Finite Group Databasebeta

← C31×#23570 #21480329

Subgroups — C31×#23570 #21480329

Order 7936.

1436 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic
2C2×C4×SD16×C31C212yesmaximal, complemented
3C4×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
4C31×#2058C211yesmaximal, characteristic, complemented
5C31×#2041C211yesmaximal, characteristic, complemented
6C31×#2034C212yesmaximal, complemented
7C31×#2024C211yesmaximal, characteristic, complemented
8C31×#2014C211yesmaximal, characteristic, complemented
9C31×#2000C212yesmaximal, complemented
10C4×C31×#102C212yesmaximal, complemented
11C4×C31×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41×C31—21no—
13C22×SD16×C31C2212yes—
14C31×#532—28no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#127C2211yescharacteristic
17C2×C31×#124—21no—
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×#121—21no—
20C2×C31×#121C2212yes—
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2212yes—
24C22×C42×C31C2211yescharacteristic
25C4×SD16×C31—28no—
26C2×C31×#106C2212yes—
27C2×C31×#103C2212yes—
28C2×C31×#102C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×C22⋊C8C2212yes—
32C2×C4×C8×C31C2212yes—
33C31×#391C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#387—21no—
36C31×#385—21no—
37C31×#380C2211yescharacteristic
38C4×C4⋊C4×C31—21no—
39C4×C4⋊C4×C31C2211yescharacteristic
40C4×C22⋊C4×C31—21no—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#336C2211yescharacteristic
44C31×#336C2211yescharacteristic
45C2×#41×C31—28no—
46C2×#41×C31—44no—
47C2×#41×C31—44no—
48C22×Q8×C31—21no—
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—22no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31C2312yes—
54C23×C4×C31—22no—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31D411yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C2×SD16×C31—24no—
59C2×SD16×C31—24no—
60C2×SD16×C31—24no—
61C22×C8×C31C2312yes—
62C22×C8×C31C2312yes—
63C31×C4⋊Q8—24no—
64C31×#127—24no—
65C31×#124—28no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×#121—28no—
69C31×#121—24no—
70C31×#121—24no—
71C4×Q8×C31—24no—
72C4×D4×C31—28no—
73C4×D4×C31—24no—
74C4×D4×C31D418yes—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C42×C31—24no—
88C2×C42×C31—21no—
89C2×C42×C31—21no—
90C2×C42×C31D411yescharacteristic
91C2×C42×C31D411yescharacteristic
92C2×C42×C31C2311yescharacteristic
93C2×C42×C31C2311yescharacteristic
94C31×#106—24no—
95C31×#103—24no—
96C31×#102—24no—
97C31×#102—24no—
98C31×#102—24no—
99C31×C22⋊C8—24no—
100C4×C8×C31—24no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—21no—
105C31×#95—21no—
106C24×C31—22no—
107C24×C31C2×D411yescharacteristic
108#41×C31—432no—
109C2×Q8×C31—24no—
110C2×Q8×C31—21no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—42no—
114C2×Q8×C31—42no—
115C2×Q8×C31#4112yescomplemented
116C2×D4×C31—24no—
117C2×D4×C31—28no—
118C2×D4×C31—24no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31C2×D414yes—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—28no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—41no—
146C22×C4×C31—44no—
147C22×C4×C31—44no—
148C22×C4×C31SD1614yescomplemented
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2411yescharacteristic
155SD16×C31—216no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31#4114yescomplemented
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31C2×D414yes—
171C4⋊C4×C31#4112yescomplemented
172C22⋊C4×C31—28no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—28no—
175C22⋊C4×C31—42no—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31—22no—
179C42×C31—42no—
180C42×C31—42no—
181C42×C31C2×D414yes—
182#23570C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—24no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31C2×SD1612yes—
197C23×C31C22×D411yescharacteristic
198Q8×C31—24no—
199Q8×C31—48no—
200Q8×C31—42no—
201D4×C31—24no—
202D4×C31—24no—
203D4×C31—48no—
204D4×C31—48no—
205D4×C31—416no—
206D4×C31—42no—
207D4×C31—42no—
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—28no—
215C2×C4×C31—21no—
216C2×C4×C31—24no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—44no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—416no—
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#12012yes—
238C2×C4×C31#12012yes—
239C2×C4×C31C2×SD1612yes—
240C2×C4×C31C2×SD1612yes—
241C2×C4×C31#13512yes—
242C2×C4×C31AGL(1,ℤ/8)12yes—
243C2×C4×C31C22×D412yes—
244C2×C4×C31C2×#4111yescharacteristic
245C2×C4×C31C2×#4112yes—
246C2×C4×C31C2×#4111yescharacteristic
247C8×C31—24no—
248C8×C31—42no—
249C2×#532C2×C3111yescharacteristic
250C2×C4×SD16C2×C3112yescomplemented
251C4×C22⋊Q8C2×C3111yescharacteristic, complemented
252#2058C2×C3111yescharacteristic, complemented
253#2041C2×C3111yescharacteristic, complemented
254#2034C2×C3112yescomplemented
255#2024C2×C3111yescharacteristic, complemented
256#2014C2×C3111yescharacteristic, complemented
257#2000C2×C3112yescomplemented
258C4×#102C2×C3112yescomplemented
259C4×C22⋊C8C2×C3111yescharacteristic, complemented
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—44no—
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—82no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31C2×#12011yescharacteristic
275C22×C31#52511yescharacteristic
276C22×C31#53211yescharacteristic
277C22×C31C22×SD1611yescharacteristic
278C22×C31C2×#13511yescharacteristic
279C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C31#57511yescharacteristic
281C4×C31—22no—
282C4×C31—22no—
283C4×C31—22no—
284C4×C31—42no—
285C4×C31—41no—
286C4×C31—42no—
287C4×C31—44no—
288C4×C31—81no—
289C4×C31—81no—
290C4×C31#45014yes—
291C4×C31#52512yes—
292C4×C31#53212yes—
293C22×#41—21no—
294C22×SD16C22×C3112yes—
295#532—28no—
296C2×C4⋊Q8C22×C3111yescharacteristic
297C2×#127C22×C3111yescharacteristic
298C2×#124—21no—
299C2×C22⋊Q8C22×C3111yescharacteristic
300C2×#121—21no—
301C2×#121C22×C3112yes—
302C2×C4×Q8C22×C3111yescharacteristic
303C2×C4×D4—21no—
304C2×C4×D4C22×C3112yes—
305C22×C42C22×C3111yescharacteristic
306C4×SD16—28no—
307C2×#106C22×C3112yes—
308C2×#103C22×C3112yes—
309C2×#102C22×C3112yes—
310C2×#102C22×C3112yes—
311C2×#102C22×C3112yes—
312C2×C22⋊C8C22×C3112yes—
313C2×C4×C8C22×C3112yes—
314#391C22×C3111yescharacteristic
315#389C22×C3111yescharacteristic
316#387—21no—
317#385—21no—
318#380C22×C3111yescharacteristic
319C4×C4⋊C4—21no—
320C4×C4⋊C4C22×C3111yescharacteristic
321C4×C22⋊C4—21no—
322#340C22×C3112yes—
323#340C22×C3112yes—
324#336C22×C3111yescharacteristic
325#336C22×C3111yescharacteristic
326C2×C31—42no—
327C2×C31—84no—
328C2×C31#238811yescharacteristic
329C2×C31C2×#45011yescharacteristic
330C2×C31#300211yescharacteristic
331C2×C31#310311yescharacteristic
332C2×C31#310611yescharacteristic
333C2×C31#316711yescharacteristic
334C2×C31#317111yescharacteristic
335C2×#41—28no—
336C2×#41—44no—
337C2×#41—44no—
338C22×Q8—21no—
339C22×Q8C23×C3111yescharacteristic
340C22×D4—22no—
341C22×D4—21no—
342C22×D4—21no—
343C22×D4C23×C3112yes—
344C23×C4—22no—
345C23×C4D4×C3111yescharacteristic
346C23×C4D4×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C2×SD16—24no—
349C2×SD16—24no—
350C2×SD16—24no—
351C22×C8C23×C3112yes—
352C22×C8C23×C3112yes—
353C4⋊Q8—24no—
354#127—24no—
355#124—28no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358#121—28no—
359#121—24no—
360#121—24no—
361C4×Q8—24no—
362C4×D4—28no—
363C4×D4—24no—
364C4×D4D4×C3118yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C23×C3111yescharacteristic
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—21no—
377C2×C42—24no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42D4×C3111yescharacteristic
381C2×C42D4×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383C2×C42C23×C3111yescharacteristic
384#106—24no—
385#103—24no—
386#102—24no—
387#102—24no—
388#102—24no—
389C22⋊C8—24no—
390C4×C8—24no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396C31#2357011yescharacteristic, direct product factor, Sylow
397C24—22no—
398C24C2×D4×C3111yescharacteristic
399#41—432no—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8#41×C3112yescomplemented
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×D4×C3114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4SD16×C3114yescomplemented
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C24×C3111yescharacteristic
446SD16—216no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8#41×C3114yescomplemented
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4C2×D4×C3114yes—
462C4⋊C4#41×C3112yescomplemented
463C22⋊C4—28no—
464C22⋊C4—24no—
465C22⋊C4—28no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42—42no—
472C42C2×D4×C3114yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C2×SD16×C3112yes—
487C23C22×D4×C3111yescharacteristic
488Q8—24no—
489Q8—48no—
490Q8—42no—
491D4—24no—
492D4—24no—
493D4—48no—
494D4—48no—
495D4—416no—
496D4—42no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4C31×#12012yes—
528C2×C4C31×#12012yes—
529C2×C4C2×SD16×C3112yes—
530C2×C4C2×SD16×C3112yes—
531C2×C4C31×#13512yes—
532C2×C4C31×AGL(1,ℤ/8)12yes—
533C2×C4C22×D4×C3112yes—
534C2×C4C2×#41×C3111yescharacteristic
535C2×C4C2×#41×C3112yes—
536C2×C4C2×#41×C3111yescharacteristic
537C8—24no—
538C8—42no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×C31×#12011yescharacteristic
554C22C31×#52511yescharacteristic
555C22C31×#53211yescharacteristic
556C22C22×SD16×C3111yescharacteristic
557C22C2×C31×#13511yescharacteristic
558C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
559C22C31×#57511yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—44no—
567C4—81no—
568C4—81no—
569C4C31×#45014yes—
570C4C31×#52512yes—
571C4C31×#53212yes—
572C2—42no—
573C2—84no—
574C2C31×#238811yescharacteristic
575C2C2×C31×#45011yescharacteristic
576C2C31×#300211yescharacteristic
577C2C31×#310311yescharacteristic
578C2C31×#310611yescharacteristic
579C2C31×#316711yescharacteristic
580C2C31×#317111yescharacteristic
subgroupG/H
centerC22×C4×C31C2×D4
commutator, Frattini, cosocleC22×C4C24×C31
socleC23×C31C22×D4
prime radicalC2×C31×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×SD16×C31C2semidirect
3. C4×C31×C22⋊Q8C2semidirect
4. C31×#2058C2semidirect
5. C31×#2041C2semidirect
6. C31×#2034C2semidirect
7. C31×#2024C2semidirect
8. C31×#2014C2semidirect
9. C31×#2000C2semidirect
10. C4×C31×#102C2semidirect
11. C4×C31×C22⋊C8C2semidirect
115. C2×Q8×C31#41semidirect
148. C22×C4×C31SD16semidirect
161. C2×C8×C31#41semidirect
171. C4⋊C4×C31#41semidirect
182. #23570C31direct (N × H)
250. C2×C4×SD16C2×C31semidirect
251. C4×C22⋊Q8C2×C31semidirect
252. #2058C2×C31semidirect
253. #2041C2×C31semidirect
254. #2034C2×C31semidirect
255. #2024C2×C31semidirect
256. #2014C2×C31semidirect
257. #2000C2×C31semidirect
258. C4×#102C2×C31semidirect
259. C4×C22⋊C8C2×C31semidirect
396. C31#23570direct (N × H)
406. C2×Q8#41×C31semidirect
439. C22×C4SD16×C31semidirect
452. C2×C8#41×C31semidirect
462. C4⋊C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C4×SD16×C312
C4×C31×C22⋊Q82
C31×#20582
C31×#20412
C31×#20342
C31×#20242
C31×#20142
C31×#20002
C4×C31×#1022
C4×C31×C22⋊C82
#2357031