Finite Group Databasebeta

← C31×#23567 #21480326

Subgroups — C31×#23567 #21480326

Order 7936.

1520 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic, complemented
2C2×C4×D8×C31C212yesmaximal, complemented
3C4×C31×#121C211yesmaximal, characteristic, complemented
4C31×#2057C211yesmaximal, characteristic, complemented
5C31×#2041C211yesmaximal, characteristic, complemented
6C31×#2033C212yesmaximal, complemented
7C31×#2023C211yesmaximal, characteristic, complemented
8C31×#2011C211yesmaximal, characteristic, complemented
9C31×#1999C212yesmaximal, complemented
10C4×C31×#102C212yesmaximal, complemented
11C4×C31×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×D8×C31C2212yes—
14C31×#532—28no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#127C2211yescharacteristic
17C2×C31×#124—21no—
18C2×C31×#121—21no—
19C2×C31×#121C2212yes—
20C2×C31×#121C2211yescharacteristic
21C2×C4×D4×C31—21no—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C2×C4×D4×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C4×D8×C31—28no—
27C2×C31×#107C2212yes—
28C2×C31×#102C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×#102C2212yes—
32C2×C31×C22⋊C8C2212yes—
33C2×C4×C8×C31C2212yes—
34C31×#390C2211yescharacteristic
35C31×#389C2211yescharacteristic
36C31×#388—21no—
37C31×#386—21no—
38C31×#380C2211yescharacteristic
39C4×C4⋊C4×C31C2211yescharacteristic
40C4×C22⋊C4×C31—21no—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#336C2211yescharacteristic
44C31×#336C2211yescharacteristic
45C2×#41×C31—28no—
46C2×#41×C31—44no—
47C2×#41×C31—44no—
48C22×Q8×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—22no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31C2312yes—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—22no—
56C23×C4×C31—21no—
57C23×C4×C31D411yescharacteristic
58C23×C4×C31D411yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×D8×C31—24no—
61C2×D8×C31—24no—
62C2×D8×C31—24no—
63C22×C8×C31C2312yes—
64C22×C8×C31C2312yes—
65C31×C4⋊Q8—24no—
66C31×#127—24no—
67C31×#124—28no—
68C31×#121—28no—
69C31×#121—24no—
70C31×#121—22no—
71C31×#121—24no—
72C31×#121—22no—
73C4×D4×C31—28no—
74C4×D4×C31—24no—
75C4×D4×C31—24no—
76C4×D4×C31—44no—
77C4×D4×C31D418yescomplemented
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C42×C31—24no—
92C2×C42×C31—21no—
93C2×C42×C31D411yescharacteristic
94C2×C42×C31D411yescharacteristic
95C2×C42×C31C2311yescharacteristic
96C2×C42×C31C2311yescharacteristic
97C31×#107—24no—
98C31×#102—24no—
99C31×#102—24no—
100C31×#102—24no—
101C31×#102—24no—
102C31×C22⋊C8—24no—
103C4×C8×C31—24no—
104C31×#95—21no—
105C31×#95—21no—
106C31×#95—21no—
107C24×C31—21no—
108C24×C31—22no—
109C24×C31C2×D411yescharacteristic
110#41×C31—432no—
111C2×Q8×C31—24no—
112C2×Q8×C31—42no—
113C2×Q8×C31—42no—
114C2×D4×C31—24no—
115C2×D4×C31—28no—
116C2×D4×C31—24no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—21no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—44no—
130C2×D4×C31—42no—
131C2×D4×C31—44no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C2×D4×C31C2×D414yes—
135C2×D4×C31#4112yescomplemented
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—28no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—41no—
150C22×C4×C31—44no—
151C22×C4×C31—44no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31—42no—
155C22×C4×C31D814yescomplemented
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2411yescharacteristic
162D8×C31—216no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31#4114yescomplemented
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—42no—
175C4⋊C4×C31C2×D414yes—
176C4⋊C4×C31#4112yescomplemented
177C22⋊C4×C31—28no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—28no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—42no—
183C22⋊C4×C31—42no—
184C42×C31—22no—
185C42×C31—22no—
186C42×C31—22no—
187C42×C31—42no—
188C42×C31C2×D414yes—
189#23567C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—24no—
197C23×C31—22no—
198C23×C31—41no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—42no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—42no—
210C23×C31C2×D812yes—
211C23×C31C22×D411yescharacteristic
212Q8×C31—48no—
213D4×C31—24no—
214D4×C31—24no—
215D4×C31—24no—
216D4×C31—48no—
217D4×C31—48no—
218D4×C31—48no—
219D4×C31—416no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—42no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—28no—
230C2×C4×C31—21no—
231C2×C4×C31—24no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—44no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—416no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—84no—
250C2×C4×C31#12012yes—
251C2×C4×C31#12012yes—
252C2×C4×C31C2×D812yes—
253C2×C4×C31C2×D812yes—
254C2×C4×C31#13512yes—
255C2×C4×C31AGL(1,ℤ/8)12yes—
256C2×C4×C31C22×D412yes—
257C2×C4×C31C2×#4111yescharacteristic
258C2×C4×C31C2×#4112yes—
259C2×C4×C31C2×#4111yescharacteristic
260C8×C31—24no—
261C8×C31—42no—
262C2×#532C2×C3111yescharacteristic, complemented
263C2×C4×D8C2×C3112yescomplemented
264C4×#121C2×C3111yescharacteristic, complemented
265#2057C2×C3111yescharacteristic, complemented
266#2041C2×C3111yescharacteristic, complemented
267#2033C2×C3112yescomplemented
268#2023C2×C3111yescharacteristic, complemented
269#2011C2×C3111yescharacteristic, complemented
270#1999C2×C3112yescomplemented
271C4×#102C2×C3112yescomplemented
272C4×C22⋊C8C2×C3111yescharacteristic, complemented
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—22no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—44no—
283C22×C31—42no—
284C22×C31—44no—
285C22×C31—44no—
286C22×C31—82no—
287C22×C31—82no—
288C22×C31—81no—
289C22×C31—82no—
290C22×C31—82no—
291C22×C31—81no—
292C22×C31—81no—
293C22×C31—81no—
294C22×C31C2×#12011yescharacteristic
295C22×C31#52511yescharacteristic
296C22×C31#53211yescharacteristic
297C22×C31C22×D811yescharacteristic
298C22×C31C2×#13511yescharacteristic
299C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C31#57511yescharacteristic
301C4×C31—22no—
302C4×C31—22no—
303C4×C31—22no—
304C4×C31—42no—
305C4×C31—41no—
306C4×C31—44no—
307C4×C31—81no—
308C4×C31—81no—
309C4×C31#44714yes—
310C4×C31#52512yes—
311C4×C31#53212yes—
312C22×#41—21no—
313C22×D8C22×C3112yes—
314#532—28no—
315C2×C4⋊Q8C22×C3111yescharacteristic
316C2×#127C22×C3111yescharacteristic
317C2×#124—21no—
318C2×#121—21no—
319C2×#121C22×C3112yes—
320C2×#121C22×C3111yescharacteristic
321C2×C4×D4—21no—
322C2×C4×D4—21no—
323C2×C4×D4C22×C3111yescharacteristic
324C2×C4×D4C22×C3112yes—
325C22×C42C22×C3111yescharacteristic
326C4×D8—28no—
327C2×#107C22×C3112yes—
328C2×#102C22×C3112yes—
329C2×#102C22×C3112yes—
330C2×#102C22×C3112yes—
331C2×#102C22×C3112yes—
332C2×C22⋊C8C22×C3112yes—
333C2×C4×C8C22×C3112yes—
334#390C22×C3111yescharacteristic
335#389C22×C3111yescharacteristic
336#388—21no—
337#386—21no—
338#380C22×C3111yescharacteristic
339C4×C4⋊C4C22×C3111yescharacteristic
340C4×C22⋊C4—21no—
341#340C22×C3112yes—
342#340C22×C3112yes—
343#336C22×C3111yescharacteristic
344#336C22×C3111yescharacteristic
345C2×C31—42no—
346C2×C31—82no—
347C2×C31—84no—
348C2×C31#238811yescharacteristic
349C2×C31C2×#44711yescharacteristic
350C2×C31#300211yescharacteristic
351C2×C31#310011yescharacteristic
352C2×C31#310611yescharacteristic
353C2×C31#316911yescharacteristic
354C2×C31#317111yescharacteristic
355C2×#41—28no—
356C2×#41—44no—
357C2×#41—44no—
358C22×Q8—21no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—21no—
363C22×D4C23×C3112yes—
364C22×D4C23×C3111yescharacteristic
365C23×C4—22no—
366C23×C4—21no—
367C23×C4D4×C3111yescharacteristic
368C23×C4D4×C3111yescharacteristic
369C23×C4C23×C3111yescharacteristic
370C2×D8—24no—
371C2×D8—24no—
372C2×D8—24no—
373C22×C8C23×C3112yes—
374C22×C8C23×C3112yes—
375C4⋊Q8—24no—
376#127—24no—
377#124—28no—
378#121—28no—
379#121—24no—
380#121—22no—
381#121—24no—
382#121—22no—
383C4×D4—28no—
384C4×D4—24no—
385C4×D4—24no—
386C4×D4—44no—
387C4×D4D4×C3118yescomplemented
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4C23×C3111yescharacteristic
391C2×C4⋊C4C23×C3111yescharacteristic
392C2×C4⋊C4C23×C3112yes—
393C2×C4⋊C4C23×C3111yescharacteristic
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—24no—
402C2×C42—21no—
403C2×C42D4×C3111yescharacteristic
404C2×C42D4×C3111yescharacteristic
405C2×C42C23×C3111yescharacteristic
406C2×C42C23×C3111yescharacteristic
407#107—24no—
408#102—24no—
409#102—24no—
410#102—24no—
411#102—24no—
412C22⋊C8—24no—
413C4×C8—24no—
414#95—21no—
415#95—21no—
416#95—21no—
417C31#2356711yescharacteristic, direct product factor, Sylow
418C24—21no—
419C24—22no—
420C24C2×D4×C3111yescharacteristic
421#41—432no—
422C2×Q8—24no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4C2×D4×C3114yes—
446C2×D4#41×C3112yescomplemented
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4D8×C3114yescomplemented
467C22×C4C2×D4×C3111yescharacteristic
468C22×C4C2×D4×C3111yescharacteristic
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C24×C3111yescharacteristic
473D8—216no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8#41×C3114yescomplemented
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4C2×D4×C3114yes—
487C4⋊C4#41×C3112yescomplemented
488C22⋊C4—28no—
489C22⋊C4—24no—
490C22⋊C4—28no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42C2×D4×C3114yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—24no—
507C23—22no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23C2×D8×C3112yes—
521C23C22×D4×C3111yescharacteristic
522Q8—48no—
523D4—24no—
524D4—24no—
525D4—24no—
526D4—48no—
527D4—48no—
528D4—48no—
529D4—416no—
530D4—42no—
531D4—42no—
532D4—42no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—416no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—84no—
560C2×C4C31×#12012yes—
561C2×C4C31×#12012yes—
562C2×C4C2×D8×C3112yes—
563C2×C4C2×D8×C3112yes—
564C2×C4C31×#13512yes—
565C2×C4C31×AGL(1,ℤ/8)12yes—
566C2×C4C22×D4×C3112yes—
567C2×C4C2×#41×C3111yescharacteristic
568C2×C4C2×#41×C3112yes—
569C2×C4C2×#41×C3111yescharacteristic
570C8—24no—
571C8—42no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22C2×C31×#12011yescharacteristic
594C22C31×#52511yescharacteristic
595C22C31×#53211yescharacteristic
596C22C22×D8×C3111yescharacteristic
597C22C2×C31×#13511yescharacteristic
598C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
599C22C31×#57511yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—44no—
606C4—81no—
607C4—81no—
608C4C31×#44714yes—
609C4C31×#52512yes—
610C4C31×#53212yes—
611C2—42no—
612C2—82no—
613C2—84no—
614C2C31×#238811yescharacteristic
615C2C2×C31×#44711yescharacteristic
616C2C31×#300211yescharacteristic
617C2C31×#310011yescharacteristic
618C2C31×#310611yescharacteristic
619C2C31×#316911yescharacteristic
620C2C31×#317111yescharacteristic
subgroupG/H
centerC22×C4×C31C2×D4
commutator, Frattini, cosocleC22×C4C24×C31
socleC23×C31C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#532C2semidirect
2. C2×C4×D8×C31C2semidirect
3. C4×C31×#121C2semidirect
4. C31×#2057C2semidirect
5. C31×#2041C2semidirect
6. C31×#2033C2semidirect
7. C31×#2023C2semidirect
8. C31×#2011C2semidirect
9. C31×#1999C2semidirect
10. C4×C31×#102C2semidirect
11. C4×C31×C22⋊C8C2semidirect
77. C4×D4×C31D4semidirect
135. C2×D4×C31#41semidirect
155. C22×C4×C31D8semidirect
168. C2×C8×C31#41semidirect
176. C4⋊C4×C31#41semidirect
189. #23567C31direct (N × H)
262. C2×#532C2×C31semidirect
263. C2×C4×D8C2×C31semidirect
264. C4×#121C2×C31semidirect
265. #2057C2×C31semidirect
266. #2041C2×C31semidirect
267. #2033C2×C31semidirect
268. #2023C2×C31semidirect
269. #2011C2×C31semidirect
270. #1999C2×C31semidirect
271. C4×#102C2×C31semidirect
272. C4×C22⋊C8C2×C31semidirect
387. C4×D4D4×C31semidirect
417. C31#23567direct (N × H)
446. C2×D4#41×C31semidirect
466. C22×C4D8×C31semidirect
479. C2×C8#41×C31semidirect
487. C4⋊C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C4×D8×C312
C4×C31×#1212
C31×#20572
C31×#20412
C31×#20332
C31×#20232
C31×#20112
C31×#19992
C4×C31×#1022
C4×C31×C22⋊C82
#2356731