Finite Group Databasebeta

← C31×#17367 #21474126

Subgroups — C31×#17367 #21474126

Order 7936.

1350 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#518C211yesmaximal, characteristic
2C31×#2973C212yesmaximal, complemented
3C31×#1466C212yesmaximal, complemented
4C31×#1465C212yesmaximal, complemented
5C31×#1464C212yesmaximal, complemented
6C31×#1462C212yesmaximal, complemented
7C31×#1459C212yesmaximal, complemented
8C2×C31×#323C212yesmaximal, complemented
9C23×D4×C31C411yescharacteristic
10C31×#518—28no—
11C2×C4×D4×C31C411yescharacteristic
12C2×C4×D4×C31C411yescharacteristic
13C2×C4×D4×C31C411yescharacteristic
14C2×C4×D4×C31C411yescharacteristic
15C2×C4×D4×C31C2211yescharacteristic
16C2×C4×D4×C31C2211yescharacteristic
17C2×C4×D4×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C2×C31×#117C411yescharacteristic
20C2×C31×#117C2211yescharacteristic
21C22×C22⋊C4×C31C411yescharacteristic
22C22×C22⋊C4×C31C411yescharacteristic
23C22×C22⋊C4×C31C2211yescharacteristic
24C22×C22⋊C4×C31C2211yescharacteristic
25C31×#435—28no—
26C31×#432C2214yescomplemented
27C31×#407C2214yescomplemented
28C2×C31×C22⋊C8C2214yescomplemented
29C31×#324C2214yescomplemented
30C31×#323C2214yes—
31C31×#323C2214yescomplemented
32C31×#323C2214yescomplemented
33C25×C31C2×C411yescharacteristic
34C25×C31C2×C411yescharacteristic
35C22×D4×C31—24no—
36C22×D4×C31—24no—
37C22×D4×C31—24no—
38C22×D4×C31C2×C411yescharacteristic
39C22×D4×C31C2×C411yescharacteristic
40C22×D4×C31C2×C411yescharacteristic
41C22×D4×C31C2×C411yescharacteristic
42C23×C4×C31C2×C411yescharacteristic
43C23×C4×C31C2×C411yescharacteristic
44C23×C4×C31C2×C411yescharacteristic
45C23×C4×C31C2×C411yescharacteristic
46C23×C4×C31C2×C411yescharacteristic
47C23×C4×C31C2×C411yescharacteristic
48C23×C4×C31C2×C411yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C2×M16×C31—24no—
52C22×C8×C31—24no—
53C4×D4×C31—24no—
54C4×D4×C31—24no—
55C4×D4×C31—24no—
56C4×D4×C31—24no—
57C4×D4×C31—24no—
58C4×D4×C31—24no—
59C4×D4×C31—24no—
60C4×D4×C31D418yes—
61C31×#117—24no—
62C31×#117—24no—
63C2×C4⋊C4×C31C2×C411yescharacteristic
64C2×C4⋊C4×C31C2×C411yescharacteristic
65C2×C4⋊C4×C31C2×C411yescharacteristic
66C2×C4⋊C4×C31C2311yescharacteristic
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31C2×C411yescharacteristic
72C2×C22⋊C4×C31C2×C411yescharacteristic
73C2×C22⋊C4×C31C2×C411yescharacteristic
74C2×C22⋊C4×C31C2×C411yescharacteristic
75C2×C22⋊C4×C31C2×C411yescharacteristic
76C2×C22⋊C4×C31C2×C411yescharacteristic
77C2×C22⋊C4×C31C2×C411yescharacteristic
78C2×C22⋊C4×C31C2×C411yescharacteristic
79C2×C22⋊C4×C31C2×C411yescharacteristic
80C2×C22⋊C4×C31C2311yescharacteristic
81C2×C22⋊C4×C31C2311yescharacteristic
82C2×C22⋊C4×C31C2311yescharacteristic
83C2×C42×C31C2×C411yescharacteristic
84C2×C42×C31C2×C411yescharacteristic
85C2×C42×C31C2×C411yescharacteristic
86C2×C42×C31C2311yescharacteristic
87C31×C4⋊C8—24no—
88C31×C22⋊C8—24no—
89C31×C22⋊C8—24no—
90C31×C22⋊C8C2318yes—
91C31×#97—24no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—81no—
98C24×C31—81no—
99C24×C31—81no—
100C24×C31—81no—
101C24×C31M1614yescomplemented
102C24×C31C22×C411yescharacteristic
103C24×C31C22×C411yescharacteristic
104C24×C31C22×C411yescharacteristic
105C24×C31C22×C411yescharacteristic
106C24×C31C22×C411yescharacteristic
107C2×D4×C31—232no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—22no—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31C22⋊C414yes—
119C2×D4×C31M1614yescomplemented
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—81no—
130C22×C4×C31—81no—
131C22×C4×C31—81no—
132C22×C4×C31—81no—
133C22×C4×C31—81no—
134C22×C4×C31—81no—
135C22×C4×C31—81no—
136C22×C4×C31—81no—
137C22×C4×C31C22×C411yescharacteristic
138C22×C4×C31C22×C411yescharacteristic
139C22×C4×C31C22×C411yescharacteristic
140C22×C4×C31C22×C411yescharacteristic
141C22×C4×C31C22×C411yescharacteristic
142C22×C4×C31C22×C411yescharacteristic
143C22×C4×C31C22×C411yescharacteristic
144C22×C4×C31C22×C411yescharacteristic
145C22×C4×C31C22×C411yescharacteristic
146C22×C4×C31C2×D414yes—
147C22×C4×C31C2×D414yes—
148C22×C4×C31C2411yescharacteristic
149M16×C31—84no—
150C2×C8×C31—28no—
151C2×C8×C31—84no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31C22⋊C414yes—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—22no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31C22⋊C414yes—
167C22⋊C4×C31C22⋊C414yes—
168C42×C31—22no—
169C42×C31—22no—
170C42×C31—22no—
171C42×C31C2×D414yes—
172#17367C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—28no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—28no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—82no—
194C23×C31—82no—
195C23×C31—82no—
196C23×C31—82no—
197C23×C31—84no—
198C23×C31—84no—
199C23×C31—84no—
200C23×C31—84no—
201C23×C31C2×C22⋊C412yes—
202C23×C31C2×C22⋊C412yes—
203C23×C31C2×M1612yes—
204C23×C31C2×M1612yes—
205C23×C31C2×M1612yes—
206C23×C31C2×M1612yes—
207C23×C31C2×M1612yes—
208C23×C31C23×C411yescharacteristic
209D4×C31—28no—
210D4×C31—28no—
211D4×C31—28no—
212D4×C31—28no—
213C2×C4×C31—28no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—44no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—82no—
233C2×C4×C31—82no—
234C2×C4×C31—82no—
235C2×C4×C31—82no—
236C2×C4×C31—82no—
237C2×C4×C31—82no—
238C2×C4×C31C2×C22⋊C412yes—
239C2×C4×C31C2×C22⋊C412yes—
240C2×C4×C31C2×C22⋊C412yes—
241C2×C4×C31C2×C22⋊C412yes—
242C2×C4×C31C2×M1612yes—
243C2×C4×C31C22×D412yes—
244C8×C31—48no—
245C2×#518C2×C3111yescharacteristic
246#2973C2×C3112yescomplemented
247#1466C2×C3112yescomplemented
248#1465C2×C3112yescomplemented
249#1464C2×C3112yescomplemented
250#1462C2×C3112yescomplemented
251#1459C2×C3112yescomplemented
252C2×#323C2×C3112yescomplemented
253C22×C31—21no—
254C22×C31—22no—
255C22×C31—21no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—22no—
259C22×C31—22no—
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—88no—
280C22×C31—88no—
281C22×C31#40714yes—
282C22×C31C22×C22⋊C411yescharacteristic
283C22×C31C22×M1611yescharacteristic
284C22×C31C22×M1611yescharacteristic
285C4×C31—22no—
286C4×C31—22no—
287C4×C31—22no—
288C4×C31—22no—
289C4×C31—22no—
290C4×C31—22no—
291C4×C31—41no—
292C4×C31—41no—
293C4×C31—41no—
294C4×C31—41no—
295C23×D4C4×C3111yescharacteristic
296#518—28no—
297C2×C4×D4C4×C3111yescharacteristic
298C2×C4×D4C4×C3111yescharacteristic
299C2×C4×D4C4×C3111yescharacteristic
300C2×C4×D4C4×C3111yescharacteristic
301C2×C4×D4C22×C3111yescharacteristic
302C2×C4×D4C22×C3111yescharacteristic
303C2×C4×D4C22×C3111yescharacteristic
304C2×C4×D4C22×C3111yescharacteristic
305C2×#117C4×C3111yescharacteristic
306C2×#117C22×C3111yescharacteristic
307C22×C22⋊C4C4×C3111yescharacteristic
308C22×C22⋊C4C4×C3111yescharacteristic
309C22×C22⋊C4C22×C3111yescharacteristic
310C22×C22⋊C4C22×C3111yescharacteristic
311#435—28no—
312#432C22×C3114yescomplemented
313#407C22×C3114yescomplemented
314C2×C22⋊C8C22×C3114yescomplemented
315#324C22×C3114yescomplemented
316#323C22×C3114yes—
317#323C22×C3114yescomplemented
318#323C22×C3114yescomplemented
319C2×C31—21no—
320C2×C31—21no—
321C2×C31—24no—
322C2×C31—42no—
323C2×C31—42no—
324C2×C31—42no—
325C2×C31—42no—
326C2×C31C2×#40711yescharacteristic
327C2×C31#288311yescharacteristic
328C2×C31#288811yescharacteristic
329C25C2×C4×C3111yescharacteristic
330C25C2×C4×C3111yescharacteristic
331C22×D4—24no—
332C22×D4—24no—
333C22×D4—24no—
334C22×D4C2×C4×C3111yescharacteristic
335C22×D4C2×C4×C3111yescharacteristic
336C22×D4C2×C4×C3111yescharacteristic
337C22×D4C2×C4×C3111yescharacteristic
338C23×C4C2×C4×C3111yescharacteristic
339C23×C4C2×C4×C3111yescharacteristic
340C23×C4C2×C4×C3111yescharacteristic
341C23×C4C2×C4×C3111yescharacteristic
342C23×C4C2×C4×C3111yescharacteristic
343C23×C4C2×C4×C3111yescharacteristic
344C23×C4C2×C4×C3111yescharacteristic
345C23×C4C23×C3111yescharacteristic
346C23×C4C23×C3111yescharacteristic
347C2×M16—24no—
348C22×C8—24no—
349C4×D4—24no—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—24no—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4D4×C3118yes—
357#117—24no—
358#117—24no—
359C2×C4⋊C4C2×C4×C3111yescharacteristic
360C2×C4⋊C4C2×C4×C3111yescharacteristic
361C2×C4⋊C4C2×C4×C3111yescharacteristic
362C2×C4⋊C4C23×C3111yescharacteristic
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4C2×C4×C3111yescharacteristic
368C2×C22⋊C4C2×C4×C3111yescharacteristic
369C2×C22⋊C4C2×C4×C3111yescharacteristic
370C2×C22⋊C4C2×C4×C3111yescharacteristic
371C2×C22⋊C4C2×C4×C3111yescharacteristic
372C2×C22⋊C4C2×C4×C3111yescharacteristic
373C2×C22⋊C4C2×C4×C3111yescharacteristic
374C2×C22⋊C4C2×C4×C3111yescharacteristic
375C2×C22⋊C4C2×C4×C3111yescharacteristic
376C2×C22⋊C4C23×C3111yescharacteristic
377C2×C22⋊C4C23×C3111yescharacteristic
378C2×C22⋊C4C23×C3111yescharacteristic
379C2×C42C2×C4×C3111yescharacteristic
380C2×C42C2×C4×C3111yescharacteristic
381C2×C42C2×C4×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383C4⋊C8—24no—
384C22⋊C8—24no—
385C22⋊C8—24no—
386C22⋊C8C23×C3118yes—
387#97—24no—
388C31#1736711yescharacteristic, direct product factor, Sylow
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—81no—
395C24—81no—
396C24—81no—
397C24—81no—
398C24M16×C3114yescomplemented
399C24C22×C4×C3111yescharacteristic
400C24C22×C4×C3111yescharacteristic
401C24C22×C4×C3111yescharacteristic
402C24C22×C4×C3111yescharacteristic
403C24C22×C4×C3111yescharacteristic
404C2×D4—232no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4C22⋊C4×C3114yes—
416C2×D4M16×C3114yescomplemented
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C22×C4×C3111yescharacteristic
435C22×C4C22×C4×C3111yescharacteristic
436C22×C4C22×C4×C3111yescharacteristic
437C22×C4C22×C4×C3111yescharacteristic
438C22×C4C22×C4×C3111yescharacteristic
439C22×C4C22×C4×C3111yescharacteristic
440C22×C4C22×C4×C3111yescharacteristic
441C22×C4C22×C4×C3111yescharacteristic
442C22×C4C22×C4×C3111yescharacteristic
443C22×C4C2×D4×C3114yes—
444C22×C4C2×D4×C3114yes—
445C22×C4C24×C3111yescharacteristic
446M16—84no—
447C2×C8—28no—
448C2×C8—84no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4C22⋊C4×C3114yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4C22⋊C4×C3114yes—
464C22⋊C4C22⋊C4×C3114yes—
465C42—22no—
466C42—22no—
467C42—22no—
468C42C2×D4×C3114yes—
469C23—28no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—28no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23C2×C22⋊C4×C3112yes—
498C23C2×C22⋊C4×C3112yes—
499C23C2×M16×C3112yes—
500C23C2×M16×C3112yes—
501C23C2×M16×C3112yes—
502C23C2×M16×C3112yes—
503C23C2×M16×C3112yes—
504C23C23×C4×C3111yescharacteristic
505D4—28no—
506D4—28no—
507D4—28no—
508D4—28no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C2×C22⋊C4×C3112yes—
535C2×C4C2×C22⋊C4×C3112yes—
536C2×C4C2×C22⋊C4×C3112yes—
537C2×C4C2×C22⋊C4×C3112yes—
538C2×C4C2×M16×C3112yes—
539C2×C4C22×D4×C3112yes—
540C8—48no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—88no—
568C22—88no—
569C22C31×#40714yes—
570C22C22×C22⋊C4×C3111yescharacteristic
571C22C22×M16×C3111yescharacteristic
572C22C22×M16×C3111yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C2—21no—
584C2—21no—
585C2—24no—
586C2—42no—
587C2—42no—
588C2—42no—
589C2—42no—
590C2C2×C31×#40711yescharacteristic
591C2C31×#288311yescharacteristic
592C2C31×#288811yescharacteristic
subgroupG/H
center, socleC22×C31C22×C22⋊C4
commutatorC23C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC23×D4×C31C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2973C2semidirect
3. C31×#1466C2semidirect
4. C31×#1465C2semidirect
5. C31×#1464C2semidirect
6. C31×#1462C2semidirect
7. C31×#1459C2semidirect
8. C2×C31×#323C2semidirect
26. C31×#432C22semidirect
27. C31×#407C22semidirect
28. C2×C31×C22⋊C8C22semidirect
29. C31×#324C22semidirect
31. C31×#323C22semidirect
32. C31×#323C22semidirect
101. C24×C31M16semidirect
119. C2×D4×C31M16semidirect
172. #17367C31direct (N × H)
246. #2973C2×C31semidirect
247. #1466C2×C31semidirect
248. #1465C2×C31semidirect
249. #1464C2×C31semidirect
250. #1462C2×C31semidirect
251. #1459C2×C31semidirect
252. C2×#323C2×C31semidirect
312. #432C22×C31semidirect
313. #407C22×C31semidirect
314. C2×C22⋊C8C22×C31semidirect
315. #324C22×C31semidirect
317. #323C22×C31semidirect
318. #323C22×C31semidirect
388. C31#17367direct (N × H)
398. C24M16×C31semidirect
416. C2×D4M16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5182
C31×#29732
C31×#14662
C31×#14652
C31×#14642
C31×#14622
C31×#14592
C2×C31×#3232
#1736731