Finite Group Databasebeta

← C31×#24122 #21480881

Subgroups — C31×#24122 #21480881

Order 7936.

1134 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
1C2×C31×#554C211yesmaximal, characteristic
2C2×C31×#477C211yesmaximal, characteristic, complemented
3C2×C4×Q16×C31C211yesmaximal, characteristic, complemented
4C31×#2538C211yesmaximal, characteristic
5C31×#2036C212yesmaximal, complemented
6C31×#2001C212yesmaximal, complemented
7C31×#1990C211yesmaximal, characteristic, complemented
8C31×#1954C211yesmaximal, characteristic
9C31×#1872C212yesmaximal, complemented
10C31×#1852C212yesmaximal, complemented
11C31×#1803C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2211yescharacteristic
13C31×#554—24no—
14C31×#554—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#125—21no—
17C2×C31×C22⋊Q8—22no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×C22⋊Q8C2212yes—
20C2×C4×Q8×C31C2211yescharacteristic
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C22×C4⋊C4×C31C2212yes—
25C31×#477—24no—
26C4×Q16×C31—24no—
27C2×C31×#107C2211yescharacteristic
28C2×C31×#107C2212yes—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×#103C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#103C2211yescharacteristic
33C2×C31×#103C2212yes—
34C2×C31×C22⋊C8C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C4×C8×C31C2211yescharacteristic
37C31×#396—22no—
38C31×#390—21no—
39C31×#389—21no—
40C31×#384C2211yescharacteristic
41C31×#380C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31—21no—
46C22×Q8×C31C2311yescharacteristic
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31D411yescharacteristic
49C23×C4×C31—22no—
50C23×C4×C31C2312yes—
51C2×Q16×C31—22no—
52C2×Q16×C31—22no—
53C2×Q16×C31—22no—
54C22×C8×C31C2311yescharacteristic
55C22×C8×C31C2312yes—
56C22×C8×C31C2311yescharacteristic
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8D414yes—
59C31×#125—28no—
60C31×C22⋊Q8—216no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C31×C22⋊Q8—42no—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31—22no—
68C4×Q8×C31D414yes—
69C4×D4×C31—28no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—24no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31D411yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C4⋊C4×C31C2312yes—
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2312yes—
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×C31—21no—
93C2×C42×C31C2311yescharacteristic
94C31×#107—22no—
95C31×#107—24no—
96C31×C4⋊C8—22no—
97C31×#103—22no—
98C31×#103—22no—
99C31×#103—24no—
100C31×#103—42no—
101C31×C22⋊C8—24no—
102C31×C22⋊C8—42no—
103C4×C8×C31—22no—
104C31×#95—22no—
105C24×C31C2×D412yes—
106C2×Q8×C31—24no—
107C2×Q8×C31—21no—
108C2×Q8×C31—21no—
109C2×Q8×C31—21no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—81no—
118C2×Q8×C31—81no—
119C2×Q8×C31C2×D412yes—
120C2×Q8×C31#4112yes—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31Q1614yescomplemented
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—28no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—21no—
136C22×C4×C31—22no—
137C22×C4×C31—82no—
138C22×C4×C31—82no—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2411yescharacteristic
143Q16×C31—28no—
144C2×C8×C31—21no—
145C2×C8×C31—21no—
146C2×C8×C31—21no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—21no—
150C2×C8×C31—22no—
151C2×C8×C31—21no—
152C2×C8×C31#4112yes—
153C4⋊C4×C31—28no—
154C4⋊C4×C31—28no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—44no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31—81no—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D414yes—
170C22⋊C4×C31—28no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—82no—
174C42×C31—24no—
175C42×C31—42no—
176C42×C31C2×D412yes—
177C42×C31C2×D412yes—
178#24122C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31C2×Q1614yes—
186C23×C31C22×D411yescharacteristic
187Q8×C31—22no—
188Q8×C31—22no—
189Q8×C31—22no—
190Q8×C31—22no—
191Q8×C31—41no—
192Q8×C31—41no—
193Q8×C31—41no—
194Q8×C31—41no—
195Q8×C31—84no—
196D4×C31—28no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—88no—
225C2×C4×C31#12012yes—
226C2×C4×C31#12012yes—
227C2×C4×C31C2×Q1612yes—
228C2×C4×C31#13712yes—
229C2×C4×C31C22×D411yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C2×#4111yescharacteristic
232C2×C4×C31C2×#4111yescharacteristic
233C2×C4×C31H(5,𝔽2)11yescharacteristic
234C2×C4×C312−1+411yescharacteristic
235C8×C31—22no—
236C8×C31—41no—
237C8×C31—42no—
238C2×#554C2×C3111yescharacteristic
239C2×#477C2×C3111yescharacteristic, complemented
240C2×C4×Q16C2×C3111yescharacteristic, complemented
241#2538C2×C3111yescharacteristic
242#2036C2×C3112yescomplemented
243#2001C2×C3112yescomplemented
244#1990C2×C3111yescharacteristic, complemented
245#1954C2×C3111yescharacteristic
246#1872C2×C3112yescomplemented
247#1852C2×C3112yescomplemented
248#1803C2×C3111yescharacteristic, complemented
249C22×C31—24no—
250C22×C31—24no—
251C22×C31—24no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31C2×#12011yescharacteristic
257C22×C31#54611yescharacteristic
258C22×C31#54711yescharacteristic
259C22×C31C22×Q1611yescharacteristic
260C22×C31C2×#13711yescharacteristic
261C22×C31#57611yescharacteristic
262C22×C31#57711yescharacteristic
263C4×C31—22no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—42no—
268C4×C31—44no—
269C4×C31—82no—
270C4×C31—81no—
271C4×C31#45114yes—
272C4×C31#52511yescharacteristic
273C4×C31#53711yescharacteristic
274C4×C31#54811yescharacteristic
275C4×C31#55011yescharacteristic
276C22×Q16C22×C3111yescharacteristic
277#554—24no—
278#554—24no—
279C2×C4⋊Q8C22×C3111yescharacteristic
280C2×#125—21no—
281C2×C22⋊Q8—22no—
282C2×C22⋊Q8C22×C3112yes—
283C2×C22⋊Q8C22×C3112yes—
284C2×C4×Q8C22×C3111yescharacteristic
285C2×C4×Q8C22×C3111yescharacteristic
286C2×C4×D4—21no—
287C2×C4×D4C22×C3111yescharacteristic
288C22×C4⋊C4C22×C3112yes—
289#477—24no—
290C4×Q16—24no—
291C2×#107C22×C3112yes—
292C2×#107C22×C3111yescharacteristic
293C2×C4⋊C8C22×C3111yescharacteristic
294C2×#103C22×C3112yes—
295C2×#103C22×C3111yescharacteristic
296C2×#103C22×C3111yescharacteristic
297C2×#103C22×C3112yes—
298C2×C22⋊C8C22×C3112yes—
299C2×C22⋊C8C22×C3112yes—
300C2×C4×C8C22×C3111yescharacteristic
301#396—22no—
302#390—21no—
303#389—21no—
304#384C22×C3111yescharacteristic
305#380C22×C3112yes—
306#340C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309C2×C31—44no—
310C2×C31#240711yescharacteristic
311C2×C31C2×#45111yescharacteristic
312C2×C31#301211yescharacteristic
313C2×C31#329911yescharacteristic
314C2×C31#333311yescharacteristic
315C2×C31#333811yescharacteristic
316C2×C31#335611yescharacteristic
317C22×Q8—21no—
318C22×Q8C23×C3111yescharacteristic
319C22×Q8C23×C3111yescharacteristic
320C22×D4D4×C3111yescharacteristic
321C23×C4—22no—
322C23×C4C23×C3112yes—
323C2×Q16—22no—
324C2×Q16—22no—
325C2×Q16—22no—
326C22×C8C23×C3111yescharacteristic
327C22×C8C23×C3112yes—
328C22×C8C23×C3111yescharacteristic
329C4⋊Q8—22no—
330C4⋊Q8D4×C3114yes—
331#125—28no—
332C22⋊Q8—216no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—42no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8D4×C3114yes—
341C4×D4—28no—
342C4×D4—22no—
343C4×D4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C4⋊C4C23×C3112yes—
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C4⋊C4C23×C3112yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4D4×C3112yes—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42C23×C3111yescharacteristic
366#107—22no—
367#107—24no—
368C4⋊C8—22no—
369#103—22no—
370#103—22no—
371#103—24no—
372#103—42no—
373C22⋊C8—24no—
374C22⋊C8—42no—
375C4×C8—22no—
376#95—22no—
377C31#2412211yescharacteristic, direct product factor, Sylow
378C24C2×D4×C3112yes—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8C2×D4×C3112yes—
393C2×Q8#41×C3112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4Q16×C3114yescomplemented
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C24×C3111yescharacteristic
416Q16—28no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8#41×C3112yes—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4C2×D4×C3112yes—
442C4⋊C4C2×D4×C3114yes—
443C22⋊C4—28no—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C42—24no—
448C42—42no—
449C42C2×D4×C3112yes—
450C42C2×D4×C3112yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23C2×Q16×C3114yes—
458C23C22×D4×C3111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—84no—
468D4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—88no—
497C2×C4C31×#12012yes—
498C2×C4C31×#12012yes—
499C2×C4C2×Q16×C3112yes—
500C2×C4C31×#13712yes—
501C2×C4C22×D4×C3111yescharacteristic
502C2×C4C22×D4×C3111yescharacteristic
503C2×C4C2×#41×C3111yescharacteristic
504C2×C4C2×#41×C3111yescharacteristic
505C2×C4C31×H(5,𝔽2)11yescharacteristic
506C2×C4C31×2−1+411yescharacteristic
507C8—22no—
508C8—41no—
509C8—42no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×C31×#12011yescharacteristic
518C22C31×#54611yescharacteristic
519C22C31×#54711yescharacteristic
520C22C22×Q16×C3111yescharacteristic
521C22C2×C31×#13711yescharacteristic
522C22C31×#57611yescharacteristic
523C22C31×#57711yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—82no—
531C4—81no—
532C4C31×#45114yes—
533C4C31×#52511yescharacteristic
534C4C31×#53711yescharacteristic
535C4C31×#54811yescharacteristic
536C4C31×#55011yescharacteristic
537C2—44no—
538C2C31×#240711yescharacteristic
539C2C2×C31×#45111yescharacteristic
540C2C31×#301211yescharacteristic
541C2C31×#329911yescharacteristic
542C2C31×#333311yescharacteristic
543C2C31×#333811yescharacteristic
544C2C31×#335611yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31D4
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×C4×Q16×C31C2semidirect
5. C31×#2036C2semidirect
6. C31×#2001C2semidirect
7. C31×#1990C2semidirect
9. C31×#1872C2semidirect
10. C31×#1852C2semidirect
11. C31×#1803C2semidirect
123. C2×D4×C31Q16semidirect
178. #24122C31direct (N × H)
239. C2×#477C2×C31semidirect
240. C2×C4×Q16C2×C31semidirect
242. #2036C2×C31semidirect
243. #2001C2×C31semidirect
244. #1990C2×C31semidirect
246. #1872C2×C31semidirect
247. #1852C2×C31semidirect
248. #1803C2×C31semidirect
377. C31#24122direct (N × H)
396. C2×D4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4772
C2×C4×Q16×C312
C31×#25382
C31×#20362
C31×#20012
C31×#19902
C31×#19542
C31×#18722
C31×#18522
C31×#18032
#2412231