Finite Group Databasebeta

← C31×#18269 #21475028

Subgroups — C31×#18269 #21475028

Order 7936.

924 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3538C211yesmaximal, characteristic
2C31×#3092C211yesmaximal, characteristic
3C2×C31×#484C211yesmaximal, characteristic, complemented
4C31×#2503C211yesmaximal, characteristic, complemented
5C31×#1617C214yesmaximal, complemented
6C31×#1615C214yesmaximal, complemented
7C31×#1508C211yesmaximal, characteristic, complemented
8C31×#1501C211yesmaximal, characteristic
9C31×#1472C211yesmaximal, characteristic, complemented
10C31×#563—22no—
11C31×#552—21no—
12C31×#552—21no—
13C31×#551—22no—
14D4×Q8×C31—21no—
15D4×Q8×C31—21no—
16C31×#544—21no—
17C31×#544C2212yes—
18C31×#541—21no—
19C31×#539—21no—
20C31×#539C2212yes—
21C31×#536—21no—
22C31×#525C2211yescharacteristic
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#484—22no—
27C31×#484C2214yes—
28C31×#484C2212yes—
29C31×#483—21no—
30C31×#481C2212yes—
31C31×#480—21no—
32C31×#428—21no—
33C2×C31×#107—21no—
34C31×#417—21no—
35C2×C31×#103—21no—
36C31×#407C2211yescharacteristic
37C2×C31×C22⋊C8C2211yescharacteristic
38C31×#400C2214yes—
39C31×#391—21no—
40C31×#382—21no—
41C31×#381—22no—
42C4×C22⋊C4×C31C2211yescharacteristic
43C31×#328C2214yes—
44C31×#328C2212yes—
45C31×#327C2212yes—
46C31×#324C2214yes—
47C2×#41×C31D411yescharacteristic
48C22×Q8×C31—21no—
49C22×Q8×C31D411yescharacteristic
50C22×Q8×C31D411yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31C2311yescharacteristic
53C2×M16×C31—21no—
54C22×C8×C31—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#125—21no—
64C31×#125—22no—
65C31×#125—22no—
66C31×#125—22no—
67C31×#125—22no—
68C31×#124—22no—
69C31×#123—22no—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—21no—
73C31×#123—22no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8D411yescharacteristic
87C31×C22⋊Q8D412yes—
88C31×C22⋊Q8C2312yes—
89C31×C22⋊Q8C2311yescharacteristic
90C31×#121D411yescharacteristic
91C31×#121C2311yescharacteristic
92C31×#120—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×D4×C31—22no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—22no—
99C31×#117—21no—
100C31×#117—22no—
101C31×#117—22no—
102C31×#117D411yescharacteristic
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—21no—
108C2×C4⋊C4×C31—21no—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C4⋊C4×C31D411yescharacteristic
111C2×C4⋊C4×C31C2312yes—
112C2×C4⋊C4×C31C2312yes—
113C2×C22⋊C4×C31—21no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31D412yes—
116C2×C42×C31—21no—
117C31×#107—24no—
118C31×#107—22no—
119C31×#106—22no—
120C31×#103—24no—
121C31×#103—22no—
122C31×#102—22no—
123C31×C22⋊C8—21no—
124C31×C22⋊C8—21no—
125C31×C22⋊C8C2312yes—
126C31×C22⋊C8C2312yes—
127C31×#95—24no—
128C31×#95—24no—
129C31×#95C2312yes—
130C24×C31C2×D411yescharacteristic
131#41×C31—42no—
132C2×Q8×C31—22no—
133C2×Q8×C31—21no—
134C2×Q8×C31—21no—
135C2×Q8×C31—21no—
136C2×Q8×C31—42no—
137C2×Q8×C31—42no—
138C2×Q8×C31—82no—
139C2×Q8×C31C2×D412yes—
140C2×Q8×C31C2×D411yescharacteristic
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31C2×D411yescharacteristic
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—42no—
152C22×C4×C31—44no—
153C22×C4×C31—41no—
154C22×C4×C31—41no—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2411yescharacteristic
163M16×C31—42no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—42no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—41no—
187C4⋊C4×C31—41no—
188C4⋊C4×C31C2×D411yescharacteristic
189C4⋊C4×C31C2×D412yes—
190C4⋊C4×C31C2×D411yescharacteristic
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—21no—
202C22⋊C4×C31—21no—
203C22⋊C4×C31—42no—
204C42×C31—22no—
205C42×C31—22no—
206C42×C31—22no—
207C42×C31—21no—
208C42×C31—42no—
209#18269C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—41no—
214C23×C31#12011yescharacteristic
215C23×C31#12011yescharacteristic
216C23×C31C22×D411yescharacteristic
217Q8×C31—42no—
218Q8×C31—41no—
219Q8×C31—42no—
220Q8×C31—42no—
221Q8×C31—81no—
222Q8×C31—82no—
223D4×C31—41no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—81no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—24no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—44no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—81no—
250C2×C4×C31—82no—
251C2×C4×C31—82no—
252C2×C4×C31—82no—
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C2×C4×C31C22×D411yescharacteristic
257C8×C31—42no—
258C8×C31—42no—
259#3538C2×C3111yescharacteristic
260#3092C2×C3111yescharacteristic
261C2×#484C2×C3111yescharacteristic, complemented
262#2503C2×C3111yescharacteristic, complemented
263#1617C2×C3114yescomplemented
264#1615C2×C3114yescomplemented
265#1508C2×C3111yescharacteristic, complemented
266#1501C2×C3111yescharacteristic
267#1472C2×C3111yescharacteristic, complemented
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—21no—
273C22×C31—21no—
274C22×C31—21no—
275C22×C31—44no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31#45412yes—
280C22×C31C2×#12011yescharacteristic
281C22×C31#57812yes—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—21no—
285C4×C31—42no—
286C4×C31—41no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—41no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—82no—
293C4×C31—82no—
294C4×C31#45812yes—
295#563—22no—
296#552—21no—
297#552—21no—
298#551—22no—
299D4×Q8—21no—
300D4×Q8—21no—
301#544—21no—
302#544C22×C3112yes—
303#541—21no—
304#539—21no—
305#539C22×C3112yes—
306#536—21no—
307#525C22×C3111yescharacteristic
308C2×C22⋊Q8C22×C3111yescharacteristic
309C2×C22⋊Q8C22×C3111yescharacteristic
310C22×C4⋊C4C22×C3111yescharacteristic
311#484—22no—
312#484C22×C3114yes—
313#484C22×C3112yes—
314#483—21no—
315#481C22×C3112yes—
316#480—21no—
317#428—21no—
318C2×#107—21no—
319#417—21no—
320C2×#103—21no—
321#407C22×C3111yescharacteristic
322C2×C22⋊C8C22×C3111yescharacteristic
323#400C22×C3114yes—
324#391—21no—
325#382—21no—
326#381—22no—
327C4×C22⋊C4C22×C3111yescharacteristic
328#328C22×C3114yes—
329#328C22×C3112yes—
330#327C22×C3112yes—
331#324C22×C3114yes—
332C2×C31—22no—
333C2×C31—22no—
334C2×C31—41no—
335C2×C31—81no—
336C2×C31#301111yescharacteristic
337C2×C31C2×#45411yescharacteristic
338C2×C31C2×#45811yescharacteristic
339C2×#41D4×C3111yescharacteristic
340C22×Q8—21no—
341C22×Q8D4×C3111yescharacteristic
342C22×Q8D4×C3111yescharacteristic
343C23×C4—21no—
344C23×C4C23×C3111yescharacteristic
345C2×M16—21no—
346C22×C8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—22no—
349C4⋊Q8—22no—
350C4⋊Q8—22no—
351#126—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#125—21no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—22no—
360#124—22no—
361#123—22no—
362#123—22no—
363#123—22no—
364#123—21no—
365#123—22no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8D4×C3112yes—
379C22⋊Q8D4×C3111yescharacteristic
380C22⋊Q8C23×C3112yes—
381C22⋊Q8C23×C3111yescharacteristic
382#121D4×C3111yescharacteristic
383#121C23×C3111yescharacteristic
384#120—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×D4—22no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—22no—
391#117—21no—
392#117—22no—
393#117—22no—
394#117D4×C3111yescharacteristic
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4D4×C3111yescharacteristic
402C2×C4⋊C4D4×C3111yescharacteristic
403C2×C4⋊C4C23×C3112yes—
404C2×C4⋊C4C23×C3112yes—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4D4×C3112yes—
408C2×C42—21no—
409#107—24no—
410#107—22no—
411#106—22no—
412#103—24no—
413#103—22no—
414#102—22no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8C23×C3112yes—
418C22⋊C8C23×C3112yes—
419#95—24no—
420#95—24no—
421#95C23×C3112yes—
422C31#1826911yescharacteristic, direct product factor, Sylow
423C24C2×D4×C3111yescharacteristic
424#41—42no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—82no—
432C2×Q8C2×D4×C3112yes—
433C2×Q8C2×D4×C3111yescharacteristic
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4C2×D4×C3111yescharacteristic
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4C2×D4×C3112yes—
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3112yes—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C24×C3111yescharacteristic
456M16—42no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—42no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×D4×C3111yescharacteristic
482C4⋊C4C2×D4×C3112yes—
483C4⋊C4C2×D4×C3111yescharacteristic
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—42no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—42no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—41no—
506C23C31×#12011yescharacteristic
507C23C31×#12011yescharacteristic
508C23C22×D4×C3111yescharacteristic
509Q8—42no—
510Q8—41no—
511Q8—42no—
512Q8—42no—
513Q8—81no—
514Q8—82no—
515D4—41no—
516D4—42no—
517D4—42no—
518D4—81no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4C31×#12011yescharacteristic
546C2×C4C31×#12011yescharacteristic
547C2×C4C22×D4×C3111yescharacteristic
548C2×C4C22×D4×C3111yescharacteristic
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—44no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22C31×#45412yes—
563C22C2×C31×#12011yescharacteristic
564C22C31×#57812yes—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4C31×#45812yes—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—81no—
582C2C31×#301111yescharacteristic
583C2C2×C31×#45411yescharacteristic
584C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#484C2semidirect
4. C31×#2503C2semidirect
5. C31×#1617C2semidirect
6. C31×#1615C2semidirect
7. C31×#1508C2semidirect
9. C31×#1472C2semidirect
209. #18269C31direct (N × H)
261. C2×#484C2×C31semidirect
262. #2503C2×C31semidirect
263. #1617C2×C31semidirect
264. #1615C2×C31semidirect
265. #1508C2×C31semidirect
267. #1472C2×C31semidirect
422. C31#18269direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35382
C31×#30922
C2×C31×#4842
C31×#25032
C31×#16172
C31×#16152
C31×#15082
C31×#15012
C31×#14722
#1826931