C23×C22⋊C4C2×C3111yescharacteristic
270C2×#385C2×C3112yescomplemented
271C2×#380C2×C3112yescomplemented
272C2×#379C2×C3111yescharacteristic
273#1439C2×C3114yescomplemented
274C22⋊#95C2×C3112yescomplemented
275C22⋊#95C2×C3112yescomplemented
276C22×C31—24no—
277C22×C31—24no—
278C22×C31—28no—
279C22×C31—28no—
280C22×C31—28no—
281C22×C31—28no—
282C22×C31—28no—
283C22×C31—28no—
284C22×C31—28no—
285C22×C31—28no—
286C22×C31—216no—
287C22×C31—216no—
288C22×C31—24no—
289C22×C31—24no—
290C22×C31—24no—
291C22×C31—24no—
292C22×C31—24no—
293C22×C31—464no—
294C22×C31—44no—
295C22×C31—44no—
296C22×C31—44no—
297C22×C31—44no—
298C22×C31—48no—
299C22×C31#38018yescomplemented
300C22×C31#38518yescomplemented
301C22×C31C22×C22⋊C411yescharacteristic
302C22×C31C22×C4⋊C411yescharacteristic
303C22×C31C2×#11711yescharacteristic
304C22×C31C2×C4×D412yes—
305C22×C31#51812yes—
306C22×C31C2×#12011yescharacteristic
307C22×C31C2×#12112yes—
308C22×C31C2×C22⋊Q812yes—
309C22×C31C2×C22⋊Q812yes—
310C22×C31C2×#12311yescharacteristic
311C22×C31#53412yes—
312C22×C31#54312yes—
313C22×C31D4214yes—
314C22×C31#54614yes—
315C22×C31#54614yes—
316C22×C31#55114yes—
317C4×C31—28no—
318C4×C31—44no—
319C4×C31—48no—
320C4×C31—82no—
321C4×C31—82no—
322C26C22×C3111yescharacteristic
323C24×C4C22×C3112yes—
324C24×C4C22×C3112yes—
325C22×C4⋊C4C22×C3114yescomplemented
326C22×C22⋊C4—24no—
327C22×C22⋊C4—28no—
328C22×C22⋊C4—28no—
329C22×C22⋊C4C4×C3118yescomplemented
330C22×C22⋊C4C22×C3112yes—
331C22×C22⋊C4C22×C3111yescharacteristic
332C22×C22⋊C4C22×C3112yes—
333C22×C22⋊C4C22×C3111yescharacteristic
334C22×C22⋊C4C22×C3111yescharacteristic
335C22×C22⋊C4C22×C3112yes—
336C22×C22⋊C4C22×C3112yes—
337C22×C22⋊C4C22×C3112yes—
338C22×C22⋊C4C22×C3111yescharacteristic
339#385—28no—
340#380—28no—
341#379—24no—
342C2×#95C22×C3114yescomplemented
343C2×#95C22×C3114yescomplemented
344C2×#95C22×C3114yescomplemented
345C2×C31—28no—
346C2×C31—28no—
347C2×C31—44no—
348C2×C31C2×#38011yescharacteristic
349C2×C31C2×#38511yescharacteristic
350C2×C31#231511yescharacteristic
351C2×C31D4×C22⋊C412yes—
352C2×C31#234212yes—
353C2×C31#241012yes—
354C2×C31#241212yes—
355C2×C31#243714yes—
356C25—24no—
357C25—24no—
358C25—24no—
359C25—24no—
360C25—44no—
361C25C2×C4×C3114yes—
362C25D4×C3114yes—
363C25Q8×C3114yes—
364C25C23×C3111yescharacteristic
365C25C23×C3111yescharacteristic
366C25C23×C3111yescharacteristic
367C23×C4—24no—
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—28no—
371C23×C4—28no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—44no—
375C23×C4—44no—
376C23×C4C2×C4×C3118yescomplemented
377C23×C4C23×C3112yes—
378C23×C4C23×C3112yes—
379C23×C4C23×C3112yes—
380C23×C4C23×C3114yes—
381C23×C4C23×C3112yes—
382C2×C4⋊C4—28no—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—44no—
385C2×C22⋊C4—232no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—232no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—44no—
402C2×C22⋊C4C2×C4×C3114yes—
403C2×C22⋊C4C2×C4×C3114yes—
404C2×C22⋊C4C2×C4×C3118yescomplemented
405C2×C22⋊C4D4×C3118yes—
406C2×C22⋊C4D4×C3118yes—
407#95—28no—
408#95—28no—
409#95—44no—
410C31#1387711yescharacteristic, direct product factor, Sylow
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊C4×C31116yescomplemented
438C24C22×C4×C3112yes—
439C24C22×C4×C3112yes—
440C24C22×C4×C3112yes—
441C24C2×D4×C3112yes—
442C24C2×D4×C3114yes—
443C24C2×D4×C3112yes—
444C24C2×D4×C3112yes—
445C24C2×D4×C3114yes—
446C24C2×Q8×C3112yes—
447C24C2×Q8×C3112yes—
448C24C2×Q8×C3112yes—
449C24#41×C3114yes—
450C24#41×C3114yes—
451C24C24×C3111yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4C22×C4×C3114yes—
486C22×C4C22×C4×C3114yes—
487C22×C4C2×D4×C3118yes—
488C22×C4C2×D4×C3118yes—
489C4⋊C4—416no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4C22⋊C4×C31116yescomplemented
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—232no—
533C23—232no—
534C23—416no—
535C23—432no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—464no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊C4×C3114yes—
545C23C2×C4⋊C4×C3114yes—
546C23C2×C4⋊C4×C3114yes—
547C23C2×C4⋊C4×C3114yes—
548C23C31×#11714yes—
549C23C4×D4×C3118yes—
550C23C31×#12014yes—
551C23C31×#12118yes—
552C23C31×C22⋊Q818yes—
553C23C31×C22⋊Q818yes—
554C23C31×#12314yes—
555C23C23×C4×C3111yescharacteristic
556C23C22×D4×C3112yes—
557C23C22×D4×C3111yescharacteristic
558C23C22×D4×C3112yes—
559C23C22×Q8×C3111yescharacteristic
560C23C2×#41×C3112yes—
561C23C2×#41×C3112yes—
562C23C31×H(5,𝔽2)14yes—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C2×C22⋊C4×C3118yescomplemented
592C22—24no—
593C22—24no—
594C22—28no—
595C22—28no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—216no—
603C22—216no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22C31×#38018yescomplemented
616C22C31×#38518yescomplemented
617C22C22×C22⋊C4×C3111yescharacteristic
618C22C22×C4⋊C4×C3111yescharacteristic
619C22C2×C31×#11711yescharacteristic
620C22C2×C4×D4×C3112yes—
621C22C31×#51812yes—
622C22C2×C31×#12011yescharacteristic
623C22C2×C31×#12112yes—
624C22C2×C31×C22⋊Q812yes—
625C22C2×C31×C22⋊Q812yes—
626C22C2×C31×#12311yescharacteristic
627C22C31×#53412yes—
628C22C31×#54312yes—
629C22D42×C3114yes—
630C22C31×#54614yes—
631C22C31×#54614yes—
632C22C31×#55114yes—
633C4—28no—
634C4—44no—
635C4—48no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×C31×#38011yescharacteristic
642C2C2×C31×#38511yescharacteristic
643C2C31×#231511yescharacteristic
644C2D4×C22⋊C4×C3112yes—
645C2C31×#234212yes—
646C2C31×#241012yes—
647C2C31×#241212yes—
648C2C31×#243714yes—
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC26×C31C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#385C2semidirect
4. C2×C31×#380C2semidirect
6. C31×#1439C2semidirect
7. C31×C22⋊#95C2semidirect
8. C31×C22⋊#95C2semidirect
12. C22×C4⋊C4×C31C22semidirect
16. C22×C22⋊C4×C31C4semidirect
29. C2×C31×#95C22semidirect
30. C2×C31×#95C22semidirect
31. C2×C31×#95C22semidirect
52. C23×C4×C31C2×C4semidirect
80. C2×C22⋊C4×C31C2×C4semidirect
112. C24×C31C4⋊C4semidirect
173. C22⋊C4×C31C22⋊C4semidirect
174. #13877C31direct (N × H)
267. C2×C4×C31C2×C22⋊C4semidirect
270. C2×#385C2×C31semidirect
271. C2×#380C2×C31semidirect
273. #1439C2×C31semidirect
274. C22⋊#95C2×C31semidirect
275. C22⋊#95C2×C31semidirect
299. C22×C31#380semidirect
300. C22×C31#385semidirect
325. C22×C4⋊C4C22×C31semidirect
329. C22×C22⋊C4C4×C31semidirect
342. C2×#95C22×C31semidirect
343. C2×#95C22×C31semidirect
344. C2×#95C22×C31semidirect
376. C23×C4C2×C4×C31semidirect
404. C2×C22⋊C4C2×C4×C31semidirect
410. C31#13877direct (N × H)
437. C24C4⋊C4×C31semidirect
498. C22⋊C4C22⋊C4×C31semidirect
591. C2×C4C2×C22⋊C4×C31semidirect
615. C22C31×#380semidirect
616. C22C31×#385semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C312
C2×C31×#3852
C2×C31×#3802
C2×C31×#3792
C31×#14392
C31×C22⋊#952
#1387731