Finite Group Databasebeta

← C31×#15783 #21472542

Subgroups — C31×#15783 #21472542

Order 7936.

1586 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
1C22×C31×C4⋊Q8C211yesmaximal, characteristic
2C2×C31×#400C211yesmaximal, characteristic
3C2×C31×#398C211yesmaximal, characteristic
4C2×C31×#391C211yesmaximal, characteristic
5C31×#1446C212yesmaximal
6C31×#1446C211yesmaximal, characteristic
7C31×#1442C212yesmaximal
8C31×#1441C212yesmaximal
9C31×#1436C212yesmaximal
10C31×#1436C211yesmaximal, characteristic
11C31×#1436C211yesmaximal, characteristic
12C23×Q8×C31C2211yescharacteristic
13C23×Q8×C31C2211yescharacteristic
14C2×C31×C4⋊Q8—24no—
15C2×C31×C4⋊Q8—24no—
16C2×C31×C4⋊Q8—24no—
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2212yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C42×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#400—24no—
26C31×#398—24no—
27C31×#391—24no—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
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—22no—
45C22×Q8×C31—22no—
46C22×Q8×C31—24no—
47C22×Q8×C31—24no—
48C22×Q8×C31—42no—
49C22×Q8×C31—42no—
50C22×Q8×C31D414yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2312yes—
57C23×C4×C31C2312yes—
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C23×C4×C31C2311yescharacteristic
62C31×C4⋊Q8—232no—
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—22no—
74C2×C4⋊C4×C31—24no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31D414yes—
80C2×C4⋊C4×C31D414yes—
81C2×C42×C31—22no—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—22no—
85C2×C42×C31—42no—
86C31×#95—22no—
87C31×#95—24no—
88C31×#95—22no—
89C31×#95—24no—
90C31×#95—24no—
91C31×#95—24no—
92C31×#95—24no—
93C31×#95—24no—
94C31×#95—24no—
95C31×#95—42no—
96C31×#95—42no—
97C31×#95—42no—
98C31×#95Q818yescomplemented
99C24×C31C2411yescharacteristic
100C2×Q8×C31—28no—
101C2×Q8×C31—28no—
102C2×Q8×C31—28no—
103C2×Q8×C31—44no—
104C2×Q8×C31—48no—
105C2×Q8×C31—48no—
106C2×Q8×C31—48no—
107C2×Q8×C31—48no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
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×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×Q814yes—
169C22×C4×C31C2×Q814yes—
170C22×C4×C31C2×Q814yes—
171C22×C4×C31#4114yes—
172C22×C4×C31#4114yes—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—28no—
175C4⋊C4×C31—28no—
176C4⋊C4×C31—28no—
177C4⋊C4×C31—44no—
178C4⋊C4×C31—48no—
179C42×C31—28no—
180C42×C31—44no—
181#15783C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31C22×D411yescharacteristic
183C23×C31C22×D411yescharacteristic
184C23×C31C22×D411yescharacteristic
185C23×C31C22×Q812yes—
186C23×C31C2×#4112yes—
187C23×C31C2×#4112yes—
188C23×C31H(5,𝔽2)12yes—
189C23×C312−1+412yes—
190C23×C312−1+412yes—
191Q8×C31—416no—
192Q8×C31—416no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—22no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
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—42no—
251C2×C4×C31—42no—
252C2×C4×C31#12014yes—
253C2×C4×C31#12314yes—
254C2×C4×C31#12414yes—
255C2×C4×C31C4⋊Q814yes—
256C22×C4⋊Q8C2×C3111yescharacteristic
257C2×#400C2×C3111yescharacteristic
258C2×#398C2×C3111yescharacteristic
259C2×#391C2×C3111yescharacteristic
260#1446C2×C3112yes—
261#1446C2×C3111yescharacteristic
262#1442C2×C3112yes—
263#1441C2×C3112yes—
264#1436C2×C3112yes—
265#1436C2×C3111yescharacteristic
266#1436C2×C3111yescharacteristic
267C22×C31C2×#12011yescharacteristic
268C22×C31C2×#12311yescharacteristic
269C22×C31C2×#12411yescharacteristic
270C22×C31C2×C4⋊Q811yescharacteristic
271C22×C31#53312yes—
272C22×C31#53411yescharacteristic
273C22×C31#53611yescharacteristic
274C22×C31#53611yescharacteristic
275C22×C31#54112yes—
276C22×C31#54212yes—
277C22×C31#54412yes—
278C22×C31#54612yes—
279C22×C31#54612yes—
280C22×C31#54812yes—
281C22×C31#54812yes—
282C22×C31D4×Q812yes—
283C22×C31D4×Q812yes—
284C22×C31#55412yes—
285C22×C31#55612yes—
286C22×C31#55712yes—
287C22×C31#56312yes—
288C4×C31—24no—
289C4×C31—24no—
290C4×C31—24no—
291C4×C31—24no—
292C4×C31—42no—
293C4×C31—42no—
294C4×C31—42no—
295C4×C31—44no—
296C4×C31—44no—
297C4×C31—82no—
298C4×C31—82no—
299C23×Q8C22×C3111yescharacteristic
300C23×Q8C22×C3111yescharacteristic
301C2×C4⋊Q8—24no—
302C2×C4⋊Q8—24no—
303C2×C4⋊Q8—24no—
304C22×C4⋊C4C22×C3111yescharacteristic
305C22×C4⋊C4C22×C3112yes—
306C22×C4⋊C4C22×C3111yescharacteristic
307C22×C4⋊C4C22×C3111yescharacteristic
308C22×C4⋊C4C22×C3111yescharacteristic
309C22×C4⋊C4C22×C3111yescharacteristic
310C22×C42C22×C3111yescharacteristic
311C22×C42C22×C3111yescharacteristic
312#400—24no—
313#398—24no—
314#391—24no—
315C2×#95C22×C3111yescharacteristic
316C2×#95C22×C3112yes—
317C2×#95C22×C3112yes—
318C2×#95C22×C3111yescharacteristic
319C2×#95C22×C3112yes—
320C2×#95C22×C3112yes—
321C2×#95C22×C3112yes—
322C2×#95C22×C3112yes—
323C2×#95C22×C3112yes—
324C2×#95C22×C3112yes—
325C2×#95C22×C3112yes—
326C2×#95C22×C3112yes—
327C2×#95C22×C3112yes—
328C2×C31#240511yescharacteristic
329C2×C31#243411yescharacteristic
330C2×C31#243611yescharacteristic
331C2×C31#244612yes—
332C2×C31#249212yes—
333C2×C31#255012yes—
334C2×C31#271312yes—
335C2×C31#271612yes—
336C2×C31#272912yes—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—24no—
343C22×Q8—24no—
344C22×Q8—42no—
345C22×Q8—42no—
346C22×Q8D4×C3114yes—
347C23×C4C23×C3112yes—
348C23×C4C23×C3112yes—
349C23×C4C23×C3111yescharacteristic
350C23×C4C23×C3111yescharacteristic
351C23×C4C23×C3111yescharacteristic
352C23×C4C23×C3112yes—
353C23×C4C23×C3112yes—
354C23×C4C23×C3111yescharacteristic
355C23×C4C23×C3111yescharacteristic
356C23×C4C23×C3111yescharacteristic
357C23×C4C23×C3111yescharacteristic
358C4⋊Q8—232no—
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—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4D4×C3114yes—
376C2×C4⋊C4D4×C3114yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—42no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—42no—
392#95—42no—
393#95—42no—
394#95Q8×C3118yescomplemented
395C31#1578311yescharacteristic, direct product factor, Sylow
396C24C24×C3111yescharacteristic
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
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×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4C2×D4×C3112yes—
457C22×C4C2×D4×C3112yes—
458C22×C4C2×D4×C3112yes—
459C22×C4C2×D4×C3112yes—
460C22×C4C2×D4×C3112yes—
461C22×C4C2×D4×C3112yes—
462C22×C4C2×D4×C3112yes—
463C22×C4C2×D4×C3112yes—
464C22×C4C2×D4×C3112yes—
465C22×C4C2×Q8×C3114yes—
466C22×C4C2×Q8×C3114yes—
467C22×C4C2×Q8×C3114yes—
468C22×C4#41×C3114yes—
469C22×C4#41×C3114yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C42—28no—
477C42—44no—
478C23C22×D4×C3111yescharacteristic
479C23C22×D4×C3111yescharacteristic
480C23C22×D4×C3111yescharacteristic
481C23C22×Q8×C3112yes—
482C23C2×#41×C3112yes—
483C23C2×#41×C3112yes—
484C23C31×H(5,𝔽2)12yes—
485C23C31×2−1+412yes—
486C23C31×2−1+412yes—
487Q8—416no—
488Q8—416no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
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×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4C31×#12014yes—
549C2×C4C31×#12314yes—
550C2×C4C31×#12414yes—
551C2×C4C31×C4⋊Q814yes—
552C22C2×C31×#12011yescharacteristic
553C22C2×C31×#12311yescharacteristic
554C22C2×C31×#12411yescharacteristic
555C22C2×C31×C4⋊Q811yescharacteristic
556C22C31×#53312yes—
557C22C31×#53411yescharacteristic
558C22C31×#53611yescharacteristic
559C22C31×#53611yescharacteristic
560C22C31×#54112yes—
561C22C31×#54212yes—
562C22C31×#54412yes—
563C22C31×#54612yes—
564C22C31×#54612yes—
565C22C31×#54812yes—
566C22C31×#54812yes—
567C22D4×Q8×C3112yes—
568C22D4×Q8×C3112yes—
569C22C31×#55412yes—
570C22C31×#55612yes—
571C22C31×#55712yes—
572C22C31×#56312yes—
573C4—24no—
574C4—24no—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—82no—
583C4—82no—
584C2C31×#240511yescharacteristic
585C2C31×#243411yescharacteristic
586C2C31×#243611yescharacteristic
587C2C31×#244612yes—
588C2C31×#249212yes—
589C2C31×#255012yes—
590C2C31×#271312yes—
591C2C31×#271612yes—
592C2C31×#272912yes—
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
181. #15783C31direct (N × H)
394. #95Q8×C31semidirect
395. C31#15783direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×C4⋊Q82
C2×C31×#4002
C2×C31×#3982
C2×C31×#3912
C31×#14462
C31×#14422
C31×#14412
C31×#14362
#1578331