Finite Group Databasebeta

← C31×#18959 #21475718

Subgroups — C31×#18959 #21475718

Order 7936.

874 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3554C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C31×#3230C211yesmaximal, characteristic, complemented
4C31×#3206C211yesmaximal, characteristic, complemented
5C31×#1637C212yesmaximal, complemented
6C31×#1636C212yesmaximal, complemented
7C31×#1631C212yesmaximal, complemented
8C31×#1630C212yesmaximal, complemented
9C31×#1553C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic, complemented
12C31×#562—22no—
13C31×#559—21no—
14C31×#559C2211yescharacteristic
15C31×#557C2211yescharacteristic
16C31×#554—21no—
17C31×#550C2211yescharacteristic
18D4×Q8×C31C2211yescharacteristic
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×C4⋊Q8C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#501—21no—
31C31×#500—21no—
32C31×#492C2212yes—
33C31×#489—21no—
34C31×#487—21no—
35C31×#487C2212yes—
36C31×#484—21no—
37C31×#484—21no—
38C31×#483—21no—
39C31×#480—21no—
40C31×#477C2212yes—
41C31×#477C2212yes—
42C31×#477C2212yes—
43C31×#476C2212yes—
44C31×#461—22no—
45C31×#448—22no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#332C2212yes—
50C31×#331C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×Q8⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31—21no—
59C2×SD16×C31—24no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2312yes—
68C31×C4⋊Q8C2311yescharacteristic
69C31×#127—21no—
70C31×#127C2311yescharacteristic
71C31×#126—22no—
72C31×#126—21no—
73C31×#125—22no—
74C31×#125—21no—
75C31×#125—21no—
76C31×#125D411yescharacteristic
77C31×#124—22no—
78C31×#124—22no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124D411yescharacteristic
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—22no—
86C31×#123—22no—
87C31×#123—22no—
88C31×#123—22no—
89C31×#123—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8D411yescharacteristic
98C31×C22⋊Q8D411yescharacteristic
99C31×C22⋊Q8D411yescharacteristic
100C31×#121—22no—
101C31×#121—22no—
102C31×#121—22no—
103C31×#121—21no—
104C31×#121—21no—
105C31×#121D412yes—
106C31×#121D411yescharacteristic
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31C2311yescharacteristic
111C4×Q8×C31C2311yescharacteristic
112C4×Q8×C31C2311yescharacteristic
113C4×D4×C31—22no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31—21no—
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31—21no—
120C4×D4×C31D412yes—
121C4×D4×C31C2311yescharacteristic
122C31×#117—21no—
123C31×#117—21no—
124C2×C4⋊C4×C31—22no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#107—22no—
130C31×#107—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#103—22no—
136C31×#103—22no—
137C31×#103—22no—
138C31×#103—24no—
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—
146#41×C31—82no—
147#41×C31—82no—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—22no—
152C2×Q8×C31—41no—
153C2×Q8×C31—82no—
154C2×Q8×C31—82no—
155C2×Q8×C31C2×D411yescharacteristic
156C2×Q8×C31C2×D411yescharacteristic
157C2×Q8×C31C2×D411yescharacteristic
158C2×D4×C31—22no—
159C2×D4×C31—22no—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31C2×D411yescharacteristic
165C2×D4×C31C2×D412yes—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—22no—
171C22×C4×C31—41no—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175SD16×C31—84no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
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—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—41no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31C2×D412yes—
200C4⋊C4×C31C2×D411yescharacteristic
201C4⋊C4×C31C2×D411yescharacteristic
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—21no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—21no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C42×C31—22no—
219C42×C31—21no—
220C42×C31—21no—
221C42×C31—21no—
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2411yescharacteristic
226#18959C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—21no—
228C23×C31—22no—
229C23×C31#12011yescharacteristic
230Q8×C31—22no—
231Q8×C31—22no—
232Q8×C31—22no—
233Q8×C31—44no—
234Q8×C31—42no—
235Q8×C31—81no—
236Q8×C31—82no—
237Q8×C31—82no—
238D4×C31—22no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—42no—
242D4×C31—81no—
243D4×C31—82no—
244D4×C31—82no—
245D4×C31—82no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—82no—
260C2×C4×C31—82no—
261C2×C4×C31—82no—
262C2×C4×C31—82no—
263C2×C4×C31—82no—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C8×C31—42no—
271C8×C31—42no—
272C8×C31—42no—
273#3554C2×C3111yescharacteristic
274#3238C2×C3111yescharacteristic, complemented
275#3230C2×C3111yescharacteristic, complemented
276#3206C2×C3111yescharacteristic, complemented
277#1637C2×C3112yescomplemented
278#1636C2×C3112yescomplemented
279#1631C2×C3112yescomplemented
280#1630C2×C3112yescomplemented
281#1553C2×C3111yescharacteristic, complemented
282#1497C2×C3111yescharacteristic, complemented
283#1482C2×C3111yescharacteristic, complemented
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—41no—
293C22×C31C2×#12011yescharacteristic
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—41no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—41no—
302C4×C31—41no—
303C4×C31—41no—
304C4×C31—41no—
305C4×C31—41no—
306C4×C31—82no—
307C4×C31#57712yes—
308C4×C31#57812yes—
309#562—22no—
310#559—21no—
311#559C22×C3111yescharacteristic
312#557C22×C3111yescharacteristic
313#554—21no—
314#550C22×C3111yescharacteristic
315D4×Q8C22×C3111yescharacteristic
316#548—21no—
317#547—21no—
318#542—21no—
319#542—21no—
320#540—21no—
321#539—22no—
322#536—21no—
323#533C22×C3111yescharacteristic
324#532C22×C3111yescharacteristic
325C2×C4⋊Q8C22×C3111yescharacteristic
326#529C22×C3111yescharacteristic
327#501—21no—
328#500—21no—
329#492C22×C3112yes—
330#489—21no—
331#487—21no—
332#487C22×C3112yes—
333#484—21no—
334#484—21no—
335#483—21no—
336#480—21no—
337#477C22×C3112yes—
338#477C22×C3112yes—
339#477C22×C3112yes—
340#476C22×C3112yes—
341#461—22no—
342#448—22no—
343#432C22×C3111yescharacteristic
344#431C22×C3111yescharacteristic
345#431C22×C3111yescharacteristic
346#332C22×C3112yes—
347#331C22×C3112yes—
348Q8⋊C8C22×C3112yes—
349Q8⋊C8C22×C3112yes—
350Q8⋊C8C22×C3112yes—
351D4⋊C8C22×C3112yes—
352C2×C31—41no—
353C2×C31—81no—
354C2×C31—82no—
355C2×C31#300911yescharacteristic
356C2×C31#301111yescharacteristic
357C2×C31#302111yescharacteristic
358C2×#41—21no—
359C2×#41—21no—
360C22×Q8—21no—
361C22×Q8—21no—
362C2×SD16—24no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8D4×C3111yescharacteristic
369C4⋊Q8D4×C3111yescharacteristic
370C4⋊Q8C23×C3112yes—
371C4⋊Q8C23×C3111yescharacteristic
372#127—21no—
373#127C23×C3111yescharacteristic
374#126—22no—
375#126—21no—
376#125—22no—
377#125—21no—
378#125—21no—
379#125D4×C3111yescharacteristic
380#124—22no—
381#124—22no—
382#124—21no—
383#124—21no—
384#124—21no—
385#124D4×C3111yescharacteristic
386#123—22no—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8D4×C3111yescharacteristic
401C22⋊Q8D4×C3111yescharacteristic
402C22⋊Q8D4×C3111yescharacteristic
403#121—22no—
404#121—22no—
405#121—22no—
406#121—21no—
407#121—21no—
408#121D4×C3111yescharacteristic
409#121D4×C3112yes—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C23×C3111yescharacteristic
414C4×Q8C23×C3111yescharacteristic
415C4×Q8C23×C3111yescharacteristic
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4D4×C3112yes—
424C4×D4C23×C3111yescharacteristic
425#117—21no—
426#117—21no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C42C23×C3111yescharacteristic
431#107—22no—
432#107—22no—
433#107—22no—
434#106—22no—
435C4⋊C8—21no—
436C4⋊C8C23×C3112yes—
437C4⋊C8C23×C3112yes—
438#103—22no—
439#103—22no—
440#103—22no—
441#103—24no—
442#102—22no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446#97—21no—
447C4×C8—21no—
448C4×C8C23×C3112yes—
449C31#1895911yescharacteristic, direct product factor, Sylow
450#41—82no—
451#41—82no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—41no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8C2×D4×C3111yescharacteristic
460C2×Q8C2×D4×C3111yescharacteristic
461C2×Q8C2×D4×C3111yescharacteristic
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×D4×C3111yescharacteristic
469C2×D4C2×D4×C3112yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479SD16—84no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×D4×C3112yes—
504C4⋊C4C2×D4×C3111yescharacteristic
505C4⋊C4C2×D4×C3111yescharacteristic
506C4⋊C4C2×D4×C3111yescharacteristic
507C4⋊C4C2×D4×C3111yescharacteristic
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42C2×D4×C3111yescharacteristic
527C42C2×D4×C3111yescharacteristic
528C42C2×D4×C3111yescharacteristic
529C42C24×C3111yescharacteristic
530C23—21no—
531C23—22no—
532C23C31×#12011yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—44no—
537Q8—42no—
538Q8—81no—
539Q8—82no—
540Q8—82no—
541D4—22no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—81no—
546D4—82no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4C31×#12011yescharacteristic
568C2×C4C31×#12011yescharacteristic
569C2×C4C31×#12011yescharacteristic
570C2×C4C22×D4×C3111yescharacteristic
571C2×C4C22×D4×C3111yescharacteristic
572C2×C4C22×D4×C3111yescharacteristic
573C8—42no—
574C8—42no—
575C8—42no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22C2×C31×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4C31×#57712yes—
600C4C31×#57812yes—
601C2—41no—
602C2—81no—
603C2—82no—
604C2C31×#300911yescharacteristic
605C2C31×#301111yescharacteristic
606C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3554C2
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×#3238C2semidirect
3. C31×#3230C2semidirect
4. C31×#3206C2semidirect
5. C31×#1637C2semidirect
6. C31×#1636C2semidirect
7. C31×#1631C2semidirect
8. C31×#1630C2semidirect
9. C31×#1553C2semidirect
10. C31×#1497C2semidirect
11. C31×#1482C2semidirect
226. #18959C31direct (N × H)
274. #3238C2×C31semidirect
275. #3230C2×C31semidirect
276. #3206C2×C31semidirect
277. #1637C2×C31semidirect
278. #1636C2×C31semidirect
279. #1631C2×C31semidirect
280. #1630C2×C31semidirect
281. #1553C2×C31semidirect
282. #1497C2×C31semidirect
283. #1482C2×C31semidirect
449. C31#18959direct (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×#32382
C31×#32302
C31×#32062
C31×#16372
C31×#16362
C31×#16312
C31×#16302
C31×#15532
C31×#14972
C31×#14822
#1895931