Finite Group Databasebeta

← C31×#18591 #21475350

Subgroups — C31×#18591 #21475350

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic
2C31×#3255C211yesmaximal, characteristic, complemented
3C31×#3165C211yesmaximal, characteristic, complemented
4C31×#3107C211yesmaximal, characteristic
5C31×#1653C212yesmaximal, complemented
6C31×#1651C212yesmaximal, complemented
7C31×#1649C212yesmaximal, complemented
8C31×#1648C212yesmaximal, complemented
9C31×#1546C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic
11C31×#1485C211yesmaximal, characteristic, complemented
12Q82×C31—21no—
13C31×#557C2212yes—
14C31×#554—21no—
15C31×#554—22no—
16D4×Q8×C31—22no—
17D4×Q8×C31C2212yes—
18C31×#544—22no—
19C31×#544—21no—
20C31×#533—21no—
21C31×#533C2211yescharacteristic
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#501—22no—
28C31×#500C2212yes—
29C31×#498C2212yes—
30C31×#485—21no—
31C31×#475—21no—
32C31×#475C2211yescharacteristic
33C31×#475C2212yes—
34C31×#474C2211yescharacteristic
35C31×#470—22no—
36C31×#462—21no—
37C31×#462C2211yescharacteristic
38C31×#462C2212yes—
39C31×#461C2211yescharacteristic
40C31×#452—21no—
41C4×Q16×C31—21no—
42C4×SD16×C31—21no—
43C31×#431C2211yescharacteristic
44C31×#428—22no—
45C31×#423C2211yescharacteristic
46C4×M16×C31C2211yescharacteristic
47C31×#332C2212yes—
48C31×#332C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×Q8⋊C8C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31D411yescharacteristic
54C22×Q8×C31—22no—
55C22×Q8×C31—21no—
56C2×Q16×C31—21no—
57C2×Q16×C31—22no—
58C2×SD16×C31—21no—
59C2×M16×C31—21no—
60C2×M16×C31—21no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—24no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D411yescharacteristic
70C31×C4⋊Q8D412yes—
71C31×C4⋊Q8C2311yescharacteristic
72C31×C4⋊Q8C2311yescharacteristic
73C31×C4⋊Q8C2312yes—
74C31×#125—22no—
75C31×#125—24no—
76C31×#125—22no—
77C31×#125—21no—
78C31×#125—21no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—24no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—24no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8D412yes—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31D411yescharacteristic
95C4×Q8×C31C2311yescharacteristic
96C4×Q8×C31C2312yes—
97C4×D4×C31—24no—
98C4×D4×C31—22no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31D411yescharacteristic
102C4×D4×C31C2311yescharacteristic
103C31×#117—22no—
104C31×#117—21no—
105C31×#117—21no—
106C31×#117—21no—
107C31×#117D411yescharacteristic
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D412yes—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—24no—
115C31×#107—21no—
116C31×#106—24no—
117C31×#106—21no—
118C31×#106—22no—
119C31×C4⋊C8—21no—
120C31×C4⋊C8C2312yes—
121C31×C4⋊C8C2311yescharacteristic
122C31×C4⋊C8C2311yescharacteristic
123C31×#103—21no—
124C31×#103—22no—
125C31×#103—21no—
126C31×#103—21no—
127C31×#103—22no—
128C31×#102—21no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31C2312yes—
133#41×C31—22no—
134#41×C31—22no—
135C2×Q8×C31—24no—
136C2×Q8×C31—21no—
137C2×Q8×C31—22no—
138C2×Q8×C31—22no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×Q8×C31—81no—
142C2×Q8×C31—81no—
143C2×Q8×C31—82no—
144C2×Q8×C31—82no—
145C2×Q8×C31C2×D412yes—
146C2×Q8×C31C2×D411yescharacteristic
147C2×D4×C31—21no—
148C2×D4×C31C2×D411yescharacteristic
149C22×C4×C31—21no—
150C22×C4×C31—24no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157Q16×C31—42no—
158Q16×C31—82no—
159SD16×C31—42no—
160M16×C31—24no—
161M16×C31—81no—
162C2×C8×C31—22no—
163C2×C8×C31—21no—
164C2×C8×C31—21no—
165C2×C8×C31—22no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31—41no—
186C4⋊C4×C31C2×D411yescharacteristic
187C4⋊C4×C31C2×D412yes—
188C4⋊C4×C31C2×D411yescharacteristic
189C4⋊C4×C31C2×D414yes—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—41no—
196C42×C31—24no—
197C42×C31—21no—
198C42×C31—21no—
199C42×C31—21no—
200C42×C31—22no—
201C42×C31—41no—
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205C42×C31C2411yescharacteristic
206#18591C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31#12011yescharacteristic
209Q8×C31—21no—
210Q8×C31—22no—
211Q8×C31—22no—
212Q8×C31—42no—
213Q8×C31—42no—
214Q8×C31—42no—
215Q8×C31—41no—
216Q8×C31—41no—
217Q8×C31—41no—
218Q8×C31—41no—
219Q8×C31—82no—
220Q8×C31—84no—
221Q8×C31#13712yes—
222D4×C31—24no—
223D4×C31—21no—
224D4×C31#13712yes—
225C2×C4×C31—24no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—24no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—82no—
243C2×C4×C31—84no—
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C8×C31—24no—
251C8×C31—41no—
252C8×C31—41no—
253C8×C31—42no—
254#3503C2×C3111yescharacteristic
255#3255C2×C3111yescharacteristic, complemented
256#3165C2×C3111yescharacteristic, complemented
257#3107C2×C3111yescharacteristic
258#1653C2×C3112yescomplemented
259#1651C2×C3112yescomplemented
260#1649C2×C3112yescomplemented
261#1648C2×C3112yescomplemented
262#1546C2×C3111yescharacteristic, complemented
263#1489C2×C3111yescharacteristic
264#1485C2×C3111yescharacteristic, complemented
265C22×C31—21no—
266C22×C31—22no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31C2×#12011yescharacteristic
272C4×C31—21no—
273C4×C31—22no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—41no—
277C4×C31—42no—
278C4×C31—44no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—81no—
282C4×C31—81no—
283C4×C31#45612yes—
284C4×C31C2×#13711yescharacteristic
285C4×C31C2×#13711yescharacteristic
286C4×C31#57611yescharacteristic
287C4×C31#57711yescharacteristic
288Q82—21no—
289#557C22×C3112yes—
290#554—21no—
291#554—22no—
292D4×Q8—22no—
293D4×Q8C22×C3112yes—
294#544—22no—
295#544—21no—
296#533—21no—
297#533C22×C3111yescharacteristic
298#533C22×C3111yescharacteristic
299C2×C4⋊Q8—21no—
300C2×C4⋊Q8C22×C3111yescharacteristic
301#520—21no—
302C4×#41C22×C3111yescharacteristic
303#501—22no—
304#500C22×C3112yes—
305#498C22×C3112yes—
306#485—21no—
307#475—21no—
308#475C22×C3111yescharacteristic
309#475C22×C3112yes—
310#474C22×C3111yescharacteristic
311#470—22no—
312#462—21no—
313#462C22×C3111yescharacteristic
314#462C22×C3112yes—
315#461C22×C3111yescharacteristic
316#452—21no—
317C4×Q16—21no—
318C4×SD16—21no—
319#431C22×C3111yescharacteristic
320#428—22no—
321#423C22×C3111yescharacteristic
322C4×M16C22×C3111yescharacteristic
323#332C22×C3112yes—
324#332C22×C3112yes—
325Q8⋊C8C22×C3112yes—
326Q8⋊C8C22×C3112yes—
327Q8⋊C8C22×C3112yes—
328D4⋊C8C22×C3112yes—
329C2×C31—41no—
330C2×C31—42no—
331C2×C31#300411yescharacteristic
332C2×C31#301211yescharacteristic
333C2×C31C2×#45611yescharacteristic
334C2×#41D4×C3111yescharacteristic
335C22×Q8—22no—
336C22×Q8—21no—
337C2×Q16—21no—
338C2×Q16—22no—
339C2×SD16—21no—
340C2×M16—21no—
341C2×M16—21no—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8—24no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8D4×C3111yescharacteristic
350C4⋊Q8D4×C3112yes—
351C4⋊Q8D4×C3111yescharacteristic
352C4⋊Q8C23×C3111yescharacteristic
353C4⋊Q8C23×C3111yescharacteristic
354C4⋊Q8C23×C3112yes—
355#125—22no—
356#125—24no—
357#125—22no—
358#125—21no—
359#125—21no—
360C22⋊Q8—22no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—24no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8D4×C3112yes—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8D4×C3111yescharacteristic
376C4×Q8C23×C3111yescharacteristic
377C4×Q8C23×C3112yes—
378C4×D4—24no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4D4×C3111yescharacteristic
383C4×D4C23×C3111yescharacteristic
384#117—22no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117D4×C3111yescharacteristic
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4D4×C3112yes—
393C2×C42—21no—
394C2×C42C23×C3111yescharacteristic
395#107—24no—
396#107—21no—
397#106—24no—
398#106—21no—
399#106—22no—
400C4⋊C8—21no—
401C4⋊C8C23×C3112yes—
402C4⋊C8C23×C3111yescharacteristic
403C4⋊C8C23×C3111yescharacteristic
404#103—21no—
405#103—22no—
406#103—21no—
407#103—21no—
408#103—22no—
409#102—21no—
410C22⋊C8—21no—
411#97—21no—
412C4×C8—21no—
413C4×C8C23×C3112yes—
414C31#1859111yescharacteristic, direct product factor, Sylow
415#41—22no—
416#41—22no—
417C2×Q8—24no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—81no—
424C2×Q8—81no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8C2×D4×C3112yes—
428C2×Q8C2×D4×C3111yescharacteristic
429C2×D4—21no—
430C2×D4C2×D4×C3111yescharacteristic
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439Q16—42no—
440Q16—82no—
441SD16—42no—
442M16—24no—
443M16—81no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×D4×C3111yescharacteristic
469C4⋊C4C2×D4×C3112yes—
470C4⋊C4C2×D4×C3111yescharacteristic
471C4⋊C4C2×D4×C3114yes—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—41no—
478C42—24no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—41no—
484C42C2×D4×C3111yescharacteristic
485C42C2×D4×C3111yescharacteristic
486C42C2×D4×C3111yescharacteristic
487C42C24×C3111yescharacteristic
488C23—21no—
489C23C31×#12011yescharacteristic
490Q8—21no—
491Q8—22no—
492Q8—22no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—82no—
501Q8—84no—
502Q8C31×#13712yes—
503D4—24no—
504D4—21no—
505D4C31×#13712yes—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#12011yescharacteristic
528C2×C4C22×D4×C3111yescharacteristic
529C2×C4C22×D4×C3111yescharacteristic
530C2×C4C22×D4×C3111yescharacteristic
531C8—24no—
532C8—41no—
533C8—41no—
534C8—42no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22C2×C31×#12011yescharacteristic
542C4—21no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—42no—
548C4—44no—
549C4—41no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4C31×#45612yes—
554C4C2×C31×#13711yescharacteristic
555C4C2×C31×#13711yescharacteristic
556C4C31×#57611yescharacteristic
557C4C31×#57711yescharacteristic
558C2—41no—
559C2—42no—
560C2C31×#300411yescharacteristic
561C2C31×#301211yescharacteristic
562C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×#41×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3255C2semidirect
3. C31×#3165C2semidirect
5. C31×#1653C2semidirect
6. C31×#1651C2semidirect
7. C31×#1649C2semidirect
8. C31×#1648C2semidirect
9. C31×#1546C2semidirect
11. C31×#1485C2semidirect
206. #18591C31direct (N × H)
255. #3255C2×C31semidirect
256. #3165C2×C31semidirect
258. #1653C2×C31semidirect
259. #1651C2×C31semidirect
260. #1649C2×C31semidirect
261. #1648C2×C31semidirect
262. #1546C2×C31semidirect
264. #1485C2×C31semidirect
414. C31#18591direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#32552
C31×#31652
C31×#31072
C31×#16532
C31×#16512
C31×#16492
C31×#16482
C31×#15462
C31×#14892
C31×#14852
#1859131