Finite Group Databasebeta

← C31×#19087 #21475846

Subgroups — C31×#19087 #21475846

Order 7936.

930 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×#3558C211yesmaximal, characteristic
2C31×#3240C211yesmaximal, characteristic, complemented
3C31×#3216C211yesmaximal, characteristic, complemented
4C31×#3215C211yesmaximal, characteristic, complemented
5C31×#1661C212yesmaximal, complemented
6C31×#1661C212yesmaximal, complemented
7C31×#1659C212yesmaximal, complemented
8C31×#1659C212yesmaximal, complemented
9C31×#1536C211yesmaximal, characteristic, complemented
10C31×#1535C211yesmaximal, characteristic, complemented
11C31×#1533C211yesmaximal, characteristic, complemented
12C31×#562C2212yes—
13C31×H(3,𝔽4)C2212yes—
14C31×#559—21no—
15C31×#548—21no—
16C31×#546—22no—
17C31×#546—21no—
18D42×C31—21no—
19C31×#542—21no—
20C31×#538—22no—
21C31×#538—21no—
22C31×#535—21no—
23C31×#534—21no—
24C31×#533C2211yescharacteristic, complemented
25C31×#532C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501C2211yescharacteristic, complemented
29C31×#501C2211yescharacteristic, complemented
30C31×#497C2212yescomplemented
31C31×#497C2212yescomplemented
32C31×#489C2211yescharacteristic, complemented
33C31×#489C2211yescharacteristic, complemented
34C31×#489C2212yescomplemented
35C31×#489C2212yescomplemented
36C31×#485—21no—
37C31×#482—21no—
38C31×#479—21no—
39C31×#478—21no—
40C31×#464—22no—
41C31×#462—21no—
42C31×#460—21no—
43C31×#452—22no—
44C31×#451—21no—
45C31×#450—21no—
46C31×#432C2211yescharacteristic, complemented
47C31×#432C2211yescharacteristic, complemented
48C31×#432C2211yescharacteristic, complemented
49C31×#330C2212yescomplemented
50C31×#329C2212yescomplemented
51C31×#329C2212yescomplemented
52C31×#329C2212yescomplemented
53C31×#329C2212yescomplemented
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31D412yes—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31—22no—
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C2×Q16×C31—22no—
62C2×Q16×C31—22no—
63C2×SD16×C31—22no—
64C2×SD16×C31—22no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×#127—21no—
69C31×#127D411yescharacteristic
70C31×#126—22no—
71C31×#126—21no—
72C31×#125—21no—
73C31×#125C2311yescharacteristic
74C31×#124—21no—
75C31×#124—22no—
76C31×#124—22no—
77C31×#124—22no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124D412yes—
81C31×#124C2311yescharacteristic
82C31×#124C2312yes—
83C31×#124C2312yes—
84C31×#123—22no—
85C31×#123—22no—
86C31×#123—21no—
87C31×#123—22no—
88C31×#123—22no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8D411yescharacteristic
95C31×#121—22no—
96C31×#121—22no—
97C31×#121—22no—
98C31×#121—21no—
99C31×#121—22no—
100C31×#121—22no—
101C31×#121—21no—
102C31×#121—21no—
103C31×#121D411yescharacteristic
104C31×#120—22no—
105C31×#120—22no—
106C31×#120—22no—
107C31×#120—22no—
108C31×#120—22no—
109C4×Q8×C31—21no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×Q8×C31D411yescharacteristic
113C4×D4×C31—22no—
114C4×D4×C31—22no—
115C4×D4×C31—21no—
116C4×D4×C31—21no—
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31D411yescharacteristic
120C31×#117—21no—
121C31×#117—21no—
122C2×C22⋊C4×C31—22no—
123C2×C22⋊C4×C31—21no—
124C2×C22⋊C4×C31—21no—
125C2×C42×C31C2311yescharacteristic
126C31×#107—21no—
127C31×#107—21no—
128C31×#106—21no—
129C31×#106—21no—
130C31×C4⋊C8—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—22no—
134C31×#103—22no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#102—22no—
138C31×#102—22no—
139C31×#102—21no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8—21no—
144C31×#97C2312yes—
145C31×#97C2312yes—
146C31×#97C2311yescharacteristic
147C31×#97C2311yescharacteristic
148C24×C31—22no—
149C24×C31—21no—
150#41×C31—82no—
151#41×C31—82no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—81no—
156C2×Q8×C31C2×D411yescharacteristic
157C2×Q8×C31C2×D412yes—
158C2×Q8×C31C2×D412yes—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—21no—
167C2×D4×C31—21no—
168C2×D4×C31—21no—
169C2×D4×C31—81no—
170C2×D4×C31—82no—
171C2×D4×C31—84no—
172C2×D4×C31C2×D411yescharacteristic
173C2×D4×C31C2×D412yes—
174C2×D4×C31C2×D412yes—
175C22×C4×C31—22no—
176C22×C4×C31—22no—
177C22×C4×C31—21no—
178C22×C4×C31—21no—
179C22×C4×C31—22no—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183Q16×C31—82no—
184Q16×C31—82no—
185SD16×C31—82no—
186SD16×C31—82no—
187C2×C8×C31—21no—
188C2×C8×C31—21no—
189C2×C8×C31—22no—
190C2×C8×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—41no—
204C4⋊C4×C31—41no—
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D411yescharacteristic
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—22no—
223C22⋊C4×C31—41no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—41no—
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2411yescharacteristic
231#19087C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31—22no—
234C23×C31—22no—
235C23×C31—42no—
236C23×C31—81no—
237C23×C31—81no—
238C23×C31—82no—
239C23×C31—82no—
240C23×C31—82no—
241C23×C31#12011yescharacteristic
242Q8×C31—42no—
243Q8×C31—42no—
244Q8×C31—41no—
245Q8×C31—42no—
246Q8×C31—41no—
247Q8×C31—42no—
248Q8×C31—41no—
249D4×C31—42no—
250D4×C31—42no—
251D4×C31—41no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—42no—
255D4×C31—44no—
256D4×C31—81no—
257D4×C31—82no—
258D4×C31—82no—
259D4×C31—82no—
260D4×C31—82no—
261D4×C31—82no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—42no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—82no—
274C2×C4×C31—81no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C8×C31—41no—
284C8×C31—41no—
285C8×C31—42no—
286C8×C31—42no—
287#3558C2×C3111yescharacteristic
288#3240C2×C3111yescharacteristic, complemented
289#3216C2×C3111yescharacteristic, complemented
290#3215C2×C3111yescharacteristic, complemented
291#1661C2×C3112yescomplemented
292#1661C2×C3112yescomplemented
293#1659C2×C3112yescomplemented
294#1659C2×C3112yescomplemented
295#1536C2×C3111yescharacteristic, complemented
296#1535C2×C3111yescharacteristic, complemented
297#1533C2×C3111yescharacteristic, complemented
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—41no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—42no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—162no—
311C22×C31—164no—
312C22×C31C2×#12011yescharacteristic
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—21no—
319C4×C31—41no—
320C4×C31—41no—
321C4×C31—42no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—81no—
325C4×C31—81no—
326#562C22×C3112yes—
327H(3,𝔽4)C22×C3112yes—
328#559—21no—
329#548—21no—
330#546—22no—
331#546—21no—
332D42—21no—
333#542—21no—
334#538—22no—
335#538—21no—
336#535—21no—
337#534—21no—
338#533C22×C3111yescharacteristic, complemented
339#532C22×C3111yescharacteristic
340#529C22×C3111yescharacteristic
341C2×#124C22×C3111yescharacteristic
342#501C22×C3111yescharacteristic, complemented
343#501C22×C3111yescharacteristic, complemented
344#497C22×C3112yescomplemented
345#497C22×C3112yescomplemented
346#489C22×C3111yescharacteristic, complemented
347#489C22×C3111yescharacteristic, complemented
348#489C22×C3112yescomplemented
349#489C22×C3112yescomplemented
350#485—21no—
351#482—21no—
352#479—21no—
353#478—21no—
354#464—22no—
355#462—21no—
356#460—21no—
357#452—22no—
358#451—21no—
359#450—21no—
360#432C22×C3111yescharacteristic, complemented
361#432C22×C3111yescharacteristic, complemented
362#432C22×C3111yescharacteristic, complemented
363#330C22×C3112yescomplemented
364#329C22×C3112yescomplemented
365#329C22×C3112yescomplemented
366#329C22×C3112yescomplemented
367#329C22×C3112yescomplemented
368#329C22×C3112yescomplemented
369C2×C31—41no—
370C2×C31—82no—
371C2×C31—81no—
372C2×C31—82no—
373C2×C31#301911yescharacteristic
374C2×C31#301911yescharacteristic
375C2×C31#302211yescharacteristic
376C2×#41—21no—
377C2×#41D4×C3112yes—
378C22×Q8D4×C3111yescharacteristic
379C22×D4—22no—
380C22×D4—21no—
381C22×D4D4×C3111yescharacteristic
382C2×Q16—22no—
383C2×Q16—22no—
384C2×SD16—22no—
385C2×SD16—22no—
386C4⋊Q8D4×C3111yescharacteristic
387C4⋊Q8C23×C3111yescharacteristic
388C4⋊Q8C23×C3111yescharacteristic
389#127—21no—
390#127D4×C3111yescharacteristic
391#126—22no—
392#126—21no—
393#125—21no—
394#125C23×C3111yescharacteristic
395#124—21no—
396#124—22no—
397#124—22no—
398#124—22no—
399#124—21no—
400#124—21no—
401#124D4×C3112yes—
402#124C23×C3112yes—
403#124C23×C3111yescharacteristic
404#124C23×C3112yes—
405#123—22no—
406#123—22no—
407#123—21no—
408#123—22no—
409#123—22no—
410C22⋊Q8—21no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8D4×C3111yescharacteristic
416#121—22no—
417#121—22no—
418#121—22no—
419#121—21no—
420#121—22no—
421#121—22no—
422#121—21no—
423#121—21no—
424#121D4×C3111yescharacteristic
425#120—22no—
426#120—22no—
427#120—22no—
428#120—22no—
429#120—22no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8D4×C3111yescharacteristic
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4D4×C3111yescharacteristic
441#117—21no—
442#117—21no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C42C23×C3111yescharacteristic
447#107—21no—
448#107—21no—
449#106—21no—
450#106—21no—
451C4⋊C8—21no—
452C4⋊C8—21no—
453C4⋊C8—21no—
454#103—22no—
455#103—22no—
456#103—21no—
457#103—21no—
458#102—22no—
459#102—22no—
460#102—21no—
461#102—21no—
462C22⋊C8—21no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465#97C23×C3111yescharacteristic
466#97C23×C3111yescharacteristic
467#97C23×C3112yes—
468#97C23×C3112yes—
469C31#1908711yescharacteristic, direct product factor, Sylow
470C24—22no—
471C24—21no—
472#41—82no—
473#41—82no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—81no—
478C2×Q8C2×D4×C3111yescharacteristic
479C2×Q8C2×D4×C3112yes—
480C2×Q8C2×D4×C3112yes—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—81no—
492C2×D4—82no—
493C2×D4—84no—
494C2×D4C2×D4×C3111yescharacteristic
495C2×D4C2×D4×C3112yes—
496C2×D4C2×D4×C3112yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4C2×D4×C3111yescharacteristic
503C22×C4C2×D4×C3111yescharacteristic
504C22×C4C2×D4×C3111yescharacteristic
505Q16—82no—
506Q16—82no—
507SD16—82no—
508SD16—82no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—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—41no—
526C4⋊C4—41no—
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—41no—
546C42—22no—
547C42—21no—
548C42—41no—
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C42C24×C3111yescharacteristic
553C23—21no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—81no—
558C23—81no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23C31×#12011yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—41no—
566Q8—42no—
567Q8—41no—
568Q8—42no—
569Q8—41no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—44no—
577D4—81no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—82no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4C31×#12011yescharacteristic
599C2×C4C31×#12011yescharacteristic
600C2×C4C31×#12011yescharacteristic
601C2×C4C22×D4×C3111yescharacteristic
602C2×C4C22×D4×C3111yescharacteristic
603C2×C4C22×D4×C3111yescharacteristic
604C8—41no—
605C8—41no—
606C8—42no—
607C8—42no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—162no—
621C22—164no—
622C22C2×C31×#12011yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C2—41no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2C31×#301911yescharacteristic
641C2C31×#301911yescharacteristic
642C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3558C2
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
2. C31×#3240C2semidirect
3. C31×#3216C2semidirect
4. C31×#3215C2semidirect
5. C31×#1661C2semidirect
6. C31×#1661C2semidirect
7. C31×#1659C2semidirect
8. C31×#1659C2semidirect
9. C31×#1536C2semidirect
10. C31×#1535C2semidirect
11. C31×#1533C2semidirect
24. C31×#533C22semidirect
28. C31×#501C22semidirect
29. C31×#501C22semidirect
30. C31×#497C22semidirect
31. C31×#497C22semidirect
32. C31×#489C22semidirect
33. C31×#489C22semidirect
34. C31×#489C22semidirect
35. C31×#489C22semidirect
46. C31×#432C22semidirect
47. C31×#432C22semidirect
48. C31×#432C22semidirect
49. C31×#330C22semidirect
50. C31×#329C22semidirect
51. C31×#329C22semidirect
52. C31×#329C22semidirect
53. C31×#329C22semidirect
54. C31×#329C22semidirect
231. #19087C31direct (N × H)
288. #3240C2×C31semidirect
289. #3216C2×C31semidirect
290. #3215C2×C31semidirect
291. #1661C2×C31semidirect
292. #1661C2×C31semidirect
293. #1659C2×C31semidirect
294. #1659C2×C31semidirect
295. #1536C2×C31semidirect
296. #1535C2×C31semidirect
297. #1533C2×C31semidirect
338. #533C22×C31semidirect
342. #501C22×C31semidirect
343. #501C22×C31semidirect
344. #497C22×C31semidirect
345. #497C22×C31semidirect
346. #489C22×C31semidirect
347. #489C22×C31semidirect
348. #489C22×C31semidirect
349. #489C22×C31semidirect
360. #432C22×C31semidirect
361. #432C22×C31semidirect
362. #432C22×C31semidirect
363. #330C22×C31semidirect
364. #329C22×C31semidirect
365. #329C22×C31semidirect
366. #329C22×C31semidirect
367. #329C22×C31semidirect
368. #329C22×C31semidirect
469. C31#19087direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35582
C31×#32402
C31×#32162
C31×#32152
C31×#16612
C31×#16592
C31×#15362
C31×#15352
C31×#15332
#1908731