Finite Group Databasebeta

← C31×#18279 #21475038

Subgroups — C31×#18279 #21475038

Order 7936.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic
2C31×#3113C211yesmaximal, characteristic, complemented
3C31×#3108C211yesmaximal, characteristic
4C31×#2584C211yesmaximal, characteristic, complemented
5C31×#1611C212yesmaximal, complemented
6C31×#1609C212yesmaximal
7C31×#1605C212yesmaximal, complemented
8C31×#1603C212yesmaximal
9C31×#1509C211yesmaximal, characteristic
10C31×#1503C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#564—21no—
13C31×#563—21no—
14C31×#557—21no—
15C31×#554—21no—
16C31×#552—21no—
17C31×#552—22no—
18C31×#552—21no—
19C31×#544—21no—
20C31×#544C2212yes—
21C31×#541—21no—
22C31×#541—22no—
23C31×#539C2212yes—
24C31×#533C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#483C2211yescharacteristic
29C31×#483C2212yes—
30C31×#480C2211yescharacteristic
31C31×#479—21no—
32C31×#475—21no—
33C31×#464—21no—
34C31×#462—21no—
35C31×#451C2211yescharacteristic
36C31×#451C2212yes—
37C31×#448C2211yescharacteristic
38C31×#441—22no—
39C4×Q16×C31—21no—
40C4×SD16×C31—21no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#398C2212yes—
44C31×#395C2212yes—
45C31×#389—21no—
46C31×#389—22no—
47C31×#389—21no—
48C4×C4⋊C4×C31C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C22×Q8×C31C2311yescharacteristic
56C2×Q16×C31—21no—
57C2×Q16×C31—22no—
58C2×SD16×C31—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—22no—
67C31×#126—22no—
68C31×#126—21no—
69C31×#125—22no—
70C31×#125—22no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—21no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—22no—
77C31×#125—22no—
78C31×#125—21no—
79C31×#125—21no—
80C31×#125—21no—
81C31×#125—21no—
82C31×#125—21no—
83C31×#125D411yescharacteristic
84C31×#124D411yescharacteristic
85C31×#123—22no—
86C31×#123—21no—
87C31×#123—22no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8C2311yescharacteristic
96C31×C22⋊Q8C2312yes—
97C31×#121C2311yescharacteristic
98C4×Q8×C31—21no—
99C4×Q8×C31—22no—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31D411yescharacteristic
107C4×Q8×C31D412yes—
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31D411yescharacteristic
111C31×#117—22no—
112C31×#117—21no—
113C31×#117—21no—
114C31×#117—22no—
115C31×#117—21no—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31—22no—
119C2×C4⋊C4×C31—21no—
120C2×C4⋊C4×C31—22no—
121C2×C4⋊C4×C31D411yescharacteristic
122C2×C4⋊C4×C31D412yes—
123C2×C4⋊C4×C31D411yescharacteristic
124C2×C4⋊C4×C31C2311yescharacteristic
125C2×C4⋊C4×C31C2312yes—
126C2×C42×C31—21no—
127C2×C42×C31—21no—
128C2×C42×C31C2311yescharacteristic
129C31×#107—21no—
130C31×#106—21no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—21no—
135C31×#103—22no—
136C31×#103—21no—
137C31×#103—21no—
138C31×#103—22no—
139C31×#102—21no—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2312yes—
143C31×#97—21no—
144C4×C8×C31—21no—
145C31×#95—22no—
146C31×#95—22no—
147C31×#95C2312yes—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—42no—
153C2×Q8×C31C2×D412yes—
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—21no—
156C2×D4×C31C2×D411yescharacteristic
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—22no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2411yescharacteristic
167Q16×C31—42no—
168Q16×C31—44no—
169SD16×C31—42no—
170C2×C8×C31—21no—
171C2×C8×C31—21no—
172C2×C8×C31—22no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—44no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31—41no—
204C4⋊C4×C31—82no—
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D412yes—
207C4⋊C4×C31C2×D411yescharacteristic
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—41no—
223C42×C31—41no—
224C42×C31—81no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229#18279C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—21no—
231C23×C31C22×D411yescharacteristic
232Q8×C31—22no—
233Q8×C31—22no—
234Q8×C31—41no—
235Q8×C31—42no—
236Q8×C31—41no—
237Q8×C31—41no—
238Q8×C31—42no—
239Q8×C31—82no—
240D4×C31—41no—
241D4×C31—41no—
242D4×C31—41no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—24no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—82no—
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C22×D411yescharacteristic
274C8×C31—41no—
275C8×C31—41no—
276C8×C31—42no—
277#3563C2×C3111yescharacteristic
278#3113C2×C3111yescharacteristic, complemented
279#3108C2×C3111yescharacteristic
280#2584C2×C3111yescharacteristic, complemented
281#1611C2×C3112yescomplemented
282#1609C2×C3112yes—
283#1605C2×C3112yescomplemented
284#1603C2×C3112yes—
285#1509C2×C3111yescharacteristic
286#1503C2×C3111yescharacteristic, complemented
287#1470C2×C3111yescharacteristic, complemented
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31C2×#12011yescharacteristic
294C22×C31#57711yescharacteristic
295C22×C31#57811yescharacteristic
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—44no—
302C4×C31—42no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—41no—
310C4×C31—81no—
311C4×C31#45612yes—
312#564—21no—
313#563—21no—
314#557—21no—
315#554—21no—
316#552—21no—
317#552—22no—
318#552—21no—
319#544—21no—
320#544C22×C3112yes—
321#541—21no—
322#541—22no—
323#539C22×C3112yes—
324#533C22×C3111yescharacteristic
325#529C22×C3111yescharacteristic
326C2×#125C22×C3111yescharacteristic
327C2×C4×Q8C22×C3111yescharacteristic
328#483C22×C3111yescharacteristic
329#483C22×C3112yes—
330#480C22×C3111yescharacteristic
331#479—21no—
332#475—21no—
333#464—21no—
334#462—21no—
335#451C22×C3111yescharacteristic
336#451C22×C3112yes—
337#448C22×C3111yescharacteristic
338#441—22no—
339C4×Q16—21no—
340C4×SD16—21no—
341#432C22×C3111yescharacteristic
342#431C22×C3111yescharacteristic
343#398C22×C3112yes—
344#395C22×C3112yes—
345#389—21no—
346#389—22no—
347#389—21no—
348C4×C4⋊C4C22×C3111yescharacteristic
349#328C22×C3112yes—
350#328C22×C3112yes—
351#328C22×C3112yes—
352#327C22×C3112yes—
353#324C22×C3112yes—
354#324C22×C3112yes—
355C2×C31—21no—
356C2×C31—21no—
357C2×C31—81no—
358C2×C31#301311yescharacteristic
359C2×C31C2×#45611yescharacteristic
360C2×C31#302511yescharacteristic
361C22×Q8C23×C3111yescharacteristic
362C2×Q16—21no—
363C2×Q16—22no—
364C2×SD16—21no—
365C4⋊Q8—21no—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368C4⋊Q8D4×C3111yescharacteristic
369C4⋊Q8D4×C3111yescharacteristic
370#126—22no—
371#126—22no—
372#126—22no—
373#126—22no—
374#126—21no—
375#125—22no—
376#125—22no—
377#125—22no—
378#125—22no—
379#125—21no—
380#125—22no—
381#125—22no—
382#125—22no—
383#125—22no—
384#125—21no—
385#125—21no—
386#125—21no—
387#125—21no—
388#125—21no—
389#125D4×C3111yescharacteristic
390#124D4×C3111yescharacteristic
391#123—22no—
392#123—21no—
393#123—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8—22no—
401C22⋊Q8C23×C3111yescharacteristic
402C22⋊Q8C23×C3112yes—
403#121C23×C3111yescharacteristic
404C4×Q8—21no—
405C4×Q8—22no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8D4×C3111yescharacteristic
413C4×Q8D4×C3112yes—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4D4×C3111yescharacteristic
417#117—22no—
418#117—21no—
419#117—21no—
420#117—22no—
421#117—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4D4×C3111yescharacteristic
428C2×C4⋊C4D4×C3112yes—
429C2×C4⋊C4D4×C3111yescharacteristic
430C2×C4⋊C4C23×C3111yescharacteristic
431C2×C4⋊C4C23×C3112yes—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42C23×C3111yescharacteristic
435#107—21no—
436#106—21no—
437#106—22no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440#103—21no—
441#103—22no—
442#103—21no—
443#103—21no—
444#103—22no—
445#102—21no—
446C22⋊C8C23×C3112yes—
447C22⋊C8C23×C3111yescharacteristic
448C22⋊C8C23×C3111yescharacteristic
449#97—21no—
450C4×C8—21no—
451#95—22no—
452#95—22no—
453#95C23×C3112yes—
454C31#1827911yescharacteristic, direct product factor, Sylow
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—42no—
460C2×Q8C2×D4×C3112yes—
461C2×Q8C2×D4×C3111yescharacteristic
462C2×D4—21no—
463C2×D4C2×D4×C3111yescharacteristic
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4C2×D4×C3112yes—
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3112yes—
473C22×C4C24×C3111yescharacteristic
474Q16—42no—
475Q16—44no—
476SD16—42no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—44no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—82no—
512C4⋊C4C2×D4×C3111yescharacteristic
513C4⋊C4C2×D4×C3112yes—
514C4⋊C4C2×D4×C3111yescharacteristic
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—41no—
530C42—41no—
531C42—81no—
532C42C2×D4×C3111yescharacteristic
533C42C2×D4×C3111yescharacteristic
534C42C2×D4×C3111yescharacteristic
535C42C2×D4×C3111yescharacteristic
536C23—21no—
537C23C22×D4×C3111yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—42no—
542Q8—41no—
543Q8—41no—
544Q8—42no—
545Q8—82no—
546D4—41no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—82no—
574C2×C4C31×#12011yescharacteristic
575C2×C4C31×#12011yescharacteristic
576C2×C4C31×#12011yescharacteristic
577C2×C4C31×#12011yescharacteristic
578C2×C4C22×D4×C3111yescharacteristic
579C2×C4C22×D4×C3111yescharacteristic
580C8—41no—
581C8—41no—
582C8—42no—
583C22—21no—
584C22—21no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C2×C31×#12011yescharacteristic
589C22C31×#57711yescharacteristic
590C22C31×#57811yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4C31×#45612yes—
607C2—21no—
608C2—21no—
609C2—81no—
610C2C31×#301311yescharacteristic
611C2C2×C31×#45611yescharacteristic
612C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3113C2semidirect
4. C31×#2584C2semidirect
5. C31×#1611C2semidirect
7. C31×#1605C2semidirect
10. C31×#1503C2semidirect
11. C31×#1470C2semidirect
229. #18279C31direct (N × H)
278. #3113C2×C31semidirect
280. #2584C2×C31semidirect
281. #1611C2×C31semidirect
283. #1605C2×C31semidirect
286. #1503C2×C31semidirect
287. #1470C2×C31semidirect
454. C31#18279direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#31132
C31×#31082
C31×#25842
C31×#16112
C31×#16092
C31×#16052
C31×#16032
C31×#15092
C31×#15032
C31×#14702
#1827931