Finite Group Databasebeta

← C31×#17386 #21474145

Subgroups — C31×#17386 #21474145

Order 7936.

1080 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#538C211yesmaximal, characteristic
2C31×#2973C211yesmaximal, characteristic, complemented
3C31×#2972C211yesmaximal, characteristic, complemented
4C31×#1473C212yesmaximal, complemented
5C31×#1471C212yesmaximal, complemented
6C31×#1469C212yesmaximal, complemented
7C31×#1468C212yesmaximal, complemented
8C31×#1459C212yesmaximal, complemented
9C2×C31×#323C212yesmaximal, complemented
10C31×#538—24no—
11C31×#538—24no—
12C2×C31×#126C411yescharacteristic
13C2×C31×#126C2211yescharacteristic
14C2×C31×#124C411yescharacteristic
15C2×C31×#124C2211yescharacteristic
16C2×C31×#123C411yescharacteristic
17C2×C31×#123C2211yescharacteristic
18C2×C31×C22⋊Q8C411yescharacteristic
19C2×C31×#121C411yescharacteristic
20C2×C31×#121C411yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C31×#120C411yescharacteristic
23C2×C31×#120C411yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C22×C22⋊C4×C31C2211yescharacteristic
27C31×#435—24no—
28C8×D4×C31—24no—
29C31×#432C2212yescomplemented
30C31×#431C2212yescomplemented
31C31×#407C2212yescomplemented
32C2×C31×C22⋊C8C2212yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×C22⋊C8C2212yescomplemented
35C31×#324C2214yescomplemented
36C31×#323C2214yes—
37C31×#323C2214yes—
38C31×#323C2214yescomplemented
39C25×C31C2×C411yescharacteristic
40C22×Q8×C31C2×C411yescharacteristic
41C22×D4×C31C2×C411yescharacteristic
42C22×D4×C31C2×C411yescharacteristic
43C22×D4×C31C2×C411yescharacteristic
44C22×D4×C31C2×C411yescharacteristic
45C22×D4×C31C2×C411yescharacteristic
46C22×D4×C31C2×C411yescharacteristic
47C22×D4×C31C2×C411yescharacteristic
48C23×C4×C31C2×C411yescharacteristic
49C23×C4×C31C2×C411yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C2×M16×C31—22no—
53C22×C8×C31—22no—
54C22×C8×C31—22no—
55C22×C8×C31—22no—
56C31×#126—24no—
57C31×#126—22no—
58C31×#126—22no—
59C31×#124—24no—
60C31×#124—22no—
61C31×#124—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—24no—
65C31×C22⋊Q8—24no—
66C31×#121—24no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—24no—
70C31×#120—24no—
71C31×#120—24no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31D414yes—
75C4×D4×C31D414yes—
76C2×C4⋊C4×C31C2×C411yescharacteristic
77C2×C4⋊C4×C31C2×C411yescharacteristic
78C2×C4⋊C4×C31C2×C411yescharacteristic
79C2×C4⋊C4×C31C2×C411yescharacteristic
80C2×C4⋊C4×C31C2×C411yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31C2×C411yescharacteristic
85C2×C22⋊C4×C31C2×C411yescharacteristic
86C2×C22⋊C4×C31C2×C411yescharacteristic
87C2×C22⋊C4×C31C2×C411yescharacteristic
88C2×C22⋊C4×C31C2×C411yescharacteristic
89C2×C22⋊C4×C31C2×C411yescharacteristic
90C2×C22⋊C4×C31C2×C411yescharacteristic
91C2×C22⋊C4×C31C2×C411yescharacteristic
92C2×C22⋊C4×C31C2×C411yescharacteristic
93C2×C22⋊C4×C31C2×C411yescharacteristic
94C2×C22⋊C4×C31C2×C411yescharacteristic
95C2×C22⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31C2311yescharacteristic
98C2×C42×C31C2×C411yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C31×C4⋊C8—22no—
101C31×C4⋊C8—22no—
102C31×C22⋊C8—22no—
103C31×C22⋊C8—22no—
104C31×C22⋊C8—22no—
105C31×C22⋊C8—22no—
106C31×C22⋊C8C2314yes—
107C31×C22⋊C8C2314yes—
108C31×#97—22no—
109C4×C8×C31—22no—
110C24×C31—22no—
111C24×C31—24no—
112C24×C31—81no—
113C24×C31—81no—
114C24×C31C22×C411yescharacteristic
115C24×C31C22×C411yescharacteristic
116C24×C31C22×C411yescharacteristic
117C24×C31C22×C411yescharacteristic
118C24×C31C22×C411yescharacteristic
119C2×Q8×C31—22no—
120C2×Q8×C31—81no—
121C2×D4×C31—24no—
122C2×D4×C31—22no—
123C2×D4×C31—24no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—81no—
130C2×D4×C31—81no—
131C2×D4×C31—81no—
132C2×D4×C31—81no—
133C2×D4×C31—81no—
134C2×D4×C31C22⋊C414yes—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—81no—
138C22×C4×C31—81no—
139C22×C4×C31—81no—
140C22×C4×C31—81no—
141C22×C4×C31C22×C411yescharacteristic
142C22×C4×C31C22×C411yescharacteristic
143C22×C4×C31C22×C411yescharacteristic
144C22×C4×C31C22×C411yescharacteristic
145C22×C4×C31C22×C411yescharacteristic
146C22×C4×C31C22×C411yescharacteristic
147C22×C4×C31C22×C411yescharacteristic
148C22×C4×C31C22×C411yescharacteristic
149C22×C4×C31C22×C411yescharacteristic
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2411yescharacteristic
155M16×C31—82no—
156C2×C8×C31—24no—
157C2×C8×C31—24no—
158C2×C8×C31—82no—
159C2×C8×C31—82no—
160C2×C8×C31—82no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31C22⋊C414yes—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31C22⋊C414yes—
184C22⋊C4×C31C22⋊C414yes—
185C42×C31—22no—
186C42×C31C2×D412yes—
187C42×C31C2×D412yes—
188#17386C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—22no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—44no—
207C23×C31—82no—
208C23×C31—82no—
209C23×C31—88no—
210C23×C31C2×C22⋊C412yes—
211C23×C31C2×C22⋊C412yes—
212C23×C31#13114yes—
213C23×C31C23×C411yescharacteristic
214Q8×C31—82no—
215D4×C31—28no—
216D4×C31—44no—
217D4×C31—82no—
218D4×C31—82no—
219D4×C31—82no—
220D4×C31—82no—
221D4×C31—82no—
222C2×C4×C31—22no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
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—41no—
240C2×C4×C31—41no—
241C2×C4×C31—82no—
242C2×C4×C31—82no—
243C2×C4×C31C2×C22⋊C412yes—
244C2×C4×C31C2×C22⋊C412yes—
245C2×C4×C31C2×C22⋊C412yes—
246C2×C4×C31C2×C22⋊C412yes—
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C8×C31—44no—
250C8×C31—44no—
251C2×#538C2×C3111yescharacteristic
252#2973C2×C3111yescharacteristic, complemented
253#2972C2×C3111yescharacteristic, complemented
254#1473C2×C3112yescomplemented
255#1471C2×C3112yescomplemented
256#1469C2×C3112yescomplemented
257#1468C2×C3112yescomplemented
258#1459C2×C3112yescomplemented
259C2×#323C2×C3112yescomplemented
260C22×C31—21no—
261C22×C31—22no—
262C22×C31—21no—
263C22×C31—22no—
264C22×C31—22no—
265C22×C31—22no—
266C22×C31—22no—
267C22×C31—44no—
268C22×C31—41no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—41no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31—88no—
282C22×C31#40814yes—
283C22×C31C22×C22⋊C411yescharacteristic
284C22×C31C2×#13112yes—
285C4×C31—22no—
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—22no—
289C4×C31—22no—
290C4×C31—22no—
291C4×C31—41no—
292C4×C31—41no—
293C4×C31—41no—
294C4×C31—41no—
295C4×C31—41no—
296#538—24no—
297#538—24no—
298C2×#126C4×C3111yescharacteristic
299C2×#126C22×C3111yescharacteristic
300C2×#124C4×C3111yescharacteristic
301C2×#124C22×C3111yescharacteristic
302C2×#123C4×C3111yescharacteristic
303C2×#123C22×C3111yescharacteristic
304C2×C22⋊Q8C4×C3111yescharacteristic
305C2×#121C4×C3111yescharacteristic
306C2×#121C4×C3111yescharacteristic
307C2×#121C22×C3111yescharacteristic
308C2×#120C4×C3111yescharacteristic
309C2×#120C4×C3111yescharacteristic
310C2×C4×D4C22×C3111yescharacteristic
311C2×C4×D4C22×C3111yescharacteristic
312C22×C22⋊C4C22×C3111yescharacteristic
313#435—24no—
314C8×D4—24no—
315#432C22×C3112yescomplemented
316#431C22×C3112yescomplemented
317#407C22×C3112yescomplemented
318C2×C22⋊C8C22×C3112yes—
319C2×C22⋊C8C22×C3112yes—
320C2×C22⋊C8C22×C3112yescomplemented
321#324C22×C3114yescomplemented
322#323C22×C3114yes—
323#323C22×C3114yes—
324#323C22×C3114yescomplemented
325C2×C31—22no—
326C2×C31—24no—
327C2×C31—42no—
328C2×C31—81no—
329C2×C31—81no—
330C2×C31—81no—
331C2×C31C2×#40811yescharacteristic
332C2×C31#288311yescharacteristic
333C2×C31#288811yescharacteristic
334C25C2×C4×C3111yescharacteristic
335C22×Q8C2×C4×C3111yescharacteristic
336C22×D4C2×C4×C3111yescharacteristic
337C22×D4C2×C4×C3111yescharacteristic
338C22×D4C2×C4×C3111yescharacteristic
339C22×D4C2×C4×C3111yescharacteristic
340C22×D4C2×C4×C3111yescharacteristic
341C22×D4C2×C4×C3111yescharacteristic
342C22×D4C2×C4×C3111yescharacteristic
343C23×C4C2×C4×C3111yescharacteristic
344C23×C4C2×C4×C3111yescharacteristic
345C23×C4C23×C3111yescharacteristic
346C23×C4C23×C3111yescharacteristic
347C2×M16—22no—
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351#126—24no—
352#126—22no—
353#126—22no—
354#124—24no—
355#124—22no—
356#124—22no—
357#123—22no—
358#123—22no—
359#123—24no—
360C22⋊Q8—24no—
361#121—24no—
362#121—22no—
363#121—22no—
364#121—24no—
365#120—24no—
366#120—24no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4D4×C3114yes—
370C4×D4D4×C3114yes—
371C2×C4⋊C4C2×C4×C3111yescharacteristic
372C2×C4⋊C4C2×C4×C3111yescharacteristic
373C2×C4⋊C4C2×C4×C3111yescharacteristic
374C2×C4⋊C4C2×C4×C3111yescharacteristic
375C2×C4⋊C4C2×C4×C3111yescharacteristic
376C2×C4⋊C4C23×C3111yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4C2×C4×C3111yescharacteristic
380C2×C22⋊C4C2×C4×C3111yescharacteristic
381C2×C22⋊C4C2×C4×C3111yescharacteristic
382C2×C22⋊C4C2×C4×C3111yescharacteristic
383C2×C22⋊C4C2×C4×C3111yescharacteristic
384C2×C22⋊C4C2×C4×C3111yescharacteristic
385C2×C22⋊C4C2×C4×C3111yescharacteristic
386C2×C22⋊C4C2×C4×C3111yescharacteristic
387C2×C22⋊C4C2×C4×C3111yescharacteristic
388C2×C22⋊C4C2×C4×C3111yescharacteristic
389C2×C22⋊C4C2×C4×C3111yescharacteristic
390C2×C22⋊C4C23×C3111yescharacteristic
391C2×C22⋊C4C23×C3111yescharacteristic
392C2×C22⋊C4C23×C3111yescharacteristic
393C2×C42C2×C4×C3111yescharacteristic
394C2×C42C23×C3111yescharacteristic
395C4⋊C8—22no—
396C4⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—22no—
399C22⋊C8—22no—
400C22⋊C8—22no—
401C22⋊C8C23×C3114yes—
402C22⋊C8C23×C3114yes—
403#97—22no—
404C4×C8—22no—
405C31#1738611yescharacteristic, direct product factor, Sylow
406C24—22no—
407C24—24no—
408C24—81no—
409C24—81no—
410C24C22×C4×C3111yescharacteristic
411C24C22×C4×C3111yescharacteristic
412C24C22×C4×C3111yescharacteristic
413C24C22×C4×C3111yescharacteristic
414C24C22×C4×C3111yescharacteristic
415C2×Q8—22no—
416C2×Q8—81no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4C22⋊C4×C3114yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
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×C4C22×C4×C3111yescharacteristic
444C22×C4C22×C4×C3111yescharacteristic
445C22×C4C22×C4×C3111yescharacteristic
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3112yes—
448C22×C4C2×D4×C3112yes—
449C22×C4C2×D4×C3112yes—
450C22×C4C24×C3111yescharacteristic
451M16—82no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—82no—
455C2×C8—82no—
456C2×C8—82no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4C22⋊C4×C3114yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4C22⋊C4×C3114yes—
480C22⋊C4C22⋊C4×C3114yes—
481C42—22no—
482C42C2×D4×C3112yes—
483C42C2×D4×C3112yes—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—44no—
502C23—82no—
503C23—82no—
504C23—88no—
505C23C2×C22⋊C4×C3112yes—
506C23C2×C22⋊C4×C3112yes—
507C23C31×#13114yes—
508C23C23×C4×C3111yescharacteristic
509Q8—82no—
510D4—28no—
511D4—44no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—82no—
516D4—82no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4C2×C22⋊C4×C3112yes—
539C2×C4C2×C22⋊C4×C3112yes—
540C2×C4C2×C22⋊C4×C3112yes—
541C2×C4C2×C22⋊C4×C3112yes—
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C8—44no—
545C8—44no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—44no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—88no—
568C22C31×#40814yes—
569C22C22×C22⋊C4×C3111yescharacteristic
570C22C2×C31×#13112yes—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C2—22no—
583C2—24no—
584C2—42no—
585C2—81no—
586C2—81no—
587C2—81no—
588C2C2×C31×#40811yescharacteristic
589C2C31×#288311yescharacteristic
590C2C31×#288811yescharacteristic
subgroupG/H
center, socleC22×C31C22×C22⋊C4
commutatorC23C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#538C2
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×#2972C2semidirect
4. C31×#1473C2semidirect
5. C31×#1471C2semidirect
6. C31×#1469C2semidirect
7. C31×#1468C2semidirect
8. C31×#1459C2semidirect
9. C2×C31×#323C2semidirect
29. C31×#432C22semidirect
30. C31×#431C22semidirect
31. C31×#407C22semidirect
34. C2×C31×C22⋊C8C22semidirect
35. C31×#324C22semidirect
38. C31×#323C22semidirect
188. #17386C31direct (N × H)
252. #2973C2×C31semidirect
253. #2972C2×C31semidirect
254. #1473C2×C31semidirect
255. #1471C2×C31semidirect
256. #1469C2×C31semidirect
257. #1468C2×C31semidirect
258. #1459C2×C31semidirect
259. C2×#323C2×C31semidirect
315. #432C22×C31semidirect
316. #431C22×C31semidirect
317. #407C22×C31semidirect
320. C2×C22⋊C8C22×C31semidirect
321. #324C22×C31semidirect
324. #323C22×C31semidirect
405. C31#17386direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5382
C31×#29732
C31×#29722
C31×#14732
C31×#14712
C31×#14692
C31×#14682
C31×#14592
C2×C31×#3232
#1738631