Finite Group Databasebeta

← C31×#24727 #21481486

Subgroups — C31×#24727 #21481486

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#463C212yesmaximal
2C2×C31×#461C211yesmaximal, characteristic, complemented
3C2×C4×D8×C31C211yesmaximal, characteristic
4C31×#2547C211yesmaximal, characteristic, complemented
5D4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
6C31×#2068C212yesmaximal, complemented
7C31×#2042C212yesmaximal, complemented
8C31×#1956C211yesmaximal, characteristic, complemented
9C31×#1939C211yesmaximal, characteristic, complemented
10C31×#1877C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C22×D8×C31C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#124C2212yes—
16C2×C4×D4×C31C2211yescharacteristic
17C2×C4×D4×C31C2212yes—
18C2×C4×D4×C31C2211yescharacteristic
19C22×C4⋊C4×C31—21no—
20C31×#463—28no—
21C31×#461—24no—
22C4×D8×C31—24no—
23C2×C31×#107C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×C4⋊C8C2212yes—
26C2×C31×#103C2212yes—
27C2×C31×#103C2212yes—
28C2×C31×#103C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2211yescharacteristic
31C2×C31×#102C2211yescharacteristic
32C2×C4×C8×C31C2211yescharacteristic
33C31×#397—22no—
34C31×#391C2212yes—
35C31×#389C2212yes—
36C31×#385—22no—
37C31×#384C2211yescharacteristic
38C31×#380—21no—
39C31×#380—21no—
40C31×#380—21no—
41C4×C4⋊C4×C31C2211yescharacteristic
42C31×#340C2212yes—
43C31×#336C2212yes—
44C22×Q8×C31C2312yes—
45C22×D4×C31C2311yescharacteristic
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31—21no—
48C23×C4×C31—21no—
49C23×C4×C31—22no—
50C2×SD16×C31—24no—
51C2×SD16×C31—24no—
52C2×SD16×C31—24no—
53C2×D8×C31—22no—
54C2×D8×C31—22no—
55C2×D8×C31D414yes—
56C22×C8×C31C2311yescharacteristic
57C22×C8×C31C2312yes—
58C22×C8×C31C2311yescharacteristic
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×#124—24no—
62C31×#124—42no—
63C4×D4×C31—24no—
64C4×D4×C31—22no—
65C4×D4×C31—22no—
66C4×D4×C31—22no—
67C4×D4×C31—42no—
68C4×D4×C31D414yes—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—44no—
75C2×C4⋊C4×C31D411yescharacteristic
76C2×C4⋊C4×C31D411yescharacteristic
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C42×C31C2312yes—
86C2×C42×C31C2311yescharacteristic
87C31×#107—22no—
88C31×C4⋊C8—24no—
89C31×C4⋊C8—22no—
90C31×#103—24no—
91C31×#103—24no—
92C31×#103—42no—
93C31×#102—22no—
94C31×#102—24no—
95C31×#102—22no—
96C4×C8×C31—22no—
97C31×#95—22no—
98C31×#95—22no—
99C31×#95—22no—
100C31×#95D412yes—
101C24×C31—21no—
102C24×C31—21no—
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×Q8×C31—22no—
106C2×Q8×C31—22no—
107C2×Q8×C31—22no—
108C2×Q8×C31—22no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—21no—
114C2×D4×C31—21no—
115C2×D4×C31—21no—
116C2×D4×C31—21no—
117C2×D4×C31C2×D412yes—
118C2×D4×C31C2×D412yes—
119C2×D4×C31C2×D412yes—
120C2×D4×C31#4112yes—
121C22×C4×C31—21no—
122C22×C4×C31—24no—
123C22×C4×C31—21no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—44no—
127C22×C4×C31—44no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31—82no—
134C22×C4×C31C2×D412yes—
135C22×C4×C31C2×D412yes—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2411yescharacteristic
139SD16×C31—48no—
140D8×C31—28no—
141C2×C8×C31—21no—
142C2×C8×C31—21no—
143C2×C8×C31—21no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—21no—
148C2×C8×C31—21no—
149C2×C8×C31C2×D412yes—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—44no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—81no—
162C4⋊C4×C31C2×D412yes—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—44no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—44no—
167C22⋊C4×C31—42no—
168C42×C31—22no—
169C42×C31—22no—
170C42×C31C2×D412yes—
171C42×C31#4112yes—
172#24727C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31C22×D411yescharacteristic
188Q8×C31—42no—
189Q8×C31—42no—
190Q8×C31—42no—
191Q8×C31—42no—
192D4×C31—22no—
193D4×C31—22no—
194D4×C31—22no—
195D4×C31—22no—
196D4×C31—22no—
197D4×C31—41no—
198D4×C31—41no—
199D4×C31#12114yes—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—48no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—88no—
222C2×C4×C31—84no—
223C2×C4×C31#12112yes—
224C2×C4×C31#12112yes—
225C2×C4×C31#13512yes—
226C2×C4×C31AGL(1,ℤ/8)12yes—
227C2×C4×C31C22×D411yescharacteristic
228C2×C4×C31C22×D411yescharacteristic
229C2×C4×C31C2×#4111yescharacteristic
230C2×C4×C31C2×#4111yescharacteristic
231C2×C4×C31H(5,𝔽2)11yescharacteristic
232C2×C4×C312−1+411yescharacteristic
233C8×C31—22no—
234C8×C31—42no—
235C8×C31—41no—
236C2×#463C2×C3112yes—
237C2×#461C2×C3111yescharacteristic, complemented
238C2×C4×D8C2×C3111yescharacteristic
239#2547C2×C3111yescharacteristic, complemented
240D4×C4⋊C4C2×C3111yescharacteristic, complemented
241#2068C2×C3112yescomplemented
242#2042C2×C3112yescomplemented
243#1956C2×C3111yescharacteristic, complemented
244#1939C2×C3111yescharacteristic, complemented
245#1877C2×C3112yescomplemented
246#1802C2×C3111yescharacteristic, complemented
247C22×C31—24no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—81no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31—81no—
261C22×C31C2×#12111yescharacteristic
262C22×C31D4211yescharacteristic
263C22×C31#54811yescharacteristic
264C22×C31C2×#13511yescharacteristic
265C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C31#57711yescharacteristic
267C22×C31#57811yescharacteristic
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—42no—
272C4×C31—82no—
273C4×C31—82no—
274C4×C31—81no—
275C4×C31#46314yescomplemented
276C4×C31C2×#12111yescharacteristic
277C4×C31#53711yescharacteristic
278C4×C31#54711yescharacteristic
279C4×C31#55311yescharacteristic
280C22×SD16C22×C3112yes—
281C22×D8C22×C3111yescharacteristic
282C2×C4⋊Q8C22×C3111yescharacteristic
283C2×#124C22×C3112yes—
284C2×C4×D4C22×C3111yescharacteristic
285C2×C4×D4C22×C3112yes—
286C2×C4×D4C22×C3111yescharacteristic
287C22×C4⋊C4—21no—
288#463—28no—
289#461—24no—
290C4×D8—24no—
291C2×#107C22×C3111yescharacteristic
292C2×C4⋊C8C22×C3111yescharacteristic
293C2×C4⋊C8C22×C3112yes—
294C2×#103C22×C3112yes—
295C2×#103C22×C3112yes—
296C2×#103C22×C3112yes—
297C2×#102C22×C3112yes—
298C2×#102C22×C3111yescharacteristic
299C2×#102C22×C3111yescharacteristic
300C2×C4×C8C22×C3111yescharacteristic
301#397—22no—
302#391C22×C3112yes—
303#389C22×C3112yes—
304#385—22no—
305#384C22×C3111yescharacteristic
306#380—21no—
307#380—21no—
308#380—21no—
309C4×C4⋊C4C22×C3111yescharacteristic
310#340C22×C3112yes—
311#336C22×C3112yes—
312C2×C31—44no—
313C2×C31—82no—
314C2×C31#241611yescharacteristic
315C2×C31C2×#46311yescharacteristic
316C2×C31#304611yescharacteristic
317C2×C31#327211yescharacteristic
318C2×C31#328311yescharacteristic
319C2×C31#336011yescharacteristic
320C2×C31#336711yescharacteristic
321C22×Q8C23×C3112yes—
322C22×D4C23×C3111yescharacteristic
323C22×D4C23×C3111yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4—22no—
327C2×SD16—24no—
328C2×SD16—24no—
329C2×SD16—24no—
330C2×D8—22no—
331C2×D8—22no—
332C2×D8D4×C3114yes—
333C22×C8C23×C3112yes—
334C22×C8C23×C3111yescharacteristic
335C22×C8C23×C3111yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338#124—24no—
339#124—42no—
340C4×D4—24no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—42no—
345C4×D4D4×C3114yes—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4D4×C3111yescharacteristic
353C2×C4⋊C4D4×C3111yescharacteristic
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3112yes—
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42C23×C3112yes—
363C2×C42C23×C3111yescharacteristic
364#107—22no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367#103—24no—
368#103—24no—
369#103—42no—
370#102—22no—
371#102—24no—
372#102—22no—
373C4×C8—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95D4×C3112yes—
378C31#2472711yescharacteristic, direct product factor, Sylow
379C24—21no—
380C24—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×D4×C3112yes—
396C2×D4C2×D4×C3112yes—
397C2×D4C2×D4×C3112yes—
398C2×D4#41×C3112yes—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3112yes—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C24×C3111yescharacteristic
417SD16—48no—
418D8—28no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×D4×C3112yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4C2×D4×C3112yes—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C42—22no—
447C42—22no—
448C42C2×D4×C3112yes—
449C42#41×C3112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22×D4×C3111yescharacteristic
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4C31×#12114yes—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—88no—
499C2×C4—84no—
500C2×C4C31×#12112yes—
501C2×C4C31×#12112yes—
502C2×C4C31×#13512yes—
503C2×C4C31×AGL(1,ℤ/8)12yes—
504C2×C4C22×D4×C3111yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C2×#41×C3111yescharacteristic
507C2×C4C2×#41×C3111yescharacteristic
508C2×C4C31×H(5,𝔽2)11yescharacteristic
509C2×C4C31×2−1+411yescharacteristic
510C8—22no—
511C8—42no—
512C8—41no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C2×C31×#12111yescharacteristic
528C22D42×C3111yescharacteristic
529C22C31×#54811yescharacteristic
530C22C2×C31×#13511yescharacteristic
531C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
532C22C31×#57711yescharacteristic
533C22C31×#57811yescharacteristic
534C4—22no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—82no—
540C4—81no—
541C4C31×#46314yescomplemented
542C4C2×C31×#12111yescharacteristic
543C4C31×#53711yescharacteristic
544C4C31×#54711yescharacteristic
545C4C31×#55311yescharacteristic
546C2—44no—
547C2—82no—
548C2C31×#241611yescharacteristic
549C2C2×C31×#46311yescharacteristic
550C2C31×#304611yescharacteristic
551C2C31×#327211yescharacteristic
552C2C31×#328311yescharacteristic
553C2C31×#336011yescharacteristic
554C2C31×#336711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D8×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#461C2semidirect
4. C31×#2547C2semidirect
5. D4×C4⋊C4×C31C2semidirect
6. C31×#2068C2semidirect
7. C31×#2042C2semidirect
8. C31×#1956C2semidirect
9. C31×#1939C2semidirect
10. C31×#1877C2semidirect
11. C31×#1802C2semidirect
172. #24727C31direct (N × H)
237. C2×#461C2×C31semidirect
239. #2547C2×C31semidirect
240. D4×C4⋊C4C2×C31semidirect
241. #2068C2×C31semidirect
242. #2042C2×C31semidirect
243. #1956C2×C31semidirect
244. #1939C2×C31semidirect
245. #1877C2×C31semidirect
246. #1802C2×C31semidirect
275. C4×C31#463semidirect
378. C31#24727direct (N × H)
541. C4C31×#463semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4632
C2×C31×#4612
C2×C4×D8×C312
C31×#25472
D4×C4⋊C4×C312
C31×#20682
C31×#20422
C31×#19562
C31×#19392
C31×#18772
C31×#18022
#2472731