Finite Group Databasebeta

← C31×#32797 #21489556

Subgroups — C31×#32797 #21489556

Order 7936.

868 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2860C211yesmaximal, characteristic, complemented
2C31×#2858C211yesmaximal, characteristic, complemented
3C31×#2043C211yesmaximal, characteristic, complemented
4C31×#2042C212yesmaximal, complemented
5C31×#2042C212yesmaximal, complemented
6C31×#2041C211yesmaximal, characteristic, complemented
7C31×#2013C211yesmaximal, characteristic, complemented
8C31×#2012C212yesmaximal, complemented
9C31×#2012C212yesmaximal, complemented
10C31×#2011C211yesmaximal, characteristic
11C31×#1839C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C22×D8×C31C2212yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#127C2211yescharacteristic
16C2×C31×#126—21no—
17C2×C31×#124C2212yes—
18C2×C31×#124C2212yes—
19C2×C31×#103C2212yes—
20C2×C31×#103C2212yes—
21C2×C31×#103C2212yes—
22C2×C31×#102C2212yes—
23C2×C31×#102C2212yes—
24C2×C31×#102C2212yes—
25C31×#390—21no—
26C31×#390—21no—
27C31×#390—21no—
28C31×#390—21no—
29C31×#390C2211yescharacteristic
30C31×#390C2211yescharacteristic
31C31×#390C2211yescharacteristic
32C31×#389C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#388—21no—
36C31×#388—21no—
37C31×#388—21no—
38C43×C31C2211yescharacteristic
39C31×#336C2211yescharacteristic
40C31×#336C2212yes—
41C31×#336C2212yes—
42C31×#336C2211yescharacteristic
43C22×Q8×C31C2312yes—
44C22×D4×C31—21no—
45C22×D4×C31—21no—
46C22×D4×C31C2312yes—
47C22×D4×C31C2311yescharacteristic
48C2×SD16×C31—42no—
49C2×SD16×C31—42no—
50C2×SD16×C31—42no—
51C2×D8×C31—42no—
52C2×D8×C31—42no—
53C2×D8×C31—42no—
54C22×C8×C31C2312yes—
55C22×C8×C31C2312yes—
56C31×C4⋊Q8—42no—
57C31×#127—42no—
58C31×#126—44no—
59C31×#124—42no—
60C31×#124—42no—
61C31×#124—42no—
62C31×#124—42no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31C2311yescharacteristic
69C2×C4⋊C4×C31C2311yescharacteristic
70C2×C4⋊C4×C31C2311yescharacteristic
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—21no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—21no—
81C2×C42×C31D411yescharacteristic
82C2×C42×C31D411yescharacteristic
83C2×C42×C31D411yescharacteristic
84C2×C42×C31D411yescharacteristic
85C2×C42×C31C2311yescharacteristic
86C2×C42×C31C2311yescharacteristic
87C2×C42×C31C2311yescharacteristic
88C31×#103—42no—
89C31×#103—42no—
90C31×#103—42no—
91C31×#102—42no—
92C31×#102—42no—
93C31×#102—42no—
94C31×#95—21no—
95C31×#95—21no—
96C31×#95—21no—
97C24×C31—21no—
98C24×C31—22no—
99C2×Q8×C31—22no—
100C2×Q8×C31—22no—
101C2×Q8×C31—22no—
102C2×Q8×C31—22no—
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—21no—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—21no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31—42no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—22no—
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2411yescharacteristic
132SD16×C31—48no—
133D8×C31—48no—
134C2×C8×C31—22no—
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C2×C8×C31—22no—
138C2×C8×C31—22no—
139C2×C8×C31—22no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—42no—
145C4⋊C4×C31—42no—
146C4⋊C4×C31—42no—
147C4⋊C4×C31—42no—
148C4⋊C4×C31—42no—
149C22⋊C4×C31—42no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31—44no—
152C22⋊C4×C31—44no—
153C22⋊C4×C31—44no—
154C22⋊C4×C31—44no—
155C22⋊C4×C31—42no—
156C22⋊C4×C31—42no—
157C22⋊C4×C31—42no—
158C22⋊C4×C31—42no—
159C42×C31—22no—
160C42×C31—22no—
161C42×C31—22no—
162C42×C31—21no—
163C42×C31—21no—
164C42×C31—22no—
165C42×C31—41no—
166C42×C31—41no—
167#32797C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—41no—
172C23×C31—41no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—42no—
176C23×C31—41no—
177C23×C31—42no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31C22×D411yescharacteristic
183Q8×C31—42no—
184Q8×C31—42no—
185Q8×C31—42no—
186Q8×C31—42no—
187D4×C31—42no—
188D4×C31—42no—
189D4×C31—42no—
190D4×C31—42no—
191D4×C31—42no—
192D4×C31—42no—
193D4×C31—84no—
194D4×C31—84no—
195C2×C4×C31—21no—
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—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
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#13512yes—
224C2×C4×C31AGL(1,ℤ/8)12yes—
225C2×C4×C31H(5,𝔽2)12yes—
226C2×C4×C31H(5,𝔽2)12yes—
227C2×C4×C31H(5,𝔽2)11yescharacteristic
228C2×C4×C31H(5,𝔽2)11yescharacteristic
229C8×C31—42no—
230C8×C31—42no—
231#2860C2×C3111yescharacteristic, complemented
232#2858C2×C3111yescharacteristic, complemented
233#2043C2×C3111yescharacteristic, complemented
234#2042C2×C3112yescomplemented
235#2042C2×C3112yescomplemented
236#2041C2×C3111yescharacteristic, complemented
237#2013C2×C3111yescharacteristic, complemented
238#2012C2×C3112yescomplemented
239#2012C2×C3112yescomplemented
240#2011C2×C3111yescharacteristic
241#1839C2×C3111yescharacteristic, complemented
242C22×C31—81no—
243C22×C31—82no—
244C22×C31—82no—
245C22×C31—81no—
246C22×C31—82no—
247C22×C31—81no—
248C22×C31—82no—
249C22×C31—81no—
250C22×C31—82no—
251C22×C31—81no—
252C22×C31—82no—
253C22×C31—81no—
254C22×C31—82no—
255C22×C31—81no—
256C22×C31#53511yescharacteristic
257C22×C31#53511yescharacteristic
258C22×C31#53511yescharacteristic
259C22×C31C2×#13511yescharacteristic
260C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C31#57511yescharacteristic
262C22×C31#57511yescharacteristic
263C4×C31—22no—
264C4×C31—22no—
265C4×C31—41no—
266C4×C31—41no—
267C4×C31—41no—
268C4×C31—41no—
269C4×C31—82no—
270C4×C31—81no—
271C4×C31—81no—
272C4×C31—81no—
273C4×C31#56012yes—
274C4×C31#56012yes—
275C22×SD16C22×C3112yes—
276C22×D8C22×C3112yes—
277C2×C4⋊Q8C22×C3111yescharacteristic
278C2×#127C22×C3111yescharacteristic
279C2×#126—21no—
280C2×#124C22×C3112yes—
281C2×#124C22×C3112yes—
282C2×#103C22×C3112yes—
283C2×#103C22×C3112yes—
284C2×#103C22×C3112yes—
285C2×#102C22×C3112yes—
286C2×#102C22×C3112yes—
287C2×#102C22×C3112yes—
288#390—21no—
289#390—21no—
290#390—21no—
291#390—21no—
292#390C22×C3111yescharacteristic
293#390C22×C3111yescharacteristic
294#390C22×C3111yescharacteristic
295#389C22×C3111yescharacteristic
296#389C22×C3111yescharacteristic
297#389C22×C3111yescharacteristic
298#388—21no—
299#388—21no—
300#388—21no—
301C43C22×C3111yescharacteristic
302#336C22×C3111yescharacteristic
303#336C22×C3112yes—
304#336C22×C3112yes—
305#336C22×C3111yescharacteristic
306C2×C31—162no—
307C2×C31—161no—
308C2×C31#281811yescharacteristic
309C2×C31#320711yescharacteristic
310C2×C31#321111yescharacteristic
311C2×C31#321211yescharacteristic
312C2×C31#321311yescharacteristic
313C2×C31#321411yescharacteristic
314C2×C31#321911yescharacteristic
315C22×Q8C23×C3112yes—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4C23×C3112yes—
319C22×D4C23×C3111yescharacteristic
320C2×SD16—42no—
321C2×SD16—42no—
322C2×SD16—42no—
323C2×D8—42no—
324C2×D8—42no—
325C2×D8—42no—
326C22×C8C23×C3112yes—
327C22×C8C23×C3112yes—
328C4⋊Q8—42no—
329#127—42no—
330#126—44no—
331#124—42no—
332#124—42no—
333#124—42no—
334#124—42no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4C23×C3111yescharacteristic
341C2×C4⋊C4C23×C3111yescharacteristic
342C2×C4⋊C4C23×C3111yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
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×C42D4×C3111yescharacteristic
354C2×C42D4×C3111yescharacteristic
355C2×C42D4×C3111yescharacteristic
356C2×C42D4×C3111yescharacteristic
357C2×C42C23×C3111yescharacteristic
358C2×C42C23×C3111yescharacteristic
359C2×C42C23×C3111yescharacteristic
360#103—42no—
361#103—42no—
362#103—42no—
363#102—42no—
364#102—42no—
365#102—42no—
366#95—21no—
367#95—21no—
368#95—21no—
369C31#3279711yescharacteristic, direct product factor, Sylow
370C24—21no—
371C24—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4C2×D4×C3111yescharacteristic
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C24×C3111yescharacteristic
405SD16—48no—
406D8—48no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—21no—
436C42—21no—
437C42—22no—
438C42—41no—
439C42—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23C22×D4×C3111yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—84no—
466D4—84no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4C31×#13512yes—
496C2×C4C31×AGL(1,ℤ/8)12yes—
497C2×C4C31×H(5,𝔽2)12yes—
498C2×C4C31×H(5,𝔽2)12yes—
499C2×C4C31×H(5,𝔽2)11yescharacteristic
500C2×C4C31×H(5,𝔽2)11yescharacteristic
501C8—42no—
502C8—42no—
503C22—82no—
504C22—81no—
505C22—82no—
506C22—81no—
507C22—82no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—82no—
512C22—81no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22C31×#53511yescharacteristic
518C22C31×#53511yescharacteristic
519C22C31×#53511yescharacteristic
520C22C2×C31×#13511yescharacteristic
521C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
522C22C31×#57511yescharacteristic
523C22C31×#57511yescharacteristic
524C4—22no—
525C4—22no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4C31×#56012yes—
535C4C31×#56012yes—
536C2—162no—
537C2—161no—
538C2C31×#281811yescharacteristic
539C2C31×#320711yescharacteristic
540C2C31×#321111yescharacteristic
541C2C31×#321211yescharacteristic
542C2C31×#321311yescharacteristic
543C2C31×#321411yescharacteristic
544C2C31×#321911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C43×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2860C2semidirect
2. C31×#2858C2semidirect
3. C31×#2043C2semidirect
4. C31×#2042C2semidirect
5. C31×#2042C2semidirect
6. C31×#2041C2semidirect
7. C31×#2013C2semidirect
8. C31×#2012C2semidirect
9. C31×#2012C2semidirect
11. C31×#1839C2semidirect
167. #32797C31direct (N × H)
231. #2860C2×C31semidirect
232. #2858C2×C31semidirect
233. #2043C2×C31semidirect
234. #2042C2×C31semidirect
235. #2042C2×C31semidirect
236. #2041C2×C31semidirect
237. #2013C2×C31semidirect
238. #2012C2×C31semidirect
239. #2012C2×C31semidirect
241. #1839C2×C31semidirect
369. C31#32797direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28602
C31×#28582
C31×#20432
C31×#20422
C31×#20412
C31×#20132
C31×#20122
C31×#20112
C31×#18392
#3279731