Finite Group Databasebeta

← C31×#18960 #21475719

Subgroups — C31×#18960 #21475719

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3555C211yesmaximal, characteristic, complemented
2C31×#3217C211yesmaximal, characteristic
3C31×#3208C211yesmaximal, characteristic, complemented
4C31×#3208C211yesmaximal, characteristic, complemented
5C31×#1628C212yesmaximal, complemented
6C31×#1626C212yesmaximal, complemented
7C31×#1623C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1554C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic, complemented
12C31×#563—22no—
13Q82×C31C2211yescharacteristic, complemented
14C31×#557—21no—
15C31×#557C2211yescharacteristic, complemented
16C31×#556—21no—
17C31×#556—21no—
18C31×#556C2211yescharacteristic, complemented
19C31×#554C2211yescharacteristic, complemented
20C31×#552—21no—
21C31×#544—21no—
22C31×#542—21no—
23C31×#541—21no—
24C31×#541—21no—
25C31×#541—22no—
26C31×#533C2211yescharacteristic
27C31×#533C2211yescharacteristic
28C2×C31×#127C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#496—21no—
31C31×#492—21no—
32C31×#492C2212yes—
33C31×#490—21no—
34C31×#486—21no—
35C31×#486C2212yes—
36C31×#460—22no—
37C31×#460C2212yes—
38C31×#460C2212yes—
39C31×#460C2212yes—
40C31×#459C2212yes—
41C31×#450—22no—
42C31×#450—21no—
43C31×#450—21no—
44C31×#450—21no—
45C31×#447—21no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#333C2212yescomplemented
50C31×#332C2212yescomplemented
51C31×Q8⋊C8C2212yescomplemented
52C31×Q8⋊C8C2212yescomplemented
53C31×Q8⋊C8C2212yescomplemented
54C31×D4⋊C8C2212yescomplemented
55C22×D4×C31—21no—
56C22×D4×C31—21no—
57C22×D4×C31—21no—
58C2×SD16×C31—24no—
59C2×SD16×C31—22no—
60C2×SD16×C31—22no—
61C2×SD16×C31—22no—
62C2×D8×C31—22no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D411yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×C4⋊Q8C2311yescharacteristic
72C31×#127—21no—
73C31×#127—21no—
74C31×#127—21no—
75C31×#127C2312yes—
76C31×#126—22no—
77C31×#126—22no—
78C31×#126—22no—
79C31×#126—22no—
80C31×#126—22no—
81C31×#126—22no—
82C31×#126—21no—
83C31×#125—22no—
84C31×#125—22no—
85C31×#125—22no—
86C31×#125—22no—
87C31×#125—21no—
88C31×#125—22no—
89C31×#125—21no—
90C31×#125—21no—
91C31×#125—21no—
92C31×#125D411yescharacteristic
93C31×#124—22no—
94C31×#124—21no—
95C31×#124D411yescharacteristic
96C31×#123—22no—
97C31×#123—21no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8—21no—
101C31×C22⋊Q8—21no—
102C31×C22⋊Q8—22no—
103C31×C22⋊Q8—21no—
104C31×C22⋊Q8—21no—
105C31×C22⋊Q8D411yescharacteristic
106C31×C22⋊Q8D411yescharacteristic
107C31×C22⋊Q8D411yescharacteristic
108C31×C22⋊Q8D412yes—
109C31×#121D411yescharacteristic
110C4×Q8×C31—22no—
111C4×Q8×C31—21no—
112C4×Q8×C31—21no—
113C4×Q8×C31—21no—
114C4×Q8×C31—21no—
115C4×Q8×C31—21no—
116C4×Q8×C31—21no—
117C4×Q8×C31—21no—
118C4×Q8×C31D412yes—
119C4×Q8×C31C2311yescharacteristic
120C4×Q8×C31C2311yescharacteristic
121C4×Q8×C31C2311yescharacteristic
122C4×D4×C31—21no—
123C4×D4×C31—21no—
124C4×D4×C31C2311yescharacteristic
125C31×#117—22no—
126C31×#117—21no—
127C31×#117—21no—
128C31×#117—21no—
129C31×#117—21no—
130C2×C4⋊C4×C31—21no—
131C2×C42×C31C2311yescharacteristic
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#102—22no—
136C31×#102—22no—
137C31×#102—24no—
138C31×#102—22no—
139C31×#102—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146C24×C31—41no—
147C2×Q8×C31—22no—
148C2×Q8×C31—21no—
149C2×Q8×C31—21no—
150C2×Q8×C31C2×D411yescharacteristic
151C2×Q8×C31C2×D411yescharacteristic
152C2×Q8×C31C2×D411yescharacteristic
153C2×Q8×C31C2×D412yes—
154C2×D4×C31—21no—
155C2×D4×C31—22no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—41no—
159C2×D4×C31—41no—
160C2×D4×C31—41no—
161C2×D4×C31—82no—
162C2×D4×C31—82no—
163C2×D4×C31—82no—
164C2×D4×C31C2×D411yescharacteristic
165C22×C4×C31—22no—
166C22×C4×C31—21no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170SD16×C31—82no—
171SD16×C31—82no—
172SD16×C31—82no—
173SD16×C31—84no—
174D8×C31—82no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D412yes—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C42×C31—22no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—22no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31C2×D411yescharacteristic
226C42×C31C2×D411yescharacteristic
227C42×C31C2×D411yescharacteristic
228C42×C31C2411yescharacteristic
229#18960C3111yesmaximal, characteristic, direct product factor, Sylow
230C23×C31—21no—
231C23×C31—42no—
232C23×C31—82no—
233C23×C31—82no—
234C23×C31—82no—
235C23×C31#12011yescharacteristic
236Q8×C31—22no—
237Q8×C31—22no—
238Q8×C31—22no—
239Q8×C31—42no—
240Q8×C31—42no—
241Q8×C31—81no—
242Q8×C31—82no—
243D4×C31—22no—
244D4×C31—44no—
245D4×C31—42no—
246D4×C31—42no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—82no—
250D4×C31—82no—
251D4×C31—82no—
252D4×C31—81no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—21no—
261C2×C4×C31—21no—
262C2×C4×C31—22no—
263C2×C4×C31—21no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—82no—
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31C22×D411yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C8×C31—42no—
274C8×C31—42no—
275C8×C31—42no—
276#3555C2×C3111yescharacteristic, complemented
277#3217C2×C3111yescharacteristic
278#3208C2×C3111yescharacteristic, complemented
279#3208C2×C3111yescharacteristic, complemented
280#1628C2×C3112yescomplemented
281#1626C2×C3112yescomplemented
282#1623C2×C3112yescomplemented
283#1622C2×C3112yescomplemented
284#1554C2×C3111yescharacteristic, complemented
285#1497C2×C3111yescharacteristic, complemented
286#1482C2×C3111yescharacteristic, complemented
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31—164no—
297C22×C31C2×#12011yescharacteristic
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—41no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—41no—
306C4×C31—42no—
307C4×C31—41no—
308C4×C31—42no—
309C4×C31—41no—
310C4×C31—41no—
311C4×C31#57712yes—
312C4×C31#57712yes—
313#563—22no—
314Q82C22×C3111yescharacteristic, complemented
315#557—21no—
316#557C22×C3111yescharacteristic, complemented
317#556—21no—
318#556—21no—
319#556C22×C3111yescharacteristic, complemented
320#554C22×C3111yescharacteristic, complemented
321#552—21no—
322#544—21no—
323#542—21no—
324#541—21no—
325#541—21no—
326#541—22no—
327#533C22×C3111yescharacteristic
328#533C22×C3111yescharacteristic
329C2×#127C22×C3111yescharacteristic
330#529C22×C3111yescharacteristic
331#496—21no—
332#492—21no—
333#492C22×C3112yes—
334#490—21no—
335#486—21no—
336#486C22×C3112yes—
337#460—22no—
338#460C22×C3112yes—
339#460C22×C3112yes—
340#460C22×C3112yes—
341#459C22×C3112yes—
342#450—22no—
343#450—21no—
344#450—21no—
345#450—21no—
346#447—21no—
347#432C22×C3111yescharacteristic
348#431C22×C3111yescharacteristic
349#431C22×C3111yescharacteristic
350#333C22×C3112yescomplemented
351#332C22×C3112yescomplemented
352Q8⋊C8C22×C3112yescomplemented
353Q8⋊C8C22×C3112yescomplemented
354Q8⋊C8C22×C3112yescomplemented
355D4⋊C8C22×C3112yescomplemented
356C2×C31—41no—
357C2×C31—81no—
358C2×C31—162no—
359C2×C31#301011yescharacteristic
360C2×C31#301011yescharacteristic
361C2×C31#302111yescharacteristic
362C22×D4—21no—
363C22×D4—21no—
364C22×D4—21no—
365C2×SD16—24no—
366C2×SD16—22no—
367C2×SD16—22no—
368C2×SD16—22no—
369C2×D8—22no—
370C4⋊Q8—22no—
371C4⋊Q8—21no—
372C4⋊Q8—22no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8D4×C3111yescharacteristic
376C4⋊Q8D4×C3111yescharacteristic
377C4⋊Q8C23×C3111yescharacteristic
378C4⋊Q8C23×C3111yescharacteristic
379#127—21no—
380#127—21no—
381#127—21no—
382#127C23×C3112yes—
383#126—22no—
384#126—22no—
385#126—22no—
386#126—22no—
387#126—22no—
388#126—22no—
389#126—21no—
390#125—22no—
391#125—22no—
392#125—22no—
393#125—22no—
394#125—21no—
395#125—22no—
396#125—21no—
397#125—21no—
398#125—21no—
399#125D4×C3111yescharacteristic
400#124—22no—
401#124—21no—
402#124D4×C3111yescharacteristic
403#123—22no—
404#123—21no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8D4×C3111yescharacteristic
413C22⋊Q8D4×C3111yescharacteristic
414C22⋊Q8D4×C3111yescharacteristic
415C22⋊Q8D4×C3112yes—
416#121D4×C3111yescharacteristic
417C4×Q8—22no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C3112yes—
426C4×Q8C23×C3111yescharacteristic
427C4×Q8C23×C3111yescharacteristic
428C4×Q8C23×C3111yescharacteristic
429C4×D4—21no—
430C4×D4—21no—
431C4×D4C23×C3111yescharacteristic
432#117—22no—
433#117—21no—
434#117—21no—
435#117—21no—
436#117—21no—
437C2×C4⋊C4—21no—
438C2×C42C23×C3111yescharacteristic
439C4⋊C8—21no—
440C4⋊C8C23×C3112yes—
441C4⋊C8C23×C3112yes—
442#102—22no—
443#102—22no—
444#102—24no—
445#102—22no—
446#102—22no—
447C22⋊C8—21no—
448C22⋊C8—21no—
449C22⋊C8—21no—
450#97—21no—
451C4×C8—21no—
452C4×C8C23×C3112yes—
453C31#1896011yescharacteristic, direct product factor, Sylow
454C24—41no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8C2×D4×C3111yescharacteristic
459C2×Q8C2×D4×C3111yescharacteristic
460C2×Q8C2×D4×C3111yescharacteristic
461C2×Q8C2×D4×C3112yes—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—82no—
470C2×D4—82no—
471C2×D4—82no—
472C2×D4C2×D4×C3111yescharacteristic
473C22×C4—22no—
474C22×C4—21no—
475C22×C4C2×D4×C3111yescharacteristic
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478SD16—82no—
479SD16—82no—
480SD16—82no—
481SD16—84no—
482D8—82no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4C2×D4×C3111yescharacteristic
511C4⋊C4C2×D4×C3111yescharacteristic
512C4⋊C4C2×D4×C3111yescharacteristic
513C4⋊C4C2×D4×C3111yescharacteristic
514C4⋊C4C2×D4×C3112yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42C2×D4×C3111yescharacteristic
534C42C2×D4×C3111yescharacteristic
535C42C2×D4×C3111yescharacteristic
536C42C24×C3111yescharacteristic
537C23—21no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23C31×#12011yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—42no—
547Q8—42no—
548Q8—81no—
549Q8—82no—
550D4—22no—
551D4—44no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—82no—
557D4—82no—
558D4—82no—
559D4—81no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4C31×#12011yescharacteristic
575C2×C4C31×#12011yescharacteristic
576C2×C4C31×#12011yescharacteristic
577C2×C4C22×D4×C3111yescharacteristic
578C2×C4C22×D4×C3111yescharacteristic
579C2×C4C22×D4×C3111yescharacteristic
580C8—42no—
581C8—42no—
582C8—42no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—164no—
593C22C2×C31×#12011yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4C31×#57712yes—
608C4C31×#57712yes—
609C2—41no—
610C2—81no—
611C2—162no—
612C2C31×#301011yescharacteristic
613C2C31×#301011yescharacteristic
614C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3217C2
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×#3555C2semidirect
3. C31×#3208C2semidirect
4. C31×#3208C2semidirect
5. C31×#1628C2semidirect
6. C31×#1626C2semidirect
7. C31×#1623C2semidirect
8. C31×#1622C2semidirect
9. C31×#1554C2semidirect
10. C31×#1497C2semidirect
11. C31×#1482C2semidirect
13. Q82×C31C22semidirect
15. C31×#557C22semidirect
18. C31×#556C22semidirect
19. C31×#554C22semidirect
49. C31×#333C22semidirect
50. C31×#332C22semidirect
51. C31×Q8⋊C8C22semidirect
52. C31×Q8⋊C8C22semidirect
53. C31×Q8⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
229. #18960C31direct (N × H)
276. #3555C2×C31semidirect
278. #3208C2×C31semidirect
279. #3208C2×C31semidirect
280. #1628C2×C31semidirect
281. #1626C2×C31semidirect
282. #1623C2×C31semidirect
283. #1622C2×C31semidirect
284. #1554C2×C31semidirect
285. #1497C2×C31semidirect
286. #1482C2×C31semidirect
314. Q82C22×C31semidirect
316. #557C22×C31semidirect
319. #556C22×C31semidirect
320. #554C22×C31semidirect
350. #333C22×C31semidirect
351. #332C22×C31semidirect
352. Q8⋊C8C22×C31semidirect
353. Q8⋊C8C22×C31semidirect
354. Q8⋊C8C22×C31semidirect
355. D4⋊C8C22×C31semidirect
453. C31#18960direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35552
C31×#32172
C31×#32082
C31×#16282
C31×#16262
C31×#16232
C31×#16222
C31×#15542
C31×#14972
C31×#14822
#1896031