Finite Group Databasebeta

← C31×#18167 #21474926

Subgroups — C31×#18167 #21474926

Order 7936.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#3007C212yesmaximal, complemented
3C31×#2388C211yesmaximal, characteristic
4C31×#1605C212yesmaximal, complemented
5C31×#1602C212yesmaximal, complemented
6C31×#1599C212yesmaximal, complemented
7C31×#1596C212yesmaximal, complemented
8C31×#1508C212yesmaximal, complemented
9C31×#1465C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C2×C31×#137—22no—
12C31×#554—24no—
13C31×#554—22no—
14D4×Q8×C31—22no—
15C31×#544C2212yes—
16C31×#543—21no—
17C31×#543C2212yes—
18C31×#533—21no—
19C2×C31×C22⋊Q8—21no—
20C2×C31×C22⋊Q8C2212yes—
21C31×#518—21no—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#452—24no—
25C31×#451C2214yes—
26C31×#450C2214yes—
27C31×#417—22no—
28C31×#407C2212yes—
29C31×#395C2211yescharacteristic
30C31×#394—21no—
31C31×#394C2211yescharacteristic
32C31×#393C2211yescharacteristic
33C31×#392—21no—
34C31×#392C2211yescharacteristic
35C4×C22⋊C4×C31—21no—
36C4×C22⋊C4×C31C2211yescharacteristic
37C31×#328C2214yes—
38C31×#328C2214yes—
39C31×#324C2212yes—
40C31×#323C2212yes—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—44no—
44C22×Q8×C31—21no—
45C22×Q8×C31D412yescomplemented
46C22×Q8×C31C2311yescharacteristic
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C31×#137—48no—
55C2×Q16×C31—24no—
56C2×SD16×C31—24no—
57C2×M16×C31—22no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—24no—
61C31×#125—22no—
62C31×#125—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—24no—
73C31×C22⋊Q8D414yescomplemented
74C31×C22⋊Q8C2314yes—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×D4×C31—24no—
78C4×D4×C31—24no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—44no—
82C31×#117—22no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D412yescomplemented
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31C2311yescharacteristic
98C2×C42×C31—21no—
99C2×C42×C31D411yescharacteristic
100C31×#103—24no—
101C31×#102—24no—
102C31×C22⋊C8—22no—
103C31×C22⋊C8C2314yes—
104C31×#95—21no—
105C31×#95—21no—
106C31×#95C2311yescharacteristic
107C31×#95C2311yescharacteristic
108C24×C31—21no—
109C24×C31C2×D411yescharacteristic
110C24×C31C2×D411yescharacteristic
111#41×C31—28no—
112#41×C31—44no—
113#41×C31—44no—
114#41×C31—44no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—42no—
120C2×Q8×C31—44no—
121C2×Q8×C31—82no—
122C2×Q8×C31C2×D414yes—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C22×C4×C31—24no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—41no—
148C22×C4×C31—41no—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2411yescharacteristic
156Q16×C31—48no—
157SD16×C31—48no—
158M16×C31—44no—
159C2×C8×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31C2×D414yes—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—42no—
179C22⋊C4×C31—42no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—42no—
182C42×C31—24no—
183C42×C31—21no—
184C42×C31—21no—
185C42×C31—42no—
186#18167C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—22no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31#12012yes—
203C23×C31#13712yescomplemented
204C23×C31C22×D411yescharacteristic
205Q8×C31—24no—
206Q8×C31—44no—
207Q8×C31—42no—
208Q8×C31—44no—
209Q8×C31—82no—
210Q8×C31—82no—
211D4×C31—28no—
212D4×C31—24no—
213D4×C31—42no—
214D4×C31—48no—
215D4×C31—48no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—21no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—21no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—44no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—44no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—81no—
241C2×C4×C31—82no—
242C2×C4×C31—82no—
243C2×C4×C31—82no—
244C2×C4×C31—82no—
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#13712yescomplemented
248C2×C4×C31C22×D412yes—
249C8×C31—44no—
250#3501C2×C3111yescharacteristic, complemented
251#3007C2×C3112yescomplemented
252#2388C2×C3111yescharacteristic
253#1605C2×C3112yescomplemented
254#1602C2×C3112yescomplemented
255#1599C2×C3112yescomplemented
256#1596C2×C3112yescomplemented
257#1508C2×C3112yescomplemented
258#1465C2×C3111yescharacteristic, complemented
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—22no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—41no—
269C22×C31—41no—
270C22×C31—42no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31C2×#12011yescharacteristic
279C22×C31C2×#13711yescharacteristic
280C22×C31C2×#13711yescharacteristic
281C4×C31—22no—
282C4×C31—22no—
283C4×C31—21no—
284C4×C31—42no—
285C4×C31—41no—
286C4×C31—44no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—44no—
290C4×C31—81no—
291C4×C31B(4,𝔽2)12yes—
292C22×#41C22×C3111yescharacteristic
293C2×#137—22no—
294#554—24no—
295#554—22no—
296D4×Q8—22no—
297#544C22×C3112yes—
298#543—21no—
299#543C22×C3112yes—
300#533—21no—
301C2×C22⋊Q8—21no—
302C2×C22⋊Q8C22×C3112yes—
303#518—21no—
304C2×C4×D4—21no—
305C2×C4×D4C22×C3111yescharacteristic
306#452—24no—
307#451C22×C3114yes—
308#450C22×C3114yes—
309#417—22no—
310#407C22×C3112yes—
311#395C22×C3111yescharacteristic
312#394—21no—
313#394C22×C3111yescharacteristic
314#393C22×C3111yescharacteristic
315#392—21no—
316#392C22×C3111yescharacteristic
317C4×C22⋊C4—21no—
318C4×C22⋊C4C22×C3111yescharacteristic
319#328C22×C3114yes—
320#328C22×C3114yes—
321#324C22×C3112yes—
322#323C22×C3112yes—
323C2×C31—21no—
324C2×C31—21no—
325C2×C31—42no—
326C2×C31—42no—
327C2×C31—82no—
328C2×C31#300411yescharacteristic
329C2×C31#301811yescharacteristic
330C2×C31C2×B(4,𝔽2)11yescharacteristic
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—44no—
334C22×Q8—21no—
335C22×Q8D4×C3112yescomplemented
336C22×Q8C23×C3111yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4D4×C3111yescharacteristic
340C22×D4C23×C3111yescharacteristic
341C23×C4—21no—
342C23×C4D4×C3111yescharacteristic
343C23×C4C23×C3111yescharacteristic
344#137—48no—
345C2×Q16—24no—
346C2×SD16—24no—
347C2×M16—22no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—24no—
351#125—22no—
352#125—24no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—24no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—24no—
363C22⋊Q8D4×C3114yescomplemented
364C22⋊Q8C23×C3114yes—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—44no—
372#117—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4D4×C3112yescomplemented
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4D4×C3111yescharacteristic
387C2×C22⋊C4C23×C3111yescharacteristic
388C2×C42—21no—
389C2×C42D4×C3111yescharacteristic
390#103—24no—
391#102—24no—
392C22⋊C8—22no—
393C22⋊C8C23×C3114yes—
394#95—21no—
395#95—21no—
396#95C23×C3111yescharacteristic
397#95C23×C3111yescharacteristic
398C31#1816711yescharacteristic, direct product factor, Sylow
399C24—21no—
400C24C2×D4×C3111yescharacteristic
401C24C2×D4×C3111yescharacteristic
402#41—28no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8C2×D4×C3114yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3112yes—
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3112yes—
446C22×C4C24×C3111yescharacteristic
447Q16—48no—
448SD16—48no—
449M16—44no—
450C2×C8—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4C2×D4×C3114yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—42no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C31×#12012yes—
493C23C31×#13712yescomplemented
494C23C22×D4×C3111yescharacteristic
495Q8—24no—
496Q8—44no—
497Q8—42no—
498Q8—44no—
499Q8—82no—
500Q8—82no—
501D4—28no—
502D4—24no—
503D4—42no—
504D4—48no—
505D4—48no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4C31×#12011yescharacteristic
536C2×C4C31×#12011yescharacteristic
537C2×C4C31×#13712yescomplemented
538C2×C4C22×D4×C3112yes—
539C8—44no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C31×#12011yescharacteristic
560C22C2×C31×#13711yescharacteristic
561C22C2×C31×#13711yescharacteristic
562C4—22no—
563C4—22no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—44no—
568C4—42no—
569C4—41no—
570C4—44no—
571C4—81no—
572C4C31×B(4,𝔽2)12yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—42no—
577C2—82no—
578C2C31×#300411yescharacteristic
579C2C31×#301811yescharacteristic
580C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2388C2
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
1. C31×#3501C2semidirect
2. C31×#3007C2semidirect
4. C31×#1605C2semidirect
5. C31×#1602C2semidirect
6. C31×#1599C2semidirect
7. C31×#1596C2semidirect
8. C31×#1508C2semidirect
9. C31×#1465C2semidirect
45. C22×Q8×C31D4semidirect
73. C31×C22⋊Q8D4semidirect
87. C2×C4⋊C4×C31D4semidirect
186. #18167C31direct (N × H)
203. C23×C31#137semidirect
247. C2×C4×C31#137semidirect
250. #3501C2×C31semidirect
251. #3007C2×C31semidirect
253. #1605C2×C31semidirect
254. #1602C2×C31semidirect
255. #1599C2×C31semidirect
256. #1596C2×C31semidirect
257. #1508C2×C31semidirect
258. #1465C2×C31semidirect
335. C22×Q8D4×C31semidirect
363. C22⋊Q8D4×C31semidirect
377. C2×C4⋊C4D4×C31semidirect
398. C31#18167direct (N × H)
493. C23C31×#137semidirect
537. C2×C4C31×#137semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#30072
C31×#23882
C31×#16052
C31×#16022
C31×#15992
C31×#15962
C31×#15082
C31×#14652
#1816731