Finite Group Databasebeta

← C31×#15177 #21471936

Subgroups — C31×#15177 #21471936

Order 7936.

2770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C212yesmaximal, complemented
2C22×C4×D4×C31C211yesmaximal, characteristic
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C2×C31×#380C212yesmaximal, complemented
5C31×#1444C212yesmaximal, complemented
6C31×#1439C214yesmaximal, complemented
7C31×#1438C212yesmaximal
8C31×C4⋊#95C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2212yes—
12C2×C31×C22⋊Q8—28no—
13C2×C31×C22⋊Q8—28no—
14C2×C31×C22⋊Q8—28no—
15C2×C4×D4×C31—28no—
16C2×C4×D4×C31—24no—
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2214yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2214yes—
23C22×C22⋊C4×C31C2214yes—
24C22×C42×C31C2211yescharacteristic
25C31×#391—24no—
26C31×#380—28no—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2214yes—
31C2×C31×#95C2212yes—
32C25×C31C2312yes—
33C22×Q8×C31—24no—
34C22×Q8×C31—24no—
35C22×Q8×C31—22no—
36C22×Q8×C31—22no—
37C22×Q8×C31D414yescomplemented
38C22×D4×C31—24no—
39C22×D4×C31—24no—
40C22×D4×C31—22no—
41C22×D4×C31—22no—
42C22×D4×C31Q814yescomplemented
43C23×C4×C31—24no—
44C23×C4×C31—24no—
45C23×C4×C31—24no—
46C23×C4×C31—24no—
47C23×C4×C31—24no—
48C23×C4×C31—24no—
49C23×C4×C31D418yes—
50C23×C4×C31C2312yes—
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2312yes—
53C23×C4×C31C2314yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C31×C22⋊Q8—264no—
58C4×D4×C31—232no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—28no—
62C2×C4⋊C4×C31—28no—
63C2×C4⋊C4×C31—28no—
64C2×C4⋊C4×C31—24no—
65C2×C4⋊C4×C31—24no—
66C2×C4⋊C4×C31—24no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31D418yescomplemented
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—28no—
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—28no—
74C2×C22⋊C4×C31—28no—
75C2×C22⋊C4×C31—28no—
76C2×C22⋊C4×C31—28no—
77C2×C22⋊C4×C31—44no—
78C2×C42×C31—24no—
79C2×C42×C31D414yes—
80C31×#95—28no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—24no—
84C31×#95—42no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31C2×D414yes—
98C24×C31C2×Q814yes—
99C24×C31#4114yes—
100C24×C31C2411yescharacteristic
101C2×Q8×C31—216no—
102C2×Q8×C31—28no—
103C2×Q8×C31—44no—
104C2×Q8×C31—48no—
105C2×Q8×C31—44no—
106C2×D4×C31—28no—
107C2×D4×C31—28no—
108C2×D4×C31—216no—
109C2×D4×C31—216no—
110C2×D4×C31—28no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—24no—
117C22×C4×C31—24no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—24no—
125C22×C4×C31—24no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—24no—
130C22×C4×C31—22no—
131C22×C4×C31—21no—
132C22×C4×C31—24no—
133C22×C4×C31—216no—
134C22×C4×C31—216no—
135C22×C4×C31—216no—
136C22×C4×C31—22no—
137C22×C4×C31—24no—
138C22×C4×C31—24no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—48no—
142C22×C4×C31—48no—
143C22×C4×C31—48no—
144C22×C4×C31—48no—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D418yes—
147C22×C4×C31C2×D414yes—
148C22×C4×C31C2×D414yes—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D414yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D414yes—
154C22×C4×C31C2×Q812yes—
155C22×C4×C31#4112yes—
156C22×C4×C31#4112yes—
157C4⋊C4×C31—28no—
158C4⋊C4×C31—232no—
159C4⋊C4×C31—216no—
160C4⋊C4×C31—44no—
161C22⋊C4×C31—216no—
162C22⋊C4×C31—232no—
163C22⋊C4×C31—416no—
164C42×C31—28no—
165#15177C3111yesmaximal, characteristic, direct product factor, Sylow
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×C31—24no—
176C23×C31—24no—
177C23×C31—24no—
178C23×C31—24no—
179C23×C31—24no—
180C23×C31—24no—
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31C22⋊Q818yescomplemented
201C23×C31C22×D412yes—
202C23×C31C22×D411yescharacteristic
203C23×C31C22×D411yescharacteristic
204C23×C31C22×D412yes—
205C23×C31C22×Q811yescharacteristic
206C23×C31C2×#4111yescharacteristic
207C23×C31C2×#4112yes—
208C23×C31C2×#4111yescharacteristic
209C23×C31H(5,𝔽2)11yescharacteristic
210C23×C312−1+412yes—
211C23×C312−1+411yescharacteristic
212Q8×C31—416no—
213D4×C31—232no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—22no—
219C2×C4×C31—28no—
220C2×C4×C31—24no—
221C2×C4×C31—28no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—28no—
228C2×C4×C31—28no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—44no—
234C2×C4×C31—44no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—44no—
241C2×C4×C31—44no—
242C2×C4×C31—432no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—44no—
247C2×C4×C31—44no—
248C2×C4×C31#12014yes—
249C2×C4×C31#12018yes—
250C2×C4×C31#12118yes—
251C2×C4×C31#12114yes—
252C2×C4×C31C22⋊Q814yescomplemented
253C2×C4×C31#12714yes—
254C22×C22⋊Q8C2×C3112yescomplemented
255C22×C4×D4C2×C3111yescharacteristic
256C2×#391C2×C3111yescharacteristic, complemented
257C2×#380C2×C3112yescomplemented
258#1444C2×C3112yescomplemented
259#1439C2×C3114yescomplemented
260#1438C2×C3112yes—
261C4⋊#95C2×C3111yescharacteristic, complemented
262C22×C31—28no—
263C22×C31—28no—
264C22×C31—28no—
265C22×C31—44no—
266C22×C31—44no—
267C22×C31—44no—
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—44no—
271C22×C31—44no—
272C22×C31—44no—
273C22×C31—44no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31C2×#12011yescharacteristic
278C22×C31C2×#12012yes—
279C22×C31C2×#12112yes—
280C22×C31C2×#12111yescharacteristic
281C22×C31C2×C22⋊Q812yes—
282C22×C31C2×C22⋊Q811yescharacteristic
283C22×C31#52512yes—
284C22×C31#52511yescharacteristic
285C22×C31C2×#12711yescharacteristic
286C22×C31#53612yes—
287C22×C31#53611yescharacteristic
288C22×C31#53712yes—
289C22×C31#53711yescharacteristic
290C22×C31#54612yes—
291C22×C31#54712yes—
292C22×C31#54812yes—
293C22×C31#54812yes—
294C22×C31D4×Q811yescharacteristic
295C22×C31D4×Q812yes—
296C22×C31#55011yescharacteristic
297C22×C31#55212yes—
298C22×C31#55411yescharacteristic
299C22×C31#55611yescharacteristic
300C4×C31—24no—
301C4×C31—42no—
302C4×C31—44no—
303C4×C31—44no—
304C4×C31—48no—
305C4×C31—82no—
306C4×C31#39218yes—
307C23×Q8C22×C3111yescharacteristic
308C23×D4C22×C3111yescharacteristic
309C24×C4C22×C3112yes—
310C2×C22⋊Q8—28no—
311C2×C22⋊Q8—28no—
312C2×C22⋊Q8—28no—
313C2×C4×D4—28no—
314C2×C4×D4—24no—
315C22×C4⋊C4C22×C3111yescharacteristic
316C22×C4⋊C4C22×C3114yes—
317C22×C4⋊C4C22×C3111yescharacteristic
318C22×C4⋊C4C22×C3112yes—
319C22×C22⋊C4C22×C3112yes—
320C22×C22⋊C4C22×C3114yes—
321C22×C22⋊C4C22×C3114yes—
322C22×C42C22×C3111yescharacteristic
323#391—24no—
324#380—28no—
325C2×#95C22×C3112yes—
326C2×#95C22×C3112yes—
327C2×#95C22×C3112yes—
328C2×#95C22×C3114yes—
329C2×#95C22×C3112yes—
330C2×C31—48no—
331C2×C31C2×#39211yescharacteristic
332C2×C31#240712yes—
333C2×C31#240912yes—
334C2×C31#241512yes—
335C2×C31#241712yes—
336C2×C31#253611yescharacteristic
337C2×C31#253811yescharacteristic
338C2×C31#254011yescharacteristic
339C2×C31#254211yescharacteristic
340C2×C31#261611yescharacteristic
341C2×C31#284811yescharacteristic
342C25C23×C3112yes—
343C22×Q8—24no—
344C22×Q8—24no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8D4×C3114yescomplemented
348C22×D4—24no—
349C22×D4—24no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4Q8×C3114yescomplemented
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4D4×C3118yes—
360C23×C4C23×C3112yes—
361C23×C4C23×C3112yes—
362C23×C4C23×C3114yes—
363C23×C4C23×C3111yescharacteristic
364C23×C4C23×C3112yes—
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C22⋊Q8—264no—
368C4×D4—232no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C3118yescomplemented
379C2×C22⋊C4—24no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—28no—
384C2×C22⋊C4—28no—
385C2×C22⋊C4—28no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—44no—
388C2×C42—24no—
389C2×C42D4×C3114yes—
390#95—28no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—42no—
395C31#1517711yescharacteristic, direct product factor, Sylow
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24C2×D4×C3114yes—
409C24C2×Q8×C3114yes—
410C24#41×C3114yes—
411C24C24×C3111yescharacteristic
412C2×Q8—216no—
413C2×Q8—28no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—44no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4C2×D4×C3112yes—
457C22×C4C2×D4×C3118yes—
458C22×C4C2×D4×C3114yes—
459C22×C4C2×D4×C3114yes—
460C22×C4C2×D4×C3112yes—
461C22×C4C2×D4×C3114yes—
462C22×C4C2×D4×C3112yes—
463C22×C4C2×D4×C3112yes—
464C22×C4C2×D4×C3114yes—
465C22×C4C2×Q8×C3112yes—
466C22×C4#41×C3112yes—
467C22×C4#41×C3112yes—
468C4⋊C4—28no—
469C4⋊C4—232no—
470C4⋊C4—216no—
471C4⋊C4—44no—
472C22⋊C4—216no—
473C22⋊C4—232no—
474C22⋊C4—416no—
475C42—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C31×C22⋊Q818yescomplemented
511C23C22×D4×C3112yes—
512C23C22×D4×C3111yescharacteristic
513C23C22×D4×C3111yescharacteristic
514C23C22×D4×C3112yes—
515C23C22×Q8×C3111yescharacteristic
516C23C2×#41×C3111yescharacteristic
517C23C2×#41×C3112yes—
518C23C2×#41×C3111yescharacteristic
519C23C31×H(5,𝔽2)11yescharacteristic
520C23C31×2−1+412yes—
521C23C31×2−1+411yescharacteristic
522Q8—416no—
523D4—232no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—432no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4C31×#12014yes—
559C2×C4C31×#12018yes—
560C2×C4C31×#12118yes—
561C2×C4C31×#12114yes—
562C2×C4C31×C22⋊Q814yescomplemented
563C2×C4C31×#12714yes—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22C2×C31×#12011yescharacteristic
580C22C2×C31×#12012yes—
581C22C2×C31×#12112yes—
582C22C2×C31×#12111yescharacteristic
583C22C2×C31×C22⋊Q812yes—
584C22C2×C31×C22⋊Q811yescharacteristic
585C22C31×#52512yes—
586C22C31×#52511yescharacteristic
587C22C2×C31×#12711yescharacteristic
588C22C31×#53612yes—
589C22C31×#53611yescharacteristic
590C22C31×#53712yes—
591C22C31×#53711yescharacteristic
592C22C31×#54612yes—
593C22C31×#54712yes—
594C22C31×#54812yes—
595C22C31×#54812yes—
596C22D4×Q8×C3111yescharacteristic
597C22D4×Q8×C3112yes—
598C22C31×#55011yescharacteristic
599C22C31×#55212yes—
600C22C31×#55411yescharacteristic
601C22C31×#55611yescharacteristic
602C4—24no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—48no—
607C4—82no—
608C4C31×#39218yes—
609C2—48no—
610C2C2×C31×#39211yescharacteristic
611C2C31×#240712yes—
612C2C31×#240912yes—
613C2C31×#241512yes—
614C2C31×#241712yes—
615C2C31×#253611yescharacteristic
616C2C31×#253811yescharacteristic
617C2C31×#254011yescharacteristic
618C2C31×#254211yescharacteristic
619C2C31×#261611yescharacteristic
620C2C31×#284811yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×C22⋊Q8C2semidirect
3. C2×C31×#391C2semidirect
4. C2×C31×#380C2semidirect
5. C31×#1444C2semidirect
6. C31×#1439C2semidirect
8. C31×C4⋊#95C2semidirect
37. C22×Q8×C31D4semidirect
42. C22×D4×C31Q8semidirect
68. C2×C4⋊C4×C31D4semidirect
165. #15177C31direct (N × H)
200. C23×C31C22⋊Q8semidirect
252. C2×C4×C31C22⋊Q8semidirect
254. C22×C22⋊Q8C2×C31semidirect
256. C2×#391C2×C31semidirect
257. C2×#380C2×C31semidirect
258. #1444C2×C31semidirect
259. #1439C2×C31semidirect
261. C4⋊#95C2×C31semidirect
347. C22×Q8D4×C31semidirect
352. C22×D4Q8×C31semidirect
378. C2×C4⋊C4D4×C31semidirect
395. C31#15177direct (N × H)
510. C23C31×C22⋊Q8semidirect
562. 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
C22×C4×D4×C312
C2×C31×#3912
C2×C31×#3802
C31×#14442
C31×#14392
C31×#14382
C31×C4⋊#952
#1517731