Finite Group Databasebeta

← C31×#24132 #21480891

Subgroups — C31×#24132 #21480891

Order 7936.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C2×C31×#477C211yesmaximal, characteristic, complemented
3C2×C31×#440C211yesmaximal, characteristic, complemented
4C31×#2536C211yesmaximal, characteristic, complemented
5C31×#2033C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1993C211yesmaximal, characteristic, complemented
8C31×#1955C211yesmaximal, characteristic, complemented
9C31×#1872C212yesmaximal, complemented
10C31×#1852C212yesmaximal, complemented
11C31×#1804C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×SD16×C31C2211yescharacteristic
14C31×#547—24no—
15C31×#547—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#123—22no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2212yes—
20C2×C31×#121C2212yes—
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C22×C4⋊C4×C31C2212yes—
25C31×#477—24no—
26C31×#440—24no—
27C2×C31×#107C2211yescharacteristic
28C2×C31×#107C2212yes—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×#103C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#102C2211yescharacteristic
33C2×C31×#102C2212yes—
34C2×C31×C22⋊C8C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×#97C2211yescharacteristic
37C31×#398—21no—
38C31×#394—21no—
39C31×#388—22no—
40C31×#382C2211yescharacteristic
41C31×#380C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C2×#41×C31—28no—
46C2×#41×C31—48no—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—22no—
50C22×D4×C31D411yescharacteristic
51C22×D4×C31C2311yescharacteristic
52C23×C4×C31—22no—
53C23×C4×C31—21no—
54C23×C4×C31C2312yes—
55C2×SD16×C31—22no—
56C2×SD16×C31—22no—
57C2×SD16×C31—22no—
58C22×C8×C31C2311yescharacteristic
59C22×C8×C31C2312yes—
60C22×C8×C31C2311yescharacteristic
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D414yes—
63C31×#123—216no—
64C31×C22⋊Q8—28no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8—42no—
67C31×#121—24no—
68C31×#121—24no—
69C4×Q8×C31—22no—
70C4×Q8×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31D414yes—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31D411yescharacteristic
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2312yes—
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31D412yes—
91C2×C42×C31—21no—
92C2×C42×C31C2311yescharacteristic
93C31×#107—22no—
94C31×#107—24no—
95C31×C4⋊C8—22no—
96C31×#103—22no—
97C31×#103—24no—
98C31×#102—22no—
99C31×#102—42no—
100C31×C22⋊C8—24no—
101C31×C22⋊C8—42no—
102C31×#97—22no—
103C31×#95—22no—
104C31×#95—21no—
105C31×#95—21no—
106C24×C31—21no—
107C24×C31C2×D412yes—
108#41×C31—432no—
109C2×Q8×C31—24no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—44no—
116C2×Q8×C31#4112yescomplemented
117C2×D4×C31—24no—
118C2×D4×C31—28no—
119C2×D4×C31—22no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—44no—
126C2×D4×C31—44no—
127C2×D4×C31C2×D412yes—
128C22×C4×C31—28no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—24no—
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—21no—
141C22×C4×C31—44no—
142C22×C4×C31—44no—
143C22×C4×C31—44no—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2411yescharacteristic
148SD16×C31—28no—
149C2×C8×C31—21no—
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—21no—
155C2×C8×C31—22no—
156C2×C8×C31—21no—
157C2×C8×C31#4112yescomplemented
158C4⋊C4×C31—28no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—44no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D414yes—
171C22⋊C4×C31—28no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—44no—
175C22⋊C4×C31—44no—
176C42×C31—42no—
177C42×C31C2×D412yes—
178C42×C31C2×D412yes—
179#24132C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31AGL(1,ℤ/8)14yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—22no—
194Q8×C31—22no—
195Q8×C31—48no—
196Q8×C31—41no—
197Q8×C31—41no—
198D4×C31—28no—
199D4×C31—22no—
200D4×C31—22no—
201D4×C31—416no—
202D4×C31—41no—
203D4×C31—41no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—24no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—48no—
213C2×C4×C31—416no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31#12012yes—
229C2×C4×C31#12012yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C2×#4111yescharacteristic
233C2×C4×C31C2×#4111yescharacteristic
234C2×C4×C31H(5,𝔽2)11yescharacteristic
235C2×C4×C312−1+411yescharacteristic
236C8×C31—22no—
237C8×C31—41no—
238C8×C31—42no—
239C2×#547C2×C3111yescharacteristic
240C2×#477C2×C3111yescharacteristic, complemented
241C2×#440C2×C3111yescharacteristic, complemented
242#2536C2×C3111yescharacteristic, complemented
243#2033C2×C3112yescomplemented
244#2000C2×C3112yescomplemented
245#1993C2×C3111yescharacteristic, complemented
246#1955C2×C3111yescharacteristic, complemented
247#1872C2×C3112yescomplemented
248#1852C2×C3112yescomplemented
249#1804C2×C3111yescharacteristic, complemented
250C22×C31—24no—
251C22×C31—24no—
252C22×C31—24no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31C2×#12011yescharacteristic
263C22×C31#54611yescharacteristic
264C22×C31#54711yescharacteristic
265C22×C31C2×AGL(1,ℤ/8)12yes—
266C22×C31#57812yes—
267C4×C31—22no—
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—42no—
272C4×C31—44no—
273C4×C31—81no—
274C4×C31—82no—
275C4×C31#52511yescharacteristic
276C4×C31#53711yescharacteristic
277C4×C31#54811yescharacteristic
278C4×C31#55011yescharacteristic
279C22×#41—21no—
280C22×SD16C22×C3111yescharacteristic
281#547—24no—
282#547—24no—
283C2×C4⋊Q8C22×C3111yescharacteristic
284C2×#123—22no—
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8C22×C3112yes—
287C2×#121C22×C3112yes—
288C2×C4×Q8C22×C3111yescharacteristic
289C2×C4×D4C22×C3111yescharacteristic
290C2×C4×D4C22×C3111yescharacteristic
291C22×C4⋊C4C22×C3112yes—
292#477—24no—
293#440—24no—
294C2×#107C22×C3112yes—
295C2×#107C22×C3111yescharacteristic
296C2×C4⋊C8C22×C3111yescharacteristic
297C2×#103C22×C3112yes—
298C2×#103C22×C3111yescharacteristic
299C2×#102C22×C3111yescharacteristic
300C2×#102C22×C3112yes—
301C2×C22⋊C8C22×C3112yes—
302C2×C22⋊C8C22×C3112yes—
303C2×#97C22×C3111yescharacteristic
304#398—21no—
305#394—21no—
306#388—22no—
307#382C22×C3111yescharacteristic
308#380C22×C3112yes—
309#340C22×C3112yes—
310#340C22×C3112yes—
311#340C22×C3112yes—
312C2×C31—44no—
313C2×C31—82no—
314C2×C31#240711yescharacteristic
315C2×C31#300311yescharacteristic
316C2×C31#301111yescharacteristic
317C2×C31#330812yes—
318C2×C31#334312yes—
319C2×#41—28no—
320C2×#41—48no—
321C22×Q8—21no—
322C22×Q8C23×C3111yescharacteristic
323C22×D4—22no—
324C22×D4D4×C3111yescharacteristic
325C22×D4C23×C3111yescharacteristic
326C23×C4—22no—
327C23×C4—21no—
328C23×C4C23×C3112yes—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×SD16—22no—
332C22×C8C23×C3111yescharacteristic
333C22×C8C23×C3112yes—
334C22×C8C23×C3111yescharacteristic
335C4⋊Q8—22no—
336C4⋊Q8D4×C3114yes—
337#123—216no—
338C22⋊Q8—28no—
339C22⋊Q8—24no—
340C22⋊Q8—42no—
341#121—24no—
342#121—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4D4×C3114yes—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3112yes—
358C2×C4⋊C4C23×C3111yescharacteristic
359C2×C4⋊C4C23×C3112yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4D4×C3112yes—
365C2×C42—21no—
366C2×C42C23×C3111yescharacteristic
367#107—22no—
368#107—24no—
369C4⋊C8—22no—
370#103—22no—
371#103—24no—
372#102—22no—
373#102—42no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376#97—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380C31#2413211yescharacteristic, direct product factor, Sylow
381C24—21no—
382C24C2×D4×C3112yes—
383#41—432no—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—44no—
391C2×Q8#41×C3112yescomplemented
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×D4×C3112yes—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4C2×D4×C3112yes—
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C24×C3111yescharacteristic
423SD16—28no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8#41×C3112yescomplemented
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4C2×D4×C3112yes—
445C4⋊C4C2×D4×C3114yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—42no—
452C42C2×D4×C3112yes—
453C42C2×D4×C3112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23C31×AGL(1,ℤ/8)14yes—
466C23C22×D4×C3111yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—48no—
470Q8—41no—
471Q8—41no—
472D4—28no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—416no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4C31×#12012yes—
503C2×C4C31×#12012yes—
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—41no—
512C8—42no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×C31×#12011yescharacteristic
526C22C31×#54611yescharacteristic
527C22C31×#54711yescharacteristic
528C22C2×C31×AGL(1,ℤ/8)12yes—
529C22C31×#57812yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—44no—
536C4—81no—
537C4—82no—
538C4C31×#52511yescharacteristic
539C4C31×#53711yescharacteristic
540C4C31×#54811yescharacteristic
541C4C31×#55011yescharacteristic
542C2—44no—
543C2—82no—
544C2C31×#240711yescharacteristic
545C2C31×#300311yescharacteristic
546C2C31×#301111yescharacteristic
547C2C31×#330812yes—
548C2C31×#334312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#477C2semidirect
3. C2×C31×#440C2semidirect
4. C31×#2536C2semidirect
5. C31×#2033C2semidirect
6. C31×#2000C2semidirect
7. C31×#1993C2semidirect
8. C31×#1955C2semidirect
9. C31×#1872C2semidirect
10. C31×#1852C2semidirect
11. C31×#1804C2semidirect
116. C2×Q8×C31#41semidirect
157. C2×C8×C31#41semidirect
179. #24132C31direct (N × H)
240. C2×#477C2×C31semidirect
241. C2×#440C2×C31semidirect
242. #2536C2×C31semidirect
243. #2033C2×C31semidirect
244. #2000C2×C31semidirect
245. #1993C2×C31semidirect
246. #1955C2×C31semidirect
247. #1872C2×C31semidirect
248. #1852C2×C31semidirect
249. #1804C2×C31semidirect
380. C31#24132direct (N × H)
391. C2×Q8#41×C31semidirect
432. C2×C8#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C2×C31×#4772
C2×C31×#4402
C31×#25362
C31×#20332
C31×#20002
C31×#19932
C31×#19552
C31×#18722
C31×#18522
C31×#18042
#2413231