Finite Group Databasebeta

← C31×#15849 #21472608

Subgroups — C31×#15849 #21472608

Order 7936.

1522 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#125C211yesmaximal, characteristic
2C2×C31×#395C211yesmaximal, characteristic
3C2×C31×#395C211yesmaximal, characteristic
4C2×C31×#391C211yesmaximal, characteristic
5C31×#1445C211yesmaximal, characteristic
6C31×#1444C212yesmaximal
7C31×#1442C212yesmaximal
8C31×#1441C212yesmaximal
9C31×#1436C212yesmaximal
10C31×#1436C211yesmaximal, characteristic
11C31×#1436C211yesmaximal, characteristic
12C23×Q8×C31C2211yescharacteristic
13C2×C31×#125—24no—
14C2×C31×#125—24no—
15C2×C31×#125—24no—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2212yes—
20C22×C4⋊C4×C31C2212yes—
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C22×C42×C31C2211yescharacteristic
26C31×#395—24no—
27C31×#395—24no—
28C31×#391—24no—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2212yes—
40C2×C31×#95C2212yes—
41C22×Q8×C31—22no—
42C22×Q8×C31—22no—
43C22×Q8×C31—22no—
44C22×Q8×C31—24no—
45C22×Q8×C31—42no—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C31×#125—232no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—24no—
74C2×C4⋊C4×C31—24no—
75C2×C4⋊C4×C31—24no—
76C2×C4⋊C4×C31—24no—
77C2×C4⋊C4×C31—42no—
78C2×C4⋊C4×C31—42no—
79C2×C4⋊C4×C31D414yes—
80C2×C4⋊C4×C31D414yes—
81C2×C4⋊C4×C31D414yes—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—22no—
85C2×C42×C31—22no—
86C2×C42×C31—42no—
87C31×#95—22no—
88C31×#95—22no—
89C31×#95—24no—
90C31×#95—24no—
91C31×#95—24no—
92C31×#95—24no—
93C31×#95—24no—
94C31×#95—24no—
95C31×#95—24no—
96C31×#95—42no—
97C31×#95—42no—
98C31×#95Q818yescomplemented
99C24×C31C2411yescharacteristic
100C2×Q8×C31—28no—
101C2×Q8×C31—48no—
102C2×Q8×C31—48no—
103C2×Q8×C31—44no—
104C2×Q8×C31—44no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×Q814yes—
164C22×C4×C31C2×Q814yes—
165C22×C4×C31C2×Q814yes—
166C22×C4×C31#4114yes—
167C22×C4×C31#4114yes—
168C4⋊C4×C31—28no—
169C4⋊C4×C31—28no—
170C4⋊C4×C31—28no—
171C4⋊C4×C31—28no—
172C4⋊C4×C31—28no—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—48no—
175C4⋊C4×C31—48no—
176C42×C31—28no—
177C42×C31—44no—
178#15849C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31C22×D411yescharacteristic
180C23×C31C22×D411yescharacteristic
181C23×C31C22×D411yescharacteristic
182C23×C31C22×Q812yes—
183C23×C31C2×#4112yes—
184C23×C31C2×#4112yes—
185C23×C31H(5,𝔽2)12yes—
186C23×C312−1+412yes—
187C23×C312−1+412yes—
188Q8×C31—416no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—24no—
214C2×C4×C31—24no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31#12014yes—
251C2×C4×C31C4⋊Q814yes—
252C22×#125C2×C3111yescharacteristic
253C2×#395C2×C3111yescharacteristic
254C2×#395C2×C3111yescharacteristic
255C2×#391C2×C3111yescharacteristic
256#1445C2×C3111yescharacteristic
257#1444C2×C3112yes—
258#1442C2×C3112yes—
259#1441C2×C3112yes—
260#1436C2×C3112yes—
261#1436C2×C3111yescharacteristic
262#1436C2×C3111yescharacteristic
263C22×C31C2×#12011yescharacteristic
264C22×C31#52511yescharacteristic
265C22×C31C2×C4⋊Q811yescharacteristic
266C22×C31#53211yescharacteristic
267C22×C31#53312yes—
268C22×C31#53511yescharacteristic
269C22×C31#53611yescharacteristic
270C22×C31#53611yescharacteristic
271C22×C31#53912yes—
272C22×C31#54112yes—
273C22×C31#54412yes—
274C22×C31#54612yes—
275C22×C31#54612yes—
276C22×C31#54712yes—
277C22×C31#54812yes—
278C22×C31D4×Q812yes—
279C22×C31D4×Q812yes—
280C22×C31#55212yes—
281C22×C31#55412yes—
282C22×C31#55712yes—
283C22×C31#56312yes—
284C4×C31—24no—
285C4×C31—24no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—44no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—44no—
293C4×C31—44no—
294C4×C31—44no—
295C23×Q8C22×C3111yescharacteristic
296C2×#125—24no—
297C2×#125—24no—
298C2×#125—24no—
299C22×C4⋊C4C22×C3111yescharacteristic
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3111yescharacteristic
302C22×C4⋊C4C22×C3112yes—
303C22×C4⋊C4C22×C3112yes—
304C22×C4⋊C4C22×C3111yescharacteristic
305C22×C4⋊C4C22×C3111yescharacteristic
306C22×C4⋊C4C22×C3111yescharacteristic
307C22×C42C22×C3111yescharacteristic
308C22×C42C22×C3111yescharacteristic
309#395—24no—
310#395—24no—
311#391—24no—
312C2×#95C22×C3112yes—
313C2×#95C22×C3111yescharacteristic
314C2×#95C22×C3111yescharacteristic
315C2×#95C22×C3112yes—
316C2×#95C22×C3112yes—
317C2×#95C22×C3112yes—
318C2×#95C22×C3112yes—
319C2×#95C22×C3112yes—
320C2×#95C22×C3112yes—
321C2×#95C22×C3112yes—
322C2×#95C22×C3112yes—
323C2×#95C22×C3112yes—
324C2×C31#240411yescharacteristic
325C2×C31#243211yescharacteristic
326C2×C31#243411yescharacteristic
327C2×C31#246212yes—
328C2×C31#255012yes—
329C2×C31#255612yes—
330C2×C31#269012yes—
331C2×C31#271312yes—
332C2×C31#272012yes—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—24no—
337C22×Q8—42no—
338C23×C4C23×C3111yescharacteristic
339C23×C4C23×C3111yescharacteristic
340C23×C4C23×C3111yescharacteristic
341C23×C4C23×C3111yescharacteristic
342C23×C4C23×C3111yescharacteristic
343C23×C4C23×C3112yes—
344C23×C4C23×C3112yes—
345C23×C4C23×C3112yes—
346C23×C4C23×C3112yes—
347C23×C4C23×C3111yescharacteristic
348C23×C4C23×C3111yescharacteristic
349#125—232no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4D4×C3114yes—
372C2×C4⋊C4D4×C3114yes—
373C2×C4⋊C4D4×C3114yes—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—42no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—42no—
389#95—42no—
390#95Q8×C3118yescomplemented
391C31#1584911yescharacteristic, direct product factor, Sylow
392C24C24×C3111yescharacteristic
393C2×Q8—28no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—44no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4C2×D4×C3112yes—
448C22×C4C2×D4×C3112yes—
449C22×C4C2×D4×C3112yes—
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3112yes—
452C22×C4C2×D4×C3112yes—
453C22×C4C2×D4×C3112yes—
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×Q8×C3114yes—
457C22×C4C2×Q8×C3114yes—
458C22×C4C2×Q8×C3114yes—
459C22×C4#41×C3114yes—
460C22×C4#41×C3114yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C42—28no—
470C42—44no—
471C23C22×D4×C3111yescharacteristic
472C23C22×D4×C3111yescharacteristic
473C23C22×D4×C3111yescharacteristic
474C23C22×Q8×C3112yes—
475C23C2×#41×C3112yes—
476C23C2×#41×C3112yes—
477C23C31×H(5,𝔽2)12yes—
478C23C31×2−1+412yes—
479C23C31×2−1+412yes—
480Q8—416no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4C31×#12014yes—
543C2×C4C31×C4⋊Q814yes—
544C22C2×C31×#12011yescharacteristic
545C22C31×#52511yescharacteristic
546C22C2×C31×C4⋊Q811yescharacteristic
547C22C31×#53211yescharacteristic
548C22C31×#53312yes—
549C22C31×#53511yescharacteristic
550C22C31×#53611yescharacteristic
551C22C31×#53611yescharacteristic
552C22C31×#53912yes—
553C22C31×#54112yes—
554C22C31×#54412yes—
555C22C31×#54612yes—
556C22C31×#54612yes—
557C22C31×#54712yes—
558C22C31×#54812yes—
559C22D4×Q8×C3112yes—
560C22D4×Q8×C3112yes—
561C22C31×#55212yes—
562C22C31×#55412yes—
563C22C31×#55712yes—
564C22C31×#56312yes—
565C4—24no—
566C4—24no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—44no—
574C4—44no—
575C4—44no—
576C2C31×#240411yescharacteristic
577C2C31×#243211yescharacteristic
578C2C31×#243411yescharacteristic
579C2C31×#246212yes—
580C2C31×#255012yes—
581C2C31×#255612yes—
582C2C31×#269012yes—
583C2C31×#271312yes—
584C2C31×#272012yes—
subgroupG/H
center, socle, prime radicalC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
98. C31×#95Q8semidirect
178. #15849C31direct (N × H)
390. #95Q8×C31semidirect
391. C31#15849direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1252
C2×C31×#3952
C2×C31×#3912
C31×#14452
C31×#14442
C31×#14422
C31×#14412
C31×#14362
#1584931