Finite Group Databasebeta

← C31×#19048 #21475807

Subgroups — C31×#19048 #21475807

Order 7936.

908 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
1C31×#3561C211yesmaximal, characteristic, complemented
2C31×#3213C211yesmaximal, characteristic, complemented
3C31×#3180C211yesmaximal, characteristic, complemented
4C31×#3135C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1662C212yesmaximal, complemented
7C31×#1657C212yesmaximal, complemented
8C31×#1656C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1531C211yesmaximal, characteristic, complemented
11C2×C31×#329C211yesmaximal, characteristic, complemented
12C31×#563C2212yes—
13C31×#562C2212yes—
14C31×#556—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17C31×#548—21no—
18C31×#546—21no—
19C31×#543—21no—
20C31×#542—21no—
21C31×#542—22no—
22C31×#538—21no—
23C31×#536—21no—
24C31×#533C2211yescharacteristic
25C31×#532C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501C2211yescharacteristic, complemented
29C31×#496C2212yes—
30C31×#496C2211yescharacteristic, complemented
31C31×#491C2212yescomplemented
32C31×#490C2212yescomplemented
33C31×#490C2211yescharacteristic, complemented
34C31×#489—21no—
35C31×#489—21no—
36C31×#489C2212yes—
37C31×#489C2211yescharacteristic, complemented
38C31×#466—21no—
39C31×#465—21no—
40C31×#463—22no—
41C31×#448—21no—
42C31×#447—21no—
43C31×#442—21no—
44C31×#440—21no—
45C31×#432C2211yescharacteristic
46C31×#416—22no—
47C2×C31×#97C2211yescharacteristic
48C2×C31×#97C2211yescharacteristic
49C31×#330C2212yescomplemented
50C31×#330C2212yescomplemented
51C31×#330C2212yescomplemented
52C31×#329C2212yes—
53C31×#329C2212yes—
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31D411yescharacteristic
60C2×SD16×C31—21no—
61C2×SD16×C31—22no—
62C2×D8×C31—21no—
63C2×D8×C31—22no—
64C22×C8×C31—21no—
65C22×C8×C31—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D411yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×#127C2311yescharacteristic
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—21no—
76C31×#125C2312yes—
77C31×#125C2311yescharacteristic
78C31×#124—21no—
79C31×#124—22no—
80C31×#124—22no—
81C31×#124—21no—
82C31×#124—21no—
83C31×#124—21no—
84C31×#124D412yes—
85C31×#124C2312yes—
86C31×#124C2311yescharacteristic
87C31×#123—22no—
88C31×#123—22no—
89C31×#123—21no—
90C31×#123—22no—
91C31×#123—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8—21no—
100C31×C22⋊Q8—22no—
101C31×C22⋊Q8—22no—
102C31×C22⋊Q8D411yescharacteristic
103C31×#121—21no—
104C31×#121—22no—
105C31×#121—22no—
106C31×#121—21no—
107C31×#121—21no—
108C31×#121D411yescharacteristic
109C31×#120—22no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×Q8×C31—21no—
113C4×Q8×C31D411yescharacteristic
114C4×D4×C31—22no—
115C4×D4×C31—21no—
116C4×D4×C31—22no—
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31—21no—
120C4×D4×C31D411yescharacteristic
121C31×#117—22no—
122C31×#117—21no—
123C31×#117D412yes—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—21no—
129C31×#106—21no—
130C31×C4⋊C8—21no—
131C31×#103—21no—
132C31×#103—22no—
133C31×#103—22no—
134C31×#103—22no—
135C31×#102—21no—
136C31×#102—22no—
137C31×#102—22no—
138C31×#102—22no—
139C31×#102—21no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C31×#97C2311yescharacteristic
145C31×#97C2312yes—
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148C24×C31—21no—
149#41×C31—44no—
150#41×C31—82no—
151C2×Q8×C31—22no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—82no—
157C2×Q8×C31—81no—
158C2×Q8×C31C2×D411yescharacteristic
159C2×Q8×C31C2×D412yes—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—21no—
166C2×D4×C31—21no—
167C2×D4×C31—41no—
168C2×D4×C31—81no—
169C2×D4×C31C2×D411yescharacteristic
170C2×D4×C31C2×D412yes—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—41no—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D411yescharacteristic
180SD16×C31—42no—
181SD16×C31—82no—
182D8×C31—42no—
183D8×C31—82no—
184C2×C8×C31—21no—
185C2×C8×C31—21no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—42no—
189C2×C8×C31—42no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31C2×D412yes—
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—41no—
224C22⋊C4×C31—41no—
225C42×C31—22no—
226C42×C31—22no—
227C42×C31—21no—
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2411yescharacteristic
232#19048C3111yesmaximal, characteristic, direct product factor, Sylow
233C23×C31—22no—
234C23×C31—21no—
235C23×C31—42no—
236C23×C31—41no—
237C23×C31—81no—
238C23×C31—81no—
239C23×C31#12011yescharacteristic
240Q8×C31—41no—
241Q8×C31—42no—
242Q8×C31—42no—
243Q8×C31—42no—
244Q8×C31—42no—
245Q8×C31—81no—
246Q8×C31—82no—
247D4×C31—41no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—41no—
251D4×C31—42no—
252D4×C31—41no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31—82no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—42no—
268C2×C4×C31—41no—
269C2×C4×C31—82no—
270C2×C4×C31—81no—
271C2×C4×C31—82no—
272C2×C4×C31—82no—
273C2×C4×C31—82no—
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C2×C4×C31C22×D411yescharacteristic
279C2×C4×C31C22×D411yescharacteristic
280C8×C31—22no—
281C8×C31—42no—
282C8×C31—41no—
283C8×C31—42no—
284#3561C2×C3111yescharacteristic, complemented
285#3213C2×C3111yescharacteristic, complemented
286#3180C2×C3111yescharacteristic, complemented
287#3135C2×C3111yescharacteristic, complemented
288#1664C2×C3112yescomplemented
289#1662C2×C3112yescomplemented
290#1657C2×C3112yescomplemented
291#1656C2×C3112yescomplemented
292#1532C2×C3111yescharacteristic, complemented
293#1531C2×C3111yescharacteristic, complemented
294C2×#329C2×C3111yescharacteristic, complemented
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—41no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—41no—
301C22×C31—41no—
302C22×C31—42no—
303C22×C31—81no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31—162no—
307C22×C31#45412yes—
308C22×C31C2×#12011yescharacteristic
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—41no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—41no—
319C4×C31—42no—
320C4×C31—42no—
321C4×C31—81no—
322#563C22×C3112yes—
323#562C22×C3112yes—
324#556—21no—
325#551—22no—
326D4×Q8—21no—
327#548—21no—
328#546—21no—
329#543—21no—
330#542—21no—
331#542—22no—
332#538—21no—
333#536—21no—
334#533C22×C3111yescharacteristic
335#532C22×C3111yescharacteristic
336#529C22×C3111yescharacteristic
337C2×#124C22×C3111yescharacteristic
338#501C22×C3111yescharacteristic, complemented
339#496C22×C3112yes—
340#496C22×C3111yescharacteristic, complemented
341#491C22×C3112yescomplemented
342#490C22×C3112yescomplemented
343#490C22×C3111yescharacteristic, complemented
344#489—21no—
345#489—21no—
346#489C22×C3112yes—
347#489C22×C3111yescharacteristic, complemented
348#466—21no—
349#465—21no—
350#463—22no—
351#448—21no—
352#447—21no—
353#442—21no—
354#440—21no—
355#432C22×C3111yescharacteristic
356#416—22no—
357C2×#97C22×C3111yescharacteristic
358C2×#97C22×C3111yescharacteristic
359#330C22×C3112yescomplemented
360#330C22×C3112yescomplemented
361#330C22×C3112yescomplemented
362#329C22×C3112yes—
363#329C22×C3112yes—
364#329C22×C3112yescomplemented
365C2×C31—22no—
366C2×C31—81no—
367C2×C31—82no—
368C2×C31—161no—
369C2×C31C2×#45411yescharacteristic
370C2×C31#301811yescharacteristic
371C2×C31#301911yescharacteristic
372C2×#41—21no—
373C2×#41—21no—
374C22×Q8—21no—
375C22×Q8D4×C3111yescharacteristic
376C22×D4D4×C3111yescharacteristic
377C2×SD16—21no—
378C2×SD16—22no—
379C2×D8—21no—
380C2×D8—22no—
381C22×C8—21no—
382C22×C8—21no—
383C4⋊Q8—21no—
384C4⋊Q8—22no—
385C4⋊Q8D4×C3111yescharacteristic
386C4⋊Q8D4×C3111yescharacteristic
387C4⋊Q8C23×C3111yescharacteristic
388#127C23×C3111yescharacteristic
389#126—22no—
390#126—22no—
391#126—22no—
392#126—21no—
393#125C23×C3111yescharacteristic
394#125C23×C3112yes—
395#124—21no—
396#124—22no—
397#124—22no—
398#124—21no—
399#124—21no—
400#124—21no—
401#124D4×C3112yes—
402#124C23×C3112yes—
403#124C23×C3111yescharacteristic
404#123—22no—
405#123—22no—
406#123—21no—
407#123—22no—
408#123—22no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8D4×C3111yescharacteristic
420#121—21no—
421#121—22no—
422#121—22no—
423#121—21no—
424#121—21no—
425#121D4×C3111yescharacteristic
426#120—22no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8D4×C3111yescharacteristic
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4D4×C3111yescharacteristic
438#117—22no—
439#117—21no—
440#117D4×C3112yes—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C42C23×C3111yescharacteristic
445#107—21no—
446#106—21no—
447C4⋊C8—21no—
448#103—21no—
449#103—22no—
450#103—22no—
451#103—22no—
452#102—21no—
453#102—22no—
454#102—22no—
455#102—22no—
456#102—21no—
457#102—21no—
458C22⋊C8—21no—
459#97—21no—
460#97—21no—
461#97C23×C3112yes—
462#97C23×C3111yescharacteristic
463#97C23×C3111yescharacteristic
464#97C23×C3112yes—
465C31#1904811yescharacteristic, direct product factor, Sylow
466C24—21no—
467#41—44no—
468#41—82no—
469C2×Q8—22no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—82no—
475C2×Q8—81no—
476C2×Q8C2×D4×C3111yescharacteristic
477C2×Q8C2×D4×C3112yes—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—41no—
486C2×D4—81no—
487C2×D4C2×D4×C3111yescharacteristic
488C2×D4C2×D4×C3112yes—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C2×D4×C3111yescharacteristic
498SD16—42no—
499SD16—82no—
500D8—42no—
501D8—82no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C2×C8—42no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4C2×D4×C3112yes—
523C4⋊C4C2×D4×C3112yes—
524C4⋊C4C2×D4×C3111yescharacteristic
525C4⋊C4C2×D4×C3111yescharacteristic
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42C2×D4×C3111yescharacteristic
547C42C2×D4×C3111yescharacteristic
548C42C2×D4×C3111yescharacteristic
549C42C24×C3111yescharacteristic
550C23—22no—
551C23—21no—
552C23—42no—
553C23—41no—
554C23—81no—
555C23—81no—
556C23C31×#12011yescharacteristic
557Q8—41no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—81no—
563Q8—82no—
564D4—41no—
565D4—42no—
566D4—42no—
567D4—41no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—42no—
572D4—82no—
573D4—82no—
574D4—82no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4C31×#12011yescharacteristic
592C2×C4C31×#12011yescharacteristic
593C2×C4C31×#12011yescharacteristic
594C2×C4C22×D4×C3111yescharacteristic
595C2×C4C22×D4×C3111yescharacteristic
596C2×C4C22×D4×C3111yescharacteristic
597C8—22no—
598C8—42no—
599C8—41no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—162no—
613C22C31×#45412yes—
614C22C2×C31×#12011yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—81no—
628C2—22no—
629C2—82no—
630C2—81no—
631C2—161no—
632C2C2×C31×#45411yescharacteristic
633C2C31×#301811yescharacteristic
634C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3561C2semidirect
2. C31×#3213C2semidirect
3. C31×#3180C2semidirect
4. C31×#3135C2semidirect
5. C31×#1664C2semidirect
6. C31×#1662C2semidirect
7. C31×#1657C2semidirect
8. C31×#1656C2semidirect
9. C31×#1532C2semidirect
10. C31×#1531C2semidirect
11. C2×C31×#329C2semidirect
28. C31×#501C22semidirect
30. C31×#496C22semidirect
31. C31×#491C22semidirect
32. C31×#490C22semidirect
33. C31×#490C22semidirect
37. C31×#489C22semidirect
49. C31×#330C22semidirect
50. C31×#330C22semidirect
51. C31×#330C22semidirect
54. C31×#329C22semidirect
232. #19048C31direct (N × H)
284. #3561C2×C31semidirect
285. #3213C2×C31semidirect
286. #3180C2×C31semidirect
287. #3135C2×C31semidirect
288. #1664C2×C31semidirect
289. #1662C2×C31semidirect
290. #1657C2×C31semidirect
291. #1656C2×C31semidirect
292. #1532C2×C31semidirect
293. #1531C2×C31semidirect
294. C2×#329C2×C31semidirect
338. #501C22×C31semidirect
340. #496C22×C31semidirect
341. #491C22×C31semidirect
342. #490C22×C31semidirect
343. #490C22×C31semidirect
347. #489C22×C31semidirect
359. #330C22×C31semidirect
360. #330C22×C31semidirect
361. #330C22×C31semidirect
364. #329C22×C31semidirect
465. C31#19048direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#32132
C31×#31802
C31×#31352
C31×#16642
C31×#16622
C31×#16572
C31×#16562
C31×#15322
C31×#15312
C2×C31×#3292
#1904831