Finite Group Databasebeta

← C31×#9117 #21465876

Subgroups — C31×#9117 #21465876

Order 7936.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#407C211yesmaximal, characteristic, complemented
2C31×#2847C211yesmaximal, characteristic
3C2×C31×#327C212yesmaximal, complemented
4C31×C22⋊#95C211yesmaximal, characteristic, complemented
5C31×#1294C212yesmaximal, complemented
6C24×C4×C31C2211yescharacteristic
7C22×M16×C31—21no—
8C2×C31×#121C411yescharacteristic, complemented
9C2×C31×#121C411yescharacteristic, complemented
10C2×C31×#121C2211yescharacteristic
11C2×C31×#120—21no—
12C22×C22⋊C4×C31—21no—
13C31×#407—24no—
14C2×C31×C22⋊C8—21no—
15C2×C31×C22⋊C8C2212yes—
16C31×#386—21no—
17C31×#386—21no—
18C31×#386—21no—
19C31×#380C411yescharacteristic, complemented
20C31×#380C411yescharacteristic, complemented
21C31×#380C2211yescharacteristic
22C2×C31×#95—21no—
23C2×C31×#95C2212yes—
24C31×#327—24no—
25C25×C31D411yescharacteristic
26C22×D4×C31—21no—
27C22×D4×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31C2×C411yescharacteristic
30C23×C4×C31—22no—
31C23×C4×C31—22no—
32C23×C4×C31—22no—
33C23×C4×C31—42no—
34C23×C4×C31—42no—
35C23×C4×C31C2×C411yescharacteristic
36C23×C4×C31C2×C411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31D411yescharacteristic
39C23×C4×C31D411yescharacteristic
40C23×C4×C31C2311yescharacteristic
41C2×M16×C31—24no—
42C2×M16×C31—42no—
43C2×M16×C31—42no—
44C22×C8×C31—22no—
45C31×#121—22no—
46C31×#121—22no—
47C31×#121—42no—
48C31×#121—42no—
49C31×#120—44no—
50C2×C4⋊C4×C31C2×C411yescharacteristic
51C2×C4⋊C4×C31C2×C411yescharacteristic
52C2×C4⋊C4×C31C2×C411yescharacteristic
53C2×C22⋊C4×C31—24no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—42no—
61C2×C22⋊C4×C31—42no—
62C31×C22⋊C8—24no—
63C31×C22⋊C8—22no—
64C31×C22⋊C8D414yescomplemented
65C31×#95—21no—
66C31×#95—21no—
67C31×#95—21no—
68C31×#95—22no—
69C31×#95—22no—
70C31×#95—42no—
71C24×C31—22no—
72C24×C31—22no—
73C24×C31—21no—
74C24×C31—21no—
75C24×C31—22no—
76C24×C31—22no—
77C24×C31—21no—
78C24×C31—41no—
79C24×C31—41no—
80C24×C31C22⋊C411yescharacteristic
81C24×C31C22⋊C411yescharacteristic
82C24×C31C2×D411yescharacteristic
83C2×D4×C31—22no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—42no—
87C2×D4×C31—42no—
88C2×D4×C31—42no—
89C2×D4×C31—42no—
90C2×D4×C31—42no—
91C2×D4×C31—42no—
92C2×D4×C31—42no—
93C2×D4×C31—42no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
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—21no—
107C22×C4×C31—21no—
108C22×C4×C31—24no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31—41no—
117C22×C4×C31—41no—
118C22×C4×C31—41no—
119C22×C4×C31—48no—
120C22×C4×C31—48no—
121C22×C4×C31—42no—
122C22×C4×C31—42no—
123C22×C4×C31—42no—
124C22×C4×C31—42no—
125C22×C4×C31—42no—
126C22×C4×C31—42no—
127C22×C4×C31C22⋊C411yescharacteristic
128C22×C4×C31C22⋊C411yescharacteristic
129C22×C4×C31C22×C411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D412yes—
132C22×C4×C31#4112yes—
133M16×C31—48no—
134C2×C8×C31—24no—
135C2×C8×C31—42no—
136C2×C8×C31—42no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—41no—
139C4⋊C4×C31—41no—
140C22⋊C4×C31—42no—
141C22⋊C4×C31—48no—
142C22⋊C4×C31—42no—
143C22⋊C4×C31—42no—
144C22⋊C4×C31—42no—
145C22⋊C4×C31—42no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—41no—
148#9117C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—21no—
152C23×C31—21no—
153C23×C31—24no—
154C23×C31—24no—
155C23×C31—21no—
156C23×C31—21no—
157C23×C31—21no—
158C23×C31—21no—
159C23×C31—21no—
160C23×C31—21no—
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—21no—
164C23×C31—21no—
165C23×C31—21no—
166C23×C31—21no—
167C23×C31—44no—
168C23×C31—44no—
169C23×C31—44no—
170C23×C31—44no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31#9912yes—
183C23×C31#10412yes—
184C23×C31C2×C22⋊C411yescharacteristic
185C23×C31#12112yes—
186C23×C31AGL(1,ℤ/8)11yescharacteristic
187C23×C31AGL(1,ℤ/8)12yes—
188C23×C31#13711yescharacteristic
189D4×C31—82no—
190D4×C31—84no—
191D4×C31—84no—
192D4×C31—82no—
193D4×C31—82no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—21no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—44no—
208C2×C4×C31—44no—
209C2×C4×C31—44no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—44no—
213C2×C4×C31—44no—
214C2×C4×C31—44no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—82no—
225C2×C4×C31—82no—
226C2×C4×C31—82no—
227C2×C4×C31—82no—
228C2×C4×C31#9912yes—
229C2×C4×C31C4×D412yes—
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12311yescharacteristic
232C8×C31—44no—
233C2×#407C2×C3111yescharacteristic, complemented
234#2847C2×C3111yescharacteristic
235C2×#327C2×C3112yescomplemented
236C22⋊#95C2×C3111yescharacteristic, complemented
237#1294C2×C3112yescomplemented
238C22×C31—22no—
239C22×C31—22no—
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—22no—
243C22×C31—22no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—22no—
252C22×C31—48no—
253C22×C31—48no—
254C22×C31—41no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31—41no—
260C22×C31—41no—
261C22×C31—41no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31#38611yescharacteristic
270C22×C31C2×#9911yescharacteristic
271C22×C31C2×#9911yescharacteristic
272C22×C31#41711yescharacteristic
273C22×C31#41811yescharacteristic
274C22×C31C2×#10411yescharacteristic
275C22×C31#42111yescharacteristic
276C22×C31#46812yes—
277C22×C31#46912yes—
278C4×C31—21no—
279C4×C31—22no—
280C4×C31—21no—
281C4×C31—22no—
282C4×C31—41no—
283C4×C31—84no—
284C4×C31—81no—
285C4×C31—81no—
286C4×C31—81no—
287C24×C4C22×C3111yescharacteristic
288C22×M16—21no—
289C2×#121C4×C3111yescharacteristic, complemented
290C2×#121C4×C3111yescharacteristic, complemented
291C2×#121C22×C3111yescharacteristic
292C2×#120—21no—
293C22×C22⋊C4—21no—
294#407—24no—
295C2×C22⋊C8—21no—
296C2×C22⋊C8C22×C3112yes—
297#386—21no—
298#386—21no—
299#386—21no—
300#380C4×C3111yescharacteristic, complemented
301#380C4×C3111yescharacteristic, complemented
302#380C22×C3111yescharacteristic
303C2×#95—21no—
304C2×#95C22×C3112yes—
305#327—24no—
306C2×C31—22no—
307C2×C31—22no—
308C2×C31—42no—
309C2×C31—42no—
310C2×C31—161no—
311C2×C31#150411yescharacteristic
312C2×C31#150711yescharacteristic
313C2×C31#151011yescharacteristic
314C2×C31#151311yescharacteristic
315C2×C31#187111yescharacteristic
316C2×C31#189511yescharacteristic
317C2×C31#189811yescharacteristic
318C25D4×C3111yescharacteristic
319C22×D4—21no—
320C22×D4—21no—
321C22×D4—21no—
322C22×D4C2×C4×C3111yescharacteristic
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—42no—
327C23×C4—42no—
328C23×C4C2×C4×C3111yescharacteristic
329C23×C4C2×C4×C3111yescharacteristic
330C23×C4D4×C3111yescharacteristic
331C23×C4D4×C3111yescharacteristic
332C23×C4D4×C3111yescharacteristic
333C23×C4C23×C3111yescharacteristic
334C2×M16—24no—
335C2×M16—42no—
336C2×M16—42no—
337C22×C8—22no—
338#121—22no—
339#121—22no—
340#121—42no—
341#121—42no—
342#120—44no—
343C2×C4⋊C4C2×C4×C3111yescharacteristic
344C2×C4⋊C4C2×C4×C3111yescharacteristic
345C2×C4⋊C4C2×C4×C3111yescharacteristic
346C2×C22⋊C4—24no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C22⋊C8—24no—
356C22⋊C8—22no—
357C22⋊C8D4×C3114yescomplemented
358#95—21no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—42no—
364C31#911711yescharacteristic, direct product factor, Sylow
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊C4×C3111yescharacteristic
375C24C22⋊C4×C3111yescharacteristic
376C24C2×D4×C3111yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C3111yescharacteristic
422C22×C4C22⋊C4×C3111yescharacteristic
423C22×C4C22×C4×C3111yescharacteristic
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3112yes—
426C22×C4#41×C3112yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23C31×#9912yes—
476C23C31×#10412yes—
477C23C2×C22⋊C4×C3111yescharacteristic
478C23C31×#12112yes—
479C23C31×AGL(1,ℤ/8)11yescharacteristic
480C23C31×AGL(1,ℤ/8)12yes—
481C23C31×#13711yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4C31×#9912yes—
522C2×C4C4×D4×C3112yes—
523C2×C4C31×#12011yescharacteristic
524C2×C4C31×#12311yescharacteristic
525C8—44no—
526C22—22no—
527C22—22no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C31×#38611yescharacteristic
558C22C2×C31×#9911yescharacteristic
559C22C2×C31×#9911yescharacteristic
560C22C31×#41711yescharacteristic
561C22C31×#41811yescharacteristic
562C22C2×C31×#10411yescharacteristic
563C22C31×#42111yescharacteristic
564C22C31×#46812yes—
565C22C31×#46912yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C31×#150411yescharacteristic
581C2C31×#150711yescharacteristic
582C2C31×#151011yescharacteristic
583C2C31×#151311yescharacteristic
584C2C31×#187111yescharacteristic
585C2C31×#189511yescharacteristic
586C2C31×#189811yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#407C2semidirect
3. C2×C31×#327C2semidirect
4. C31×C22⋊#95C2semidirect
5. C31×#1294C2semidirect
8. C2×C31×#121C4semidirect
9. C2×C31×#121C4semidirect
19. C31×#380C4semidirect
20. C31×#380C4semidirect
64. C31×C22⋊C8D4semidirect
148. #9117C31direct (N × H)
233. C2×#407C2×C31semidirect
235. C2×#327C2×C31semidirect
236. C22⋊#95C2×C31semidirect
237. #1294C2×C31semidirect
289. C2×#121C4×C31semidirect
290. C2×#121C4×C31semidirect
300. #380C4×C31semidirect
301. #380C4×C31semidirect
357. C22⋊C8D4×C31semidirect
364. C31#9117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4072
C31×#28472
C2×C31×#3272
C31×C22⋊#952
C31×#12942
#911731