Finite Group Databasebeta

← C31×#18335 #21475094

Subgroups — C31×#18335 #21475094

Order 7936.

1194 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic, complemented
2C31×#3217C212yesmaximal, complemented
3C31×#2541C211yesmaximal, characteristic, complemented
4C31×#1600C214yesmaximal, complemented
5C31×#1596C212yesmaximal, complemented
6C31×#1595C212yesmaximal, complemented
7C31×#1509C212yesmaximal, complemented
8C31×#1466C211yesmaximal, characteristic, complemented
9C31×#543C2211yescharacteristic
10C31×#539—22no—
11C31×#539—21no—
12C31×#538—22no—
13C31×#538C2212yes—
14C31×#534—21no—
15C31×#534C2211yescharacteristic
16C2×C31×#127—21no—
17C2×C31×#127C2211yescharacteristic, complemented
18C31×#529—22no—
19C31×#529C2212yes—
20C31×#525—21no—
21C31×#525—22no—
22C2×C31×#121—21no—
23C2×C31×#121—21no—
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#496—22no—
27C31×#490—22no—
28C31×#460—22no—
29C31×#459—22no—
30C31×#450C2214yescomplemented
31C31×#447C2214yes—
32C31×#432C2212yescomplemented
33C31×#392C2212yes—
34C31×#392C2212yescomplemented
35C31×#390—21no—
36C4×C4⋊C4×C31C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#323C2212yes—
40C31×#323C2212yescomplemented
41C2×#41×C31—21no—
42C22×D4×C31—21no—
43C22×D4×C31—22no—
44C22×D4×C31—22no—
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31D411yescharacteristic
48C22×D4×C31C2312yes—
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C2×SD16×C31—24no—
53C2×D8×C31—24no—
54C31×#127—22no—
55C31×#127—22no—
56C31×#127—44no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#125—22no—
61C31×#125D412yescomplemented
62C31×#124—22no—
63C31×#124D412yescomplemented
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×#123—22no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8C2312yes—
76C31×#121—22no—
77C31×#121—21no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—21no—
81C31×#121—44no—
82C31×#121—44no—
83C31×#121C2312yes—
84C31×#120—22no—
85C31×#120—22no—
86C31×#120—22no—
87C4×Q8×C31D412yescomplemented
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31D412yescomplemented
96C31×#117—21no—
97C31×#117—22no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31D411yescharacteristic
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31C2311yescharacteristic
106C2×C22⋊C4×C31C2311yescharacteristic
107C2×C42×C31—21no—
108C2×C42×C31—21no—
109C2×C42×C31C2311yescharacteristic
110C31×C4⋊C8—22no—
111C31×#102—24no—
112C31×#102—24no—
113C31×C22⋊C8C2314yes—
114C31×#97—22no—
115C31×#95C2312yes—
116C24×C31—21no—
117C24×C31—22no—
118C24×C31C2×D411yescharacteristic
119C24×C31C2×D411yescharacteristic
120#41×C31—82no—
121C2×Q8×C31—21no—
122C2×Q8×C31C2×D412yes—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—41no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C2×D4×C31—44no—
135C2×D4×C31—44no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—42no—
140C2×D4×C31—44no—
141C2×D4×C31—44no—
142C2×D4×C31—42no—
143C2×D4×C31—41no—
144C2×D4×C31—41no—
145C2×D4×C31—82no—
146C2×D4×C31—82no—
147C2×D4×C31—82no—
148C2×D4×C31C2×D412yes—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—82no—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2411yescharacteristic
167SD16×C31—84no—
168D8×C31—84no—
169C2×C8×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D412yes—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—44no—
195C22⋊C4×C31—44no—
196C42×C31—22no—
197C42×C31—22no—
198C42×C31—42no—
199C42×C31C2×D412yes—
200C42×C31C2×D412yes—
201#18335C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—22no—
206C23×C31—21no—
207C23×C31—21no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—41no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31—42no—
220C23×C31—81no—
221C23×C31—81no—
222C23×C31AGL(1,ℤ/8)12yes—
223C23×C31C22×D411yescharacteristic
224Q8×C31—42no—
225Q8×C31—81no—
226D4×C31—44no—
227D4×C31—42no—
228D4×C31—44no—
229D4×C31—44no—
230D4×C31—44no—
231D4×C31—42no—
232D4×C31—42no—
233D4×C31—84no—
234D4×C31—82no—
235D4×C31—82no—
236D4×C31—81no—
237D4×C31—82no—
238D4×C31—88no—
239D4×C31—88no—
240D4×C31—82no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—22no—
244C2×C4×C31—24no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—22no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—44no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—81no—
260C2×C4×C31—81no—
261C2×C4×C31—81no—
262C2×C4×C31—82no—
263C2×C4×C31—81no—
264C2×C4×C31#12012yes—
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31C22×D412yes—
268C8×C31—44no—
269#3459C2×C3111yescharacteristic, complemented
270#3217C2×C3112yescomplemented
271#2541C2×C3111yescharacteristic, complemented
272#1600C2×C3114yescomplemented
273#1596C2×C3112yescomplemented
274#1595C2×C3112yescomplemented
275#1509C2×C3112yescomplemented
276#1466C2×C3111yescharacteristic, complemented
277C22×C31—21no—
278C22×C31—21no—
279C22×C31—22no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—42no—
286C22×C31—41no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—81no—
291C22×C31—81no—
292C22×C31—82no—
293C22×C31—82no—
294C22×C31—82no—
295C22×C31—82no—
296C22×C31—84no—
297C22×C31—82no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31C2×#12011yescharacteristic
301C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C31#57511yescharacteristic
303C4×C31—22no—
304C4×C31—21no—
305C4×C31—22no—
306C4×C31—44no—
307C4×C31—42no—
308C4×C31—41no—
309C4×C31—41no—
310C4×C31—42no—
311C4×C31—42no—
312C4×C31#45312yes—
313#543C22×C3111yescharacteristic
314#539—22no—
315#539—21no—
316#538—22no—
317#538C22×C3112yes—
318#534—21no—
319#534C22×C3111yescharacteristic
320C2×#127—21no—
321C2×#127C22×C3111yescharacteristic, complemented
322#529—22no—
323#529C22×C3112yes—
324#525—21no—
325#525—22no—
326C2×#121—21no—
327C2×#121—21no—
328C2×C4×D4—21no—
329C2×C4×D4C22×C3111yescharacteristic
330#496—22no—
331#490—22no—
332#460—22no—
333#459—22no—
334#450C22×C3114yescomplemented
335#447C22×C3114yes—
336#432C22×C3112yescomplemented
337#392C22×C3112yes—
338#392C22×C3112yescomplemented
339#390—21no—
340C4×C4⋊C4C22×C3111yescharacteristic, complemented
341#328C22×C3114yescomplemented
342#327C22×C3114yes—
343#323C22×C3112yes—
344#323C22×C3112yescomplemented
345C2×C31—21no—
346C2×C31—21no—
347C2×C31—42no—
348C2×C31—81no—
349C2×C31—82no—
350C2×C31—162no—
351C2×C31#300611yescharacteristic
352C2×C31C2×#45311yescharacteristic
353C2×C31#302711yescharacteristic
354C2×#41—21no—
355C22×D4—21no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4D4×C3111yescharacteristic
361C22×D4C23×C3112yes—
362C23×C4—21no—
363C23×C4D4×C3111yescharacteristic
364C23×C4D4×C3111yescharacteristic
365C2×SD16—24no—
366C2×D8—24no—
367#127—22no—
368#127—22no—
369#127—44no—
370#126—22no—
371#126—22no—
372#126—22no—
373#125—22no—
374#125D4×C3112yescomplemented
375#124—22no—
376#124D4×C3112yescomplemented
377#123—22no—
378#123—22no—
379#123—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8C23×C3112yes—
389#121—22no—
390#121—21no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—44no—
395#121—44no—
396#121C23×C3112yes—
397#120—22no—
398#120—22no—
399#120—22no—
400C4×Q8D4×C3112yescomplemented
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4D4×C3112yescomplemented
409#117—21no—
410#117—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4D4×C3111yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4C23×C3111yescharacteristic
419C2×C22⋊C4C23×C3111yescharacteristic
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C23×C3111yescharacteristic
423C4⋊C8—22no—
424#102—24no—
425#102—24no—
426C22⋊C8C23×C3114yes—
427#97—22no—
428#95C23×C3112yes—
429C31#1833511yescharacteristic, direct product factor, Sylow
430C24—21no—
431C24—22no—
432C24C2×D4×C3111yescharacteristic
433C24C2×D4×C3111yescharacteristic
434#41—82no—
435C2×Q8—21no—
436C2×Q8C2×D4×C3112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4C2×D4×C3112yes—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C24×C3111yescharacteristic
481SD16—84no—
482D8—84no—
483C2×C8—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4C2×D4×C3112yes—
495C4⋊C4C2×D4×C3112yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C42—22no—
511C42—22no—
512C42—42no—
513C42C2×D4×C3112yes—
514C42C2×D4×C3112yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—81no—
534C23—81no—
535C23C31×AGL(1,ℤ/8)12yes—
536C23C22×D4×C3111yescharacteristic
537Q8—42no—
538Q8—81no—
539D4—44no—
540D4—42no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—82no—
548D4—82no—
549D4—81no—
550D4—82no—
551D4—88no—
552D4—88no—
553D4—82no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4C31×#12012yes—
578C2×C4C31×#12011yescharacteristic
579C2×C4C31×#12011yescharacteristic
580C2×C4C22×D4×C3112yes—
581C8—44no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×C31×#12011yescharacteristic
606C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
607C22C31×#57511yescharacteristic
608C4—22no—
609C4—21no—
610C4—22no—
611C4—44no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4C31×#45312yes—
618C2—21no—
619C2—21no—
620C2—42no—
621C2—81no—
622C2—82no—
623C2—162no—
624C2C31×#300611yescharacteristic
625C2C2×C31×#45311yescharacteristic
626C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ 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×#3459C2semidirect
2. C31×#3217C2semidirect
3. C31×#2541C2semidirect
4. C31×#1600C2semidirect
5. C31×#1596C2semidirect
6. C31×#1595C2semidirect
7. C31×#1509C2semidirect
8. C31×#1466C2semidirect
17. C2×C31×#127C22semidirect
30. C31×#450C22semidirect
32. C31×#432C22semidirect
34. C31×#392C22semidirect
36. C4×C4⋊C4×C31C22semidirect
37. C31×#328C22semidirect
40. C31×#323C22semidirect
61. C31×#125D4semidirect
63. C31×#124D4semidirect
87. C4×Q8×C31D4semidirect
95. C4×D4×C31D4semidirect
201. #18335C31direct (N × H)
269. #3459C2×C31semidirect
270. #3217C2×C31semidirect
271. #2541C2×C31semidirect
272. #1600C2×C31semidirect
273. #1596C2×C31semidirect
274. #1595C2×C31semidirect
275. #1509C2×C31semidirect
276. #1466C2×C31semidirect
321. C2×#127C22×C31semidirect
334. #450C22×C31semidirect
336. #432C22×C31semidirect
338. #392C22×C31semidirect
340. C4×C4⋊C4C22×C31semidirect
341. #328C22×C31semidirect
344. #323C22×C31semidirect
374. #125D4×C31semidirect
376. #124D4×C31semidirect
400. C4×Q8D4×C31semidirect
408. C4×D4D4×C31semidirect
429. C31#18335direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34592
C31×#32172
C31×#25412
C31×#16002
C31×#15962
C31×#15952
C31×#15092
C31×#14662
#1833531