Finite Group Databasebeta

← C31×#18958 #21475717

Subgroups — C31×#18958 #21475717

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3554C211yesmaximal, characteristic, complemented
2C31×#3217C211yesmaximal, characteristic, complemented
3C31×#3208C211yesmaximal, characteristic, complemented
4C31×#3205C211yesmaximal, characteristic, complemented
5C31×#1627C212yesmaximal, complemented
6C31×#1626C212yesmaximal, complemented
7C31×#1621C212yesmaximal, complemented
8C31×#1620C212yesmaximal, complemented
9C31×#1553C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic, complemented
12C31×#562—22no—
13C31×#559—21no—
14C31×#559C2211yescharacteristic, complemented
15C31×#557C2211yescharacteristic, complemented
16C31×#554—21no—
17C31×#550C2211yescharacteristic, complemented
18D4×Q8×C31C2211yescharacteristic, complemented
19C31×#548—21no—
20C31×#547—21no—
21C31×#542—21no—
22C31×#542—21no—
23C31×#540—21no—
24C31×#539—22no—
25C31×#536—21no—
26C31×#533C2211yescharacteristic
27C31×#532C2211yescharacteristic
28C2×C31×#127C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#496—21no—
31C31×#493C2212yes—
32C31×#492—21no—
33C31×#490—21no—
34C31×#486—21no—
35C31×#486C2212yes—
36C31×#460C2212yes—
37C31×#460C2212yes—
38C31×#460C2212yes—
39C31×#459—22no—
40C31×#459C2212yes—
41C31×#450—21no—
42C31×#450—21no—
43C31×#450—21no—
44C31×#447—22no—
45C31×#447—21no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#332C2212yescomplemented
50C31×#331C2212yescomplemented
51C31×Q8⋊C8C2212yescomplemented
52C31×Q8⋊C8C2212yescomplemented
53C31×Q8⋊C8C2212yescomplemented
54C31×D4⋊C8C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×D4×C31—21no—
59C22×D4×C31—21no—
60C22×D4×C31—21no—
61C2×SD16×C31—22no—
62C2×SD16×C31—22no—
63C2×SD16×C31—22no—
64C2×D8×C31—24no—
65C2×D8×C31—22no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D411yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×#127—21no—
72C31×#127—21no—
73C31×#127—21no—
74C31×#127—21no—
75C31×#127C2312yes—
76C31×#127C2311yescharacteristic
77C31×#126—22no—
78C31×#126—21no—
79C31×#125—22no—
80C31×#125—21no—
81C31×#125—21no—
82C31×#125D411yescharacteristic
83C31×#124—22no—
84C31×#124—22no—
85C31×#124—21no—
86C31×#124—21no—
87C31×#124—21no—
88C31×#124D411yescharacteristic
89C31×#123—22no—
90C31×#123—22no—
91C31×#123—22no—
92C31×#123—22no—
93C31×#123—22no—
94C31×#123—22no—
95C31×#123—21no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8—21no—
101C31×C22⋊Q8—22no—
102C31×C22⋊Q8—21no—
103C31×C22⋊Q8D411yescharacteristic
104C31×C22⋊Q8D411yescharacteristic
105C31×C22⋊Q8D411yescharacteristic
106C31×#121—22no—
107C31×#121—22no—
108C31×#121—22no—
109C31×#121—21no—
110C31×#121—21no—
111C31×#121D411yescharacteristic
112C31×#121D412yes—
113C4×Q8×C31—21no—
114C4×Q8×C31—21no—
115C4×Q8×C31—21no—
116C4×Q8×C31C2311yescharacteristic
117C4×Q8×C31C2311yescharacteristic
118C4×Q8×C31C2311yescharacteristic
119C4×D4×C31—22no—
120C4×D4×C31—21no—
121C4×D4×C31—21no—
122C4×D4×C31—21no—
123C4×D4×C31—21no—
124C4×D4×C31—21no—
125C4×D4×C31—21no—
126C4×D4×C31D412yes—
127C4×D4×C31C2311yescharacteristic
128C31×#117—21no—
129C31×#117—21no—
130C2×C4⋊C4×C31—22no—
131C2×C42×C31C2311yescharacteristic
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#102—24no—
136C31×#102—22no—
137C31×#102—22no—
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—
147#41×C31—82no—
148#41×C31—82no—
149C2×Q8×C31—22no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—82no—
153C2×Q8×C31C2×D411yescharacteristic
154C2×Q8×C31C2×D411yescharacteristic
155C2×Q8×C31C2×D411yescharacteristic
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—41no—
166C2×D4×C31—41no—
167C2×D4×C31—41no—
168C2×D4×C31—82no—
169C2×D4×C31—82no—
170C2×D4×C31—82no—
171C2×D4×C31C2×D411yescharacteristic
172C2×D4×C31C2×D412yes—
173C22×C4×C31—22no—
174C22×C4×C31—22no—
175C22×C4×C31—22no—
176C22×C4×C31—21no—
177C22×C4×C31—22no—
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D411yescharacteristic
181SD16×C31—82no—
182SD16×C31—82no—
183SD16×C31—82no—
184D8×C31—82no—
185D8×C31—84no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
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—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D412yes—
209C4⋊C4×C31C2×D411yescharacteristic
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—21no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—21no—
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2411yescharacteristic
232#18958C3111yesmaximal, characteristic, direct product factor, Sylow
233C23×C31—21no—
234C23×C31—22no—
235C23×C31—42no—
236C23×C31—82no—
237C23×C31—82no—
238C23×C31—82no—
239C23×C31#12011yescharacteristic
240Q8×C31—22no—
241Q8×C31—22no—
242Q8×C31—22no—
243Q8×C31—42no—
244Q8×C31—81no—
245Q8×C31—82no—
246D4×C31—22no—
247D4×C31—42no—
248D4×C31—44no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31—82no—
258D4×C31—81no—
259D4×C31—82no—
260D4×C31—82no—
261D4×C31—82no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—22no—
270C2×C4×C31—21no—
271C2×C4×C31—21no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—82no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31—82no—
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C8×C31—42no—
285C8×C31—42no—
286C8×C31—42no—
287#3554C2×C3111yescharacteristic, complemented
288#3217C2×C3111yescharacteristic, complemented
289#3208C2×C3111yescharacteristic, complemented
290#3205C2×C3111yescharacteristic, complemented
291#1627C2×C3112yescomplemented
292#1626C2×C3112yescomplemented
293#1621C2×C3112yescomplemented
294#1620C2×C3112yescomplemented
295#1553C2×C3111yescharacteristic, complemented
296#1497C2×C3111yescharacteristic, complemented
297#1482C2×C3111yescharacteristic, complemented
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—41no—
302C22×C31—42no—
303C22×C31—41no—
304C22×C31—42no—
305C22×C31—42no—
306C22×C31—41no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31—164no—
311C22×C31C2×#12011yescharacteristic
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—41no—
317C4×C31—41no—
318C4×C31—42no—
319C4×C31—41no—
320C4×C31—41no—
321C4×C31—41no—
322C4×C31—41no—
323C4×C31—41no—
324C4×C31#57612yes—
325C4×C31#57712yes—
326#562—22no—
327#559—21no—
328#559C22×C3111yescharacteristic, complemented
329#557C22×C3111yescharacteristic, complemented
330#554—21no—
331#550C22×C3111yescharacteristic, complemented
332D4×Q8C22×C3111yescharacteristic, complemented
333#548—21no—
334#547—21no—
335#542—21no—
336#542—21no—
337#540—21no—
338#539—22no—
339#536—21no—
340#533C22×C3111yescharacteristic
341#532C22×C3111yescharacteristic
342C2×#127C22×C3111yescharacteristic
343#529C22×C3111yescharacteristic
344#496—21no—
345#493C22×C3112yes—
346#492—21no—
347#490—21no—
348#486—21no—
349#486C22×C3112yes—
350#460C22×C3112yes—
351#460C22×C3112yes—
352#460C22×C3112yes—
353#459—22no—
354#459C22×C3112yes—
355#450—21no—
356#450—21no—
357#450—21no—
358#447—22no—
359#447—21no—
360#432C22×C3111yescharacteristic
361#431C22×C3111yescharacteristic
362#431C22×C3111yescharacteristic
363#332C22×C3112yescomplemented
364#331C22×C3112yescomplemented
365Q8⋊C8C22×C3112yescomplemented
366Q8⋊C8C22×C3112yescomplemented
367Q8⋊C8C22×C3112yescomplemented
368D4⋊C8C22×C3112yescomplemented
369C2×C31—41no—
370C2×C31—81no—
371C2×C31—82no—
372C2×C31—162no—
373C2×C31#300811yescharacteristic
374C2×C31#301011yescharacteristic
375C2×C31#302111yescharacteristic
376C2×#41—21no—
377C2×#41—21no—
378C22×Q8—21no—
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—21no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×SD16—22no—
385C2×D8—24no—
386C2×D8—22no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8D4×C3111yescharacteristic
390C4⋊Q8D4×C3111yescharacteristic
391C4⋊Q8C23×C3111yescharacteristic
392#127—21no—
393#127—21no—
394#127—21no—
395#127—21no—
396#127C23×C3112yes—
397#127C23×C3111yescharacteristic
398#126—22no—
399#126—21no—
400#125—22no—
401#125—21no—
402#125—21no—
403#125D4×C3111yescharacteristic
404#124—22no—
405#124—22no—
406#124—21no—
407#124—21no—
408#124—21no—
409#124D4×C3111yescharacteristic
410#123—22no—
411#123—22no—
412#123—22no—
413#123—22no—
414#123—22no—
415#123—22no—
416#123—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—21no—
424C22⋊Q8D4×C3111yescharacteristic
425C22⋊Q8D4×C3111yescharacteristic
426C22⋊Q8D4×C3111yescharacteristic
427#121—22no—
428#121—22no—
429#121—22no—
430#121—21no—
431#121—21no—
432#121D4×C3111yescharacteristic
433#121D4×C3112yes—
434C4×Q8—21no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8C23×C3111yescharacteristic
438C4×Q8C23×C3111yescharacteristic
439C4×Q8C23×C3111yescharacteristic
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4D4×C3112yes—
448C4×D4C23×C3111yescharacteristic
449#117—21no—
450#117—21no—
451C2×C4⋊C4—22no—
452C2×C42C23×C3111yescharacteristic
453C4⋊C8—21no—
454C4⋊C8C23×C3112yes—
455C4⋊C8C23×C3112yes—
456#102—24no—
457#102—22no—
458#102—22no—
459#102—22no—
460#102—22no—
461C22⋊C8—21no—
462C22⋊C8—21no—
463C22⋊C8—21no—
464#97—21no—
465C4×C8—21no—
466C4×C8C23×C3112yes—
467C31#1895811yescharacteristic, direct product factor, Sylow
468C24—41no—
469#41—82no—
470#41—82no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—82no—
475C2×Q8C2×D4×C3111yescharacteristic
476C2×Q8C2×D4×C3111yescharacteristic
477C2×Q8C2×D4×C3111yescharacteristic
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4—82no—
493C2×D4C2×D4×C3111yescharacteristic
494C2×D4C2×D4×C3112yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4C2×D4×C3111yescharacteristic
501C22×C4C2×D4×C3111yescharacteristic
502C22×C4C2×D4×C3111yescharacteristic
503SD16—82no—
504SD16—82no—
505SD16—82no—
506D8—82no—
507D8—84no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C4⋊C4C2×D4×C3111yescharacteristic
530C4⋊C4C2×D4×C3112yes—
531C4⋊C4C2×D4×C3111yescharacteristic
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C42C2×D4×C3111yescharacteristic
553C42C24×C3111yescharacteristic
554C23—21no—
555C23—22no—
556C23—42no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23C31×#12011yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—42no—
565Q8—81no—
566Q8—82no—
567D4—22no—
568D4—42no—
569D4—44no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—82no—
577D4—82no—
578D4—82no—
579D4—81no—
580D4—82no—
581D4—82no—
582D4—82no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4C31×#12011yescharacteristic
600C2×C4C31×#12011yescharacteristic
601C2×C4C31×#12011yescharacteristic
602C2×C4C22×D4×C3111yescharacteristic
603C2×C4C22×D4×C3111yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C8—42no—
606C8—42no—
607C8—42no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22C2×C31×#12011yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4C31×#57612yes—
635C4C31×#57712yes—
636C2—41no—
637C2—81no—
638C2—82no—
639C2—162no—
640C2C31×#300811yescharacteristic
641C2C31×#301011yescharacteristic
642C2C31×#302111yescharacteristic
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×#3554C2semidirect
2. C31×#3217C2semidirect
3. C31×#3208C2semidirect
4. C31×#3205C2semidirect
5. C31×#1627C2semidirect
6. C31×#1626C2semidirect
7. C31×#1621C2semidirect
8. C31×#1620C2semidirect
9. C31×#1553C2semidirect
10. C31×#1497C2semidirect
11. C31×#1482C2semidirect
14. C31×#559C22semidirect
15. C31×#557C22semidirect
17. C31×#550C22semidirect
18. D4×Q8×C31C22semidirect
49. C31×#332C22semidirect
50. C31×#331C22semidirect
51. C31×Q8⋊C8C22semidirect
52. C31×Q8⋊C8C22semidirect
53. C31×Q8⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
232. #18958C31direct (N × H)
287. #3554C2×C31semidirect
288. #3217C2×C31semidirect
289. #3208C2×C31semidirect
290. #3205C2×C31semidirect
291. #1627C2×C31semidirect
292. #1626C2×C31semidirect
293. #1621C2×C31semidirect
294. #1620C2×C31semidirect
295. #1553C2×C31semidirect
296. #1497C2×C31semidirect
297. #1482C2×C31semidirect
328. #559C22×C31semidirect
329. #557C22×C31semidirect
331. #550C22×C31semidirect
332. D4×Q8C22×C31semidirect
363. #332C22×C31semidirect
364. #331C22×C31semidirect
365. Q8⋊C8C22×C31semidirect
366. Q8⋊C8C22×C31semidirect
367. Q8⋊C8C22×C31semidirect
368. D4⋊C8C22×C31semidirect
467. C31#18958direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35542
C31×#32172
C31×#32082
C31×#32052
C31×#16272
C31×#16262
C31×#16212
C31×#16202
C31×#15532
C31×#14972
C31×#14822
#1895831