Finite Group Databasebeta

← C31×#16652 #21473411

Subgroups — C31×#16652 #21473411

Order 7936.

1654 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#394C211yesmaximal, characteristic
2C2×C31×#393C212yesmaximal, complemented
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C31×#1443C212yesmaximal, complemented
5C31×#1441C211yesmaximal, characteristic, complemented
6C31×#1440C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1438C212yesmaximal
9C31×#1437C212yesmaximal, complemented
10C23×Q8×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C42×C31C2211yescharacteristic
21C31×#394—24no—
22C31×#393—28no—
23C31×#391—24no—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2211yescharacteristic
34C25×C31C2312yes—
35C22×Q8×C31—22no—
36C22×Q8×C31—22no—
37C22×Q8×C31—41no—
38C22×Q8×C31—41no—
39C22×Q8×C31—41no—
40C22×Q8×C31—41no—
41C22×Q8×C31D414yescomplemented
42C22×D4×C31—24no—
43C22×D4×C31—22no—
44C22×D4×C31—42no—
45C22×D4×C31—41no—
46C22×D4×C31—41no—
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2312yes—
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2312yes—
54C23×C4×C31C2311yescharacteristic
55C2×C4⋊C4×C31—41no—
56C2×C4⋊C4×C31—41no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—41no—
59C2×C4⋊C4×C31D414yes—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31—42no—
76C2×C22⋊C4×C31—42no—
77C2×C22⋊C4×C31—42no—
78C2×C42×C31—22no—
79C2×C42×C31—41no—
80C2×C42×C31—41no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—24no—
84C31×#95—24no—
85C31×#95—24no—
86C31×#95—22no—
87C31×#95—24no—
88C31×#95—42no—
89C31×#95—41no—
90C31×#95—42no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31#4114yes—
106C24×C31C2411yescharacteristic
107C2×Q8×C31—28no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C2×Q8×C31—44no—
111C2×Q8×C31—44no—
112C2×Q8×C31—44no—
113C2×Q8×C31—44no—
114C2×D4×C31—48no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—48no—
118C2×D4×C31—44no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
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—21no—
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—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
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×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D414yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31#4114yes—
170C22×C4×C31#4112yes—
171C4⋊C4×C31—44no—
172C4⋊C4×C31—44no—
173C22⋊C4×C31—48no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—48no—
179C42×C31—44no—
180#16652C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31C22×D411yescharacteristic
217C23×C31C22×D411yescharacteristic
218C23×C31C2×#4111yescharacteristic
219C23×C31C2×#4112yes—
220C23×C31C2×#4112yes—
221C23×C31H(5,𝔽2)11yescharacteristic
222C23×C31H(5,𝔽2)11yescharacteristic
223C23×C31H(5,𝔽2)11yescharacteristic
224C23×C31H(5,𝔽2)11yescharacteristic
225C23×C31H(5,𝔽2)12yes—
226C23×C312−1+412yes—
227Q8×C31—416no—
228D4×C31—416no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—24no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—24no—
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—42no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31#12114yes—
281C2×#394C2×C3111yescharacteristic
282C2×#393C2×C3112yescomplemented
283C2×#391C2×C3111yescharacteristic, complemented
284#1443C2×C3112yescomplemented
285#1441C2×C3111yescharacteristic, complemented
286#1440C2×C3112yescomplemented
287#1439C2×C3112yescomplemented
288#1438C2×C3112yes—
289#1437C2×C3112yescomplemented
290C22×C31—44no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—44no—
296C22×C31—44no—
297C22×C31—82no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31C2×#12111yescharacteristic
306C22×C31#52912yes—
307C22×C31#53411yescharacteristic
308C22×C31#53411yescharacteristic
309C22×C31#53411yescharacteristic
310C22×C31#53511yescharacteristic
311C22×C31#53811yescharacteristic
312C22×C31#53811yescharacteristic
313C22×C31#53812yes—
314C22×C31#53812yes—
315C22×C31#53912yes—
316C22×C31#54212yes—
317C22×C31#54212yes—
318C22×C31#54612yes—
319C22×C31#54812yes—
320C22×C31#54812yes—
321C22×C31#55012yes—
322C22×C31#55111yescharacteristic
323C22×C31#55112yes—
324C22×C31#55511yescharacteristic
325C22×C31H(3,𝔽4)11yescharacteristic
326C22×C31#56212yes—
327C22×C31#56311yescharacteristic
328C4×C31—24no—
329C4×C31—44no—
330C4×C31—42no—
331C4×C31—42no—
332C4×C31—44no—
333C4×C31—82no—
334C4×C31—82no—
335C4×C31—82no—
336C23×Q8C22×C3111yescharacteristic
337C23×D4C22×C3111yescharacteristic
338C22×C4⋊C4C22×C3111yescharacteristic
339C22×C4⋊C4C22×C3111yescharacteristic
340C22×C22⋊C4C22×C3112yes—
341C22×C22⋊C4C22×C3112yes—
342C22×C22⋊C4C22×C3112yes—
343C22×C22⋊C4C22×C3112yes—
344C22×C22⋊C4C22×C3112yes—
345C22×C22⋊C4C22×C3112yes—
346C22×C42C22×C3111yescharacteristic
347#394—24no—
348#393—28no—
349#391—24no—
350C2×#95C22×C3112yes—
351C2×#95C22×C3112yes—
352C2×#95C22×C3112yes—
353C2×#95C22×C3111yescharacteristic
354C2×#95C22×C3112yes—
355C2×#95C22×C3112yes—
356C2×#95C22×C3112yes—
357C2×#95C22×C3112yes—
358C2×#95C22×C3112yes—
359C2×#95C22×C3111yescharacteristic
360C2×C31—84no—
361C2×C31#242612yes—
362C2×C31#257211yescharacteristic
363C2×C31#261711yescharacteristic
364C2×C31#265312yes—
365C2×C31#268411yescharacteristic
366C2×C31#269612yes—
367C2×C31#272811yescharacteristic
368C2×C31#273012yes—
369C2×C31#273111yescharacteristic
370C2×C31#273311yescharacteristic
371C2×C31#283211yescharacteristic
372C25C23×C3112yes—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—41no—
376C22×Q8—41no—
377C22×Q8—41no—
378C22×Q8—41no—
379C22×Q8D4×C3114yescomplemented
380C22×D4—24no—
381C22×D4—22no—
382C22×D4—42no—
383C22×D4—41no—
384C22×D4—41no—
385C23×C4C23×C3112yes—
386C23×C4C23×C3112yes—
387C23×C4C23×C3112yes—
388C23×C4C23×C3111yescharacteristic
389C23×C4C23×C3112yes—
390C23×C4C23×C3111yescharacteristic
391C23×C4C23×C3112yes—
392C23×C4C23×C3111yescharacteristic
393C2×C4⋊C4—41no—
394C2×C4⋊C4—41no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4D4×C3114yes—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C42—22no—
417C2×C42—41no—
418C2×C42—41no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—22no—
425#95—24no—
426#95—42no—
427#95—41no—
428#95—42no—
429C31#1665211yescharacteristic, direct product factor, Sylow
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24#41×C3114yes—
445C24C24×C3111yescharacteristic
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×D4—48no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—48no—
457C2×D4—44no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4C2×D4×C3112yes—
504C22×C4C2×D4×C3112yes—
505C22×C4C2×D4×C3114yes—
506C22×C4C2×D4×C3112yes—
507C22×C4C2×D4×C3112yes—
508C22×C4#41×C3114yes—
509C22×C4#41×C3112yes—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—44no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C22×D4×C3111yescharacteristic
555C23C22×D4×C3111yescharacteristic
556C23C2×#41×C3111yescharacteristic
557C23C2×#41×C3112yes—
558C23C2×#41×C3112yes—
559C23C31×H(5,𝔽2)11yescharacteristic
560C23C31×H(5,𝔽2)11yescharacteristic
561C23C31×H(5,𝔽2)11yescharacteristic
562C23C31×H(5,𝔽2)11yescharacteristic
563C23C31×H(5,𝔽2)12yes—
564C23C31×2−1+412yes—
565Q8—416no—
566D4—416no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4C31×#12114yes—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×C31×#12111yescharacteristic
635C22C31×#52912yes—
636C22C31×#53411yescharacteristic
637C22C31×#53411yescharacteristic
638C22C31×#53411yescharacteristic
639C22C31×#53511yescharacteristic
640C22C31×#53811yescharacteristic
641C22C31×#53811yescharacteristic
642C22C31×#53812yes—
643C22C31×#53812yes—
644C22C31×#53912yes—
645C22C31×#54212yes—
646C22C31×#54212yes—
647C22C31×#54612yes—
648C22C31×#54812yes—
649C22C31×#54812yes—
650C22C31×#55012yes—
651C22C31×#55111yescharacteristic
652C22C31×#55112yes—
653C22C31×#55511yescharacteristic
654C22C31×H(3,𝔽4)11yescharacteristic
655C22C31×#56212yes—
656C22C31×#56311yescharacteristic
657C4—24no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—84no—
666C2C31×#242612yes—
667C2C31×#257211yescharacteristic
668C2C31×#261711yescharacteristic
669C2C31×#265312yes—
670C2C31×#268411yescharacteristic
671C2C31×#269612yes—
672C2C31×#272811yescharacteristic
673C2C31×#273012yes—
674C2C31×#273111yescharacteristic
675C2C31×#273311yescharacteristic
676C2C31×#283211yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#393C2semidirect
3. C2×C31×#391C2semidirect
4. C31×#1443C2semidirect
5. C31×#1441C2semidirect
6. C31×#1440C2semidirect
7. C31×#1439C2semidirect
9. C31×#1437C2semidirect
41. C22×Q8×C31D4semidirect
180. #16652C31direct (N × H)
282. C2×#393C2×C31semidirect
283. C2×#391C2×C31semidirect
284. #1443C2×C31semidirect
285. #1441C2×C31semidirect
286. #1440C2×C31semidirect
287. #1439C2×C31semidirect
289. #1437C2×C31semidirect
379. C22×Q8D4×C31semidirect
429. C31#16652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3942
C2×C31×#3932
C2×C31×#3912
C31×#14432
C31×#14412
C31×#14402
C31×#14392
C31×#14382
C31×#14372
#1665231