Finite Group Databasebeta

← C31×#15182 #21471941

Subgroups — C31×#15182 #21471941

Order 7936.

1990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C211yesmaximal, characteristic
2C2×C31×#391C211yesmaximal, characteristic, complemented
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C2×C31×#384C211yesmaximal, characteristic, complemented
5C2×C31×#380C211yesmaximal, characteristic
6C2×C4×C22⋊C4×C31C211yesmaximal, characteristic
7C31×#1441C214yesmaximal, complemented
8C31×#1439C214yesmaximal
9C31×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Q8×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C2×C31×C22⋊Q8—24no—
13C2×C31×C22⋊Q8—24no—
14C2×C31×C22⋊Q8—24no—
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2214yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2212yes—
23C22×C42×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#391—24no—
26C31×#391—24no—
27C31×#384—24no—
28C31×#380—24no—
29C4×C22⋊C4×C31—24no—
30C2×C31×#95C2214yes—
31C2×C31×#95C2214yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2214yes—
35C25×C31C2311yescharacteristic
36C22×Q8×C31—22no—
37C22×Q8×C31—22no—
38C22×Q8×C31—22no—
39C22×Q8×C31—22no—
40C22×Q8×C31—22no—
41C22×Q8×C31—22no—
42C22×Q8×C31D414yes—
43C23×C4×C31—22no—
44C23×C4×C31—22no—
45C23×C4×C31—22no—
46C23×C4×C31—22no—
47C23×C4×C31—22no—
48C23×C4×C31—22no—
49C23×C4×C31D414yes—
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2312yes—
56C23×C4×C31C2312yes—
57C23×C4×C31C2311yescharacteristic
58C31×C22⋊Q8—232no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—28no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—28no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—44no—
73C2×C4⋊C4×C31D414yes—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—24no—
78C2×C22⋊C4×C31—24no—
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—44no—
81C2×C22⋊C4×C31Q818yescomplemented
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—22no—
85C2×C42×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31D414yes—
88C31×#95—24no—
89C31×#95—24no—
90C31×#95—28no—
91C31×#95—28no—
92C31×#95—44no—
93C24×C31—21no—
94C24×C31—21no—
95C24×C31—21no—
96C24×C31—22no—
97C24×C31—21no—
98C24×C31—21no—
99C24×C31—21no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31C2×D412yes—
103C24×C31C2×Q814yes—
104C24×C31C2411yescharacteristic
105C2×Q8×C31—28no—
106C2×Q8×C31—28no—
107C2×Q8×C31—28no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C2×Q8×C31—44no—
111C2×Q8×C31—44no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
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—22no—
127C22×C4×C31—22no—
128C22×C4×C31—24no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—24no—
137C22×C4×C31—22no—
138C22×C4×C31—28no—
139C22×C4×C31—28no—
140C22×C4×C31—28no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—44no—
152C22×C4×C31—44no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31C2×D414yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D414yes—
158C22×C4×C31C2×D414yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×Q814yes—
164C22×C4×C31C2×Q814yes—
165C22×C4×C31#4114yes—
166C22×C4×C31#4112yes—
167C22×C4×C31#4112yes—
168C22×C4×C31#4112yes—
169C4⋊C4×C31—216no—
170C4⋊C4×C31—28no—
171C4⋊C4×C31—416no—
172C4⋊C4×C31—44no—
173C4⋊C4×C31—44no—
174C22⋊C4×C31—216no—
175C22⋊C4×C31—216no—
176C22⋊C4×C31—48no—
177C42×C31—28no—
178C42×C31—44no—
179#15182C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—24no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—24no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—24no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31C4⋊Q814yes—
203C23×C31C22×D411yescharacteristic
204C23×C31C22×D411yescharacteristic
205C23×C31C22×D411yescharacteristic
206C23×C31C22×D411yescharacteristic
207C23×C31C22×Q812yes—
208C23×C31C2×#4111yescharacteristic
209C23×C31C2×#4111yescharacteristic
210C23×C31C2×#4111yescharacteristic
211C23×C31C2×#4112yes—
212C23×C31H(5,𝔽2)12yes—
213C23×C312−1+412yes—
214Q8×C31—416no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—22no—
228C2×C4×C31—24no—
229C2×C4×C31—28no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—44no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—416no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—44no—
255C2×C4×C31—44no—
256C2×C4×C31—44no—
257C2×C4×C31—44no—
258C2×C4×C31#12014yes—
259C2×C4×C31#12114yes—
260C2×C4×C31#12114yes—
261C2×C4×C31#12114yes—
262C2×C4×C31C22⋊Q818yescomplemented
263C2×C4×C31#12314yes—
264C2×C4×C31#12314yes—
265C2×C4×C31#12414yes—
266C22×C22⋊Q8C2×C3111yescharacteristic
267C2×#391C2×C3111yescharacteristic, complemented
268C2×#391C2×C3111yescharacteristic, complemented
269C2×#384C2×C3111yescharacteristic, complemented
270C2×#380C2×C3111yescharacteristic
271C2×C4×C22⋊C4C2×C3111yescharacteristic
272#1441C2×C3114yescomplemented
273#1439C2×C3114yes—
274C4⋊#95C2×C3111yescharacteristic, complemented
275C22×C31—24no—
276C22×C31—24no—
277C22×C31—24no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—42no—
283C22×C31—44no—
284C22×C31—42no—
285C22×C31—44no—
286C22×C31C2×#12011yescharacteristic
287C22×C31C2×#12111yescharacteristic
288C22×C31C2×#12111yescharacteristic
289C22×C31C2×#12111yescharacteristic
290C22×C31C2×C22⋊Q812yes—
291C22×C31C2×#12311yescharacteristic
292C22×C31C2×#12311yescharacteristic
293C22×C31C2×#12411yescharacteristic
294C22×C31C2×C4⋊Q811yescharacteristic
295C22×C31#53211yescharacteristic
296C22×C31#53312yes—
297C22×C31#53511yescharacteristic
298C22×C31#53611yescharacteristic
299C22×C31#53712yes—
300C22×C31#54212yes—
301C22×C31#54612yes—
302C22×C31D4×Q812yes—
303C22×C31D4×Q812yes—
304C22×C31#55012yes—
305C22×C31#55212yes—
306C22×C31#55412yes—
307C22×C31#55412yes—
308C22×C31#55612yes—
309C4×C31—28no—
310C4×C31—42no—
311C4×C31—44no—
312C4×C31—44no—
313C4×C31—44no—
314C4×C31—82no—
315C4×C31—82no—
316C4×C31#39418yes—
317C23×Q8C22×C3111yescharacteristic
318C24×C4C22×C3111yescharacteristic
319C2×C22⋊Q8—24no—
320C2×C22⋊Q8—24no—
321C2×C22⋊Q8—24no—
322C22×C4⋊C4C22×C3112yes—
323C22×C4⋊C4C22×C3111yescharacteristic
324C22×C4⋊C4C22×C3111yescharacteristic
325C22×C4⋊C4C22×C3114yes—
326C22×C4⋊C4C22×C3111yescharacteristic
327C22×C22⋊C4C22×C3112yes—
328C22×C22⋊C4C22×C3112yes—
329C22×C22⋊C4C22×C3112yes—
330C22×C42C22×C3111yescharacteristic
331C22×C42C22×C3111yescharacteristic
332#391—24no—
333#391—24no—
334#384—24no—
335#380—24no—
336C4×C22⋊C4—24no—
337C2×#95C22×C3114yes—
338C2×#95C22×C3114yes—
339C2×#95C22×C3112yes—
340C2×#95C22×C3112yes—
341C2×#95C22×C3114yes—
342C2×C31—44no—
343C2×C31C2×#39411yescharacteristic
344C2×C31#241712yes—
345C2×C31#242312yes—
346C2×C31#243111yescharacteristic
347C2×C31#243411yescharacteristic
348C2×C31#249212yes—
349C2×C31#249911yescharacteristic
350C2×C31#250211yescharacteristic
351C2×C31#253812yes—
352C2×C31#262011yescharacteristic
353C2×C31#285711yescharacteristic
354C25C23×C3111yescharacteristic
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8D4×C3114yes—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4D4×C3114yes—
369C23×C4C23×C3112yes—
370C23×C4C23×C3112yes—
371C23×C4C23×C3112yes—
372C23×C4C23×C3112yes—
373C23×C4C23×C3111yescharacteristic
374C23×C4C23×C3112yes—
375C23×C4C23×C3112yes—
376C23×C4C23×C3111yescharacteristic
377C22⋊Q8—232no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—28no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—44no—
392C2×C4⋊C4D4×C3114yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—44no—
400C2×C22⋊C4Q8×C3118yescomplemented
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42D4×C3114yes—
407#95—24no—
408#95—24no—
409#95—28no—
410#95—28no—
411#95—44no—
412C31#1518211yescharacteristic, direct product factor, Sylow
413C24—21no—
414C24—21no—
415C24—21no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24C2×D4×C3112yes—
423C24C2×Q8×C3114yes—
424C24C24×C3111yescharacteristic
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4C2×D4×C3114yes—
476C22×C4C2×D4×C3112yes—
477C22×C4C2×D4×C3114yes—
478C22×C4C2×D4×C3114yes—
479C22×C4C2×D4×C3112yes—
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3112yes—
483C22×C4C2×Q8×C3114yes—
484C22×C4C2×Q8×C3114yes—
485C22×C4#41×C3114yes—
486C22×C4#41×C3112yes—
487C22×C4#41×C3112yes—
488C22×C4#41×C3112yes—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C42—28no—
498C42—44no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C31×C4⋊Q814yes—
522C23C22×D4×C3111yescharacteristic
523C23C22×D4×C3111yescharacteristic
524C23C22×D4×C3111yescharacteristic
525C23C22×D4×C3111yescharacteristic
526C23C22×Q8×C3112yes—
527C23C2×#41×C3111yescharacteristic
528C23C2×#41×C3111yescharacteristic
529C23C2×#41×C3111yescharacteristic
530C23C2×#41×C3112yes—
531C23C31×H(5,𝔽2)12yes—
532C23C31×2−1+412yes—
533Q8—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4C31×#12014yes—
578C2×C4C31×#12114yes—
579C2×C4C31×#12114yes—
580C2×C4C31×#12114yes—
581C2×C4C31×C22⋊Q818yescomplemented
582C2×C4C31×#12314yes—
583C2×C4C31×#12314yes—
584C2×C4C31×#12414yes—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22C2×C31×#12011yescharacteristic
597C22C2×C31×#12111yescharacteristic
598C22C2×C31×#12111yescharacteristic
599C22C2×C31×#12111yescharacteristic
600C22C2×C31×C22⋊Q812yes—
601C22C2×C31×#12311yescharacteristic
602C22C2×C31×#12311yescharacteristic
603C22C2×C31×#12411yescharacteristic
604C22C2×C31×C4⋊Q811yescharacteristic
605C22C31×#53211yescharacteristic
606C22C31×#53312yes—
607C22C31×#53511yescharacteristic
608C22C31×#53611yescharacteristic
609C22C31×#53712yes—
610C22C31×#54212yes—
611C22C31×#54612yes—
612C22D4×Q8×C3112yes—
613C22D4×Q8×C3112yes—
614C22C31×#55012yes—
615C22C31×#55212yes—
616C22C31×#55412yes—
617C22C31×#55412yes—
618C22C31×#55612yes—
619C4—28no—
620C4—42no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—82no—
625C4—82no—
626C4C31×#39418yes—
627C2—44no—
628C2C2×C31×#39411yescharacteristic
629C2C31×#241712yes—
630C2C31×#242312yes—
631C2C31×#243111yescharacteristic
632C2C31×#243411yescharacteristic
633C2C31×#249212yes—
634C2C31×#249911yescharacteristic
635C2C31×#250211yescharacteristic
636C2C31×#253812yes—
637C2C31×#262011yescharacteristic
638C2C31×#285711yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#391C2semidirect
3. C2×C31×#391C2semidirect
4. C2×C31×#384C2semidirect
7. C31×#1441C2semidirect
9. C31×C4⋊#95C2semidirect
81. C2×C22⋊C4×C31Q8semidirect
179. #15182C31direct (N × H)
262. C2×C4×C31C22⋊Q8semidirect
267. C2×#391C2×C31semidirect
268. C2×#391C2×C31semidirect
269. C2×#384C2×C31semidirect
272. #1441C2×C31semidirect
274. C4⋊#95C2×C31semidirect
400. C2×C22⋊C4Q8×C31semidirect
412. C31#15182direct (N × H)
581. C2×C4C31×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C2×C31×#3912
C2×C31×#3842
C2×C31×#3802
C2×C4×C22⋊C4×C312
C31×#14412
C31×#14392
C31×C4⋊#952
#1518231