Finite Group Databasebeta

← C31×#25875 #21482634

Subgroups — C31×#25875 #21482634

Order 7936.

1430 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3498C211yesmaximal, characteristic
2C31×#3141C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic, complemented
4C31×#2858C211yesmaximal, characteristic, complemented
5C31×#2037C214yesmaximal, complemented
6C31×#1965C211yesmaximal, characteristic, complemented
7C31×#1962C211yesmaximal, characteristic, complemented
8C31×#1792C214yesmaximal, complemented
9C4×C31×#104C211yesmaximal, characteristic, complemented
10C2×C31×H(5,𝔽2)C2212yes—
11C31×#550—21no—
12C31×#550—21no—
13D42×C31—22no—
14D42×C31—22no—
15D42×C31—21no—
16D42×C31—21no—
17C31×#535—22no—
18C31×#535C2212yes—
19C31×#532—21no—
20C31×#532C2211yescharacteristic
21C2×C31×C4⋊Q8C2211yescharacteristic, complemented
22C2×C31×#127—21no—
23C2×C31×#127C2211yescharacteristic
24C2×C31×#124—22no—
25C4×#41×C31C2211yescharacteristic
26C31×#489—22no—
27C31×#487—21no—
28C31×#486—21no—
29C31×#477—21no—
30C31×#476—21no—
31C31×#475—21no—
32C31×#474—21no—
33C31×#428C2212yescomplemented
34C31×#423C2211yescharacteristic, complemented
35C2×C31×#104C2212yes—
36C2×C31×#104C2214yescomplemented
37C31×#417C2212yes—
38C31×#417C2214yescomplemented
39C2×C31×#100C2214yescomplemented
40C4×M16×C31C2211yescharacteristic, complemented
41C31×#390—22no—
42C31×#390C2214yescomplemented
43C43×C31C2211yescharacteristic, complemented
44C31×#339C2212yescomplemented
45C31×H(5,𝔽2)—28no—
46C31×H(5,𝔽2)—44no—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31C2312yes—
50C2×#41×C31C2311yescharacteristic
51C22×Q8×C31—21no—
52C22×D4×C31—24no—
53C22×D4×C31—22no—
54C22×D4×C31—22no—
55C22×D4×C31—22no—
56C22×D4×C31—22no—
57C22×D4×C31D412yes—
58C22×D4×C31D412yes—
59C22×D4×C31C2312yes—
60C2×M16×C31C2312yes—
61C2×M16×C31C2312yes—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8D411yescharacteristic
67C31×#127—22no—
68C31×#127—22no—
69C31×#127—22no—
70C31×#127—21no—
71C31×#127—21no—
72C31×#127—21no—
73C31×#127—21no—
74C31×#127D411yescharacteristic
75C31×#124—22no—
76C31×#124—48no—
77C31×#124D412yes—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—24no—
81C31×#121—24no—
82C31×#121—21no—
83C31×#121—21no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#120—24no—
87C31×#120—24no—
88C31×#120—24no—
89C4×Q8×C31D411yescharacteristic
90C4×Q8×C31D411yescharacteristic
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31D411yescharacteristic
98C4×D4×C31D411yescharacteristic
99C31×#117—21no—
100C31×#117C2311yescharacteristic
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31C2312yes—
103C2×C22⋊C4×C31—28no—
104C2×C42×C31—21no—
105C2×C42×C31—21no—
106C2×C42×C31—21no—
107C2×C42×C31C2312yes—
108C2×C42×C31C2311yescharacteristic
109C31×#107—22no—
110C31×#106—22no—
111C31×C4⋊C8—21no—
112C31×C4⋊C8—21no—
113C31×#104—24no—
114C31×#104—44no—
115C31×#103—22no—
116C31×#103—24no—
117C31×#102—22no—
118C31×#102—24no—
119C31×#100—44no—
120C31×#97—21no—
121C4×C8×C31—21no—
122C24×C31—24no—
123C24×C31C2×D414yes—
124#41×C31—22no—
125#41×C31—24no—
126#41×C31—24no—
127#41×C31—48no—
128#41×C31—44no—
129#41×C31—44no—
130#41×C31#4114yes—
131C2×Q8×C31—44no—
132C2×Q8×C31—41no—
133C2×Q8×C31—41no—
134C2×Q8×C31C2×D412yes—
135C2×Q8×C31C2×D411yescharacteristic
136C2×D4×C31—21no—
137C2×D4×C31—24no—
138C2×D4×C31—24no—
139C2×D4×C31—22no—
140C2×D4×C31—24no—
141C2×D4×C31—24no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—24no—
151C2×D4×C31—24no—
152C2×D4×C31—48no—
153C2×D4×C31—44no—
154C2×D4×C31—44no—
155C2×D4×C31—416no—
156C2×D4×C31—44no—
157C2×D4×C31—44no—
158C2×D4×C31—84no—
159C2×D4×C31—84no—
160C2×D4×C31C2×D412yes—
161C2×D4×C31C2×D411yescharacteristic
162C22×C4×C31—21no—
163C22×C4×C31—24no—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2411yescharacteristic
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—24no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31C2×D411yescharacteristic
186C4⋊C4×C31C2×D411yescharacteristic
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—416no—
195C42×C31—24no—
196C42×C31—21no—
197C42×C31—21no—
198C42×C31—21no—
199C42×C31—22no—
200C42×C31—22no—
201C42×C31—42no—
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206#25875C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—24no—
209C23×C31—22no—
210C23×C31—28no—
211C23×C31—48no—
212C23×C31—44no—
213C23×C31—44no—
214C23×C31—84no—
215C23×C31—84no—
216C23×C31C22×D411yescharacteristic
217Q8×C31—21no—
218Q8×C31—24no—
219Q8×C31—82no—
220Q8×C31—82no—
221Q8×C31#12412yescomplemented
222D4×C31—24no—
223D4×C31—21no—
224D4×C31—24no—
225D4×C31—48no—
226D4×C31—48no—
227D4×C31—44no—
228D4×C31—42no—
229D4×C31—42no—
230D4×C31—44no—
231D4×C31—44no—
232D4×C31—42no—
233D4×C31—44no—
234D4×C31—42no—
235D4×C31—82no—
236D4×C31—82no—
237D4×C31—84no—
238D4×C31—84no—
239D4×C31#12412yescomplemented
240C2×C4×C31—21no—
241C2×C4×C31—24no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—24no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—48no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—44no—
256C2×C4×C31—42no—
257C2×C4×C31#12011yescharacteristic
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C2×#4112yes—
264C2×C4×C31H(5,𝔽2)12yes—
265C8×C31—42no—
266C8×C31—42no—
267#3498C2×C3111yescharacteristic
268#3141C2×C3111yescharacteristic, complemented
269#3079C2×C3111yescharacteristic, complemented
270#2858C2×C3111yescharacteristic, complemented
271#2037C2×C3114yescomplemented
272#1965C2×C3111yescharacteristic, complemented
273#1962C2×C3111yescharacteristic, complemented
274#1792C2×C3114yescomplemented
275C4×#104C2×C3111yescharacteristic, complemented
276C22×C31—21no—
277C22×C31—21no—
278C22×C31—22no—
279C22×C31—41no—
280C22×C31—41no—
281C22×C31—42no—
282C22×C31—44no—
283C22×C31—44no—
284C22×C31—42no—
285C22×C31—44no—
286C22×C31—42no—
287C22×C31—88no—
288C22×C31—168no—
289C22×C31C2×#12011yescharacteristic
290C22×C31#54612yes—
291C4×C31—22no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—44no—
295C4×C31—42no—
296C4×C31—41no—
297C4×C31—41no—
298C4×C31—82no—
299C4×C31#45312yes—
300C4×C31#45512yes—
301C4×C31C2×#12411yescharacteristic
302C4×C31#53511yescharacteristic
303C4×C31#54612yes—
304C2×H(5,𝔽2)C22×C3112yes—
305#550—21no—
306#550—21no—
307D42—22no—
308D42—22no—
309D42—21no—
310D42—21no—
311#535—22no—
312#535C22×C3112yes—
313#532—21no—
314#532C22×C3111yescharacteristic
315C2×C4⋊Q8C22×C3111yescharacteristic, complemented
316C2×#127—21no—
317C2×#127C22×C3111yescharacteristic
318C2×#124—22no—
319C4×#41C22×C3111yescharacteristic
320#489—22no—
321#487—21no—
322#486—21no—
323#477—21no—
324#476—21no—
325#475—21no—
326#474—21no—
327#428C22×C3112yescomplemented
328#423C22×C3111yescharacteristic, complemented
329C2×#104C22×C3112yes—
330C2×#104C22×C3114yescomplemented
331#417C22×C3112yes—
332#417C22×C3114yescomplemented
333C2×#100C22×C3114yescomplemented
334C4×M16C22×C3111yescharacteristic, complemented
335#390—22no—
336#390C22×C3114yescomplemented
337C43C22×C3111yescharacteristic, complemented
338#339C22×C3112yescomplemented
339C2×C31—22no—
340C2×C31—42no—
341C2×C31—84no—
342C2×C31—82no—
343C2×C31#243511yescharacteristic
344C2×C31C2×#45311yescharacteristic
345C2×C31C2×#45511yescharacteristic
346H(5,𝔽2)—28no—
347H(5,𝔽2)—44no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C23×C3112yes—
351C2×#41C23×C3111yescharacteristic
352C22×Q8—21no—
353C22×D4—24no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4D4×C3112yes—
359C22×D4D4×C3112yes—
360C22×D4C23×C3112yes—
361C2×M16C23×C3112yes—
362C2×M16C23×C3112yes—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8D4×C3111yescharacteristic
368#127—22no—
369#127—22no—
370#127—22no—
371#127—21no—
372#127—21no—
373#127—21no—
374#127—21no—
375#127D4×C3111yescharacteristic
376#124—22no—
377#124—48no—
378#124D4×C3112yes—
379#121—22no—
380#121—22no—
381#121—24no—
382#121—24no—
383#121—21no—
384#121—21no—
385#121—22no—
386#121—22no—
387#120—24no—
388#120—24no—
389#120—24no—
390C4×Q8D4×C3111yescharacteristic
391C4×Q8D4×C3111yescharacteristic
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4D4×C3111yescharacteristic
399C4×D4D4×C3111yescharacteristic
400#117—21no—
401#117C23×C3111yescharacteristic
402C2×C4⋊C4—21no—
403C2×C4⋊C4C23×C3112yes—
404C2×C22⋊C4—28no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C23×C3112yes—
409C2×C42C23×C3111yescharacteristic
410#107—22no—
411#106—22no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414#104—24no—
415#104—44no—
416#103—22no—
417#103—24no—
418#102—22no—
419#102—24no—
420#100—44no—
421#97—21no—
422C4×C8—21no—
423C31#2587511yescharacteristic, direct product factor, Sylow
424C24—24no—
425C24C2×D4×C3114yes—
426#41—22no—
427#41—24no—
428#41—24no—
429#41—48no—
430#41—44no—
431#41—44no—
432#41#41×C3114yes—
433C2×Q8—44no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×Q8C2×D4×C3112yes—
437C2×Q8C2×D4×C3111yescharacteristic
438C2×D4—21no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—48no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—416no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—84no—
461C2×D4—84no—
462C2×D4C2×D4×C3112yes—
463C2×D4C2×D4×C3111yescharacteristic
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3111yescharacteristic
473C22×C4C24×C3111yescharacteristic
474M16—22no—
475M16—22no—
476M16—24no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4C2×D4×C3111yescharacteristic
488C4⋊C4C2×D4×C3111yescharacteristic
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—416no—
497C42—24no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42C2×D4×C3111yescharacteristic
505C42C2×D4×C3111yescharacteristic
506C42C2×D4×C3111yescharacteristic
507C42C2×D4×C3111yescharacteristic
508C23—21no—
509C23—24no—
510C23—22no—
511C23—28no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—84no—
516C23—84no—
517C23C22×D4×C3111yescharacteristic
518Q8—21no—
519Q8—24no—
520Q8—82no—
521Q8—82no—
522Q8C31×#12412yescomplemented
523D4—24no—
524D4—21no—
525D4—24no—
526D4—48no—
527D4—48no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—44no—
536D4—82no—
537D4—82no—
538D4—84no—
539D4—84no—
540D4C31×#12412yescomplemented
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4C31×#12011yescharacteristic
559C2×C4C31×#12011yescharacteristic
560C2×C4C31×#12011yescharacteristic
561C2×C4C31×#12011yescharacteristic
562C2×C4C22×D4×C3111yescharacteristic
563C2×C4C22×D4×C3111yescharacteristic
564C2×C4C2×#41×C3112yes—
565C2×C4C31×H(5,𝔽2)12yes—
566C8—42no—
567C8—42no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—88no—
580C22—168no—
581C22C2×C31×#12011yescharacteristic
582C22C31×#54612yes—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—44no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—82no—
591C4C31×#45312yes—
592C4C31×#45512yes—
593C4C2×C31×#12411yescharacteristic
594C4C31×#53511yescharacteristic
595C4C31×#54612yes—
596C2—22no—
597C2—42no—
598C2—84no—
599C2—82no—
600C2C31×#243511yescharacteristic
601C2C2×C31×#45311yescharacteristic
602C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3498C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3141C2semidirect
3. C31×#3079C2semidirect
4. C31×#2858C2semidirect
5. C31×#2037C2semidirect
6. C31×#1965C2semidirect
7. C31×#1962C2semidirect
8. C31×#1792C2semidirect
9. C4×C31×#104C2semidirect
21. C2×C31×C4⋊Q8C22semidirect
33. C31×#428C22semidirect
34. C31×#423C22semidirect
36. C2×C31×#104C22semidirect
38. C31×#417C22semidirect
39. C2×C31×#100C22semidirect
40. C4×M16×C31C22semidirect
42. C31×#390C22semidirect
43. C43×C31C22semidirect
44. C31×#339C22semidirect
206. #25875C31direct (N × H)
221. Q8×C31#124semidirect
239. D4×C31#124semidirect
268. #3141C2×C31semidirect
269. #3079C2×C31semidirect
270. #2858C2×C31semidirect
271. #2037C2×C31semidirect
272. #1965C2×C31semidirect
273. #1962C2×C31semidirect
274. #1792C2×C31semidirect
275. C4×#104C2×C31semidirect
315. C2×C4⋊Q8C22×C31semidirect
327. #428C22×C31semidirect
328. #423C22×C31semidirect
330. C2×#104C22×C31semidirect
332. #417C22×C31semidirect
333. C2×#100C22×C31semidirect
334. C4×M16C22×C31semidirect
336. #390C22×C31semidirect
337. C43C22×C31semidirect
338. #339C22×C31semidirect
423. C31#25875direct (N × H)
522. Q8C31×#124semidirect
540. D4C31×#124semidirect

Maximal subgroups

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

subgroupindex
C31×#34982
C31×#31412
C31×#30792
C31×#28582
C31×#20372
C31×#19652
C31×#19622
C31×#17922
C4×C31×#1042
#2587531