Finite Group Databasebeta

← C31×#14719 #21471478

Subgroups — C31×#14719 #21471478

Order 7936.

2982 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#121C211yesmaximal, characteristic
2C23×C4⋊C4×C31C211yesmaximal, characteristic, complemented
3C2×C31×#394C212yesmaximal, complemented
4C2×C31×#386C212yesmaximal, complemented
5C2×C31×#386C212yesmaximal, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C214yesmaximal, complemented
8C31×C22⋊#95C211yesmaximal, characteristic, complemented
9C23×D4×C31C2212yes—
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C24×C4×C31C2211yescharacteristic
13C24×C4×C31C2211yescharacteristic
14C2×C31×#121—28no—
15C2×C31×#121—24no—
16C22×C4⋊C4×C31—24no—
17C22×C4⋊C4×C31—28no—
18C22×C4⋊C4×C31C2212yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2214yes—
22C22×C22⋊C4×C31C2212yes—
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2214yes—
25C31×#394—28no—
26C31×#386—28no—
27C31×#386—28no—
28C2×C31×#95C2212yes—
29C2×C31×#95C2214yes—
30C2×C31×#95C2214yes—
31C2×C31×#95C2212yes—
32C25×C31C2312yes—
33C25×C31C2311yescharacteristic
34C22×D4×C31—24no—
35C22×D4×C31—24no—
36C22×D4×C31—24no—
37C22×D4×C31—24no—
38C22×D4×C31—24no—
39C22×D4×C31—24no—
40C22×D4×C31—24no—
41C22×D4×C31—22no—
42C22×D4×C31—24no—
43C22×D4×C31—42no—
44C22×D4×C31D418yes—
45C23×C4×C31—22no—
46C23×C4×C31—22no—
47C23×C4×C31—24no—
48C23×C4×C31—24no—
49C23×C4×C31—22no—
50C23×C4×C31—24no—
51C23×C4×C31—24no—
52C23×C4×C31—24no—
53C23×C4×C31—24no—
54C23×C4×C31—42no—
55C23×C4×C31—42no—
56C23×C4×C31D414yes—
57C23×C4×C31D414yescomplemented
58C23×C4×C31C2314yes—
59C23×C4×C31C2312yes—
60C23×C4×C31C2311yescharacteristic
61C23×C4×C31C2312yes—
62C23×C4×C31C2311yescharacteristic
63C23×C4×C31C2312yes—
64C31×#121—232no—
65C2×C4⋊C4×C31—232no—
66C2×C4⋊C4×C31—24no—
67C2×C4⋊C4×C31—24no—
68C2×C4⋊C4×C31—24no—
69C2×C4⋊C4×C31—24no—
70C2×C4⋊C4×C31—24no—
71C2×C4⋊C4×C31—24no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31D418yescomplemented
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—28no—
78C2×C22⋊C4×C31—24no—
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—24no—
81C2×C22⋊C4×C31—28no—
82C2×C22⋊C4×C31—28no—
83C2×C22⋊C4×C31—28no—
84C2×C22⋊C4×C31—28no—
85C2×C22⋊C4×C31—44no—
86C31×#95—28no—
87C31×#95—28no—
88C31×#95—24no—
89C31×#95—42no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—24no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—24no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—24no—
103C24×C31—24no—
104C24×C31C2×D414yes—
105C24×C31C2×D414yes—
106C24×C31C2×D412yes—
107C24×C31C2×D412yes—
108C24×C31#4112yes—
109C24×C31#4114yes—
110C24×C31C2411yescharacteristic
111C2×D4×C31—28no—
112C2×D4×C31—216no—
113C2×D4×C31—216no—
114C2×D4×C31—216no—
115C2×D4×C31—416no—
116C2×D4×C31—44no—
117C2×D4×C31—48no—
118C2×D4×C31—48no—
119C2×D4×C31—48no—
120C2×D4×C31—48no—
121C2×D4×C31—44no—
122C22×C4×C31—28no—
123C22×C4×C31—216no—
124C22×C4×C31—28no—
125C22×C4×C31—28no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—216no—
133C22×C4×C31—28no—
134C22×C4×C31—28no—
135C22×C4×C31—22no—
136C22×C4×C31—24no—
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—24no—
148C22×C4×C31—28no—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—24no—
155C22×C4×C31—24no—
156C22×C4×C31—44no—
157C22×C4×C31—416no—
158C22×C4×C31—416no—
159C22×C4×C31—44no—
160C22×C4×C31—44no—
161C22×C4×C31C2×D414yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D414yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D414yes—
167C22×C4×C31C2×D414yes—
168C22×C4×C31#4114yes—
169C4⋊C4×C31—216no—
170C4⋊C4×C31—28no—
171C4⋊C4×C31—216no—
172C22⋊C4×C31—216no—
173C22⋊C4×C31—416no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—416no—
176#14719C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—22no—
178C23×C31—24no—
179C23×C31—22no—
180C23×C31—24no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—28no—
186C23×C31—28no—
187C23×C31—28no—
188C23×C31—28no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—24no—
192C23×C31—28no—
193C23×C31—28no—
194C23×C31—42no—
195C23×C31—44no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—44no—
205C23×C31—44no—
206C23×C31—44no—
207C23×C31—44no—
208C23×C31—44no—
209C23×C31—44no—
210C23×C31—44no—
211C23×C31—44no—
212C23×C31#12014yes—
213C23×C31#12118yes—
214C23×C31#12114yes—
215C23×C31#12318yes—
216C23×C31#12414yes—
217C23×C31C22×D411yescharacteristic
218C23×C31C22×D412yes—
219C23×C31C22×D412yes—
220C23×C31C22×D411yescharacteristic
221C23×C31C2×#4111yescharacteristic
222C23×C31C2×#4112yes—
223C23×C31C2×#4112yes—
224C23×C31H(5,𝔽2)12yes—
225C23×C312−1+412yes—
226D4×C31—416no—
227D4×C31—432no—
228C2×C4×C31—232no—
229C2×C4×C31—24no—
230C2×C4×C31—22no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—22no—
237C2×C4×C31—28no—
238C2×C4×C31—22no—
239C2×C4×C31—24no—
240C2×C4×C31—22no—
241C2×C4×C31—24no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—22no—
245C2×C4×C31—28no—
246C2×C4×C31—416no—
247C2×C4×C31—416no—
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—44no—
255C2×C4×C31—44no—
256C2×C4×C31—44no—
257C2×C4×C31—44no—
258C2×C4×C31—44no—
259C2×C4×C31—44no—
260C2×C4×C31—44no—
261C2×C4×C31—44no—
262C2×C4×C31—44no—
263C2×C4×C31#12014yes—
264C2×C4×C31#12118yescomplemented
265C22×#121C2×C3111yescharacteristic
266C23×C4⋊C4C2×C3111yescharacteristic, complemented
267C2×#394C2×C3112yescomplemented
268C2×#386C2×C3112yescomplemented
269C2×#386C2×C3112yescomplemented
270#1441C2×C3112yescomplemented
271#1439C2×C3114yescomplemented
272C22⋊#95C2×C3111yescharacteristic, complemented
273C22×C31—24no—
274C22×C31—24no—
275C22×C31—24no—
276C22×C31—24no—
277C22×C31—24no—
278C22×C31—216no—
279C22×C31—24no—
280C22×C31—216no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—44no—
284C22×C31—48no—
285C22×C31—48no—
286C22×C31—82no—
287C22×C31—82no—
288C22×C31—82no—
289C22×C31—84no—
290C22×C31—82no—
291C22×C31—84no—
292C22×C31#39418yescomplemented
293C22×C31C2×#12011yescharacteristic
294C22×C31C2×#12011yescharacteristic
295C22×C31C2×#12111yescharacteristic
296C22×C31C2×#12112yes—
297C22×C31C2×#12112yes—
298C22×C31C2×#12312yes—
299C22×C31#52511yescharacteristic
300C22×C31#52512yes—
301C22×C31C2×#12411yescharacteristic
302C22×C31#53411yescharacteristic
303C22×C31#53611yescharacteristic
304C22×C31#53712yes—
305C22×C31#53912yes—
306C22×C31D4214yes—
307C22×C31#54714yes—
308C22×C31#54814yes—
309C22×C31#55314yes—
310C4×C31—28no—
311C4×C31—24no—
312C4×C31—44no—
313C4×C31—42no—
314C4×C31—44no—
315C4×C31—84no—
316C23×D4C22×C3112yes—
317C23×D4C22×C3111yescharacteristic
318C24×C4C22×C3111yescharacteristic
319C24×C4C22×C3111yescharacteristic
320C24×C4C22×C3111yescharacteristic
321C2×#121—28no—
322C2×#121—24no—
323C22×C4⋊C4—24no—
324C22×C4⋊C4—28no—
325C22×C4⋊C4C22×C3112yes—
326C22×C4⋊C4C22×C3111yescharacteristic
327C22×C4⋊C4C22×C3112yes—
328C22×C22⋊C4C22×C3114yes—
329C22×C22⋊C4C22×C3112yes—
330C22×C22⋊C4C22×C3112yes—
331C22×C22⋊C4C22×C3114yes—
332#394—28no—
333#386—28no—
334#386—28no—
335C2×#95C22×C3112yes—
336C2×#95C22×C3114yes—
337C2×#95C22×C3114yes—
338C2×#95C22×C3112yes—
339C2×C31—28no—
340C2×C31—84no—
341C2×C31C2×#39411yescharacteristic
342C2×C31#240311yescharacteristic
343C2×C31#240611yescharacteristic
344C2×C31#241614yes—
345C2×C31#243611yescharacteristic
346C2×C31#244314yes—
347C2×C31#259111yescharacteristic
348C2×C31#261811yescharacteristic
349C2×C31#284711yescharacteristic
350C25C23×C3112yes—
351C25C23×C3111yescharacteristic
352C22×D4—24no—
353C22×D4—24no—
354C22×D4—24no—
355C22×D4—24no—
356C22×D4—24no—
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—24no—
361C22×D4—42no—
362C22×D4D4×C3118yes—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—24no—
366C23×C4—24no—
367C23×C4—22no—
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—42no—
373C23×C4—42no—
374C23×C4D4×C3114yes—
375C23×C4D4×C3114yescomplemented
376C23×C4C23×C3114yes—
377C23×C4C23×C3112yes—
378C23×C4C23×C3111yescharacteristic
379C23×C4C23×C3112yes—
380C23×C4C23×C3111yescharacteristic
381C23×C4C23×C3112yes—
382#121—232no—
383C2×C4⋊C4—232no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4D4×C3118yescomplemented
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—28no—
401C2×C22⋊C4—28no—
402C2×C22⋊C4—28no—
403C2×C22⋊C4—44no—
404#95—28no—
405#95—28no—
406#95—24no—
407#95—42no—
408C31#1471911yescharacteristic, direct product factor, Sylow
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—24no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—24no—
422C24—24no—
423C24C2×D4×C3114yes—
424C24C2×D4×C3114yes—
425C24C2×D4×C3112yes—
426C24C2×D4×C3112yes—
427C24#41×C3112yes—
428C24#41×C3114yes—
429C24C24×C3111yescharacteristic
430C2×D4—28no—
431C2×D4—216no—
432C2×D4—216no—
433C2×D4—216no—
434C2×D4—416no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—44no—
441C22×C4—28no—
442C22×C4—216no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—24no—
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×C4—24no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—44no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4C2×D4×C3114yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3114yes—
483C22×C4C2×D4×C3114yes—
484C22×C4C2×D4×C3112yes—
485C22×C4C2×D4×C3114yes—
486C22×C4C2×D4×C3114yes—
487C22×C4#41×C3114yes—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23C31×#12014yes—
531C23C31×#12118yes—
532C23C31×#12114yes—
533C23C31×#12318yes—
534C23C31×#12414yes—
535C23C22×D4×C3111yescharacteristic
536C23C22×D4×C3112yes—
537C23C22×D4×C3112yes—
538C23C22×D4×C3111yescharacteristic
539C23C2×#41×C3111yescharacteristic
540C23C2×#41×C3112yes—
541C23C2×#41×C3112yes—
542C23C31×H(5,𝔽2)12yes—
543C23C31×2−1+412yes—
544D4—416no—
545D4—432no—
546C2×C4—232no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4C31×#12014yes—
582C2×C4C31×#12118yescomplemented
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—216no—
589C22—24no—
590C22—216no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—84no—
600C22—82no—
601C22—84no—
602C22C31×#39418yescomplemented
603C22C2×C31×#12011yescharacteristic
604C22C2×C31×#12011yescharacteristic
605C22C2×C31×#12111yescharacteristic
606C22C2×C31×#12112yes—
607C22C2×C31×#12112yes—
608C22C2×C31×#12312yes—
609C22C31×#52511yescharacteristic
610C22C31×#52512yes—
611C22C2×C31×#12411yescharacteristic
612C22C31×#53411yescharacteristic
613C22C31×#53611yescharacteristic
614C22C31×#53712yes—
615C22C31×#53912yes—
616C22D42×C3114yes—
617C22C31×#54714yes—
618C22C31×#54814yes—
619C22C31×#55314yes—
620C4—28no—
621C4—24no—
622C4—44no—
623C4—42no—
624C4—44no—
625C4—84no—
626C2—28no—
627C2—84no—
628C2C2×C31×#39411yescharacteristic
629C2C31×#240311yescharacteristic
630C2C31×#240611yescharacteristic
631C2C31×#241614yes—
632C2C31×#243611yescharacteristic
633C2C31×#244314yes—
634C2C31×#259111yescharacteristic
635C2C31×#261811yescharacteristic
636C2C31×#284711yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×C31×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C4⋊C4×C31C2semidirect
3. C2×C31×#394C2semidirect
4. C2×C31×#386C2semidirect
5. C2×C31×#386C2semidirect
6. C31×#1441C2semidirect
7. C31×#1439C2semidirect
8. C31×C22⋊#95C2semidirect
57. C23×C4×C31D4semidirect
73. C2×C4⋊C4×C31D4semidirect
176. #14719C31direct (N × H)
264. C2×C4×C31#121semidirect
266. C23×C4⋊C4C2×C31semidirect
267. C2×#394C2×C31semidirect
268. C2×#386C2×C31semidirect
269. C2×#386C2×C31semidirect
270. #1441C2×C31semidirect
271. #1439C2×C31semidirect
272. C22⋊#95C2×C31semidirect
292. C22×C31#394semidirect
375. C23×C4D4×C31semidirect
391. C2×C4⋊C4D4×C31semidirect
408. C31#14719direct (N × H)
582. C2×C4C31×#121semidirect
602. C22C31×#394semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1212
C23×C4⋊C4×C312
C2×C31×#3942
C2×C31×#3862
C31×#14412
C31×#14392
C31×C22⋊#952
#1471931