Finite Group Databasebeta

← C31×#23458 #21480217

Subgroups — C31×#23458 #21480217

Order 7936.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic, complemented
2C2×C31×#417C211yesmaximal, characteristic
3C2×C31×#416C211yesmaximal, characteristic
4C31×#2311C211yesmaximal, characteristic
5C31×#2026C212yesmaximal, complemented
6C31×#2025C212yesmaximal, complemented
7C31×#2024C212yesmaximal, complemented
8C31×#2023C212yesmaximal, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C2×C31×#340C211yesmaximal, characteristic, complemented
11C31×#1729C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×M16×C31C2211yescharacteristic
14C23×C8×C31C2211yescharacteristic
15C31×#544—24no—
16C31×#544—24no—
17C2×C31×C4⋊Q8—21no—
18C2×C31×C4⋊Q8C2212yes—
19C2×C31×#125—21no—
20C2×C31×#125C2212yes—
21C2×C31×C22⋊Q8—22no—
22C2×C31×#117C2211yescharacteristic
23C2×C31×#117C2211yescharacteristic
24C22×C4⋊C4×C31C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#417—24no—
27C31×#416—24no—
28C2×C31×#103C2211yescharacteristic
29C2×C31×#103C2211yescharacteristic
30C2×C31×#103C2212yes—
31C2×C31×#102C2211yescharacteristic
32C2×C31×#102C2211yescharacteristic
33C2×C31×#102C2212yes—
34C31×#391C2212yes—
35C31×#390C2212yes—
36C31×#386—22no—
37C31×#380—21no—
38C31×#380—21no—
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#336C2212yes—
44C31×#336C2212yes—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31Q814yescomplemented
51C22×Q8×C31—22no—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31C2311yescharacteristic
59C2×M16×C31—22no—
60C2×M16×C31—22no—
61C2×M16×C31—22no—
62C22×C8×C31—22no—
63C22×C8×C31—22no—
64C22×C8×C31—22no—
65C22×C8×C31C2311yescharacteristic
66C22×C8×C31C2311yescharacteristic
67C22×C8×C31C2312yes—
68C31×C4⋊Q8—28no—
69C31×C4⋊Q8—24no—
70C31×C4⋊Q8—24no—
71C31×#125—28no—
72C31×#125—24no—
73C31×#125—24no—
74C31×C22⋊Q8—216no—
75C31×#117—22no—
76C31×#117—22no—
77C31×#117—22no—
78C31×#117D414yes—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31D414yes—
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31D412yes—
98C2×C42×C31—21no—
99C2×C42×C31C2312yes—
100C31×#103—22no—
101C31×#103—22no—
102C31×#103—24no—
103C31×#102—22no—
104C31×#102—22no—
105C31×#102—24no—
106C31×#95—22no—
107C31×#95—22no—
108C24×C31—21no—
109C24×C31C2×D411yescharacteristic
110#41×C31—28no—
111#41×C31—28no—
112#41×C31—44no—
113#41×C31—44no—
114C2×Q8×C31—28no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31—82no—
121C2×Q8×C31—82no—
122C2×Q8×C31C2×Q812yes—
123C2×D4×C31—24no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31C2×Q812yes—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—24no—
137C22×C4×C31—24no—
138C22×C4×C31—22no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—81no—
143C22×C4×C31—81no—
144C22×C4×C31—81no—
145C22×C4×C31—81no—
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×Q812yes—
152C22×C4×C31#4112yes—
153C22×C4×C31#4112yes—
154C22×C4×C31#4112yes—
155C22×C4×C31C2411yescharacteristic
156M16×C31—44no—
157C2×C8×C31—21no—
158C2×C8×C31—21no—
159C2×C8×C31—22no—
160C2×C8×C31—21no—
161C2×C8×C31—21no—
162C2×C8×C31—21no—
163C2×C8×C31—21no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—44no—
167C4⋊C4×C31—28no—
168C4⋊C4×C31—28no—
169C4⋊C4×C31—28no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—42no—
179C4⋊C4×C31C2×D414yes—
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D412yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—82no—
185C22⋊C4×C31—82no—
186C42×C31—24no—
187C42×C31—22no—
188C42×C31—22no—
189#23458C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31#12012yes—
202C23×C31C22⋊Q812yes—
203C23×C31#12412yes—
204C23×C31C22×D411yescharacteristic
205Q8×C31—22no—
206Q8×C31—22no—
207Q8×C31—41no—
208Q8×C31—41no—
209Q8×C31—88no—
210D4×C31—22no—
211D4×C31—22no—
212D4×C31—48no—
213D4×C31—41no—
214D4×C31—41no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—48no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—84no—
238C2×C4×C31—84no—
239C2×C4×C31#12012yes—
240C2×C4×C31C22⋊Q814yes—
241C2×C4×C31C22⋊Q814yes—
242C2×C4×C31C22⋊Q812yes—
243C2×C4×C31#12414yes—
244C2×C4×C31#12414yes—
245C2×C4×C31#12412yes—
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×Q811yescharacteristic
249C2×C4×C31C2×#4111yescharacteristic
250C2×C4×C31C2×#4111yescharacteristic
251C2×C4×C31C2×#4111yescharacteristic
252C8×C31—41no—
253C8×C31—41no—
254C8×C31—42no—
255C2×#544C2×C3111yescharacteristic, complemented
256C2×#417C2×C3111yescharacteristic
257C2×#416C2×C3111yescharacteristic
258#2311C2×C3111yescharacteristic
259#2026C2×C3112yescomplemented
260#2025C2×C3112yescomplemented
261#2024C2×C3112yescomplemented
262#2023C2×C3112yescomplemented
263#1736C2×C3111yescharacteristic, complemented
264C2×#340C2×C3111yescharacteristic, complemented
265#1729C2×C3111yescharacteristic, complemented
266C22×C31—22no—
267C22×C31—22no—
268C22×C31—22no—
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31#39314yes—
280C22×C31C2×#12011yescharacteristic
281C22×C31C2×C22⋊Q811yescharacteristic
282C22×C31C2×#12411yescharacteristic
283C22×C31#57611yescharacteristic
284C22×C31#57711yescharacteristic
285C22×C31#57711yescharacteristic
286C22×C31#57811yescharacteristic
287C4×C31—42no—
288C4×C31—42no—
289C4×C31—44no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—42no—
293C4×C31—82no—
294C4×C31#39314yes—
295C4×C31C2×C22⋊Q811yescharacteristic
296C4×C31C2×C22⋊Q811yescharacteristic
297C4×C31C2×#12411yescharacteristic
298C4×C31C2×#12411yescharacteristic
299C22×#41C22×C3111yescharacteristic
300C22×M16C22×C3111yescharacteristic
301C23×C8C22×C3111yescharacteristic
302#544—24no—
303#544—24no—
304C2×C4⋊Q8—21no—
305C2×C4⋊Q8C22×C3112yes—
306C2×#125—21no—
307C2×#125C22×C3112yes—
308C2×C22⋊Q8—22no—
309C2×#117C22×C3111yescharacteristic
310C2×#117C22×C3111yescharacteristic
311C22×C4⋊C4C22×C3111yescharacteristic
312C22×C4⋊C4C22×C3111yescharacteristic
313#417—24no—
314#416—24no—
315C2×#103C22×C3111yescharacteristic
316C2×#103C22×C3111yescharacteristic
317C2×#103C22×C3112yes—
318C2×#102C22×C3111yescharacteristic
319C2×#102C22×C3111yescharacteristic
320C2×#102C22×C3112yes—
321#391C22×C3112yes—
322#390C22×C3112yes—
323#386—22no—
324#380—21no—
325#380—21no—
326#340C22×C3112yes—
327#340C22×C3112yes—
328#340C22×C3112yes—
329#340C22×C3112yes—
330#336C22×C3112yes—
331#336C22×C3112yes—
332C2×C31—24no—
333C2×C31—82no—
334C2×C31C2×#39311yescharacteristic
335C2×C31#301211yescharacteristic
336C2×C31#301311yescharacteristic
337C2×C31#308311yescharacteristic
338C2×C31#308411yescharacteristic
339C2×C31#313611yescharacteristic
340C2×C31#313711yescharacteristic
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41Q8×C3114yescomplemented
347C22×Q8—22no—
348C22×Q8C23×C3111yescharacteristic
349C22×D4—21no—
350C22×D4C23×C3111yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4C23×C3111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—22no—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8C23×C3111yescharacteristic
362C22×C8C23×C3111yescharacteristic
363C22×C8C23×C3112yes—
364C4⋊Q8—28no—
365C4⋊Q8—24no—
366C4⋊Q8—24no—
367#125—28no—
368#125—24no—
369#125—24no—
370C22⋊Q8—216no—
371#117—22no—
372#117—22no—
373#117—22no—
374#117D4×C3114yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4D4×C3111yescharacteristic
384C2×C4⋊C4D4×C3111yescharacteristic
385C2×C4⋊C4D4×C3114yes—
386C2×C4⋊C4C23×C3112yes—
387C2×C4⋊C4C23×C3112yes—
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C4⋊C4C23×C3111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4D4×C3112yes—
394C2×C42—21no—
395C2×C42C23×C3112yes—
396#103—22no—
397#103—22no—
398#103—24no—
399#102—22no—
400#102—22no—
401#102—24no—
402#95—22no—
403#95—22no—
404C31#2345811yescharacteristic, direct product factor, Sylow
405C24—21no—
406C24C2×D4×C3111yescharacteristic
407#41—28no—
408#41—28no—
409#41—44no—
410#41—44no—
411C2×Q8—28no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8C2×Q8×C3112yes—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×Q8×C3112yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4C2×D4×C3112yes—
444C22×C4C2×D4×C3112yes—
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3112yes—
448C22×C4C2×Q8×C3112yes—
449C22×C4#41×C3112yes—
450C22×C4#41×C3112yes—
451C22×C4#41×C3112yes—
452C22×C4C24×C3111yescharacteristic
453M16—44no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4C2×D4×C3114yes—
477C4⋊C4C2×D4×C3112yes—
478C4⋊C4C2×D4×C3112yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—22no—
485C42—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C31×#12012yes—
498C23C31×C22⋊Q812yes—
499C23C31×#12412yes—
500C23C22×D4×C3111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—88no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4C31×#12012yes—
536C2×C4C31×C22⋊Q814yes—
537C2×C4C31×C22⋊Q814yes—
538C2×C4C31×C22⋊Q812yes—
539C2×C4C31×#12414yes—
540C2×C4C31×#12414yes—
541C2×C4C31×#12412yes—
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C22×Q8×C3111yescharacteristic
545C2×C4C2×#41×C3111yescharacteristic
546C2×C4C2×#41×C3111yescharacteristic
547C2×C4C2×#41×C3111yescharacteristic
548C8—41no—
549C8—41no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C31×#39314yes—
565C22C2×C31×#12011yescharacteristic
566C22C2×C31×C22⋊Q811yescharacteristic
567C22C2×C31×#12411yescharacteristic
568C22C31×#57611yescharacteristic
569C22C31×#57711yescharacteristic
570C22C31×#57711yescharacteristic
571C22C31×#57811yescharacteristic
572C4—42no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C31×#39314yes—
580C4C2×C31×C22⋊Q811yescharacteristic
581C4C2×C31×C22⋊Q811yescharacteristic
582C4C2×C31×#12411yescharacteristic
583C4C2×C31×#12411yescharacteristic
584C2—24no—
585C2—82no—
586C2C2×C31×#39311yescharacteristic
587C2C31×#301211yescharacteristic
588C2C31×#301311yescharacteristic
589C2C31×#308311yescharacteristic
590C2C31×#308411yescharacteristic
591C2C31×#313611yescharacteristic
592C2C31×#313711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#544C2semidirect
5. C31×#2026C2semidirect
6. C31×#2025C2semidirect
7. C31×#2024C2semidirect
8. C31×#2023C2semidirect
9. C31×#1736C2semidirect
10. C2×C31×#340C2semidirect
11. C31×#1729C2semidirect
50. C2×#41×C31Q8semidirect
189. #23458C31direct (N × H)
255. C2×#544C2×C31semidirect
259. #2026C2×C31semidirect
260. #2025C2×C31semidirect
261. #2024C2×C31semidirect
262. #2023C2×C31semidirect
263. #1736C2×C31semidirect
264. C2×#340C2×C31semidirect
265. #1729C2×C31semidirect
346. C2×#41Q8×C31semidirect
404. C31#23458direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C2×C31×#4172
C2×C31×#4162
C31×#23112
C31×#20262
C31×#20252
C31×#20242
C31×#20232
C31×#17362
C2×C31×#3402
C31×#17292
#2345831