Finite Group Databasebeta

← C31×#18291 #21475050

Subgroups — C31×#18291 #21475050

Order 7936.

984 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic
2C31×#3190C212yesmaximal, complemented
3C31×#2638C211yesmaximal, characteristic, complemented
4C31×#1609C212yesmaximal, complemented
5C31×#1608C211yesmaximal, characteristic, complemented
6C31×#1607C211yesmaximal, characteristic, complemented
7C31×#1603C212yesmaximal, complemented
8C31×#1602C211yesmaximal, characteristic, complemented
9C31×#1601C211yesmaximal, characteristic, complemented
10C31×#1516C212yesmaximal, complemented
11C31×#1459C211yesmaximal, characteristic, complemented
12C31×#543C2211yescharacteristic
13C31×#539—22no—
14C31×#539C2212yes—
15C31×#538—22no—
16C31×#538—21no—
17C31×#534—21no—
18C31×#534C2211yescharacteristic
19C31×#529—22no—
20C31×#529—21no—
21C31×#525—22no—
22C31×#525C2212yes—
23C2×C31×C22⋊Q8C2211yescharacteristic, complemented
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#485—22no—
27C31×#483C2212yescomplemented
28C31×#480C2212yes—
29C31×#467—22no—
30C31×#465—22no—
31C31×#452—22no—
32C31×#451C2212yescomplemented
33C31×#448C2212yes—
34C2×C31×C22⋊C8C2212yescomplemented
35C31×#398C2211yescharacteristic, complemented
36C31×#397—21no—
37C31×#397C2211yescharacteristic
38C31×#395C2211yescharacteristic, complemented
39C31×#394—21no—
40C31×#393C2211yescharacteristic
41C31×#387—21no—
42C31×#385—21no—
43C31×#381C2211yescharacteristic, complemented
44C31×#328C2212yescomplemented
45C31×#328C2212yescomplemented
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yescomplemented
49C31×#323C2211yescharacteristic
50C31×#323C2211yescharacteristic
51C2×#41×C31D412yes—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31D411yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31D411yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C2×Q16×C31—22no—
59C2×SD16×C31—22no—
60C22×C8×C31—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#126—22no—
64C31×#125—21no—
65C31×#125—22no—
66C31×#124—22no—
67C31×#124—21no—
68C31×#123—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8D412yes—
81C31×C22⋊Q8C2312yes—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—21no—
86C31×#121D412yes—
87C31×#121C2312yes—
88C31×#120—22no—
89C31×#120—22no—
90C31×#120—22no—
91C4×Q8×C31—21no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—21no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C31×#117—22no—
101C31×#117D412yes—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31C2311yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—21no—
112C2×C22⋊C4×C31—21no—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31D411yescharacteristic
115C2×C22⋊C4×C31C2311yescharacteristic
116C2×C42×C31—21no—
117C2×C42×C31D411yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#103—22no—
123C31×#102—22no—
124C31×C22⋊C8—22no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C31×#95—21no—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134C24×C31—21no—
135C24×C31C2×D411yescharacteristic
136C24×C31C2×D411yescharacteristic
137#41×C31—82no—
138C2×Q8×C31—21no—
139C2×Q8×C31—22no—
140C2×Q8×C31—81no—
141C2×Q8×C31C2×D412yes—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—41no—
150C2×D4×C31—42no—
151C2×D4×C31—42no—
152C2×D4×C31C2×D412yes—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—81no—
165C22×C4×C31—81no—
166C22×C4×C31—82no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D412yes—
173C22×C4×C31C2411yescharacteristic
174Q16×C31—82no—
175SD16×C31—82no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—82no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—42no—
190C4⋊C4×C31—41no—
191C4⋊C4×C31—41no—
192C4⋊C4×C31—81no—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D412yes—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—42no—
208C22⋊C4×C31—42no—
209C22⋊C4×C31—81no—
210C22⋊C4×C31—81no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31—21no—
214C42×C31—21no—
215#18291C3111yesmaximal, characteristic, direct product factor, Sylow
216C23×C31—21no—
217C23×C31—21no—
218C23×C31—21no—
219C23×C31—21no—
220C23×C31—22no—
221C23×C31—41no—
222C23×C31—41no—
223C23×C31—41no—
224C23×C31—41no—
225C23×C31—42no—
226C23×C31—81no—
227C23×C31—81no—
228C23×C31#12012yes—
229C23×C31#13512yes—
230C23×C31C22×D411yescharacteristic
231Q8×C31—42no—
232Q8×C31—42no—
233Q8×C31—42no—
234Q8×C31—81no—
235D4×C31—42no—
236D4×C31—44no—
237D4×C31—44no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—82no—
241D4×C31—82no—
242D4×C31—81no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—42no—
256C2×C4×C31—44no—
257C2×C4×C31—42no—
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—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—81no—
273C2×C4×C31—81no—
274C2×C4×C31—82no—
275C2×C4×C31—81no—
276C2×C4×C31#12011yescharacteristic
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31C22×D412yes—
279C8×C31—42no—
280C8×C31—42no—
281#3459C2×C3111yescharacteristic
282#3190C2×C3112yescomplemented
283#2638C2×C3111yescharacteristic, complemented
284#1609C2×C3112yescomplemented
285#1608C2×C3111yescharacteristic, complemented
286#1607C2×C3111yescharacteristic, complemented
287#1603C2×C3112yescomplemented
288#1602C2×C3111yescharacteristic, complemented
289#1601C2×C3111yescharacteristic, complemented
290#1516C2×C3112yescomplemented
291#1459C2×C3111yescharacteristic, complemented
292C22×C31—21no—
293C22×C31—21no—
294C22×C31—22no—
295C22×C31—22no—
296C22×C31—22no—
297C22×C31—22no—
298C22×C31—42no—
299C22×C31—41no—
300C22×C31—41no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31—81no—
307C22×C31—84no—
308C22×C31C2×#12011yescharacteristic
309C22×C31C2×#13511yescharacteristic
310C22×C31#57511yescharacteristic
311C4×C31—22no—
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—42no—
318C4×C31—41no—
319C4×C31—42no—
320C4×C31—81no—
321C4×C31—81no—
322C4×C31—81no—
323C4×C31—81no—
324#543C22×C3111yescharacteristic
325#539—22no—
326#539C22×C3112yes—
327#538—22no—
328#538—21no—
329#534—21no—
330#534C22×C3111yescharacteristic
331#529—22no—
332#529—21no—
333#525—22no—
334#525C22×C3112yes—
335C2×C22⋊Q8C22×C3111yescharacteristic, complemented
336C2×C4×D4—21no—
337C2×C4×D4C22×C3111yescharacteristic
338#485—22no—
339#483C22×C3112yescomplemented
340#480C22×C3112yes—
341#467—22no—
342#465—22no—
343#452—22no—
344#451C22×C3112yescomplemented
345#448C22×C3112yes—
346C2×C22⋊C8C22×C3112yescomplemented
347#398C22×C3111yescharacteristic, complemented
348#397—21no—
349#397C22×C3111yescharacteristic
350#395C22×C3111yescharacteristic, complemented
351#394—21no—
352#393C22×C3111yescharacteristic
353#387—21no—
354#385—21no—
355#381C22×C3111yescharacteristic, complemented
356#328C22×C3112yescomplemented
357#328C22×C3112yescomplemented
358#327C22×C3112yes—
359#327C22×C3112yes—
360#324C22×C3112yescomplemented
361#323C22×C3111yescharacteristic
362#323C22×C3111yescharacteristic
363C2×C31—21no—
364C2×C31—21no—
365C2×C31—42no—
366C2×C31—42no—
367C2×C31—82no—
368C2×C31#300211yescharacteristic
369C2×C31#302211yescharacteristic
370C2×C31#302611yescharacteristic
371C2×#41D4×C3112yes—
372C22×Q8C23×C3111yescharacteristic
373C22×D4—21no—
374C22×D4D4×C3111yescharacteristic
375C23×C4—21no—
376C23×C4D4×C3111yescharacteristic
377C23×C4C23×C3111yescharacteristic
378C2×Q16—22no—
379C2×SD16—22no—
380C22×C8—22no—
381#126—22no—
382#126—22no—
383#126—22no—
384#125—21no—
385#125—22no—
386#124—22no—
387#124—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8D4×C3112yes—
401C22⋊Q8C23×C3112yes—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121D4×C3112yes—
407#121C23×C3112yes—
408#120—22no—
409#120—22no—
410#120—22no—
411C4×Q8—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420#117—22no—
421#117D4×C3112yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4C23×C3111yescharacteristic
427C2×C4⋊C4C23×C3111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4D4×C3111yescharacteristic
435C2×C22⋊C4C23×C3111yescharacteristic
436C2×C42—21no—
437C2×C42D4×C3111yescharacteristic
438#107—22no—
439#106—22no—
440#103—22no—
441#103—22no—
442#103—22no—
443#102—22no—
444C22⋊C8—22no—
445C22⋊C8C23×C3112yes—
446C22⋊C8C23×C3112yes—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95C23×C3111yescharacteristic
453#95C23×C3111yescharacteristic
454C31#1829111yescharacteristic, direct product factor, Sylow
455C24—21no—
456C24C2×D4×C3111yescharacteristic
457C24C2×D4×C3111yescharacteristic
458#41—82no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—81no—
462C2×Q8C2×D4×C3112yes—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4C2×D4×C3112yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—82no—
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490C22×C4C2×D4×C3111yescharacteristic
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C2×D4×C3112yes—
493C22×C4C2×D4×C3112yes—
494C22×C4C24×C3111yescharacteristic
495Q16—82no—
496SD16—82no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—82no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—81no—
514C4⋊C4C2×D4×C3112yes—
515C4⋊C4C2×D4×C3112yes—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—81no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—81no—
547C23—81no—
548C23C31×#12012yes—
549C23C31×#13512yes—
550C23C22×D4×C3111yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—81no—
555D4—42no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—82no—
561D4—82no—
562D4—81no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4C31×#12011yescharacteristic
597C2×C4C31×#12011yescharacteristic
598C2×C4C22×D4×C3112yes—
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—84no—
617C22C2×C31×#12011yescharacteristic
618C22C2×C31×#13511yescharacteristic
619C22C31×#57511yescharacteristic
620C4—22no—
621C4—21no—
622C4—21no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—21no—
634C2—21no—
635C2—42no—
636C2—42no—
637C2—82no—
638C2C31×#300211yescharacteristic
639C2C31×#302211yescharacteristic
640C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3190C2semidirect
3. C31×#2638C2semidirect
4. C31×#1609C2semidirect
5. C31×#1608C2semidirect
6. C31×#1607C2semidirect
7. C31×#1603C2semidirect
8. C31×#1602C2semidirect
9. C31×#1601C2semidirect
10. C31×#1516C2semidirect
11. C31×#1459C2semidirect
23. C2×C31×C22⋊Q8C22semidirect
27. C31×#483C22semidirect
32. C31×#451C22semidirect
34. C2×C31×C22⋊C8C22semidirect
35. C31×#398C22semidirect
38. C31×#395C22semidirect
43. C31×#381C22semidirect
44. C31×#328C22semidirect
45. C31×#328C22semidirect
48. C31×#324C22semidirect
215. #18291C31direct (N × H)
282. #3190C2×C31semidirect
283. #2638C2×C31semidirect
284. #1609C2×C31semidirect
285. #1608C2×C31semidirect
286. #1607C2×C31semidirect
287. #1603C2×C31semidirect
288. #1602C2×C31semidirect
289. #1601C2×C31semidirect
290. #1516C2×C31semidirect
291. #1459C2×C31semidirect
335. C2×C22⋊Q8C22×C31semidirect
339. #483C22×C31semidirect
344. #451C22×C31semidirect
346. C2×C22⋊C8C22×C31semidirect
347. #398C22×C31semidirect
350. #395C22×C31semidirect
355. #381C22×C31semidirect
356. #328C22×C31semidirect
357. #328C22×C31semidirect
360. #324C22×C31semidirect
454. C31#18291direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34592
C31×#31902
C31×#26382
C31×#16092
C31×#16082
C31×#16072
C31×#16032
C31×#16022
C31×#16012
C31×#15162
C31×#14592
#1829131