Finite Group Databasebeta

← C31×#24376 #21481135

Subgroups — C31×#24376 #21481135

Order 7936.

1424 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#137C211yesmaximal, characteristic
2C2×C31×#461C212yesmaximal, complemented
3C2×C31×#452C211yesmaximal, characteristic
4C31×#2434C211yesmaximal, characteristic, complemented
5C31×#2339C211yesmaximal, characteristic
6C31×#2056C212yesmaximal, complemented
7C31×#2016C212yesmaximal, complemented
8C31×#1894C211yesmaximal, characteristic, complemented
9C31×#1877C212yesmaximal, complemented
10C31×#1851C211yesmaximal, characteristic, complemented
11C31×#1787C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C23×Q8×C31C2211yescharacteristic
14C2×C31×#137—24no—
15C2×C31×#137—28no—
16C22×Q16×C31C2212yes—
17C22×Q16×C31C2211yescharacteristic
18C22×SD16×C31C2211yescharacteristic
19C22×SD16×C31C2212yes—
20C22×M16×C31C2211yescharacteristic
21C2×C31×C4⋊Q8C2212yes—
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C4×D4×C31C2212yes—
24C31×#461—28no—
25C31×#452—24no—
26C2×C31×C4⋊C8C2212yes—
27C2×C31×#103C2211yescharacteristic
28C2×C31×#103C2212yes—
29C2×C31×#103C2212yes—
30C2×C31×#103C2212yes—
31C2×C31×#102C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C31×#395—21no—
34C31×#395—21no—
35C31×#391C2212yes—
36C31×#391C2212yes—
37C31×#386—22no—
38C31×#385—22no—
39C31×#381—21no—
40C31×#380C2211yescharacteristic
41C4×C22⋊C4×C31C2211yescharacteristic
42C2×C31×#95—21no—
43C31×#340C2212yes—
44C31×#336C2212yes—
45C2×#41×C31—22no—
46C2×#41×C31—24no—
47C2×#41×C31—42no—
48C2×#41×C31D414yes—
49C22×Q8×C31—21no—
50C22×Q8×C31—24no—
51C22×Q8×C31—22no—
52C22×Q8×C31—22no—
53C22×Q8×C31—42no—
54C22×Q8×C31C2311yescharacteristic
55C22×Q8×C31C2312yes—
56C22×Q8×C31C2311yescharacteristic
57C22×D4×C31—21no—
58C22×D4×C31C2311yescharacteristic
59C23×C4×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31—22no—
62C23×C4×C31C2311yescharacteristic
63C31×#137—232no—
64C2×Q16×C31—24no—
65C2×Q16×C31—24no—
66C2×Q16×C31—22no—
67C2×Q16×C31—24no—
68C2×Q16×C31—42no—
69C2×SD16×C31—24no—
70C2×SD16×C31—24no—
71C2×SD16×C31—22no—
72C2×SD16×C31—24no—
73C2×SD16×C31D418yes—
74C2×M16×C31—22no—
75C2×M16×C31—24no—
76C22×C8×C31C2311yescharacteristic
77C22×C8×C31C2311yescharacteristic
78C22×C8×C31C2312yes—
79C31×C4⋊Q8—24no—
80C31×C4⋊Q8—42no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C4×D4×C31—24no—
84C4×D4×C31—42no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31D412yes—
94C2×C42×C31C2312yes—
95C31×C4⋊C8—24no—
96C31×#103—22no—
97C31×#103—24no—
98C31×#103—24no—
99C31×#103—42no—
100C31×#102—22no—
101C31×C22⋊C8—22no—
102C31×#95—22no—
103C31×#95—22no—
104C31×#95—21no—
105C31×#95—22no—
106C31×#95—22no—
107C31×#95D411yescharacteristic
108C31×#95D411yescharacteristic
109C24×C31—21no—
110C24×C31C2×D411yescharacteristic
111#41×C31—28no—
112#41×C31—44no—
113#41×C31—48no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—28no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31—22no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—44no—
128C2×Q8×C31—48no—
129C2×Q8×C31—44no—
130C2×Q8×C31—42no—
131C2×Q8×C31C2×D414yes—
132C2×Q8×C31C2×D412yes—
133C2×D4×C31—24no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—44no—
137C2×D4×C31C2×D412yes—
138C2×D4×C31C2×D414yes—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31—24no—
144C22×C4×C31—21no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—44no—
148C22×C4×C31—44no—
149C22×C4×C31—44no—
150C22×C4×C31—42no—
151C22×C4×C31—44no—
152C22×C4×C31—82no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31#4112yes—
158C22×C4×C31C2411yescharacteristic
159Q16×C31—216no—
160Q16×C31—44no—
161SD16×C31—216no—
162SD16×C31—44no—
163M16×C31—28no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—21no—
167C2×C8×C31—21no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31C2×D414yes—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—44no—
176C4⋊C4×C31—44no—
177C4⋊C4×C31—42no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—81no—
182C42×C31—22no—
183C42×C31—22no—
184#24376C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31—24no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—42no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31#12112yes—
195C23×C31C22×D411yescharacteristic
196Q8×C31—24no—
197Q8×C31—22no—
198Q8×C31—22no—
199Q8×C31—24no—
200Q8×C31—48no—
201Q8×C31—42no—
202Q8×C31—42no—
203Q8×C31—42no—
204Q8×C31—41no—
205Q8×C31—41no—
206Q8×C31—42no—
207D4×C31—22no—
208D4×C31—24no—
209D4×C31—48no—
210D4×C31#12014yes—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—24no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—48no—
223C2×C4×C31—48no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—88no—
236C2×C4×C31#12112yes—
237C2×C4×C31#13714yescomplemented
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C22×D412yes—
240C2×C4×C31C2×#4111yescharacteristic
241C2×C4×C31H(5,𝔽2)12yes—
242C8×C31—24no—
243C8×C31—41no—
244C8×C31—41no—
245C22×#137C2×C3111yescharacteristic
246C2×#461C2×C3112yescomplemented
247C2×#452C2×C3111yescharacteristic
248#2434C2×C3111yescharacteristic, complemented
249#2339C2×C3111yescharacteristic
250#2056C2×C3112yescomplemented
251#2016C2×C3112yescomplemented
252#1894C2×C3111yescharacteristic, complemented
253#1877C2×C3112yescomplemented
254#1851C2×C3111yescharacteristic, complemented
255#1787C2×C3111yescharacteristic, complemented
256C22×C31—22no—
257C22×C31—24no—
258C22×C31—22no—
259C22×C31—22no—
260C22×C31—44no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—44no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31C2×#12111yescharacteristic
267C22×C31D4212yes—
268C22×C31C2×#13712yes—
269C22×C31#57711yescharacteristic
270C22×C31#57811yescharacteristic
271C4×C31—22no—
272C4×C31—24no—
273C4×C31—42no—
274C4×C31—44no—
275C4×C31—42no—
276C4×C31—81no—
277C4×C31—82no—
278C4×C31—81no—
279C4×C31C2×#12011yescharacteristic
280C4×C31#53411yescharacteristic
281C4×C31#54612yes—
282C22×#41C22×C3111yescharacteristic
283C23×Q8C22×C3111yescharacteristic
284C2×#137—24no—
285C2×#137—28no—
286C22×Q16C22×C3112yes—
287C22×Q16C22×C3111yescharacteristic
288C22×SD16C22×C3111yescharacteristic
289C22×SD16C22×C3112yes—
290C22×M16C22×C3111yescharacteristic
291C2×C4⋊Q8C22×C3112yes—
292C2×C22⋊Q8C22×C3111yescharacteristic
293C2×C4×D4C22×C3112yes—
294#461—28no—
295#452—24no—
296C2×C4⋊C8C22×C3112yes—
297C2×#103C22×C3111yescharacteristic
298C2×#103C22×C3112yes—
299C2×#103C22×C3112yes—
300C2×#103C22×C3112yes—
301C2×#102C22×C3111yescharacteristic
302C2×C22⋊C8C22×C3111yescharacteristic
303#395—21no—
304#395—21no—
305#391C22×C3112yes—
306#391C22×C3112yes—
307#386—22no—
308#385—22no—
309#381—21no—
310#380C22×C3111yescharacteristic
311C4×C22⋊C4C22×C3111yescharacteristic
312C2×#95—21no—
313#340C22×C3112yes—
314#336C22×C3112yes—
315C2×C31—42no—
316C2×C31—44no—
317C2×C31#241011yescharacteristic
318C2×C31#303911yescharacteristic
319C2×C31#304611yescharacteristic
320C2×C31#326612yes—
321C2×C31#327612yes—
322C2×#41—22no—
323C2×#41—24no—
324C2×#41—42no—
325C2×#41D4×C3114yes—
326C22×Q8—21no—
327C22×Q8—24no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—42no—
331C22×Q8C23×C3111yescharacteristic
332C22×Q8C23×C3112yes—
333C22×Q8C23×C3111yescharacteristic
334C22×D4—21no—
335C22×D4C23×C3111yescharacteristic
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—22no—
339C23×C4C23×C3111yescharacteristic
340#137—232no—
341C2×Q16—24no—
342C2×Q16—24no—
343C2×Q16—22no—
344C2×Q16—24no—
345C2×Q16—42no—
346C2×SD16—24no—
347C2×SD16—24no—
348C2×SD16—22no—
349C2×SD16—24no—
350C2×SD16D4×C3118yes—
351C2×M16—22no—
352C2×M16—24no—
353C22×C8C23×C3111yescharacteristic
354C22×C8C23×C3111yescharacteristic
355C22×C8C23×C3112yes—
356C4⋊Q8—24no—
357C4⋊Q8—42no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C4×D4—24no—
361C4×D4—42no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4C23×C3112yes—
366C2×C4⋊C4C23×C3111yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4D4×C3112yes—
371C2×C42C23×C3112yes—
372C4⋊C8—24no—
373#103—22no—
374#103—24no—
375#103—24no—
376#103—42no—
377#102—22no—
378C22⋊C8—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95D4×C3111yescharacteristic
385#95D4×C3111yescharacteristic
386C31#2437611yescharacteristic, direct product factor, Sylow
387C24—21no—
388C24C2×D4×C3111yescharacteristic
389#41—28no—
390#41—44no—
391#41—48no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8C2×D4×C3114yes—
410C2×Q8C2×D4×C3112yes—
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4C2×D4×C3112yes—
416C2×D4C2×D4×C3114yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3112yes—
435C22×C4#41×C3112yes—
436C22×C4C24×C3111yescharacteristic
437Q16—216no—
438Q16—44no—
439SD16—216no—
440SD16—44no—
441M16—28no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8C2×D4×C3114yes—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C22⋊C4—24no—
457C22⋊C4—44no—
458C22⋊C4—42no—
459C22⋊C4—81no—
460C42—22no—
461C42—22no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23C31×#12112yes—
472C23C22×D4×C3111yescharacteristic
473Q8—24no—
474Q8—22no—
475Q8—22no—
476Q8—24no—
477Q8—48no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—41no—
482Q8—41no—
483Q8—42no—
484D4—22no—
485D4—24no—
486D4—48no—
487D4C31×#12014yes—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—88no—
513C2×C4C31×#12112yes—
514C2×C4C31×#13714yescomplemented
515C2×C4C22×D4×C3111yescharacteristic
516C2×C4C22×D4×C3112yes—
517C2×C4C2×#41×C3111yescharacteristic
518C2×C4C31×H(5,𝔽2)12yes—
519C8—24no—
520C8—41no—
521C8—41no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22C2×C31×#12111yescharacteristic
533C22D42×C3112yes—
534C22C2×C31×#13712yes—
535C22C31×#57711yescharacteristic
536C22C31×#57811yescharacteristic
537C4—22no—
538C4—24no—
539C4—42no—
540C4—44no—
541C4—42no—
542C4—81no—
543C4—82no—
544C4—81no—
545C4C2×C31×#12011yescharacteristic
546C4C31×#53411yescharacteristic
547C4C31×#54612yes—
548C2—42no—
549C2—44no—
550C2C31×#241011yescharacteristic
551C2C31×#303911yescharacteristic
552C2C31×#304611yescharacteristic
553C2C31×#326612yes—
554C2C31×#327612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#461C2semidirect
4. C31×#2434C2semidirect
6. C31×#2056C2semidirect
7. C31×#2016C2semidirect
8. C31×#1894C2semidirect
9. C31×#1877C2semidirect
10. C31×#1851C2semidirect
11. C31×#1787C2semidirect
184. #24376C31direct (N × H)
237. C2×C4×C31#137semidirect
246. C2×#461C2×C31semidirect
248. #2434C2×C31semidirect
250. #2056C2×C31semidirect
251. #2016C2×C31semidirect
252. #1894C2×C31semidirect
253. #1877C2×C31semidirect
254. #1851C2×C31semidirect
255. #1787C2×C31semidirect
386. C31#24376direct (N × H)
514. C2×C4C31×#137semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1372
C2×C31×#4612
C2×C31×#4522
C31×#24342
C31×#23392
C31×#20562
C31×#20162
C31×#18942
C31×#18772
C31×#18512
C31×#17872
#2437631