Finite Group Databasebeta

← C31×#26118 #21482877

Subgroups — C31×#26118 #21482877

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2570C211yesmaximal, characteristic, complemented
2D4×C4⋊C4×C31C211yesmaximal, characteristic
3C31×#2101C211yesmaximal, characteristic, complemented
4C31×#2099C211yesmaximal, characteristic, complemented
5C31×#2088C211yesmaximal, characteristic, complemented
6C31×#2088C211yesmaximal, characteristic, complemented
7C31×#1948C211yesmaximal, characteristic, complemented
8C31×#1940C211yesmaximal, characteristic, complemented
9C31×#1864C211yesmaximal, characteristic
10C31×#1864C211yesmaximal, characteristic
11C31×#1836C211yesmaximal, characteristic, complemented
12C31×#1804C211yesmaximal, characteristic
13C31×#1804C211yesmaximal, characteristic
14C31×#1802C211yesmaximal, characteristic
15C31×#1762C211yesmaximal, characteristic
16C2×C4×D4×C31C2211yescharacteristic
17C2×C4×D4×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C22×C4⋊C4×C31—21no—
20C2×C31×#106C2211yescharacteristic
21C2×C31×#106C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#102C2211yescharacteristic
26C2×C31×#102C2211yescharacteristic
27C2×C31×#102C2211yescharacteristic
28C2×C31×#102C2211yescharacteristic
29C2×C31×#97C2211yescharacteristic
30C31×#400—21no—
31C31×#398—21no—
32C31×#389C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#389C2211yescharacteristic
36C31×#387—21no—
37C31×#387—21no—
38C31×#385—21no—
39C31×#385—21no—
40C31×#384C2211yescharacteristic
41C31×#384C2211yescharacteristic
42C31×#382C2211yescharacteristic
43C31×#382C2211yescharacteristic
44C31×#380—21no—
45C4×C4⋊C4×C31C2211yescharacteristic
46C4×C4⋊C4×C31C2211yescharacteristic
47C31×#340C2211yescharacteristic
48C31×#340C2211yescharacteristic
49C31×#340C2211yescharacteristic
50C31×#340C2211yescharacteristic
51C31×#340C2211yescharacteristic
52C31×#340C2211yescharacteristic
53C31×#340C2211yescharacteristic
54C31×#340C2211yescharacteristic
55C31×#340C2211yescharacteristic
56C31×#340C2211yescharacteristic
57C31×#336C2211yescharacteristic
58C31×#336C2211yescharacteristic
59C22×D4×C31C2311yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31—21no—
62C23×C4×C31—21no—
63C22×C8×C31C2311yescharacteristic
64C22×C8×C31C2311yescharacteristic
65C22×C8×C31C2311yescharacteristic
66C22×C8×C31C2311yescharacteristic
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—41no—
71C4×D4×C31—41no—
72C4×D4×C31Q814yes—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—44no—
82C2×C4⋊C4×C31D411yescharacteristic
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C42×C31—21no—
95C2×C42×C31C2311yescharacteristic
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2311yescharacteristic
98C31×#106—41no—
99C31×#106—41no—
100C31×C4⋊C8—22no—
101C31×C4⋊C8—41no—
102C31×C4⋊C8—41no—
103C31×#102—22no—
104C31×#102—22no—
105C31×#102—22no—
106C31×#102—22no—
107C31×#97—22no—
108C31×#95—21no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95D411yescharacteristic
118C31×#95D411yescharacteristic
119C24×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31C2×Q812yes—
124C2×D4×C31#4112yes—
125C2×D4×C31#4112yes—
126C22×C4×C31—21no—
127C22×C4×C31—24no—
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—42no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—81no—
141C22×C4×C31—81no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2411yescharacteristic
149C2×C8×C31—21no—
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—21no—
158C2×C8×C31—21no—
159C2×C8×C31—21no—
160C2×C8×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—41no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31—42no—
178C4⋊C4×C31—42no—
179C4⋊C4×C31—42no—
180C4⋊C4×C31—42no—
181C4⋊C4×C31—81no—
182C4⋊C4×C31—81no—
183C4⋊C4×C31—81no—
184C4⋊C4×C31C2×Q812yes—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—42no—
187C22⋊C4×C31—42no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—21no—
191C42×C31—21no—
192C42×C31—81no—
193C42×C31C2×Q812yes—
194C42×C31#4112yes—
195#26118C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31C22×D411yescharacteristic
204D4×C31—22no—
205D4×C31—22no—
206D4×C31—22no—
207D4×C31#12514yes—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—22no—
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—21no—
223C2×C4×C31—21no—
224C2×C4×C31—48no—
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—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—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—84no—
246C2×C4×C31—84no—
247C2×C4×C31C22⋊Q812yes—
248C2×C4×C31C22⋊Q812yes—
249C2×C4×C31AGL(1,ℤ/8)12yes—
250C2×C4×C31C22×Q811yescharacteristic
251C2×C4×C31C2×#4111yescharacteristic
252C2×C4×C31C2×#4111yescharacteristic
253C2×C4×C31C2×#4111yescharacteristic
254C2×C4×C31H(5,𝔽2)11yescharacteristic
255C2×C4×C312−1+411yescharacteristic
256C8×C31—41no—
257C8×C31—41no—
258C8×C31—41no—
259C8×C31—41no—
260#2570C2×C3111yescharacteristic, complemented
261D4×C4⋊C4C2×C3111yescharacteristic
262#2101C2×C3111yescharacteristic, complemented
263#2099C2×C3111yescharacteristic, complemented
264#2088C2×C3111yescharacteristic, complemented
265#2088C2×C3111yescharacteristic, complemented
266#1948C2×C3111yescharacteristic, complemented
267#1940C2×C3111yescharacteristic, complemented
268#1864C2×C3111yescharacteristic
269#1864C2×C3111yescharacteristic
270#1836C2×C3111yescharacteristic, complemented
271#1804C2×C3111yescharacteristic
272#1804C2×C3111yescharacteristic
273#1802C2×C3111yescharacteristic
274#1762C2×C3111yescharacteristic
275C22×C31—24no—
276C22×C31—42no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31C2×C22⋊Q811yescharacteristic
283C22×C31#54611yescharacteristic
284C22×C31#54811yescharacteristic
285C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C31#57511yescharacteristic
287C22×C31#57711yescharacteristic
288C22×C31#57711yescharacteristic
289C4×C31—22no—
290C4×C31—22no—
291C4×C31—22no—
292C4×C31—42no—
293C4×C31—41no—
294C4×C31—41no—
295C4×C31—41no—
296C4×C31—81no—
297C4×C31—81no—
298C4×C31—81no—
299C4×C31—81no—
300C4×C31—81no—
301C4×C31—81no—
302C4×C31C2×#12511yescharacteristic
303C4×C31#54411yescharacteristic
304C4×C31#55211yescharacteristic
305C4×C31#55311yescharacteristic
306C2×C4×D4C22×C3111yescharacteristic
307C2×C4×D4C22×C3111yescharacteristic
308C2×C4×D4C22×C3111yescharacteristic
309C22×C4⋊C4—21no—
310C2×#106C22×C3111yescharacteristic
311C2×#106C22×C3111yescharacteristic
312C2×C4⋊C8C22×C3111yescharacteristic
313C2×C4⋊C8C22×C3111yescharacteristic
314C2×C4⋊C8C22×C3111yescharacteristic
315C2×#102C22×C3111yescharacteristic
316C2×#102C22×C3111yescharacteristic
317C2×#102C22×C3111yescharacteristic
318C2×#102C22×C3111yescharacteristic
319C2×#97C22×C3111yescharacteristic
320#400—21no—
321#398—21no—
322#389C22×C3111yescharacteristic
323#389C22×C3111yescharacteristic
324#389C22×C3111yescharacteristic
325#389C22×C3111yescharacteristic
326#387—21no—
327#387—21no—
328#385—21no—
329#385—21no—
330#384C22×C3111yescharacteristic
331#384C22×C3111yescharacteristic
332#382C22×C3111yescharacteristic
333#382C22×C3111yescharacteristic
334#380—21no—
335C4×C4⋊C4C22×C3111yescharacteristic
336C4×C4⋊C4C22×C3111yescharacteristic
337#340C22×C3111yescharacteristic
338#340C22×C3111yescharacteristic
339#340C22×C3111yescharacteristic
340#340C22×C3111yescharacteristic
341#340C22×C3111yescharacteristic
342#340C22×C3111yescharacteristic
343#340C22×C3111yescharacteristic
344#340C22×C3111yescharacteristic
345#340C22×C3111yescharacteristic
346#340C22×C3111yescharacteristic
347#336C22×C3111yescharacteristic
348#336C22×C3111yescharacteristic
349C2×C31—44no—
350C2×C31#244211yescharacteristic
351C2×C31#307711yescharacteristic
352C2×C31#308411yescharacteristic
353C2×C31#330711yescharacteristic
354C2×C31#331011yescharacteristic
355C2×C31#337311yescharacteristic
356C2×C31#337611yescharacteristic
357C22×D4C23×C3111yescharacteristic
358C23×C4—21no—
359C23×C4—21no—
360C23×C4—21no—
361C22×C8C23×C3111yescharacteristic
362C22×C8C23×C3111yescharacteristic
363C22×C8C23×C3111yescharacteristic
364C22×C8C23×C3111yescharacteristic
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—41no—
369C4×D4—41no—
370C4×D4Q8×C3114yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—44no—
380C2×C4⋊C4D4×C3111yescharacteristic
381C2×C4⋊C4D4×C3111yescharacteristic
382C2×C4⋊C4C23×C3111yescharacteristic
383C2×C4⋊C4C23×C3111yescharacteristic
384C2×C4⋊C4C23×C3111yescharacteristic
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C4⋊C4C23×C3111yescharacteristic
387C2×C4⋊C4C23×C3111yescharacteristic
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42C23×C3111yescharacteristic
394C2×C42C23×C3111yescharacteristic
395C2×C42C23×C3111yescharacteristic
396#106—41no—
397#106—41no—
398C4⋊C8—22no—
399C4⋊C8—41no—
400C4⋊C8—41no—
401#102—22no—
402#102—22no—
403#102—22no—
404#102—22no—
405#97—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95D4×C3111yescharacteristic
416#95D4×C3111yescharacteristic
417C31#2611811yescharacteristic, direct product factor, Sylow
418C24—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Q8×C3112yes—
423C2×D4#41×C3112yes—
424C2×D4#41×C3112yes—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C24×C3111yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4C2×Q8×C3112yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—81no—
492C42C2×Q8×C3112yes—
493C42#41×C3112yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×D4×C3111yescharacteristic
502D4—22no—
503D4—22no—
504D4—22no—
505D4C31×#12514yes—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
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—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4C31×C22⋊Q812yes—
546C2×C4C31×C22⋊Q812yes—
547C2×C4C31×AGL(1,ℤ/8)12yes—
548C2×C4C22×Q8×C3111yescharacteristic
549C2×C4C2×#41×C3111yescharacteristic
550C2×C4C2×#41×C3111yescharacteristic
551C2×C4C2×#41×C3111yescharacteristic
552C2×C4C31×H(5,𝔽2)11yescharacteristic
553C2×C4C31×2−1+411yescharacteristic
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C22—24no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22C2×C31×C22⋊Q811yescharacteristic
566C22C31×#54611yescharacteristic
567C22C31×#54811yescharacteristic
568C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
569C22C31×#57511yescharacteristic
570C22C31×#57711yescharacteristic
571C22C31×#57711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4C2×C31×#12511yescharacteristic
586C4C31×#54411yescharacteristic
587C4C31×#55211yescharacteristic
588C4C31×#55311yescharacteristic
589C2—44no—
590C2C31×#244211yescharacteristic
591C2C31×#307711yescharacteristic
592C2C31×#308411yescharacteristic
593C2C31×#330711yescharacteristic
594C2C31×#331011yescharacteristic
595C2C31×#337311yescharacteristic
596C2C31×#337611yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2570C2semidirect
3. C31×#2101C2semidirect
4. C31×#2099C2semidirect
5. C31×#2088C2semidirect
6. C31×#2088C2semidirect
7. C31×#1948C2semidirect
8. C31×#1940C2semidirect
11. C31×#1836C2semidirect
195. #26118C31direct (N × H)
260. #2570C2×C31semidirect
262. #2101C2×C31semidirect
263. #2099C2×C31semidirect
264. #2088C2×C31semidirect
265. #2088C2×C31semidirect
266. #1948C2×C31semidirect
267. #1940C2×C31semidirect
270. #1836C2×C31semidirect
417. C31#26118direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25702
D4×C4⋊C4×C312
C31×#21012
C31×#20992
C31×#20882
C31×#19482
C31×#19402
C31×#18642
C31×#18362
C31×#18042
C31×#18022
C31×#17622
#2611831