Finite Group Databasebeta

← C31×#18454 #21475213

Subgroups — C31×#18454 #21475213

Order 7936.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3538C211yesmaximal, characteristic, complemented
2C31×#3194C211yesmaximal, characteristic
3C31×#3187C211yesmaximal, characteristic, complemented
4C31×#2816C211yesmaximal, characteristic, complemented
5C31×#1611C212yesmaximal, complemented
6C31×#1609C212yesmaximal, complemented
7C31×#1599C212yesmaximal, complemented
8C31×#1597C212yesmaximal, complemented
9C31×#1516C211yesmaximal, characteristic, complemented
10C31×#1511C211yesmaximal, characteristic, complemented
11C31×#1472C211yesmaximal, characteristic, complemented
12C31×#563—22no—
13C31×#552—21no—
14C31×#552—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17D4×Q8×C31—21no—
18C31×#544—21no—
19C31×#544C2212yes—
20C31×#541—21no—
21C31×#539—21no—
22C31×#539C2212yes—
23C31×#536—21no—
24C31×#525C2211yescharacteristic
25C2×C31×C22⋊Q8C2211yescharacteristic
26C2×C31×C22⋊Q8C2211yescharacteristic
27C2×C31×#121C2211yescharacteristic
28C31×#483C2212yes—
29C31×#483C2211yescharacteristic
30C31×#482—21no—
31C31×#481—21no—
32C31×#480C2211yescharacteristic
33C31×#469—21no—
34C31×#468—21no—
35C31×#465—22no—
36C31×#450C2212yes—
37C31×#450C2211yescharacteristic
38C31×#449—21no—
39C31×#448—21no—
40C31×#447C2211yescharacteristic
41C31×#407C2211yescharacteristic
42C2×C31×C22⋊C8C2211yescharacteristic
43C31×#399—21no—
44C31×#398C2212yes—
45C31×#394—22no—
46C31×#394C2212yes—
47C31×#393—21no—
48C31×#381C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31D411yescharacteristic
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31D411yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31C2311yescharacteristic
61C23×C4×C31C2311yescharacteristic
62C2×SD16×C31—22no—
63C2×SD16×C31—21no—
64C2×D8×C31—21no—
65C2×M16×C31—21no—
66C22×C8×C31—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—22no—
71C31×#126—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#125—21no—
76C31×#125—22no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#124—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8D412yes—
99C31×C22⋊Q8D411yescharacteristic
100C31×C22⋊Q8C2312yes—
101C31×C22⋊Q8C2311yescharacteristic
102C31×#121—21no—
103C31×#121—21no—
104C31×#121—21no—
105C31×#121—21no—
106C31×#121D411yescharacteristic
107C31×#121C2311yescharacteristic
108C31×#120—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31—21no—
111C4×D4×C31—22no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—22no—
115C31×#117—21no—
116C31×#117—22no—
117C31×#117—22no—
118C31×#117D411yescharacteristic
119C2×C4⋊C4×C31—21no—
120C2×C4⋊C4×C31—22no—
121C2×C4⋊C4×C31D411yescharacteristic
122C2×C4⋊C4×C31D411yescharacteristic
123C2×C4⋊C4×C31C2311yescharacteristic
124C2×C4⋊C4×C31C2312yes—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C22⋊C4×C31—21no—
128C2×C22⋊C4×C31—22no—
129C2×C22⋊C4×C31—22no—
130C2×C22⋊C4×C31D412yes—
131C31×#107—21no—
132C31×#106—22no—
133C31×#106—21no—
134C31×#103—22no—
135C31×#103—21no—
136C31×#102—21no—
137C31×#102—21no—
138C31×#102—22no—
139C31×#102—21no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8C2312yes—
143C31×C22⋊C8C2311yescharacteristic
144C31×C22⋊C8C2311yescharacteristic
145C31×#95—22no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—21no—
149C24×C31C2×D411yescharacteristic
150#41×C31—81no—
151C2×Q8×C31—22no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—81no—
156C2×Q8×C31—82no—
157C2×Q8×C31—81no—
158C2×Q8×C31C2×D412yes—
159C2×Q8×C31C2×D411yescharacteristic
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—41no—
166C2×D4×C31—41no—
167C2×D4×C31—81no—
168C2×D4×C31—82no—
169C2×D4×C31C2×D411yescharacteristic
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—22no—
175C22×C4×C31—21no—
176C22×C4×C31—81no—
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2×D412yes—
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2411yescharacteristic
185SD16×C31—82no—
186SD16×C31—81no—
187D8×C31—81no—
188M16×C31—81no—
189C2×C8×C31—21no—
190C2×C8×C31—22no—
191C2×C8×C31—21no—
192C2×C8×C31—81no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—22no—
208C4⋊C4×C31—22no—
209C4⋊C4×C31—21no—
210C4⋊C4×C31—21no—
211C4⋊C4×C31—21no—
212C4⋊C4×C31—82no—
213C4⋊C4×C31C2×D412yes—
214C4⋊C4×C31C2×D411yescharacteristic
215C4⋊C4×C31C2×D411yescharacteristic
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—21no—
227C22⋊C4×C31—21no—
228C22⋊C4×C31—41no—
229C22⋊C4×C31—41no—
230C22⋊C4×C31—82no—
231C22⋊C4×C31—82no—
232C22⋊C4×C31—81no—
233C42×C31—22no—
234C42×C31—22no—
235C42×C31—22no—
236C42×C31—21no—
237#18454C3111yesmaximal, characteristic, direct product factor, Sylow
238C23×C31—22no—
239C23×C31—21no—
240C23×C31—41no—
241C23×C31—41no—
242C23×C31—42no—
243C23×C31—42no—
244C23×C31—41no—
245C23×C31—42no—
246C23×C31#12011yescharacteristic
247C23×C31#12011yescharacteristic
248C23×C31C22×D411yescharacteristic
249Q8×C31—42no—
250Q8×C31—41no—
251Q8×C31—42no—
252Q8×C31—42no—
253Q8×C31—81no—
254Q8×C31—82no—
255D4×C31—41no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—42no—
259D4×C31—42no—
260D4×C31—42no—
261D4×C31—81no—
262D4×C31—82no—
263D4×C31—82no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—21no—
269C2×C4×C31—22no—
270C2×C4×C31—22no—
271C2×C4×C31—22no—
272C2×C4×C31—22no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—41no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—42no—
283C2×C4×C31—42no—
284C2×C4×C31—41no—
285C2×C4×C31—42no—
286C2×C4×C31—42no—
287C2×C4×C31—81no—
288C2×C4×C31—82no—
289C2×C4×C31#12011yescharacteristic
290C2×C4×C31#12011yescharacteristic
291C2×C4×C31C22×D411yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C8×C31—42no—
294C8×C31—41no—
295C8×C31—41no—
296#3538C2×C3111yescharacteristic, complemented
297#3194C2×C3111yescharacteristic
298#3187C2×C3111yescharacteristic, complemented
299#2816C2×C3111yescharacteristic, complemented
300#1611C2×C3112yescomplemented
301#1609C2×C3112yescomplemented
302#1599C2×C3112yescomplemented
303#1597C2×C3112yescomplemented
304#1516C2×C3111yescharacteristic, complemented
305#1511C2×C3111yescharacteristic, complemented
306#1472C2×C3111yescharacteristic, complemented
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—21no—
313C22×C31—21no—
314C22×C31—21no—
315C22×C31—41no—
316C22×C31—41no—
317C22×C31—41no—
318C22×C31—82no—
319C22×C31—84no—
320C22×C31—81no—
321C22×C31—81no—
322C22×C31—81no—
323C22×C31C2×#12011yescharacteristic
324C22×C31#57712yes—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—21no—
329C4×C31—42no—
330C4×C31—41no—
331C4×C31—42no—
332C4×C31—41no—
333C4×C31—41no—
334C4×C31—42no—
335C4×C31—42no—
336C4×C31—82no—
337C4×C31—81no—
338#563—22no—
339#552—21no—
340#552—21no—
341#551—22no—
342D4×Q8—21no—
343D4×Q8—21no—
344#544—21no—
345#544C22×C3112yes—
346#541—21no—
347#539—21no—
348#539C22×C3112yes—
349#536—21no—
350#525C22×C3111yescharacteristic
351C2×C22⋊Q8C22×C3111yescharacteristic
352C2×C22⋊Q8C22×C3111yescharacteristic
353C2×#121C22×C3111yescharacteristic
354#483C22×C3112yes—
355#483C22×C3111yescharacteristic
356#482—21no—
357#481—21no—
358#480C22×C3111yescharacteristic
359#469—21no—
360#468—21no—
361#465—22no—
362#450C22×C3112yes—
363#450C22×C3111yescharacteristic
364#449—21no—
365#448—21no—
366#447C22×C3111yescharacteristic
367#407C22×C3111yescharacteristic
368C2×C22⋊C8C22×C3111yescharacteristic
369#399—21no—
370#398C22×C3112yes—
371#394—22no—
372#394C22×C3112yes—
373#393—21no—
374#381C22×C3111yescharacteristic
375#328C22×C3112yes—
376#328C22×C3112yes—
377#328C22×C3112yes—
378#327C22×C3112yes—
379#324C22×C3112yes—
380#324C22×C3112yes—
381C2×C31—22no—
382C2×C31—41no—
383C2×C31—41no—
384C2×C31—81no—
385C2×C31—161no—
386C2×C31#301011yescharacteristic
387C2×C31#302011yescharacteristic
388C2×C31#302611yescharacteristic
389C2×#41D4×C3111yescharacteristic
390C22×Q8—21no—
391C22×Q8D4×C3111yescharacteristic
392C22×Q8D4×C3111yescharacteristic
393C22×D4—21no—
394C22×D4C23×C3111yescharacteristic
395C23×C4C23×C3111yescharacteristic
396C2×SD16—22no—
397C2×SD16—21no—
398C2×D8—21no—
399C2×M16—21no—
400C22×C8—21no—
401C4⋊Q8—21no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#126—22no—
406#126—22no—
407#126—22no—
408#126—22no—
409#125—21no—
410#125—22no—
411#125—22no—
412#125—22no—
413#125—22no—
414#124—22no—
415#123—22no—
416#123—22no—
417#123—22no—
418#123—21no—
419#123—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—21no—
431C22⋊Q8—22no—
432C22⋊Q8D4×C3112yes—
433C22⋊Q8D4×C3111yescharacteristic
434C22⋊Q8C23×C3112yes—
435C22⋊Q8C23×C3111yescharacteristic
436#121—21no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121D4×C3111yescharacteristic
441#121C23×C3111yescharacteristic
442#120—21no—
443C4×Q8—21no—
444C4×Q8—21no—
445C4×D4—22no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—22no—
449#117—21no—
450#117—22no—
451#117—22no—
452#117D4×C3111yescharacteristic
453C2×C4⋊C4—21no—
454C2×C4⋊C4—22no—
455C2×C4⋊C4D4×C3111yescharacteristic
456C2×C4⋊C4D4×C3111yescharacteristic
457C2×C4⋊C4C23×C3111yescharacteristic
458C2×C4⋊C4C23×C3112yes—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—21no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—22no—
463C2×C22⋊C4—22no—
464C2×C22⋊C4D4×C3112yes—
465#107—21no—
466#106—22no—
467#106—21no—
468#103—22no—
469#103—21no—
470#102—21no—
471#102—21no—
472#102—22no—
473#102—21no—
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8C23×C3112yes—
477C22⋊C8C23×C3111yescharacteristic
478C22⋊C8C23×C3111yescharacteristic
479#95—21no—
480#95—22no—
481#95C23×C3112yes—
482C31#1845411yescharacteristic, direct product factor, Sylow
483C24—21no—
484C24C2×D4×C3111yescharacteristic
485#41—81no—
486C2×Q8—22no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—81no—
491C2×Q8—82no—
492C2×Q8—81no—
493C2×Q8C2×D4×C3112yes—
494C2×Q8C2×D4×C3111yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—41no—
501C2×D4—41no—
502C2×D4—81no—
503C2×D4—82no—
504C2×D4C2×D4×C3111yescharacteristic
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—81no—
512C22×C4C2×D4×C3112yes—
513C22×C4C2×D4×C3112yes—
514C22×C4C2×D4×C3111yescharacteristic
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4C24×C3111yescharacteristic
520SD16—82no—
521SD16—81no—
522D8—81no—
523M16—81no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—81no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—82no—
548C4⋊C4C2×D4×C3112yes—
549C4⋊C4C2×D4×C3111yescharacteristic
550C4⋊C4C2×D4×C3111yescharacteristic
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—81no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—21no—
572C23—22no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23C31×#12011yescharacteristic
581C23C31×#12011yescharacteristic
582C23C22×D4×C3111yescharacteristic
583Q8—42no—
584Q8—41no—
585Q8—42no—
586Q8—42no—
587Q8—81no—
588Q8—82no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596D4—82no—
597D4—82no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4C31×#12011yescharacteristic
624C2×C4C31×#12011yescharacteristic
625C2×C4C22×D4×C3111yescharacteristic
626C2×C4C22×D4×C3111yescharacteristic
627C8—42no—
628C8—41no—
629C8—41no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—82no—
642C22—84no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C2×C31×#12011yescharacteristic
647C22C31×#57712yes—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—81no—
661C2—22no—
662C2—41no—
663C2—41no—
664C2—81no—
665C2—161no—
666C2C31×#301011yescharacteristic
667C2C31×#302011yescharacteristic
668C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3194C2
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
1. C31×#3538C2semidirect
3. C31×#3187C2semidirect
4. C31×#2816C2semidirect
5. C31×#1611C2semidirect
6. C31×#1609C2semidirect
7. C31×#1599C2semidirect
8. C31×#1597C2semidirect
9. C31×#1516C2semidirect
10. C31×#1511C2semidirect
11. C31×#1472C2semidirect
237. #18454C31direct (N × H)
296. #3538C2×C31semidirect
298. #3187C2×C31semidirect
299. #2816C2×C31semidirect
300. #1611C2×C31semidirect
301. #1609C2×C31semidirect
302. #1599C2×C31semidirect
303. #1597C2×C31semidirect
304. #1516C2×C31semidirect
305. #1511C2×C31semidirect
306. #1472C2×C31semidirect
482. C31#18454direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35382
C31×#31942
C31×#31872
C31×#28162
C31×#16112
C31×#16092
C31×#15992
C31×#15972
C31×#15162
C31×#15112
C31×#14722
#1845431