Finite Group Databasebeta

← C31×#18801 #21475560

Subgroups — C31×#18801 #21475560

Order 7936.

936 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C31C211yesmaximal, characteristic
2C31×#3230C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic
4C2×C31×#462C211yesmaximal, characteristic, complemented
5C31×#1649C212yesmaximal, complemented
6C31×#1648C212yesmaximal, complemented
7C31×#1645C212yesmaximal, complemented
8C31×#1644C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1487C211yesmaximal, characteristic, complemented
11C31×#1477C211yesmaximal, characteristic
12C22×Q16×C31—21no—
13Q82×C31C2211yescharacteristic
14C31×#557—21no—
15C31×#557C2211yescharacteristic
16C31×#554—22no—
17C31×#554—21no—
18C31×#554—21no—
19C31×#554C2211yescharacteristic
20D4×Q8×C31—21no—
21D4×Q8×C31C2211yescharacteristic
22C31×#533—22no—
23C31×#533—21no—
24C31×#533—21no—
25C31×#533C2211yescharacteristic
26C2×C31×C4⋊Q8C2211yescharacteristic
27C4×#41×C31—21no—
28C4×#41×C31C2211yescharacteristic
29C2×C4×Q8×C31—21no—
30C2×C4×Q8×C31C2211yescharacteristic
31C31×#500C2212yes—
32C31×#487C2212yes—
33C31×#484—22no—
34C31×#477—21no—
35C31×#477—22no—
36C31×#476—21no—
37C31×#475C2212yes—
38C31×#474C2212yes—
39C31×#462—21no—
40C31×#462—21no—
41C31×#462C2212yes—
42C31×#462C2212yes—
43C31×#431C2211yescharacteristic
44C31×#428—21no—
45C31×#423C2211yescharacteristic
46C2×C31×C4⋊C8C2211yescharacteristic
47C31×#417—21no—
48C2×C31×#103—21no—
49C31×#333C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×Q8⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31D411yescharacteristic
56C22×Q8×C31—21no—
57C22×Q8×C31—21no—
58C22×Q8×C31D411yescharacteristic
59C2×Q16×C31—22no—
60C2×Q16×C31—22no—
61C2×Q16×C31—44no—
62C2×M16×C31—21no—
63C22×C8×C31—21no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8C2312yes—
74C31×C4⋊Q8C2311yescharacteristic
75C31×C4⋊Q8C2311yescharacteristic
76C31×#125—22no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125—21no—
80C31×#125—21no—
81C31×#125—21no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8D412yes—
93C4×Q8×C31—22no—
94C4×Q8×C31—22no—
95C4×Q8×C31—22no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31D411yescharacteristic
104C4×Q8×C31D411yescharacteristic
105C4×Q8×C31D411yescharacteristic
106C4×Q8×C31D412yes—
107C4×Q8×C31C2311yescharacteristic
108C4×Q8×C31C2311yescharacteristic
109C4×Q8×C31C2311yescharacteristic
110C4×D4×C31—22no—
111C4×D4×C31—22no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31D411yescharacteristic
117C4×D4×C31C2311yescharacteristic
118C31×#117—22no—
119C31×#117—21no—
120C31×#117—21no—
121C31×#117D411yescharacteristic
122C2×C4⋊C4×C31—22no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31D411yescharacteristic
128C2×C42×C31—22no—
129C2×C42×C31—21no—
130C2×C42×C31—21no—
131C2×C42×C31C2311yescharacteristic
132C31×#107—22no—
133C31×#107—24no—
134C31×#106—22no—
135C31×C4⋊C8—21no—
136C31×C4⋊C8—21no—
137C31×C4⋊C8—21no—
138C31×C4⋊C8C2312yes—
139C31×C4⋊C8C2312yes—
140C31×#103—22no—
141C31×#103—24no—
142C31×#103—22no—
143C31×#103—22no—
144C31×#102—22no—
145C31×C22⋊C8—21no—
146C4×C8×C31C2312yes—
147#41×C31—22no—
148#41×C31Q1614yescomplemented
149C2×Q8×C31—22no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—22no—
154C2×Q8×C31—22no—
155C2×Q8×C31—41no—
156C2×Q8×C31—42no—
157C2×Q8×C31—42no—
158C2×Q8×C31—42no—
159C2×Q8×C31—42no—
160C2×Q8×C31—41no—
161C2×Q8×C31C2×D411yescharacteristic
162C2×Q8×C31C2×D411yescharacteristic
163C2×Q8×C31C2×D411yescharacteristic
164C2×Q8×C31C2×D412yes—
165C2×D4×C31—21no—
166C2×D4×C31C2×D411yescharacteristic
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—22no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—41no—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178Q16×C31—44no—
179Q16×C31—44no—
180M16×C31—42no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×C31—42no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—41no—
208C4⋊C4×C31—41no—
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D411yescharacteristic
212C4⋊C4×C31C2×D411yescharacteristic
213C4⋊C4×C31C2×D412yes—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C42×C31—22no—
221C42×C31—22no—
222C42×C31—21no—
223C42×C31—22no—
224C42×C31—21no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31—21no—
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233C42×C31C2411yescharacteristic
234#18801C3111yesmaximal, characteristic, direct product factor, Sylow
235C23×C31—21no—
236C23×C31#12011yescharacteristic
237Q8×C31—21no—
238Q8×C31—21no—
239Q8×C31—21no—
240Q8×C31—21no—
241Q8×C31—42no—
242Q8×C31—42no—
243Q8×C31—44no—
244Q8×C31—41no—
245Q8×C31—41no—
246Q8×C31—42no—
247Q8×C31—42no—
248Q8×C31—82no—
249Q8×C31—82no—
250Q8×C31C2×Q1612yescomplemented
251Q8×C31AGL(1,ℤ/8)12yes—
252D4×C31—24no—
253D4×C31—21no—
254D4×C31C2×Q1612yescomplemented
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—24no—
260C2×C4×C31—21no—
261C2×C4×C31—21no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—44no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—42no—
273C2×C4×C31—81no—
274C2×C4×C31—81no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31C2×Q1612yes—
281C2×C4×C31#13712yes—
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C8×C31—42no—
286C8×C31—42no—
287C8×C31—42no—
288Q8×#41C2×C3111yescharacteristic
289#3230C2×C3111yescharacteristic, complemented
290#3079C2×C3111yescharacteristic
291C2×#462C2×C3111yescharacteristic, complemented
292#1649C2×C3112yescomplemented
293#1648C2×C3112yescomplemented
294#1645C2×C3112yescomplemented
295#1644C2×C3112yescomplemented
296#1550C2×C3111yescharacteristic, complemented
297#1487C2×C3111yescharacteristic, complemented
298#1477C2×C3111yescharacteristic
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—22no—
302C22×C31—41no—
303C22×C31—41no—
304C22×C31#45112yes—
305C22×C31C2×#12011yescharacteristic
306C4×C31—22no—
307C4×C31—21no—
308C4×C31—22no—
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—41no—
312C4×C31—42no—
313C4×C31—41no—
314C4×C31—41no—
315C4×C31—42no—
316C4×C31—41no—
317C4×C31—41no—
318C4×C31—82no—
319C4×C31#45112yes—
320C4×C31C22×Q1611yescharacteristic
321C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
322C4×C31C2×#13711yescharacteristic
323C4×C31#57511yescharacteristic
324C22×Q16—21no—
325Q82C22×C3111yescharacteristic
326#557—21no—
327#557C22×C3111yescharacteristic
328#554—22no—
329#554—21no—
330#554—21no—
331#554C22×C3111yescharacteristic
332D4×Q8—21no—
333D4×Q8C22×C3111yescharacteristic
334#533—22no—
335#533—21no—
336#533—21no—
337#533C22×C3111yescharacteristic
338C2×C4⋊Q8C22×C3111yescharacteristic
339C4×#41—21no—
340C4×#41C22×C3111yescharacteristic
341C2×C4×Q8—21no—
342C2×C4×Q8C22×C3111yescharacteristic
343#500C22×C3112yes—
344#487C22×C3112yes—
345#484—22no—
346#477—21no—
347#477—22no—
348#476—21no—
349#475C22×C3112yes—
350#474C22×C3112yes—
351#462—21no—
352#462—21no—
353#462C22×C3112yes—
354#462C22×C3112yes—
355#431C22×C3111yescharacteristic
356#428—21no—
357#423C22×C3111yescharacteristic
358C2×C4⋊C8C22×C3111yescharacteristic
359#417—21no—
360C2×#103—21no—
361#333C22×C3112yes—
362#332C22×C3112yes—
363Q8⋊C8C22×C3112yes—
364Q8⋊C8C22×C3112yes—
365Q8⋊C8C22×C3112yes—
366D4⋊C8C22×C3112yes—
367C2×C31—22no—
368C2×C31—42no—
369C2×C31C2×#45111yescharacteristic
370C2×C31#300611yescharacteristic
371C2×C31#302211yescharacteristic
372C2×#41D4×C3111yescharacteristic
373C22×Q8—21no—
374C22×Q8—21no—
375C22×Q8D4×C3111yescharacteristic
376C2×Q16—22no—
377C2×Q16—22no—
378C2×Q16—44no—
379C2×M16—21no—
380C22×C8—21no—
381C4⋊Q8—22no—
382C4⋊Q8—21no—
383C4⋊Q8—22no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C23×C3112yes—
391C4⋊Q8C23×C3111yescharacteristic
392C4⋊Q8C23×C3111yescharacteristic
393#125—22no—
394#125—22no—
395#125—22no—
396#125—21no—
397#125—21no—
398#125—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—22no—
409C22⋊Q8D4×C3112yes—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8D4×C3111yescharacteristic
421C4×Q8D4×C3111yescharacteristic
422C4×Q8D4×C3111yescharacteristic
423C4×Q8D4×C3112yes—
424C4×Q8C23×C3111yescharacteristic
425C4×Q8C23×C3111yescharacteristic
426C4×Q8C23×C3111yescharacteristic
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4D4×C3111yescharacteristic
434C4×D4C23×C3111yescharacteristic
435#117—22no—
436#117—21no—
437#117—21no—
438#117D4×C3111yescharacteristic
439C2×C4⋊C4—22no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4D4×C3111yescharacteristic
445C2×C42—22no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42C23×C3111yescharacteristic
449#107—22no—
450#107—24no—
451#106—22no—
452C4⋊C8—21no—
453C4⋊C8—21no—
454C4⋊C8—21no—
455C4⋊C8C23×C3112yes—
456C4⋊C8C23×C3112yes—
457#103—22no—
458#103—24no—
459#103—22no—
460#103—22no—
461#102—22no—
462C22⋊C8—21no—
463C4×C8C23×C3112yes—
464C31#1880111yescharacteristic, direct product factor, Sylow
465#41—22no—
466#41Q16×C3114yescomplemented
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—41no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—41no—
479C2×Q8C2×D4×C3111yescharacteristic
480C2×Q8C2×D4×C3111yescharacteristic
481C2×Q8C2×D4×C3111yescharacteristic
482C2×Q8C2×D4×C3112yes—
483C2×D4—21no—
484C2×D4C2×D4×C3111yescharacteristic
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496Q16—44no—
497Q16—44no—
498M16—42no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C4⋊C4C2×D4×C3111yescharacteristic
530C4⋊C4C2×D4×C3111yescharacteristic
531C4⋊C4C2×D4×C3112yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42C2×D4×C3111yescharacteristic
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C24×C3111yescharacteristic
552C23—21no—
553C23C31×#12011yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—21no—
558Q8—42no—
559Q8—42no—
560Q8—44no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—82no—
566Q8—82no—
567Q8C2×Q16×C3112yescomplemented
568Q8C31×AGL(1,ℤ/8)12yes—
569D4—24no—
570D4—21no—
571D4C2×Q16×C3112yescomplemented
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4C31×#12011yescharacteristic
595C2×C4C31×#12011yescharacteristic
596C2×C4C31×#12011yescharacteristic
597C2×C4C2×Q16×C3112yes—
598C2×C4C31×#13712yes—
599C2×C4C22×D4×C3111yescharacteristic
600C2×C4C22×D4×C3111yescharacteristic
601C2×C4C22×D4×C3111yescharacteristic
602C8—42no—
603C8—42no—
604C8—42no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22C31×#45112yes—
611C22C2×C31×#12011yescharacteristic
612C4—22no—
613C4—21no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—82no—
625C4C31×#45112yes—
626C4C22×Q16×C3111yescharacteristic
627C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
628C4C2×C31×#13711yescharacteristic
629C4C31×#57511yescharacteristic
630C2—22no—
631C2—42no—
632C2C2×C31×#45111yescharacteristic
633C2C31×#300611yescharacteristic
634C2C31×#302211yescharacteristic
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×#3230C2semidirect
4. C2×C31×#462C2semidirect
5. C31×#1649C2semidirect
6. C31×#1648C2semidirect
7. C31×#1645C2semidirect
8. C31×#1644C2semidirect
9. C31×#1550C2semidirect
10. C31×#1487C2semidirect
148. #41×C31Q16semidirect
234. #18801C31direct (N × H)
250. Q8×C31C2×Q16semidirect
254. D4×C31C2×Q16semidirect
289. #3230C2×C31semidirect
291. C2×#462C2×C31semidirect
292. #1649C2×C31semidirect
293. #1648C2×C31semidirect
294. #1645C2×C31semidirect
295. #1644C2×C31semidirect
296. #1550C2×C31semidirect
297. #1487C2×C31semidirect
464. C31#18801direct (N × H)
466. #41Q16×C31semidirect
567. Q8C2×Q16×C31semidirect
571. D4C2×Q16×C31semidirect

Maximal subgroups

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

subgroupindex
Q8×#41×C312
C31×#32302
C31×#30792
C2×C31×#4622
C31×#16492
C31×#16482
C31×#16452
C31×#16442
C31×#15502
C31×#14872
C31×#14772
#1880131