Finite Group Databasebeta

← C31×#16668 #21473427

Subgroups — C31×#16668 #21473427

Order 7936.

1398 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#399C211yesmaximal, characteristic
2C2×C31×#381C211yesmaximal, characteristic
3C31×#1445C212yesmaximal, complemented
4C31×#1442C212yesmaximal, complemented
5C31×#1441C212yesmaximal, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal
8C31×#1439C211yesmaximal, characteristic
9C31×C22⋊#95C212yesmaximal
10C24×C4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2211yescharacteristic
19C31×#399—24no—
20C31×#381—24no—
21C2×C31×#95C2211yescharacteristic
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C25×C31C2311yescharacteristic
34C23×C4×C31—22no—
35C23×C4×C31—22no—
36C23×C4×C31—41no—
37C23×C4×C31—41no—
38C23×C4×C31—41no—
39C23×C4×C31—41no—
40C23×C4×C31Q814yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2312yes—
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—41no—
57C2×C4⋊C4×C31—41no—
58C2×C4⋊C4×C31—41no—
59C2×C4⋊C4×C31—42no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—41no—
66C2×C22⋊C4×C31—41no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—41no—
70C2×C22⋊C4×C31—41no—
71C2×C22⋊C4×C31D414yes—
72C31×#95—22no—
73C31×#95—24no—
74C31×#95—24no—
75C31×#95—24no—
76C31×#95—41no—
77C31×#95—42no—
78C31×#95—42no—
79C31×#95—42no—
80C31×#95—42no—
81C31×#95—42no—
82C31×#95—42no—
83C31×#95—42no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—22no—
92C24×C31C2×D412yes—
93C24×C31C2×Q812yes—
94C24×C31#4112yes—
95C24×C31C2411yescharacteristic
96C22×C4×C31—28no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
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—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
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—44no—
145C22×C4×C31—44no—
146C22×C4×C31—44no—
147C22×C4×C31—44no—
148C22×C4×C31—44no—
149C22×C4×C31—44no—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×Q814yes—
153C22×C4×C31#4114yes—
154C22×C4×C31#4114yes—
155C4⋊C4×C31—48no—
156C4⋊C4×C31—48no—
157C4⋊C4×C31—44no—
158C4⋊C4×C31—44no—
159C22⋊C4×C31—44no—
160C22⋊C4×C31—48no—
161C22⋊C4×C31—48no—
162C22⋊C4×C31—44no—
163#16668C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—24no—
165C23×C31—22no—
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—22no—
169C23×C31—22no—
170C23×C31—22no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—42no—
185C23×C31—41no—
186C23×C31C22⋊Q814yes—
187C23×C31C22×D411yescharacteristic
188C23×C31C22×Q811yescharacteristic
189C23×C31C2×#4111yescharacteristic
190C23×C31C2×#4112yes—
191C23×C31C2×#4112yes—
192C23×C31H(5,𝔽2)11yescharacteristic
193C23×C31H(5,𝔽2)11yescharacteristic
194C23×C31H(5,𝔽2)11yescharacteristic
195C23×C31H(5,𝔽2)11yescharacteristic
196C23×C31H(5,𝔽2)12yes—
197C23×C312−1+412yes—
198C2×C4×C31—24no—
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—24no—
210C2×C4×C31—24no—
211C2×C4×C31—416no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
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×#399C2×C3111yescharacteristic
252C2×#381C2×C3111yescharacteristic
253#1445C2×C3112yescomplemented
254#1442C2×C3112yescomplemented
255#1441C2×C3112yescomplemented
256#1441C2×C3112yescomplemented
257#1439C2×C3112yes—
258#1439C2×C3111yescharacteristic
259C22⋊#95C2×C3112yes—
260C22×C31—24no—
261C22×C31—28no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—44no—
265C22×C31—44no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—44no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31C2×C22⋊Q811yescharacteristic
272C22×C31#53312yes—
273C22×C31#53411yescharacteristic
274C22×C31#53511yescharacteristic
275C22×C31#53811yescharacteristic
276C22×C31#53811yescharacteristic
277C22×C31#53912yes—
278C22×C31#53912yes—
279C22×C31#54012yes—
280C22×C31#54012yes—
281C22×C31#54311yescharacteristic
282C22×C31#54311yescharacteristic
283C22×C31#54412yes—
284C22×C31#54612yes—
285C22×C31#54712yes—
286C22×C31#55112yes—
287C22×C31#55112yes—
288C22×C31#55212yes—
289C22×C31#55911yescharacteristic
290C22×C31#55911yescharacteristic
291C22×C31#56011yescharacteristic
292C22×C31#56212yes—
293C22×C31#56311yescharacteristic
294C4×C31—44no—
295C4×C31—44no—
296C4×C31—42no—
297C4×C31—42no—
298C4×C31—44no—
299C4×C31—82no—
300C4×C31—82no—
301C4×C31—82no—
302C24×C4C22×C3111yescharacteristic
303C22×C4⋊C4C22×C3112yes—
304C22×C4⋊C4C22×C3112yes—
305C22×C4⋊C4C22×C3111yescharacteristic
306C22×C4⋊C4C22×C3111yescharacteristic
307C22×C22⋊C4C22×C3112yes—
308C22×C22⋊C4C22×C3112yes—
309C22×C22⋊C4C22×C3111yescharacteristic
310C22×C22⋊C4C22×C3111yescharacteristic
311#399—24no—
312#381—24no—
313C2×#95C22×C3111yescharacteristic
314C2×#95C22×C3112yes—
315C2×#95C22×C3112yes—
316C2×#95C22×C3112yes—
317C2×#95C22×C3112yes—
318C2×#95C22×C3112yes—
319C2×#95C22×C3112yes—
320C2×#95C22×C3112yes—
321C2×#95C22×C3111yescharacteristic
322C2×#95C22×C3112yes—
323C2×#95C22×C3112yes—
324C2×#95C22×C3112yes—
325C2×C31—44no—
326C2×C31#243812yes—
327C2×C31#260711yescharacteristic
328C2×C31#262611yescharacteristic
329C2×C31#266712yes—
330C2×C31#270612yes—
331C2×C31#270911yescharacteristic
332C2×C31#272411yescharacteristic
333C2×C31#274312yes—
334C2×C31#275211yescharacteristic
335C2×C31#277111yescharacteristic
336C2×C31#284111yescharacteristic
337C25C23×C3111yescharacteristic
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—41no—
341C23×C4—41no—
342C23×C4—41no—
343C23×C4—41no—
344C23×C4Q8×C3114yes—
345C23×C4C23×C3112yes—
346C23×C4C23×C3112yes—
347C23×C4C23×C3112yes—
348C23×C4C23×C3111yescharacteristic
349C23×C4C23×C3111yescharacteristic
350C23×C4C23×C3112yes—
351C23×C4C23×C3112yes—
352C23×C4C23×C3112yes—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—41no—
362C2×C4⋊C4—41no—
363C2×C4⋊C4—42no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—41no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4D4×C3114yes—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95—41no—
381#95—42no—
382#95—42no—
383#95—42no—
384#95—42no—
385#95—42no—
386#95—42no—
387#95—42no—
388C31#1666811yescharacteristic, direct product factor, Sylow
389C24—22no—
390C24—22no—
391C24—22no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—22no—
397C24C2×D4×C3112yes—
398C24C2×Q8×C3112yes—
399C24#41×C3112yes—
400C24C24×C3111yescharacteristic
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
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—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×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×D4×C3112yes—
457C22×C4C2×Q8×C3114yes—
458C22×C4#41×C3114yes—
459C22×C4#41×C3114yes—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23C31×C22⋊Q814yes—
491C23C22×D4×C3111yescharacteristic
492C23C22×Q8×C3111yescharacteristic
493C23C2×#41×C3111yescharacteristic
494C23C2×#41×C3112yes—
495C23C2×#41×C3112yes—
496C23C31×H(5,𝔽2)11yescharacteristic
497C23C31×H(5,𝔽2)11yescharacteristic
498C23C31×H(5,𝔽2)11yescharacteristic
499C23C31×H(5,𝔽2)11yescharacteristic
500C23C31×H(5,𝔽2)12yes—
501C23C31×2−1+412yes—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—416no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
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×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C22—24no—
556C22—28no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22C2×C31×C22⋊Q811yescharacteristic
567C22C31×#53312yes—
568C22C31×#53411yescharacteristic
569C22C31×#53511yescharacteristic
570C22C31×#53811yescharacteristic
571C22C31×#53811yescharacteristic
572C22C31×#53912yes—
573C22C31×#53912yes—
574C22C31×#54012yes—
575C22C31×#54012yes—
576C22C31×#54311yescharacteristic
577C22C31×#54311yescharacteristic
578C22C31×#54412yes—
579C22C31×#54612yes—
580C22C31×#54712yes—
581C22C31×#55112yes—
582C22C31×#55112yes—
583C22C31×#55212yes—
584C22C31×#55911yescharacteristic
585C22C31×#55911yescharacteristic
586C22C31×#56011yescharacteristic
587C22C31×#56212yes—
588C22C31×#56311yescharacteristic
589C4—44no—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—44no—
594C4—82no—
595C4—82no—
596C4—82no—
597C2—44no—
598C2C31×#243812yes—
599C2C31×#260711yescharacteristic
600C2C31×#262611yescharacteristic
601C2C31×#266712yes—
602C2C31×#270612yes—
603C2C31×#270911yescharacteristic
604C2C31×#272411yescharacteristic
605C2C31×#274312yes—
606C2C31×#275211yescharacteristic
607C2C31×#277111yescharacteristic
608C2C31×#284111yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1445C2semidirect
4. C31×#1442C2semidirect
5. C31×#1441C2semidirect
6. C31×#1441C2semidirect
163. #16668C31direct (N × H)
253. #1445C2×C31semidirect
254. #1442C2×C31semidirect
255. #1441C2×C31semidirect
256. #1441C2×C31semidirect
388. C31#16668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3992
C2×C31×#3812
C31×#14452
C31×#14422
C31×#14412
C31×#14392
C31×C22⋊#952
#1666831