Finite Group Databasebeta

← C31×#23642 #21480401

Subgroups — C31×#23642 #21480401

Order 7936.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#428C211yesmaximal, characteristic, complemented
2C2×C31×#407C211yesmaximal, characteristic
3C31×#2280C211yesmaximal, characteristic
4C2×C31×#380C211yesmaximal, characteristic
5C31×#2077C212yesmaximal, complemented
6C31×#2077C212yesmaximal, complemented
7C31×#2076C212yesmaximal, complemented
8C31×#2075C212yesmaximal, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C31×#1736C211yesmaximal, characteristic, complemented
11C31×#1736C211yesmaximal, characteristic, complemented
12C24×C4×C31C2211yescharacteristic
13C22×M16×C31C2211yescharacteristic
14C22×M16×C31C2211yescharacteristic
15C2×C31×#117C2211yescharacteristic
16C2×C31×#117C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31—21no—
20C31×#428—24no—
21C2×C31×#107C2212yescomplemented
22C2×C31×#106C2212yescomplemented
23C31×#407—24no—
24C2×C31×C22⋊C8C2212yes—
25C2×C31×C22⋊C8C2212yes—
26C31×#381—21no—
27C31×#380—21no—
28C31×#380—21no—
29C31×#380C2211yescharacteristic
30C31×#380C2212yes—
31C31×#380C2211yescharacteristic
32C31×#380C2212yes—
33C31×#380C2211yescharacteristic
34C31×#380C2211yescharacteristic
35C31×#379—21no—
36C4×C22⋊C4×C31—21no—
37C4×C22⋊C4×C31—21no—
38C2×C31×#95—21no—
39C31×#340C2212yescomplemented
40C31×#340C2212yescomplemented
41C31×#340C2212yescomplemented
42C31×#340C2212yescomplemented
43C31×#340C2212yescomplemented
44C31×#340C2212yescomplemented
45C25×C31D411yescharacteristic
46C23×C4×C31—22no—
47C23×C4×C31—22no—
48C23×C4×C31—22no—
49C23×C4×C31—22no—
50C23×C4×C31—22no—
51C23×C4×C31—22no—
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C23×C4×C31D411yescharacteristic
55C23×C4×C31D411yescharacteristic
56C23×C4×C31D411yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×M16×C31—22no—
61C2×M16×C31—22no—
62C2×M16×C31—22no—
63C2×M16×C31—22no—
64C2×M16×C31—22no—
65C2×M16×C31D414yes—
66C22×C8×C31C2312yes—
67C22×C8×C31C2312yes—
68C31×#117—22no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117—22no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—44no—
93C2×C42×C31—21no—
94C2×C42×C31—21no—
95C31×#107—24no—
96C31×#106—24no—
97C31×C22⋊C8—24no—
98C31×C22⋊C8—24no—
99C31×#95—22no—
100C31×#95—22no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—21no—
105C24×C31—22no—
106C24×C31—22no—
107C24×C31—22no—
108C24×C31—22no—
109C24×C31—22no—
110C24×C31—41no—
111C24×C31—41no—
112C24×C31C2×D411yescharacteristic
113C24×C31C2×D411yescharacteristic
114C24×C31C2×D411yescharacteristic
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—44no—
143C22×C4×C31—44no—
144C22×C4×C31—44no—
145C22×C4×C31—44no—
146C22×C4×C31—42no—
147C22×C4×C31—42no—
148C22×C4×C31—42no—
149C22×C4×C31—42no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D412yes—
156C22×C4×C31#4112yes—
157C22×C4×C31#4112yes—
158C22×C4×C31#4112yes—
159C22×C4×C31#4112yes—
160C22×C4×C31#4112yes—
161C22×C4×C31C2411yescharacteristic
162M16×C31—28no—
163M16×C31—44no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31C2×D414yes—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
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—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31—42no—
184C22⋊C4×C31—42no—
185C22⋊C4×C31—42no—
186C22⋊C4×C31—44no—
187C22⋊C4×C31—42no—
188C22⋊C4×C31—42no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—44no—
191C22⋊C4×C31—42no—
192C42×C31—42no—
193C42×C31—42no—
194#23642C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—28no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—21no—
206C23×C31—21no—
207C23×C31—21no—
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—44no—
213C23×C31—44no—
214C23×C31—44no—
215C23×C31—44no—
216C23×C31—44no—
217C23×C31—44no—
218C23×C31#12112yes—
219C23×C31#12312yes—
220C23×C31#12312yes—
221C23×C31AGL(1,ℤ/8)12yes—
222C23×C31#13712yes—
223C23×C31C22×D411yescharacteristic
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—44no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—44no—
240C2×C4×C31—44no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—84no—
260C2×C4×C31—84no—
261C2×C4×C31#12112yes—
262C2×C4×C31#12314yes—
263C2×C4×C31#12312yes—
264C2×C4×C31#12312yes—
265C2×C4×C31#12414yes—
266C2×C4×C31#12614yes—
267C2×C4×C31#12614yes—
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C2×#4111yescharacteristic
270C2×C4×C31C2×#4111yescharacteristic
271C2×C4×C31C2×#4111yescharacteristic
272C2×C4×C31C2×#4111yescharacteristic
273C2×C4×C31C2×#4111yescharacteristic
274C8×C31—24no—
275C8×C31—42no—
276C2×#428C2×C3111yescharacteristic, complemented
277C2×#407C2×C3111yescharacteristic
278#2280C2×C3111yescharacteristic
279C2×#380C2×C3111yescharacteristic
280#2077C2×C3112yescomplemented
281#2077C2×C3112yescomplemented
282#2076C2×C3112yescomplemented
283#2075C2×C3112yescomplemented
284#1736C2×C3111yescharacteristic, complemented
285#1736C2×C3111yescharacteristic, complemented
286#1736C2×C3111yescharacteristic, complemented
287C22×C31—22no—
288C22×C31—22no—
289C22×C31—22no—
290C22×C31—22no—
291C22×C31—22no—
292C22×C31—22no—
293C22×C31—22no—
294C22×C31—22no—
295C22×C31—22no—
296C22×C31—22no—
297C22×C31—22no—
298C22×C31—22no—
299C22×C31—416no—
300C22×C31—41no—
301C22×C31—41no—
302C22×C31—41no—
303C22×C31—41no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31#39714yes—
309C22×C31C2×#12111yescharacteristic
310C22×C31C2×#12311yescharacteristic
311C22×C31C2×#12311yescharacteristic
312C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C31C2×#13711yescharacteristic
314C22×C31#57511yescharacteristic
315C22×C31#57511yescharacteristic
316C4×C31—41no—
317C4×C31—41no—
318C4×C31—81no—
319C4×C31—81no—
320C4×C31—82no—
321C4×C31—81no—
322C4×C31—82no—
323C4×C31—81no—
324C4×C31#39714yes—
325C4×C31C2×#12311yescharacteristic
326C4×C31C2×#12411yescharacteristic
327C4×C31C2×#12611yescharacteristic
328C4×C31C2×#12611yescharacteristic
329C24×C4C22×C3111yescharacteristic
330C22×M16C22×C3111yescharacteristic
331C22×M16C22×C3111yescharacteristic
332C2×#117C22×C3111yescharacteristic
333C2×#117C22×C3111yescharacteristic
334C22×C4⋊C4C22×C3111yescharacteristic
335C22×C4⋊C4C22×C3111yescharacteristic
336C22×C22⋊C4—21no—
337#428—24no—
338C2×#107C22×C3112yescomplemented
339C2×#106C22×C3112yescomplemented
340#407—24no—
341C2×C22⋊C8C22×C3112yes—
342C2×C22⋊C8C22×C3112yes—
343#381—21no—
344#380—21no—
345#380—21no—
346#380C22×C3111yescharacteristic
347#380C22×C3112yes—
348#380C22×C3111yescharacteristic
349#380C22×C3112yes—
350#380C22×C3111yescharacteristic
351#380C22×C3111yescharacteristic
352#379—21no—
353C4×C22⋊C4—21no—
354C4×C22⋊C4—21no—
355C2×#95—21no—
356#340C22×C3112yescomplemented
357#340C22×C3112yescomplemented
358#340C22×C3112yescomplemented
359#340C22×C3112yescomplemented
360#340C22×C3112yescomplemented
361#340C22×C3112yescomplemented
362C2×C31—24no—
363C2×C31—42no—
364C2×C31—42no—
365C2×C31C2×#39711yescharacteristic
366C2×C31#305511yescharacteristic
367C2×C31#305611yescharacteristic
368C2×C31#309211yescharacteristic
369C2×C31#309311yescharacteristic
370C2×C31#309411yescharacteristic
371C2×C31#309511yescharacteristic
372C25D4×C3111yescharacteristic
373C23×C4—22no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—21no—
380C23×C4—21no—
381C23×C4D4×C3111yescharacteristic
382C23×C4D4×C3111yescharacteristic
383C23×C4D4×C3111yescharacteristic
384C23×C4C23×C3111yescharacteristic
385C23×C4C23×C3111yescharacteristic
386C23×C4C23×C3111yescharacteristic
387C2×M16—22no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16D4×C3114yes—
393C22×C8C23×C3112yes—
394C22×C8C23×C3112yes—
395#117—22no—
396#117—22no—
397#117—22no—
398#117—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4C23×C3112yes—
406C2×C4⋊C4C23×C3112yes—
407C2×C4⋊C4C23×C3111yescharacteristic
408C2×C4⋊C4C23×C3111yescharacteristic
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C4⋊C4C23×C3111yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—44no—
420C2×C42—21no—
421C2×C42—21no—
422#107—24no—
423#106—24no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426#95—22no—
427#95—22no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432C31#2364211yescharacteristic, direct product factor, Sylow
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—41no—
439C24—41no—
440C24C2×D4×C3111yescharacteristic
441C24C2×D4×C3111yescharacteristic
442C24C2×D4×C3111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3111yescharacteristic
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3112yes—
484C22×C4#41×C3112yes—
485C22×C4#41×C3112yes—
486C22×C4#41×C3112yes—
487C22×C4#41×C3112yes—
488C22×C4#41×C3112yes—
489C22×C4C24×C3111yescharacteristic
490M16—28no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8C2×D4×C3114yes—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C42—42no—
521C42—42no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23C31×#12112yes—
546C23C31×#12312yes—
547C23C31×#12312yes—
548C23C31×AGL(1,ℤ/8)12yes—
549C23C31×#13712yes—
550C23C22×D4×C3111yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C31×#12112yes—
589C2×C4C31×#12312yes—
590C2×C4C31×#12312yes—
591C2×C4C31×#12314yes—
592C2×C4C31×#12414yes—
593C2×C4C31×#12614yes—
594C2×C4C31×#12614yes—
595C2×C4C22×D4×C3111yescharacteristic
596C2×C4C2×#41×C3111yescharacteristic
597C2×C4C2×#41×C3111yescharacteristic
598C2×C4C2×#41×C3111yescharacteristic
599C2×C4C2×#41×C3111yescharacteristic
600C2×C4C2×#41×C3111yescharacteristic
601C8—24no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22C31×#39714yes—
625C22C2×C31×#12111yescharacteristic
626C22C2×C31×#12311yescharacteristic
627C22C2×C31×#12311yescharacteristic
628C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
629C22C2×C31×#13711yescharacteristic
630C22C31×#57511yescharacteristic
631C22C31×#57511yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4C31×#39714yes—
641C4C2×C31×#12311yescharacteristic
642C4C2×C31×#12411yescharacteristic
643C4C2×C31×#12611yescharacteristic
644C4C2×C31×#12611yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×C31×#39711yescharacteristic
649C2C31×#305511yescharacteristic
650C2C31×#305611yescharacteristic
651C2C31×#309211yescharacteristic
652C2C31×#309311yescharacteristic
653C2C31×#309411yescharacteristic
654C2C31×#309511yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#428C2semidirect
5. C31×#2077C2semidirect
6. C31×#2077C2semidirect
7. C31×#2076C2semidirect
8. C31×#2075C2semidirect
9. C31×#1736C2semidirect
10. C31×#1736C2semidirect
11. C31×#1736C2semidirect
21. C2×C31×#107C22semidirect
22. C2×C31×#106C22semidirect
39. C31×#340C22semidirect
40. C31×#340C22semidirect
41. C31×#340C22semidirect
42. C31×#340C22semidirect
43. C31×#340C22semidirect
44. C31×#340C22semidirect
194. #23642C31direct (N × H)
276. C2×#428C2×C31semidirect
280. #2077C2×C31semidirect
281. #2077C2×C31semidirect
282. #2076C2×C31semidirect
283. #2075C2×C31semidirect
284. #1736C2×C31semidirect
285. #1736C2×C31semidirect
286. #1736C2×C31semidirect
338. C2×#107C22×C31semidirect
339. C2×#106C22×C31semidirect
356. #340C22×C31semidirect
357. #340C22×C31semidirect
358. #340C22×C31semidirect
359. #340C22×C31semidirect
360. #340C22×C31semidirect
361. #340C22×C31semidirect
432. C31#23642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4282
C2×C31×#4072
C31×#22802
C2×C31×#3802
C31×#20772
C31×#20762
C31×#20752
C31×#17362
#2364231