Finite Group Databasebeta

← C31×#25602 #21482361

Subgroups — C31×#25602 #21482361

Order 7936.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic, complemented
2C2×C31×#496C211yesmaximal, characteristic, complemented
3C2×C31×#489C211yesmaximal, characteristic, complemented
4C31×#2850C211yesmaximal, characteristic, complemented
5C31×#2000C212yesmaximal, complemented
6C31×#1999C212yesmaximal, complemented
7C31×#1993C211yesmaximal, characteristic, complemented
8C31×#1961C211yesmaximal, characteristic, complemented
9C31×#1879C212yesmaximal, complemented
10C31×#1877C212yesmaximal, complemented
11C31×#1749C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×SD16×C31C2212yes—
14C22×D8×C31C2212yes—
15C31×#532—24no—
16C31×#532—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#127C2211yescharacteristic
19C2×C31×#124—21no—
20C2×C31×#124C2211yescharacteristic
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121—21no—
23C2×C31×#121C2212yes—
24C2×C31×#121C2211yescharacteristic
25C2×C4×D4×C31—21no—
26C2×C4×D4×C31C2212yes—
27C22×C42×C31C2211yescharacteristic
28C31×#496—24no—
29C31×#489—24no—
30C2×C31×#103C2212yes—
31C2×C31×#102C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×C22⋊C8C2212yes—
35C2×C31×#97C2211yescharacteristic
36C2×C31×#97C2211yescharacteristic
37C31×#391C2211yescharacteristic
38C31×#390C2211yescharacteristic
39C31×#388—21no—
40C31×#388—21no—
41C31×#386—21no—
42C31×#386—21no—
43C31×#382C2211yescharacteristic
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—28no—
48C2×#41×C31—44no—
49C2×#41×C31—44no—
50C22×Q8×C31—21no—
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—22no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31—21no—
56C22×D4×C31C2311yescharacteristic
57C22×D4×C31C2312yes—
58C23×C4×C31—22no—
59C23×C4×C31D411yescharacteristic
60C23×C4×C31D411yescharacteristic
61C23×C4×C31C2311yescharacteristic
62C2×SD16×C31—24no—
63C2×SD16×C31—24no—
64C2×SD16×C31—42no—
65C2×D8×C31—24no—
66C2×D8×C31—24no—
67C2×D8×C31—42no—
68C22×C8×C31C2312yes—
69C22×C8×C31C2312yes—
70C31×C4⋊Q8—22no—
71C31×C4⋊Q8D414yescomplemented
72C31×#127—22no—
73C31×#127D414yescomplemented
74C31×#124—28no—
75C31×#124—22no—
76C31×#124—22no—
77C31×C22⋊Q8—42no—
78C31×#121—28no—
79C31×#121—24no—
80C31×#121—24no—
81C31×#121—42no—
82C4×D4×C31—28no—
83C4×D4×C31—24no—
84C4×D4×C31—24no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C42×C31—22no—
99C2×C42×C31—22no—
100C2×C42×C31D411yescharacteristic
101C2×C42×C31D411yescharacteristic
102C2×C42×C31C2311yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×#103—24no—
105C31×#102—24no—
106C31×#102—42no—
107C31×#102—42no—
108C31×C22⋊C8—24no—
109C31×#97—22no—
110C31×#97—22no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—21no—
115C24×C31—22no—
116C24×C31—21no—
117C24×C31C2×D411yescharacteristic
118#41×C31—432no—
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—42no—
125C2×Q8×C31—42no—
126C2×D4×C31—28no—
127C2×D4×C31—24no—
128C2×D4×C31—24no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—42no—
139C2×D4×C31—42no—
140C2×D4×C31—42no—
141C2×D4×C31—42no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31C2×D414yes—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—28no—
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×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—41no—
162C22×C4×C31—41no—
163C22×C4×C31—44no—
164C22×C4×C31—44no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—48no—
172D8×C31—48no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31#4114yescomplemented
179C4⋊C4×C31—24no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—42no—
184C4⋊C4×C31C2×D414yes—
185C22⋊C4×C31—28no—
186C22⋊C4×C31—28no—
187C22⋊C4×C31—42no—
188C22⋊C4×C31—42no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—42no—
191C22⋊C4×C31—42no—
192C22⋊C4×C31—42no—
193C42×C31—22no—
194C42×C31—22no—
195C42×C31—21no—
196C42×C31—21no—
197C42×C31C2×D412yes—
198C42×C31C2×D412yes—
199#25602C3111yesmaximal, characteristic, direct product factor, Sylow
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—21no—
206C23×C31—24no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—41no—
214C23×C31—41no—
215C23×C31—41no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31AGL(1,ℤ/8)12yes—
221C23×C31C22×D411yescharacteristic
222Q8×C31—48no—
223Q8×C31—42no—
224Q8×C31—42no—
225D4×C31—24no—
226D4×C31—24no—
227D4×C31—48no—
228D4×C31—48no—
229D4×C31—416no—
230D4×C31—42no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—42no—
234D4×C31—84no—
235C2×C4×C31—28no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—21no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—24no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—44no—
250C2×C4×C31—416no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31#12012yes—
265C2×C4×C31#12012yes—
266C2×C4×C31AGL(1,ℤ/8)12yes—
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C2×#4112yes—
270C2×C4×C31H(5,𝔽2)12yes—
271C8×C31—24no—
272C8×C31—42no—
273C2×#532C2×C3111yescharacteristic, complemented
274C2×#496C2×C3111yescharacteristic, complemented
275C2×#489C2×C3111yescharacteristic, complemented
276#2850C2×C3111yescharacteristic, complemented
277#2000C2×C3112yescomplemented
278#1999C2×C3112yescomplemented
279#1993C2×C3111yescharacteristic, complemented
280#1961C2×C3111yescharacteristic, complemented
281#1879C2×C3112yescomplemented
282#1877C2×C3112yescomplemented
283#1749C2×C3111yescharacteristic, complemented
284C22×C31—22no—
285C22×C31—22no—
286C22×C31—22no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31—82no—
297C22×C31—82no—
298C22×C31—81no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—81no—
303C22×C31—81no—
304C22×C31—81no—
305C22×C31C2×#12011yescharacteristic
306C22×C31#53211yescharacteristic
307C22×C31#53511yescharacteristic
308C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C31#57511yescharacteristic
311C22×C31#57511yescharacteristic
312C4×C31—22no—
313C4×C31—22no—
314C4×C31—22no—
315C4×C31—41no—
316C4×C31—44no—
317C4×C31—41no—
318C4×C31—81no—
319C4×C31—81no—
320C4×C31—81no—
321C4×C31#54612yes—
322C4×C31#55012yes—
323C22×#41—21no—
324C22×SD16C22×C3112yes—
325C22×D8C22×C3112yes—
326#532—24no—
327#532—24no—
328C2×C4⋊Q8C22×C3111yescharacteristic
329C2×#127C22×C3111yescharacteristic
330C2×#124—21no—
331C2×#124C22×C3111yescharacteristic
332C2×C22⋊Q8C22×C3111yescharacteristic
333C2×#121—21no—
334C2×#121C22×C3112yes—
335C2×#121C22×C3111yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4C22×C3112yes—
338C22×C42C22×C3111yescharacteristic
339#496—24no—
340#489—24no—
341C2×#103C22×C3112yes—
342C2×#102C22×C3112yes—
343C2×#102C22×C3112yes—
344C2×#102C22×C3112yes—
345C2×C22⋊C8C22×C3112yes—
346C2×#97C22×C3111yescharacteristic
347C2×#97C22×C3111yescharacteristic
348#391C22×C3111yescharacteristic
349#390C22×C3111yescharacteristic
350#388—21no—
351#388—21no—
352#386—21no—
353#386—21no—
354#382C22×C3111yescharacteristic
355#340C22×C3112yes—
356#340C22×C3112yes—
357#336C22×C3112yes—
358C2×C31—42no—
359C2×C31—84no—
360C2×C31—161no—
361C2×C31#243111yescharacteristic
362C2×C31#300311yescharacteristic
363C2×C31#300511yescharacteristic
364C2×C31#317111yescharacteristic
365C2×C31#318011yescharacteristic
366C2×C31#321311yescharacteristic
367C2×C31#321411yescharacteristic
368C2×#41—28no—
369C2×#41—44no—
370C2×#41—44no—
371C22×Q8—21no—
372C22×Q8C23×C3111yescharacteristic
373C22×D4—22no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4C23×C3111yescharacteristic
378C22×D4C23×C3112yes—
379C23×C4—22no—
380C23×C4D4×C3111yescharacteristic
381C23×C4D4×C3111yescharacteristic
382C23×C4C23×C3111yescharacteristic
383C2×SD16—24no—
384C2×SD16—24no—
385C2×SD16—42no—
386C2×D8—24no—
387C2×D8—24no—
388C2×D8—42no—
389C22×C8C23×C3112yes—
390C22×C8C23×C3112yes—
391C4⋊Q8—22no—
392C4⋊Q8D4×C3114yescomplemented
393#127—22no—
394#127D4×C3114yescomplemented
395#124—28no—
396#124—22no—
397#124—22no—
398C22⋊Q8—42no—
399#121—28no—
400#121—24no—
401#121—24no—
402#121—42no—
403C4×D4—28no—
404C4×D4—24no—
405C4×D4—24no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4C23×C3111yescharacteristic
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C4⋊C4C23×C3112yes—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42D4×C3111yescharacteristic
422C2×C42D4×C3111yescharacteristic
423C2×C42C23×C3111yescharacteristic
424C2×C42C23×C3111yescharacteristic
425#103—24no—
426#102—24no—
427#102—42no—
428#102—42no—
429C22⋊C8—24no—
430#97—22no—
431#97—22no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436C31#2560211yescharacteristic, direct product factor, Sylow
437C24—22no—
438C24—21no—
439C24C2×D4×C3111yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4C2×D4×C3114yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C24×C3111yescharacteristic
493SD16—48no—
494D8—48no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8#41×C3114yescomplemented
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4C2×D4×C3114yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42C2×D4×C3112yes—
520C42C2×D4×C3112yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C31×AGL(1,ℤ/8)12yes—
542C23C22×D4×C3111yescharacteristic
543Q8—48no—
544Q8—42no—
545Q8—42no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—48no—
550D4—416no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4C31×#12012yes—
586C2×C4C31×#12012yes—
587C2×C4C31×AGL(1,ℤ/8)12yes—
588C2×C4C22×D4×C3111yescharacteristic
589C2×C4C22×D4×C3111yescharacteristic
590C2×C4C2×#41×C3112yes—
591C2×C4C31×H(5,𝔽2)12yes—
592C8—24no—
593C8—42no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×C31×#12011yescharacteristic
616C22C31×#53211yescharacteristic
617C22C31×#53511yescharacteristic
618C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
619C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
620C22C31×#57511yescharacteristic
621C22C31×#57511yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—44no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4C31×#54612yes—
632C4C31×#55012yes—
633C2—42no—
634C2—84no—
635C2—161no—
636C2C31×#243111yescharacteristic
637C2C31×#300311yescharacteristic
638C2C31×#300511yescharacteristic
639C2C31×#317111yescharacteristic
640C2C31×#318011yescharacteristic
641C2C31×#321311yescharacteristic
642C2C31×#321411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#532C2semidirect
2. C2×C31×#496C2semidirect
3. C2×C31×#489C2semidirect
4. C31×#2850C2semidirect
5. C31×#2000C2semidirect
6. C31×#1999C2semidirect
7. C31×#1993C2semidirect
8. C31×#1961C2semidirect
9. C31×#1879C2semidirect
10. C31×#1877C2semidirect
11. C31×#1749C2semidirect
71. C31×C4⋊Q8D4semidirect
73. C31×#127D4semidirect
178. C2×C8×C31#41semidirect
199. #25602C31direct (N × H)
273. C2×#532C2×C31semidirect
274. C2×#496C2×C31semidirect
275. C2×#489C2×C31semidirect
276. #2850C2×C31semidirect
277. #2000C2×C31semidirect
278. #1999C2×C31semidirect
279. #1993C2×C31semidirect
280. #1961C2×C31semidirect
281. #1879C2×C31semidirect
282. #1877C2×C31semidirect
283. #1749C2×C31semidirect
392. C4⋊Q8D4×C31semidirect
394. #127D4×C31semidirect
436. C31#25602direct (N × H)
500. C2×C8#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C31×#4962
C2×C31×#4892
C31×#28502
C31×#20002
C31×#19992
C31×#19932
C31×#19612
C31×#18792
C31×#18772
C31×#17492
#2560231