Finite Group Databasebeta

← C31×#24157 #21480916

Subgroups — C31×#24157 #21480916

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#533C211yesmaximal, characteristic
2C2×C31×#475C211yesmaximal, characteristic, complemented
3C2×C31×#462C211yesmaximal, characteristic, complemented
4C4×C31×C22⋊Q8C211yesmaximal, characteristic
5C31×#2035C212yesmaximal
6C31×#2001C212yesmaximal
7C31×#1988C211yesmaximal, characteristic, complemented
8C31×#1981C211yesmaximal, characteristic, complemented
9C31×#1878C212yesmaximal, complemented
10C31×#1865C212yesmaximal, complemented
11C31×#1797C211yesmaximal, characteristic
12C22×Q16×C31C2212yes—
13C31×#533—24no—
14C31×#533—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#125—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2212yes—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8×C31—21no—
22C2×C4×Q8×C31C2212yes—
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C31×#117—21no—
25C22×C42×C31C2211yescharacteristic
26C31×#475—24no—
27C31×#462—24no—
28C2×C31×#106C2212yes—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C31×#391C2211yescharacteristic
37C31×#389C2211yescharacteristic
38C31×#387—21no—
39C31×#385—21no—
40C31×#380C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2312yes—
49C22×Q8×C31C2311yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×Q16×C31—24no—
54C2×Q16×C31—24no—
55C2×Q16×C31—42no—
56C22×C8×C31C2312yes—
57C22×C8×C31C2312yes—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8D414yes—
61C31×C4⋊Q8D414yes—
62C31×#125—28no—
63C31×C22⋊Q8—28no—
64C31×C22⋊Q8—24no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—22no—
68C4×Q8×C31—28no—
69C4×Q8×C31—24no—
70C4×Q8×C31—24no—
71C4×Q8×C31—22no—
72C4×Q8×C31—22no—
73C31×#117—28no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C4⋊C4×C31C2312yes—
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C42×C31—22no—
90C2×C42×C31—22no—
91C2×C42×C31—22no—
92C2×C42×C31—21no—
93C2×C42×C31—21no—
94C2×C42×C31D411yescharacteristic
95C2×C42×C31D411yescharacteristic
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2311yescharacteristic
98C31×#106—24no—
99C31×C4⋊C8—22no—
100C31×C4⋊C8—22no—
101C31×#103—24no—
102C31×#103—24no—
103C31×#103—24no—
104C31×#103—42no—
105C31×C22⋊C8—42no—
106C31×#95—21no—
107C31×#95—21no—
108C31×#95—21no—
109C31×#95—21no—
110C31×#95—21no—
111C24×C31C2×D411yescharacteristic
112C2×Q8×C31—24no—
113C2×Q8×C31—22no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—21no—
123C2×Q8×C31—42no—
124C2×Q8×C31—42no—
125C2×Q8×C31C2×D414yes—
126C2×Q8×C31#4112yes—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2411yescharacteristic
151Q16×C31—48no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C4⋊C4×C31—28no—
159C4⋊C4×C31—28no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31C2×D414yes—
173C4⋊C4×C31#4112yes—
174C22⋊C4×C31—28no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—42no—
177C42×C31—28no—
178C42×C31—22no—
179C42×C31—22no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—42no—
183C42×C31—42no—
184C42×C31C2×D412yes—
185C42×C31C2×D412yes—
186#24157C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31#13712yes—
194C23×C31C22×D411yescharacteristic
195Q8×C31—22no—
196Q8×C31—24no—
197Q8×C31—24no—
198Q8×C31—22no—
199Q8×C31—41no—
200Q8×C31—42no—
201Q8×C31—42no—
202Q8×C31—41no—
203Q8×C31—48no—
204C2×C4×C31—28no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—44no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31#12012yes—
239C2×C4×C31#12012yes—
240C2×C4×C31#13512yes—
241C2×C4×C31#13712yes—
242C2×C4×C31#13712yes—
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C2×#4111yescharacteristic
246C2×C4×C31C2×#4111yescharacteristic
247C2×C4×C31H(5,𝔽2)11yescharacteristic
248C2×C4×C312−1+411yescharacteristic
249C8×C31—42no—
250C8×C31—42no—
251C2×#533C2×C3111yescharacteristic
252C2×#475C2×C3111yescharacteristic, complemented
253C2×#462C2×C3111yescharacteristic, complemented
254C4×C22⋊Q8C2×C3111yescharacteristic
255#2035C2×C3112yes—
256#2001C2×C3112yes—
257#1988C2×C3111yescharacteristic, complemented
258#1981C2×C3111yescharacteristic, complemented
259#1878C2×C3112yescomplemented
260#1865C2×C3112yescomplemented
261#1797C2×C3111yescharacteristic
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—22no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—41no—
268C22×C31—41no—
269C22×C31C2×#12011yescharacteristic
270C22×C31#52511yescharacteristic
271C22×C31#53711yescharacteristic
272C22×C31C2×#13511yescharacteristic
273C22×C31C2×#13711yescharacteristic
274C22×C31C2×#13711yescharacteristic
275C22×C31C2×#13711yescharacteristic
276C4×C31—22no—
277C4×C31—22no—
278C4×C31—42no—
279C4×C31—41no—
280C4×C31—44no—
281C4×C31—42no—
282C4×C31—41no—
283C4×C31—44no—
284C4×C31—81no—
285C4×C31—81no—
286C4×C31#45214yes—
287C4×C31#54611yescharacteristic
288C4×C31#54711yescharacteristic
289C4×C31#54811yescharacteristic
290C4×C31#55011yescharacteristic
291C22×Q16C22×C3112yes—
292#533—24no—
293#533—24no—
294C2×C4⋊Q8C22×C3111yescharacteristic
295C2×C4⋊Q8C22×C3111yescharacteristic
296C2×#125—21no—
297C2×C22⋊Q8—21no—
298C2×C22⋊Q8C22×C3112yes—
299C2×C22⋊Q8C22×C3111yescharacteristic
300C2×C4×Q8—21no—
301C2×C4×Q8C22×C3111yescharacteristic
302C2×C4×Q8C22×C3112yes—
303C2×#117—21no—
304C22×C42C22×C3111yescharacteristic
305#475—24no—
306#462—24no—
307C2×#106C22×C3112yes—
308C2×C4⋊C8C22×C3111yescharacteristic
309C2×C4⋊C8C22×C3111yescharacteristic
310C2×#103C22×C3112yes—
311C2×#103C22×C3112yes—
312C2×#103C22×C3112yes—
313C2×#103C22×C3112yes—
314C2×C22⋊C8C22×C3112yes—
315#391C22×C3111yescharacteristic
316#389C22×C3111yescharacteristic
317#387—21no—
318#385—21no—
319#380C22×C3111yescharacteristic
320C4×C4⋊C4—21no—
321C4×C4⋊C4C22×C3111yescharacteristic
322C4×C22⋊C4—21no—
323#340C22×C3112yes—
324#340C22×C3112yes—
325#336C22×C3112yes—
326C2×C31—42no—
327C2×C31#240711yescharacteristic
328C2×C31C2×#45211yescharacteristic
329C2×C31#300411yescharacteristic
330C2×C31#310711yescharacteristic
331C2×C31#311311yescharacteristic
332C2×C31#324811yescharacteristic
333C2×C31#325511yescharacteristic
334C22×Q8—21no—
335C22×Q8C23×C3112yes—
336C22×Q8C23×C3111yescharacteristic
337C23×C4D4×C3111yescharacteristic
338C23×C4D4×C3111yescharacteristic
339C23×C4C23×C3111yescharacteristic
340C2×Q16—24no—
341C2×Q16—24no—
342C2×Q16—42no—
343C22×C8C23×C3112yes—
344C22×C8C23×C3112yes—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347C4⋊Q8D4×C3114yes—
348C4⋊Q8D4×C3114yes—
349#125—28no—
350C22⋊Q8—28no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—22no—
355C4×Q8—28no—
356C4×Q8—24no—
357C4×Q8—24no—
358C4×Q8—22no—
359C4×Q8—22no—
360#117—28no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C23×C3111yescharacteristic
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42D4×C3111yescharacteristic
382C2×C42D4×C3111yescharacteristic
383C2×C42C23×C3111yescharacteristic
384C2×C42C23×C3111yescharacteristic
385#106—24no—
386C4⋊C8—22no—
387C4⋊C8—22no—
388#103—24no—
389#103—24no—
390#103—24no—
391#103—42no—
392C22⋊C8—42no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398C31#2415711yescharacteristic, direct product factor, Sylow
399C24C2×D4×C3111yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8C2×D4×C3114yes—
414C2×Q8#41×C3112yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C24×C3111yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×D4×C3114yes—
461C4⋊C4#41×C3112yes—
462C22⋊C4—28no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—42no—
471C42—42no—
472C42C2×D4×C3112yes—
473C42C2×D4×C3112yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23C31×#13712yes—
481C23C22×D4×C3111yescharacteristic
482Q8—22no—
483Q8—24no—
484Q8—24no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—42no—
489Q8—42no—
490Q8—41no—
491C2×C4—28no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4C31×#12012yes—
526C2×C4C31×#12012yes—
527C2×C4C31×#13512yes—
528C2×C4C31×#13712yes—
529C2×C4C31×#13712yes—
530C2×C4C22×D4×C3111yescharacteristic
531C2×C4C22×D4×C3111yescharacteristic
532C2×C4C2×#41×C3111yescharacteristic
533C2×C4C2×#41×C3111yescharacteristic
534C2×C4C31×H(5,𝔽2)11yescharacteristic
535C2×C4C31×2−1+411yescharacteristic
536C8—42no—
537C8—42no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×C31×#12011yescharacteristic
546C22C31×#52511yescharacteristic
547C22C31×#53711yescharacteristic
548C22C2×C31×#13511yescharacteristic
549C22C2×C31×#13711yescharacteristic
550C22C2×C31×#13711yescharacteristic
551C22C2×C31×#13711yescharacteristic
552C4—22no—
553C4—22no—
554C4—42no—
555C4—41no—
556C4—44no—
557C4—42no—
558C4—41no—
559C4—44no—
560C4—81no—
561C4—81no—
562C4C31×#45214yes—
563C4C31×#54611yescharacteristic
564C4C31×#54711yescharacteristic
565C4C31×#54811yescharacteristic
566C4C31×#55011yescharacteristic
567C2—42no—
568C2C31×#240711yescharacteristic
569C2C2×C31×#45211yescharacteristic
570C2C31×#300411yescharacteristic
571C2C31×#310711yescharacteristic
572C2C31×#311311yescharacteristic
573C2C31×#324811yescharacteristic
574C2C31×#325511yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#475C2semidirect
3. C2×C31×#462C2semidirect
7. C31×#1988C2semidirect
8. C31×#1981C2semidirect
9. C31×#1878C2semidirect
10. C31×#1865C2semidirect
186. #24157C31direct (N × H)
252. C2×#475C2×C31semidirect
253. C2×#462C2×C31semidirect
257. #1988C2×C31semidirect
258. #1981C2×C31semidirect
259. #1878C2×C31semidirect
260. #1865C2×C31semidirect
398. C31#24157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5332
C2×C31×#4752
C2×C31×#4622
C4×C31×C22⋊Q82
C31×#20352
C31×#20012
C31×#19882
C31×#19812
C31×#18782
C31×#18652
C31×#17972
#2415731