Finite Group Databasebeta

← C31×#18789 #21475548

Subgroups — C31×#18789 #21475548

Order 7936.

916 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C31C211yesmaximal, characteristic
2C31×#3209C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic
4C2×C31×#477C211yesmaximal, characteristic, complemented
5C31×#1637C212yesmaximal, complemented
6C31×#1636C212yesmaximal, complemented
7C31×#1633C212yesmaximal, complemented
8C31×#1632C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1487C211yesmaximal, characteristic, complemented
11C31×#1477C211yesmaximal, characteristic
12Q82×C31C2211yescharacteristic
13C31×#557—21no—
14C31×#557C2211yescharacteristic
15C31×#554—22no—
16C31×#554—21no—
17C31×#554—21no—
18C31×#554C2211yescharacteristic
19D4×Q8×C31—21no—
20D4×Q8×C31C2211yescharacteristic
21C31×#533—22no—
22C31×#533—21no—
23C31×#533—21no—
24C31×#533C2211yescharacteristic
25C2×C31×C4⋊Q8C2211yescharacteristic
26C4×#41×C31—21no—
27C4×#41×C31C2211yescharacteristic
28C2×C4×Q8×C31—21no—
29C2×C4×Q8×C31C2211yescharacteristic
30C31×#494C2212yes—
31C31×#487C2212yes—
32C31×#477—21no—
33C31×#477—21no—
34C31×#477C2212yes—
35C31×#477C2212yes—
36C31×#477C2212yes—
37C31×#476C2212yes—
38C31×#475—21no—
39C31×#474—21no—
40C31×#462—22no—
41C31×#451—22no—
42C31×#431C2211yescharacteristic
43C31×#428—21no—
44C2×C31×#107—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—24no—
60C2×M16×C31—21no—
61C22×C8×C31—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—21no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8C2312yes—
72C31×C4⋊Q8C2311yescharacteristic
73C31×C4⋊Q8C2311yescharacteristic
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—22no—
77C31×#125—21no—
78C31×#125—21no—
79C31×#125—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8D412yes—
91C4×Q8×C31—22no—
92C4×Q8×C31—22no—
93C4×Q8×C31—22no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31—21no—
101C4×Q8×C31D412yes—
102C4×Q8×C31D411yescharacteristic
103C4×Q8×C31D411yescharacteristic
104C4×Q8×C31D411yescharacteristic
105C4×Q8×C31C2311yescharacteristic
106C4×Q8×C31C2311yescharacteristic
107C4×Q8×C31C2311yescharacteristic
108C4×D4×C31—22no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31D411yescharacteristic
115C4×D4×C31C2311yescharacteristic
116C31×#117—22no—
117C31×#117—21no—
118C31×#117—21no—
119C31×#117D411yescharacteristic
120C2×C4⋊C4×C31—22no—
121C2×C4⋊C4×C31—21no—
122C2×C4⋊C4×C31—21no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31D411yescharacteristic
126C2×C42×C31—22no—
127C2×C42×C31—21no—
128C2×C42×C31—21no—
129C2×C42×C31C2311yescharacteristic
130C31×#107—22no—
131C31×#107—22no—
132C31×#107—22no—
133C31×#106—22no—
134C31×C4⋊C8—21no—
135C31×C4⋊C8—21no—
136C31×C4⋊C8—21no—
137C31×C4⋊C8C2312yes—
138C31×C4⋊C8C2312yes—
139C31×#103—22no—
140C31×#103—22no—
141C31×#103—22no—
142C31×#103—24no—
143C31×#102—22no—
144C31×C22⋊C8—21no—
145C4×C8×C31C2312yes—
146#41×C31—22no—
147#41×C31Q1614yescomplemented
148C2×Q8×C31—22no—
149C2×Q8×C31—22no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—22no—
153C2×Q8×C31—22no—
154C2×Q8×C31—41no—
155C2×Q8×C31—42no—
156C2×Q8×C31—42no—
157C2×Q8×C31—41no—
158C2×Q8×C31—82no—
159C2×Q8×C31C2×D411yescharacteristic
160C2×Q8×C31C2×D411yescharacteristic
161C2×Q8×C31C2×D411yescharacteristic
162C2×Q8×C31C2×D412yes—
163C2×D4×C31—21no—
164C2×D4×C31C2×D411yescharacteristic
165C22×C4×C31—21no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—41no—
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176Q16×C31—84no—
177M16×C31—42no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31—42no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—41no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31C2×D412yes—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D411yescharacteristic
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—21no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2411yescharacteristic
231#18789C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31#12011yescharacteristic
234Q8×C31—21no—
235Q8×C31—21no—
236Q8×C31—21no—
237Q8×C31—21no—
238Q8×C31—42no—
239Q8×C31—42no—
240Q8×C31—44no—
241Q8×C31—44no—
242Q8×C31—41no—
243Q8×C31—41no—
244Q8×C31—82no—
245Q8×C31—82no—
246Q8×C31C2×Q1612yes—
247Q8×C31#13712yes—
248D4×C31—24no—
249D4×C31—21no—
250D4×C31C2×Q1612yes—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—24no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—22no—
259C2×C4×C31—22no—
260C2×C4×C31—21no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—44no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—81no—
270C2×C4×C31—81no—
271C2×C4×C31—82no—
272C2×C4×C31—82no—
273C2×C4×C31#12011yescharacteristic
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31C2×Q1612yes—
277C2×C4×C31#13712yes—
278C2×C4×C31C22×D411yescharacteristic
279C2×C4×C31C22×D411yescharacteristic
280C2×C4×C31C22×D411yescharacteristic
281C8×C31—42no—
282C8×C31—42no—
283C8×C31—42no—
284Q8×#41C2×C3111yescharacteristic
285#3209C2×C3111yescharacteristic, complemented
286#3079C2×C3111yescharacteristic
287C2×#477C2×C3111yescharacteristic, complemented
288#1637C2×C3112yescomplemented
289#1636C2×C3112yescomplemented
290#1633C2×C3112yescomplemented
291#1632C2×C3112yescomplemented
292#1550C2×C3111yescharacteristic, complemented
293#1487C2×C3111yescharacteristic, complemented
294#1477C2×C3111yescharacteristic
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—22no—
298C22×C31—41no—
299C22×C31—41no—
300C22×C31#45112yes—
301C22×C31C2×#12011yescharacteristic
302C4×C31—22no—
303C4×C31—21no—
304C4×C31—22no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—41no—
308C4×C31—42no—
309C4×C31—41no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—41no—
313C4×C31—41no—
314C4×C31—82no—
315C4×C31#45112yes—
316C4×C31C22×Q1611yescharacteristic
317C4×C31C2×#13711yescharacteristic
318C4×C31C2×#13711yescharacteristic
319C4×C31#57511yescharacteristic
320Q82C22×C3111yescharacteristic
321#557—21no—
322#557C22×C3111yescharacteristic
323#554—22no—
324#554—21no—
325#554—21no—
326#554C22×C3111yescharacteristic
327D4×Q8—21no—
328D4×Q8C22×C3111yescharacteristic
329#533—22no—
330#533—21no—
331#533—21no—
332#533C22×C3111yescharacteristic
333C2×C4⋊Q8C22×C3111yescharacteristic
334C4×#41—21no—
335C4×#41C22×C3111yescharacteristic
336C2×C4×Q8—21no—
337C2×C4×Q8C22×C3111yescharacteristic
338#494C22×C3112yes—
339#487C22×C3112yes—
340#477—21no—
341#477—21no—
342#477C22×C3112yes—
343#477C22×C3112yes—
344#477C22×C3112yes—
345#476C22×C3112yes—
346#475—21no—
347#474—21no—
348#462—22no—
349#451—22no—
350#431C22×C3111yescharacteristic
351#428—21no—
352C2×#107—21no—
353#423C22×C3111yescharacteristic
354C2×C4⋊C8C22×C3111yescharacteristic
355#417—21no—
356C2×#103—21no—
357#333C22×C3112yes—
358#332C22×C3112yes—
359Q8⋊C8C22×C3112yes—
360Q8⋊C8C22×C3112yes—
361Q8⋊C8C22×C3112yes—
362D4⋊C8C22×C3112yes—
363C2×C31—22no—
364C2×C31—42no—
365C2×C31C2×#45111yescharacteristic
366C2×C31#300711yescharacteristic
367C2×C31#302111yescharacteristic
368C2×#41D4×C3111yescharacteristic
369C22×Q8—21no—
370C22×Q8—21no—
371C22×Q8D4×C3111yescharacteristic
372C2×Q16—24no—
373C2×M16—21no—
374C22×C8—21no—
375C4⋊Q8—22no—
376C4⋊Q8—21no—
377C4⋊Q8—22no—
378C4⋊Q8—21no—
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—21no—
383C4⋊Q8—21no—
384C4⋊Q8C23×C3112yes—
385C4⋊Q8C23×C3111yescharacteristic
386C4⋊Q8C23×C3111yescharacteristic
387#125—22no—
388#125—22no—
389#125—22no—
390#125—21no—
391#125—21no—
392#125—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8D4×C3112yes—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C3111yescharacteristic
415C4×Q8D4×C3111yescharacteristic
416C4×Q8D4×C3111yescharacteristic
417C4×Q8D4×C3112yes—
418C4×Q8C23×C3111yescharacteristic
419C4×Q8C23×C3111yescharacteristic
420C4×Q8C23×C3111yescharacteristic
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4D4×C3111yescharacteristic
428C4×D4C23×C3111yescharacteristic
429#117—22no—
430#117—21no—
431#117—21no—
432#117D4×C3111yescharacteristic
433C2×C4⋊C4—22no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4D4×C3111yescharacteristic
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—21no—
442C2×C42C23×C3111yescharacteristic
443#107—22no—
444#107—22no—
445#107—22no—
446#106—22no—
447C4⋊C8—21no—
448C4⋊C8—21no—
449C4⋊C8—21no—
450C4⋊C8C23×C3112yes—
451C4⋊C8C23×C3112yes—
452#103—22no—
453#103—22no—
454#103—22no—
455#103—24no—
456#102—22no—
457C22⋊C8—21no—
458C4×C8C23×C3112yes—
459C31#1878911yescharacteristic, direct product factor, Sylow
460#41—22no—
461#41Q16×C3114yescomplemented
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—41no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—41no—
472C2×Q8—82no—
473C2×Q8C2×D4×C3111yescharacteristic
474C2×Q8C2×D4×C3111yescharacteristic
475C2×Q8C2×D4×C3111yescharacteristic
476C2×Q8C2×D4×C3112yes—
477C2×D4—21no—
478C2×D4C2×D4×C3111yescharacteristic
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4C2×D4×C3111yescharacteristic
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3111yescharacteristic
490Q16—84no—
491M16—42no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—42no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C3112yes—
521C4⋊C4C2×D4×C3111yescharacteristic
522C4⋊C4C2×D4×C3111yescharacteristic
523C4⋊C4C2×D4×C3111yescharacteristic
524C4⋊C4C2×D4×C3111yescharacteristic
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42C2×D4×C3111yescharacteristic
542C42C2×D4×C3111yescharacteristic
543C42C2×D4×C3111yescharacteristic
544C42C24×C3111yescharacteristic
545C23—21no—
546C23C31×#12011yescharacteristic
547Q8—21no—
548Q8—21no—
549Q8—21no—
550Q8—21no—
551Q8—42no—
552Q8—42no—
553Q8—44no—
554Q8—44no—
555Q8—41no—
556Q8—41no—
557Q8—82no—
558Q8—82no—
559Q8C2×Q16×C3112yes—
560Q8C31×#13712yes—
561D4—24no—
562D4—21no—
563D4C2×Q16×C3112yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4C31×#12011yescharacteristic
587C2×C4C31×#12011yescharacteristic
588C2×C4C31×#12011yescharacteristic
589C2×C4C2×Q16×C3112yes—
590C2×C4C31×#13712yes—
591C2×C4C22×D4×C3111yescharacteristic
592C2×C4C22×D4×C3111yescharacteristic
593C2×C4C22×D4×C3111yescharacteristic
594C8—42no—
595C8—42no—
596C8—42no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22C31×#45112yes—
603C22C2×C31×#12011yescharacteristic
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—21no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—82no—
617C4C31×#45112yes—
618C4C22×Q16×C3111yescharacteristic
619C4C2×C31×#13711yescharacteristic
620C4C2×C31×#13711yescharacteristic
621C4C31×#57511yescharacteristic
622C2—22no—
623C2—42no—
624C2C2×C31×#45111yescharacteristic
625C2C31×#300711yescharacteristic
626C2C31×#302111yescharacteristic
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×#3209C2semidirect
4. C2×C31×#477C2semidirect
5. C31×#1637C2semidirect
6. C31×#1636C2semidirect
7. C31×#1633C2semidirect
8. C31×#1632C2semidirect
9. C31×#1550C2semidirect
10. C31×#1487C2semidirect
147. #41×C31Q16semidirect
231. #18789C31direct (N × H)
285. #3209C2×C31semidirect
287. C2×#477C2×C31semidirect
288. #1637C2×C31semidirect
289. #1636C2×C31semidirect
290. #1633C2×C31semidirect
291. #1632C2×C31semidirect
292. #1550C2×C31semidirect
293. #1487C2×C31semidirect
459. C31#18789direct (N × H)
461. #41Q16×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×#32092
C31×#30792
C2×C31×#4772
C31×#16372
C31×#16362
C31×#16332
C31×#16322
C31×#15502
C31×#14872
C31×#14772
#1878931