Finite Group Databasebeta

← C31×#14130 #21470889

Subgroups — C31×#14130 #21470889

Order 7936.

2366 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C31C211yesmaximal, characteristic
2C2×C31×#389C212yesmaximal, complemented
3C2×C31×#385C212yesmaximal
4C2×C31×#380C211yesmaximal, characteristic
5C2×C4×C22⋊C4×C31C211yesmaximal, characteristic
6C31×#1444C212yesmaximal, complemented
7C31×#1442C212yesmaximal, complemented
8C31×C4⋊#95C212yesmaximal, complemented
9C31×C22⋊#95C212yesmaximal
10C24×C4×C31C2212yes—
11C24×C4×C31C2211yescharacteristic
12C22×C4⋊C4×C31—24no—
13C22×C4⋊C4×C31—28no—
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2214yes—
16C22×C4⋊C4×C31C2212yes—
17C22×C4⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#389—28no—
24C31×#385—28no—
25C31×#380C418yescomplemented
26C4×C22⋊C4×C31—24no—
27C2×C31×#95C2214yes—
28C2×C31×#95C2214yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2214yes—
32C2×C31×#95C2212yes—
33C25×C31C2311yescharacteristic
34C23×C4×C31—24no—
35C23×C4×C31—24no—
36C23×C4×C31—24no—
37C23×C4×C31—28no—
38C23×C4×C31—24no—
39C23×C4×C31—24no—
40C23×C4×C31—44no—
41C23×C4×C31C2×C414yes—
42C23×C4×C31D414yes—
43C23×C4×C31Q814yes—
44C23×C4×C31C2314yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2314yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C2×C4⋊C4×C31—232no—
53C2×C4⋊C4×C31—28no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—28no—
60C2×C4⋊C4×C31—28no—
61C2×C4⋊C4×C31—44no—
62C2×C4⋊C4×C31C2×C418yescomplemented
63C2×C4⋊C4×C31D418yes—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31C2×C418yes—
71C2×C22⋊C4×C31D414yes—
72C2×C22⋊C4×C31Q814yes—
73C2×C42×C31—22no—
74C2×C42×C31—22no—
75C2×C42×C31—24no—
76C2×C42×C31—24no—
77C31×#95—22no—
78C31×#95—22no—
79C31×#95—28no—
80C31×#95—28no—
81C31×#95—44no—
82C31×#95C2×C418yescomplemented
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31C22×C412yes—
88C24×C31C2×D412yes—
89C24×C31C2×D412yes—
90C24×C31C2×Q812yes—
91C24×C31C2×Q812yes—
92C24×C31#4112yes—
93C24×C31#4112yes—
94C24×C31C2411yescharacteristic
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—216no—
98C22×C4×C31—28no—
99C22×C4×C31—216no—
100C22×C4×C31—28no—
101C22×C4×C31—216no—
102C22×C4×C31—22no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—24no—
108C22×C4×C31—24no—
109C22×C4×C31—24no—
110C22×C4×C31—22no—
111C22×C4×C31—24no—
112C22×C4×C31—24no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—24no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—22no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—24no—
127C22×C4×C31—24no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—416no—
131C22×C4×C31—416no—
132C22×C4×C31—48no—
133C22×C4×C31—48no—
134C22×C4×C31C4⋊C4116yescomplemented
135C22×C4×C31C22×C418yes—
136C22×C4×C31C22×C412yes—
137C22×C4×C31C22×C412yes—
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D418yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D414yes—
144C22×C4×C31C2×Q812yes—
145C22×C4×C31C2×Q812yes—
146C22×C4×C31C2×Q812yes—
147C22×C4×C31C2×Q812yes—
148C22×C4×C31#4114yes—
149C4⋊C4×C31—216no—
150C4⋊C4×C31—216no—
151C4⋊C4×C31—216no—
152C4⋊C4×C31—416no—
153C22⋊C4×C31—216no—
154C22⋊C4×C31—28no—
155C22⋊C4×C31—28no—
156C42×C31—44no—
157C42×C31—44no—
158#14130C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—24no—
160C23×C31—24no—
161C23×C31—22no—
162C23×C31—24no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—24no—
167C23×C31—24no—
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31C2×C4⋊C414yes—
176C23×C31C4×D414yes—
177C23×C31C4×Q814yes—
178C23×C31#12114yes—
179C23×C31C22⋊Q814yes—
180C23×C31#12514yes—
181C23×C31C4⋊Q814yes—
182C23×C31C23×C411yescharacteristic
183C23×C31C22×D411yescharacteristic
184C23×C31C22×D412yes—
185C23×C31C22×D411yescharacteristic
186C23×C31C22×Q811yescharacteristic
187C23×C31C22×Q811yescharacteristic
188C23×C31C2×#4112yes—
189C23×C31C2×#4111yescharacteristic
190C23×C31C2×#4111yescharacteristic
191C23×C31H(5,𝔽2)12yes—
192C23×C312−1+412yes—
193C2×C4×C31—24no—
194C2×C4×C31—22no—
195C2×C4×C31—24no—
196C2×C4×C31—22no—
197C2×C4×C31—232no—
198C2×C4×C31—28no—
199C2×C4×C31—28no—
200C2×C4×C31—28no—
201C2×C4×C31—28no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—28no—
208C2×C4×C31—28no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—432no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—44no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—44no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—44no—
224C2×C4×C31—44no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—44no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—44no—
232C2×C4×C31—42no—
233C2×C4×C31C2×C4⋊C414yes—
234C2×C4×C31C2×C4⋊C414yes—
235C2×C4×C31C4×D418yes—
236C2×C4×C31#12014yes—
237C2×C4×C31C22⋊Q818yes—
238C2×C4×C31#12314yes—
239C23×C4⋊C4C2×C3111yescharacteristic
240C2×#389C2×C3112yescomplemented
241C2×#385C2×C3112yes—
242C2×#380C2×C3111yescharacteristic
243C2×C4×C22⋊C4C2×C3111yescharacteristic
244#1444C2×C3112yescomplemented
245#1442C2×C3112yescomplemented
246C4⋊#95C2×C3112yescomplemented
247C22⋊#95C2×C3112yes—
248C22×C31—24no—
249C22×C31—28no—
250C22×C31—28no—
251C22×C31—24no—
252C22×C31—24no—
253C22×C31—28no—
254C22×C31—28no—
255C22×C31—24no—
256C22×C31—24no—
257C22×C31—24no—
258C22×C31#38918yescomplemented
259C22×C31C22×C4⋊C411yescharacteristic
260C22×C31C2×C4×D412yes—
261C22×C31C2×C4×D411yescharacteristic
262C22×C31C2×C4×Q811yescharacteristic
263C22×C31#52012yes—
264C22×C31C2×#12011yescharacteristic
265C22×C31C2×#12111yescharacteristic
266C22×C31C2×C22⋊Q811yescharacteristic
267C22×C31C2×C22⋊Q812yes—
268C22×C31C2×#12311yescharacteristic
269C22×C31C2×#12511yescharacteristic
270C22×C31C2×C4⋊Q811yescharacteristic
271C22×C31#53511yescharacteristic
272C22×C31#53611yescharacteristic
273C22×C31#54412yes—
274C22×C31D4212yes—
275C22×C31#54612yes—
276C22×C31#54712yes—
277C22×C31#54812yes—
278C22×C31D4×Q812yes—
279C22×C31#55212yes—
280C22×C31#55312yes—
281C22×C31#55412yes—
282C4×C31—24no—
283C4×C31—48no—
284C4×C31—42no—
285C4×C31—48no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—42no—
289C4×C31#38518yescomplemented
290C24×C4C22×C3112yes—
291C24×C4C22×C3111yescharacteristic
292C22×C4⋊C4—24no—
293C22×C4⋊C4—28no—
294C22×C4⋊C4C22×C3112yes—
295C22×C4⋊C4C22×C3114yes—
296C22×C4⋊C4C22×C3112yes—
297C22×C4⋊C4C22×C3112yes—
298C22×C22⋊C4C22×C3112yes—
299C22×C22⋊C4C22×C3111yescharacteristic
300C22×C22⋊C4C22×C3111yescharacteristic
301C22×C42C22×C3111yescharacteristic
302C22×C42C22×C3111yescharacteristic
303#389—28no—
304#385—28no—
305#380C4×C3118yescomplemented
306C4×C22⋊C4—24no—
307C2×#95C22×C3114yes—
308C2×#95C22×C3114yes—
309C2×#95C22×C3111yescharacteristic
310C2×#95C22×C3111yescharacteristic
311C2×#95C22×C3114yes—
312C2×#95C22×C3112yes—
313C2×C31—28no—
314C2×C31C2×#38511yescharacteristic
315C2×C31C2×#38911yescharacteristic
316C2×C31#231711yescharacteristic
317C2×C31#236012yes—
318C2×C31#236112yes—
319C2×C31#240811yescharacteristic
320C2×C31#240911yescharacteristic
321C2×C31#244012yes—
322C2×C31#244212yes—
323C2×C31#248611yescharacteristic
324C2×C31#248711yescharacteristic
325C25C23×C3111yescharacteristic
326C23×C4—24no—
327C23×C4—24no—
328C23×C4—24no—
329C23×C4—28no—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—44no—
333C23×C4C2×C4×C3114yes—
334C23×C4D4×C3114yes—
335C23×C4Q8×C3114yes—
336C23×C4C23×C3114yes—
337C23×C4C23×C3111yescharacteristic
338C23×C4C23×C3111yescharacteristic
339C23×C4C23×C3114yes—
340C23×C4C23×C3111yescharacteristic
341C23×C4C23×C3111yescharacteristic
342C23×C4C23×C3111yescharacteristic
343C23×C4C23×C3111yescharacteristic
344C2×C4⋊C4—232no—
345C2×C4⋊C4—28no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4C2×C4×C3118yescomplemented
355C2×C4⋊C4D4×C3118yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4C2×C4×C3118yes—
363C2×C22⋊C4D4×C3114yes—
364C2×C22⋊C4Q8×C3114yes—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—24no—
369#95—22no—
370#95—22no—
371#95—28no—
372#95—28no—
373#95—44no—
374#95C2×C4×C3118yescomplemented
375C31#1413011yescharacteristic, direct product factor, Sylow
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24C22×C4×C3112yes—
381C24C2×D4×C3112yes—
382C24C2×D4×C3112yes—
383C24C2×Q8×C3112yes—
384C24C2×Q8×C3112yes—
385C24#41×C3112yes—
386C24#41×C3112yes—
387C24C24×C3111yescharacteristic
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4C4⋊C4×C31116yescomplemented
428C22×C4C22×C4×C3118yes—
429C22×C4C22×C4×C3112yes—
430C22×C4C22×C4×C3112yes—
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3118yes—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3112yes—
435C22×C4C2×D4×C3112yes—
436C22×C4C2×D4×C3114yes—
437C22×C4C2×Q8×C3112yes—
438C22×C4C2×Q8×C3112yes—
439C22×C4C2×Q8×C3112yes—
440C22×C4C2×Q8×C3112yes—
441C22×C4#41×C3114yes—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—416no—
446C22⋊C4—216no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C42—44no—
450C42—44no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23C2×C4⋊C4×C3114yes—
468C23C4×D4×C3114yes—
469C23C4×Q8×C3114yes—
470C23C31×#12114yes—
471C23C31×C22⋊Q814yes—
472C23C31×#12514yes—
473C23C31×C4⋊Q814yes—
474C23C23×C4×C3111yescharacteristic
475C23C22×D4×C3111yescharacteristic
476C23C22×D4×C3112yes—
477C23C22×D4×C3111yescharacteristic
478C23C22×Q8×C3111yescharacteristic
479C23C22×Q8×C3111yescharacteristic
480C23C2×#41×C3112yes—
481C23C2×#41×C3111yescharacteristic
482C23C2×#41×C3111yescharacteristic
483C23C31×H(5,𝔽2)12yes—
484C23C31×2−1+412yes—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—232no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—432no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4C2×C4⋊C4×C3114yes—
526C2×C4C2×C4⋊C4×C3114yes—
527C2×C4C4×D4×C3118yes—
528C2×C4C31×#12014yes—
529C2×C4C31×C22⋊Q818yes—
530C2×C4C31×#12314yes—
531C22—24no—
532C22—28no—
533C22—28no—
534C22—24no—
535C22—24no—
536C22—28no—
537C22—28no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22C31×#38918yescomplemented
542C22C22×C4⋊C4×C3111yescharacteristic
543C22C2×C4×D4×C3112yes—
544C22C2×C4×D4×C3111yescharacteristic
545C22C2×C4×Q8×C3111yescharacteristic
546C22C31×#52012yes—
547C22C2×C31×#12011yescharacteristic
548C22C2×C31×#12111yescharacteristic
549C22C2×C31×C22⋊Q811yescharacteristic
550C22C2×C31×C22⋊Q812yes—
551C22C2×C31×#12311yescharacteristic
552C22C2×C31×#12511yescharacteristic
553C22C2×C31×C4⋊Q811yescharacteristic
554C22C31×#53511yescharacteristic
555C22C31×#53611yescharacteristic
556C22C31×#54412yes—
557C22D42×C3112yes—
558C22C31×#54612yes—
559C22C31×#54712yes—
560C22C31×#54812yes—
561C22D4×Q8×C3112yes—
562C22C31×#55212yes—
563C22C31×#55312yes—
564C22C31×#55412yes—
565C4—24no—
566C4—48no—
567C4—48no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4C31×#38518yescomplemented
573C2—28no—
574C2C2×C31×#38511yescharacteristic
575C2C2×C31×#38911yescharacteristic
576C2C31×#231711yescharacteristic
577C2C31×#236012yes—
578C2C31×#236112yes—
579C2C31×#240811yescharacteristic
580C2C31×#240911yescharacteristic
581C2C31×#244012yes—
582C2C31×#244212yes—
583C2C31×#248611yescharacteristic
584C2C31×#248711yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#389C2semidirect
6. C31×#1444C2semidirect
7. C31×#1442C2semidirect
8. C31×C4⋊#95C2semidirect
25. C31×#380C4semidirect
62. C2×C4⋊C4×C31C2×C4semidirect
82. C31×#95C2×C4semidirect
134. C22×C4×C31C4⋊C4semidirect
158. #14130C31direct (N × H)
240. C2×#389C2×C31semidirect
244. #1444C2×C31semidirect
245. #1442C2×C31semidirect
246. C4⋊#95C2×C31semidirect
258. C22×C31#389semidirect
289. C4×C31#385semidirect
305. #380C4×C31semidirect
354. C2×C4⋊C4C2×C4×C31semidirect
374. #95C2×C4×C31semidirect
375. C31#14130direct (N × H)
427. C22×C4C4⋊C4×C31semidirect
541. C22C31×#389semidirect
572. C4C31×#385semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C312
C2×C31×#3892
C2×C31×#3852
C2×C31×#3802
C2×C4×C22⋊C4×C312
C31×#14442
C31×#14422
C31×C4⋊#952
C31×C22⋊#952
#1413031