Finite Group Databasebeta

← C31×#16704 #21473463

Subgroups — C31×#16704 #21473463

Order 7936.

1298 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#388C211yesmaximal, characteristic
2C2×C31×#382C211yesmaximal, characteristic, complemented
3C31×#1444C211yesmaximal, characteristic, complemented
4C31×#1442C211yesmaximal, characteristic, complemented
5C31×#1442C212yesmaximal, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal
8C31×#1439C211yesmaximal, characteristic
9C31×#1439C211yesmaximal, characteristic
10C31×#1439C212yesmaximal
11C31×C4⋊#95C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2212yes—
22C22×C42×C31C2211yescharacteristic
23C31×#388—24no—
24C31×#382—24no—
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2311yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C23×C4×C31C2312yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2311yescharacteristic
42C23×C4×C31C2312yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—22no—
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—42no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—42no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31Q814yes—
73C2×C42×C31—24no—
74C2×C42×C31D414yes—
75C31×#95—24no—
76C31×#95—24no—
77C31×#95—22no—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—24no—
81C31×#95—22no—
82C31×#95—42no—
83C31×#95—42no—
84C31×#95—42no—
85C31×#95—42no—
86C31×#95—42no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—21no—
92C24×C31—21no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—21no—
96C24×C31—21no—
97C24×C31C2×Q812yes—
98C24×C31C2411yescharacteristic
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
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—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—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×Q812yes—
153C22×C4×C31C2×Q812yes—
154C22×C4×C31#4114yes—
155C22×C4×C31#4112yes—
156C22×C4×C31#4114yes—
157C4⋊C4×C31—48no—
158C4⋊C4×C31—44no—
159C4⋊C4×C31—44no—
160C4⋊C4×C31—44no—
161C4⋊C4×C31—48no—
162C22⋊C4×C31—28no—
163C22⋊C4×C31—44no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—48no—
166C22⋊C4×C31—44no—
167C42×C31—28no—
168#16704C3111yesmaximal, characteristic, direct product factor, Sylow
169C23×C31—22no—
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—42no—
181C23×C31—41no—
182C23×C31—42no—
183C23×C31—41no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31C22×D411yescharacteristic
193C23×C31C22×Q811yescharacteristic
194C23×C31C2×#4111yescharacteristic
195C23×C31C2×#4112yes—
196C23×C31C2×#4112yes—
197C23×C31H(5,𝔽2)12yes—
198C23×C31H(5,𝔽2)12yes—
199C23×C31H(5,𝔽2)11yescharacteristic
200C23×C31H(5,𝔽2)11yescharacteristic
201C23×C312−1+411yescharacteristic
202C23×C312−1+411yescharacteristic
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
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—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31C22⋊Q814yes—
260C2×C4×C31#12314yes—
261C2×#388C2×C3111yescharacteristic
262C2×#382C2×C3111yescharacteristic, complemented
263#1444C2×C3111yescharacteristic, complemented
264#1442C2×C3111yescharacteristic, complemented
265#1442C2×C3112yescomplemented
266#1441C2×C3112yescomplemented
267#1439C2×C3112yes—
268#1439C2×C3111yescharacteristic
269#1439C2×C3111yescharacteristic
270#1439C2×C3112yes—
271C4⋊#95C2×C3111yescharacteristic, complemented
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31—81no—
280C22×C31—82no—
281C22×C31—81no—
282C22×C31—82no—
283C22×C31C2×C22⋊Q811yescharacteristic
284C22×C31C2×#12311yescharacteristic
285C22×C31#53211yescharacteristic
286C22×C31#53511yescharacteristic
287C22×C31#53511yescharacteristic
288C22×C31#53711yescharacteristic
289C22×C31#53711yescharacteristic
290C22×C31#53812yes—
291C22×C31#54012yes—
292C22×C31#54011yescharacteristic
293C22×C31#54111yescharacteristic
294C22×C31#54212yes—
295C22×C31#54212yes—
296C22×C31#54411yescharacteristic
297C22×C31#54411yescharacteristic
298C22×C31#55112yes—
299C22×C31#55212yes—
300C22×C31#55312yes—
301C22×C31#55312yes—
302C22×C31#55412yes—
303C22×C31#55412yes—
304C22×C31H(3,𝔽4)12yes—
305C22×C31#56212yes—
306C4×C31—24no—
307C4×C31—24no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—44no—
311C4×C31—81no—
312C4×C31—82no—
313C4×C31—81no—
314C4×C31—82no—
315C4×C31—82no—
316C22×C4⋊C4C22×C3112yes—
317C22×C4⋊C4C22×C3111yescharacteristic
318C22×C4⋊C4C22×C3111yescharacteristic
319C22×C4⋊C4C22×C3111yescharacteristic
320C22×C4⋊C4C22×C3112yes—
321C22×C22⋊C4C22×C3111yescharacteristic
322C22×C22⋊C4C22×C3111yescharacteristic
323C22×C22⋊C4C22×C3112yes—
324C22×C22⋊C4C22×C3111yescharacteristic
325C22×C22⋊C4C22×C3112yes—
326C22×C42C22×C3111yescharacteristic
327#388—24no—
328#382—24no—
329C2×#95C22×C3111yescharacteristic
330C2×#95C22×C3111yescharacteristic
331C2×#95C22×C3112yes—
332C2×#95C22×C3112yes—
333C2×#95C22×C3112yes—
334C2×#95C22×C3111yescharacteristic
335C2×#95C22×C3111yescharacteristic
336C2×#95C22×C3112yes—
337C2×#95C22×C3112yes—
338C2×#95C22×C3112yes—
339C2×#95C22×C3112yes—
340C2×#95C22×C3112yes—
341C2×C31—82no—
342C2×C31#248711yescharacteristic
343C2×C31#249811yescharacteristic
344C2×C31#256611yescharacteristic
345C2×C31#262411yescharacteristic
346C2×C31#262911yescharacteristic
347C2×C31#266211yescharacteristic
348C2×C31#275312yes—
349C2×C31#276312yes—
350C2×C31#276812yes—
351C2×C31#277212yes—
352C2×C31#281911yescharacteristic
353C25C23×C3111yescharacteristic
354C23×C4C23×C3111yescharacteristic
355C23×C4C23×C3112yes—
356C23×C4C23×C3112yes—
357C23×C4C23×C3111yescharacteristic
358C23×C4C23×C3112yes—
359C23×C4C23×C3111yescharacteristic
360C23×C4C23×C3111yescharacteristic
361C23×C4C23×C3112yes—
362C23×C4C23×C3111yescharacteristic
363C23×C4C23×C3111yescharacteristic
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—42no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4Q8×C3114yes—
389C2×C42—24no—
390C2×C42D4×C3114yes—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—22no—
398#95—42no—
399#95—42no—
400#95—42no—
401#95—42no—
402#95—42no—
403C31#1670411yescharacteristic, direct product factor, Sylow
404C24—21no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24C2×Q8×C3112yes—
415C24C24×C3111yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
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×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4C2×D4×C3112yes—
467C22×C4C2×D4×C3112yes—
468C22×C4C2×D4×C3112yes—
469C22×C4C2×Q8×C3112yes—
470C22×C4C2×Q8×C3112yes—
471C22×C4#41×C3114yes—
472C22×C4#41×C3112yes—
473C22×C4#41×C3114yes—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C22⋊C4—28no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C42—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C22×D4×C3111yescharacteristic
509C23C22×Q8×C3111yescharacteristic
510C23C2×#41×C3111yescharacteristic
511C23C2×#41×C3112yes—
512C23C2×#41×C3112yes—
513C23C31×H(5,𝔽2)12yes—
514C23C31×H(5,𝔽2)12yes—
515C23C31×H(5,𝔽2)11yescharacteristic
516C23C31×H(5,𝔽2)11yescharacteristic
517C23C31×2−1+411yescharacteristic
518C23C31×2−1+411yescharacteristic
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4C31×C22⋊Q814yes—
576C2×C4C31×#12314yes—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—81no—
587C22—82no—
588C22C2×C31×C22⋊Q811yescharacteristic
589C22C2×C31×#12311yescharacteristic
590C22C31×#53211yescharacteristic
591C22C31×#53511yescharacteristic
592C22C31×#53511yescharacteristic
593C22C31×#53711yescharacteristic
594C22C31×#53711yescharacteristic
595C22C31×#53812yes—
596C22C31×#54012yes—
597C22C31×#54011yescharacteristic
598C22C31×#54111yescharacteristic
599C22C31×#54212yes—
600C22C31×#54212yes—
601C22C31×#54411yescharacteristic
602C22C31×#54411yescharacteristic
603C22C31×#55112yes—
604C22C31×#55212yes—
605C22C31×#55312yes—
606C22C31×#55312yes—
607C22C31×#55412yes—
608C22C31×#55412yes—
609C22C31×H(3,𝔽4)12yes—
610C22C31×#56212yes—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—82no—
621C2—82no—
622C2C31×#248711yescharacteristic
623C2C31×#249811yescharacteristic
624C2C31×#256611yescharacteristic
625C2C31×#262411yescharacteristic
626C2C31×#262911yescharacteristic
627C2C31×#266211yescharacteristic
628C2C31×#275312yes—
629C2C31×#276312yes—
630C2C31×#276812yes—
631C2C31×#277212yes—
632C2C31×#281911yescharacteristic
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
2. C2×C31×#382C2semidirect
3. C31×#1444C2semidirect
4. C31×#1442C2semidirect
5. C31×#1442C2semidirect
6. C31×#1441C2semidirect
11. C31×C4⋊#95C2semidirect
168. #16704C31direct (N × H)
262. C2×#382C2×C31semidirect
263. #1444C2×C31semidirect
264. #1442C2×C31semidirect
265. #1442C2×C31semidirect
266. #1441C2×C31semidirect
271. C4⋊#95C2×C31semidirect
403. C31#16704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3882
C2×C31×#3822
C31×#14442
C31×#14422
C31×#14412
C31×#14392
C31×C4⋊#952
#1670431