Finite Group Databasebeta

← C31×#16810 #21473569

Subgroups — C31×#16810 #21473569

Order 7936.

2478 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#397C212yesmaximal, complemented
2C2×C31×#392C212yesmaximal, complemented
3C2×C31×#392C212yesmaximal, complemented
4C2×C31×#390C211yesmaximal, characteristic, complemented
5C2×C31×#390C211yesmaximal, characteristic, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1438C214yesmaximal, complemented
8C31×#1434C211yesmaximal, characteristic, complemented
9C23×D4×C31C2211yescharacteristic
10C23×D4×C31C2214yes—
11C23×D4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2214yes—
17C22×C22⋊C4×C31C2214yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C42×C31C2211yescharacteristic
21C31×#397—28no—
22C31×#392—28no—
23C31×#392—28no—
24C31×#390—24no—
25C31×#390—24no—
26C2×C31×#95C2212yes—
27C2×C31×#95C2214yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C25×C31C2312yes—
31C25×C31C2312yes—
32C22×D4×C31—24no—
33C22×D4×C31—28no—
34C22×D4×C31—28no—
35C22×D4×C31—22no—
36C22×D4×C31—22no—
37C22×D4×C31—28no—
38C22×D4×C31—24no—
39C22×D4×C31—22no—
40C22×D4×C31—42no—
41C22×D4×C31—42no—
42C22×D4×C31—42no—
43C22×D4×C31—44no—
44C22×D4×C31—44no—
45C22×D4×C31—44no—
46C22×D4×C31—44no—
47C22×D4×C31D414yescomplemented
48C23×C4×C31C2312yes—
49C23×C4×C31C2312yes—
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2314yes—
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—42no—
58C2×C22⋊C4×C31—28no—
59C2×C22⋊C4×C31—28no—
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—28no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—44no—
70C2×C22⋊C4×C31—44no—
71C2×C22⋊C4×C31—44no—
72C2×C22⋊C4×C31—44no—
73C2×C22⋊C4×C31—44no—
74C2×C42×C31—42no—
75C2×C42×C31D414yescomplemented
76C31×#95—28no—
77C31×#95—24no—
78C31×#95—42no—
79C31×#95D418yescomplemented
80C24×C31—22no—
81C24×C31—22no—
82C24×C31—22no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—24no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—24no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—24no—
93C24×C31—24no—
94C24×C31—24no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—22no—
98C24×C31—24no—
99C24×C31—22no—
100C24×C31C2×D414yes—
101C24×C31#4114yes—
102C24×C31C2411yescharacteristic
103C2×D4×C31—28no—
104C2×D4×C31—28no—
105C2×D4×C31—48no—
106C2×D4×C31—416no—
107C2×D4×C31—416no—
108C2×D4×C31—416no—
109C2×D4×C31—44no—
110C2×D4×C31—44no—
111C2×D4×C31—44no—
112C2×D4×C31—48no—
113C2×D4×C31—416no—
114C2×D4×C31—416no—
115C2×D4×C31—416no—
116C2×D4×C31—416no—
117C2×D4×C31—416no—
118C2×D4×C31—44no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—24no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—24no—
130C22×C4×C31—24no—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—24no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—24no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31C2×D418yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D414yes—
147C22×C4×C31C2×D412yes—
148C22×C4×C31#4112yes—
149C4⋊C4×C31—44no—
150C4⋊C4×C31—44no—
151C22⋊C4×C31—416no—
152C22⋊C4×C31—48no—
153C22⋊C4×C31—48no—
154C22⋊C4×C31—416no—
155C22⋊C4×C31—48no—
156C22⋊C4×C31—48no—
157C42×C31—44no—
158#16810C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—24no—
160C23×C31—24no—
161C23×C31—28no—
162C23×C31—28no—
163C23×C31—24no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—28no—
167C23×C31—28no—
168C23×C31—24no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—44no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—44no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—44no—
181C23×C31—44no—
182C23×C31—42no—
183C23×C31—44no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—44no—
187C23×C31—44no—
188C23×C31—44no—
189C23×C31—44no—
190C23×C31—44no—
191C23×C31—44no—
192C23×C31—44no—
193C23×C31—44no—
194C23×C31—44no—
195C23×C31—42no—
196C23×C31—44no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31—44no—
202C23×C31—44no—
203C23×C31—44no—
204C23×C31—44no—
205C23×C31C22×D411yescharacteristic
206C23×C31C22×D411yescharacteristic
207C23×C31C22×D412yes—
208C23×C31C2×#4111yescharacteristic
209C23×C31C2×#4112yes—
210C23×C31H(5,𝔽2)12yes—
211C23×C31H(5,𝔽2)12yes—
212C23×C31H(5,𝔽2)12yes—
213C23×C31H(5,𝔽2)12yes—
214D4×C31—416no—
215D4×C31—416no—
216D4×C31—464no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—28no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—28no—
228C2×C4×C31—28no—
229C2×C4×C31—28no—
230C2×C4×C31—44no—
231C2×C4×C31—44no—
232C2×C4×C31—44no—
233C2×C4×C31—44no—
234C2×C4×C31—44no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—44no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—44no—
243C2×C4×C31—44no—
244C2×C4×C31—44no—
245C2×C4×C31#12114yes—
246C2×C4×C31#12714yes—
247C2×#397C2×C3112yescomplemented
248C2×#392C2×C3112yescomplemented
249C2×#392C2×C3112yescomplemented
250C2×#390C2×C3111yescharacteristic, complemented
251C2×#390C2×C3111yescharacteristic, complemented
252#1439C2×C3112yescomplemented
253#1438C2×C3114yescomplemented
254#1434C2×C3111yescharacteristic, complemented
255C22×C31—44no—
256C22×C31—44no—
257C22×C31—48no—
258C22×C31—48no—
259C22×C31—48no—
260C22×C31—48no—
261C22×C31—44no—
262C22×C31—44no—
263C22×C31—44no—
264C22×C31—44no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—82no—
268C22×C31—82no—
269C22×C31—84no—
270C22×C31—84no—
271C22×C31—84no—
272C22×C31—84no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31C2×#12111yescharacteristic
278C22×C31#52511yescharacteristic
279C22×C31C2×#12711yescharacteristic
280C22×C31#53412yes—
281C22×C31#53411yescharacteristic
282C22×C31#53511yescharacteristic
283C22×C31#53511yescharacteristic
284C22×C31#53514yes—
285C22×C31#53511yescharacteristic
286C22×C31#53812yes—
287C22×C31#53814yes—
288C22×C31D4212yes—
289C22×C31#54612yes—
290C22×C31#54614yes—
291C22×C31#55012yes—
292C22×C31#55312yes—
293C22×C31#56014yes—
294C4×C31—24no—
295C4×C31—24no—
296C4×C31—48no—
297C4×C31—42no—
298C4×C31—82no—
299C4×C31—82no—
300C23×D4C22×C3111yescharacteristic
301C23×D4C22×C3114yes—
302C23×D4C22×C3111yescharacteristic
303C22×C4⋊C4C22×C3111yescharacteristic
304C22×C4⋊C4C22×C3111yescharacteristic
305C22×C22⋊C4C22×C3112yes—
306C22×C22⋊C4C22×C3112yes—
307C22×C22⋊C4C22×C3114yes—
308C22×C22⋊C4C22×C3114yes—
309C22×C22⋊C4C22×C3112yes—
310C22×C22⋊C4C22×C3112yes—
311C22×C42C22×C3111yescharacteristic
312#397—28no—
313#392—28no—
314#392—28no—
315#390—24no—
316#390—24no—
317C2×#95C22×C3112yes—
318C2×#95C22×C3114yes—
319C2×#95C22×C3112yes—
320C2×#95C22×C3112yes—
321C2×C31—84no—
322C2×C31—84no—
323C2×C31#253511yescharacteristic
324C2×C31#253911yescharacteristic
325C2×C31#254111yescharacteristic
326C2×C31#261711yescharacteristic
327C2×C31#263111yescharacteristic
328C2×C31#265711yescharacteristic
329C2×C31#267814yes—
330C2×C31#271414yes—
331C2×C31#281511yescharacteristic
332C25C23×C3112yes—
333C25C23×C3112yes—
334C22×D4—24no—
335C22×D4—28no—
336C22×D4—28no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—28no—
340C22×D4—24no—
341C22×D4—22no—
342C22×D4—42no—
343C22×D4—42no—
344C22×D4—42no—
345C22×D4—44no—
346C22×D4—44no—
347C22×D4—44no—
348C22×D4—44no—
349C22×D4D4×C3114yescomplemented
350C23×C4C23×C3112yes—
351C23×C4C23×C3112yes—
352C23×C4C23×C3111yescharacteristic
353C23×C4C23×C3114yes—
354C23×C4C23×C3111yescharacteristic
355C23×C4C23×C3111yescharacteristic
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4—44no—
376C2×C42—42no—
377C2×C42D4×C3114yescomplemented
378#95—24no—
379#95—28no—
380#95—42no—
381#95D4×C3118yescomplemented
382C31#1681011yescharacteristic, direct product factor, Sylow
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—22no—
401C24—24no—
402C24—22no—
403C24C2×D4×C3114yes—
404C24#41×C3114yes—
405C24C24×C3111yescharacteristic
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—416no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—44no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4C2×D4×C3118yes—
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3112yes—
448C22×C4C2×D4×C3112yes—
449C22×C4C2×D4×C3114yes—
450C22×C4C2×D4×C3112yes—
451C22×C4#41×C3112yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C42—44no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C22×D4×C3111yescharacteristic
508C23C22×D4×C3111yescharacteristic
509C23C22×D4×C3112yes—
510C23C2×#41×C3111yescharacteristic
511C23C2×#41×C3112yes—
512C23C31×H(5,𝔽2)12yes—
513C23C31×H(5,𝔽2)12yes—
514C23C31×H(5,𝔽2)12yes—
515C23C31×H(5,𝔽2)12yes—
516D4—416no—
517D4—416no—
518D4—464no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4C31×#12114yes—
548C2×C4C31×#12714yes—
549C22—44no—
550C22—44no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22C2×C31×#12111yescharacteristic
572C22C31×#52511yescharacteristic
573C22C2×C31×#12711yescharacteristic
574C22C31×#53412yes—
575C22C31×#53411yescharacteristic
576C22C31×#53511yescharacteristic
577C22C31×#53511yescharacteristic
578C22C31×#53514yes—
579C22C31×#53511yescharacteristic
580C22C31×#53812yes—
581C22C31×#53814yes—
582C22D42×C3112yes—
583C22C31×#54612yes—
584C22C31×#54614yes—
585C22C31×#55012yes—
586C22C31×#55312yes—
587C22C31×#56014yes—
588C4—24no—
589C4—24no—
590C4—48no—
591C4—42no—
592C4—82no—
593C4—82no—
594C2—84no—
595C2—84no—
596C2C31×#253511yescharacteristic
597C2C31×#253911yescharacteristic
598C2C31×#254111yescharacteristic
599C2C31×#261711yescharacteristic
600C2C31×#263111yescharacteristic
601C2C31×#265711yescharacteristic
602C2C31×#267814yes—
603C2C31×#271414yes—
604C2C31×#281511yescharacteristic
subgroupG/H
center, socleC24×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
1. C2×C31×#397C2semidirect
2. C2×C31×#392C2semidirect
3. C2×C31×#392C2semidirect
4. C2×C31×#390C2semidirect
5. C2×C31×#390C2semidirect
6. C31×#1439C2semidirect
7. C31×#1438C2semidirect
8. C31×#1434C2semidirect
47. C22×D4×C31D4semidirect
75. C2×C42×C31D4semidirect
79. C31×#95D4semidirect
158. #16810C31direct (N × H)
247. C2×#397C2×C31semidirect
248. C2×#392C2×C31semidirect
249. C2×#392C2×C31semidirect
250. C2×#390C2×C31semidirect
251. C2×#390C2×C31semidirect
252. #1439C2×C31semidirect
253. #1438C2×C31semidirect
254. #1434C2×C31semidirect
349. C22×D4D4×C31semidirect
377. C2×C42D4×C31semidirect
381. #95D4×C31semidirect
382. C31#16810direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3972
C2×C31×#3922
C2×C31×#3902
C31×#14392
C31×#14382
C31×#14342
#1681031