Finite Group Databasebeta

← C31×#23494 #21480253

Subgroups — C31×#23494 #21480253

Order 7936.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic
2C31×#2911C211yesmaximal, characteristic, complemented
3C2×C31×#421C211yesmaximal, characteristic
4C31×#2896C211yesmaximal, characteristic
5C22×C31×#101C211yesmaximal, characteristic, complemented
6C31×#2310C211yesmaximal, characteristic, complemented
7C31×#2038C218yesmaximal, complemented
8C31×#1733C211yesmaximal, characteristic, complemented
9C22×#41×C31C2211yescharacteristic
10C23×Q8×C31C2211yescharacteristic
11C22×M16×C31C2211yescharacteristic
12C22×M16×C31C2211yescharacteristic
13C31×#536—22no—
14C31×#536—22no—
15C31×#536—22no—
16C31×#536C2214yes—
17C2×C31×C4⋊Q8—21no—
18C2×C31×#124—21no—
19C2×C31×#123—22no—
20C2×C31×C22⋊Q8—22no—
21C2×C31×#117C2211yescharacteristic
22C2×C31×#117C2211yescharacteristic
23C2×C31×#117C2211yescharacteristic
24C31×#428—22no—
25C31×#428C2214yescomplemented
26C31×#427—22no—
27C31×#421—24no—
28C2×C31×#104C2214yes—
29C31×#419—21no—
30C31×#418—21no—
31C31×#417C2214yes—
32C31×#416—22no—
33C2×C31×#101—24no—
34C2×C31×#101C2214yescomplemented
35C31×#391—22no—
36C31×#391C2214yescomplemented
37C31×#381—22no—
38C31×#339C2214yescomplemented
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31D412yes—
46C2×#41×C31C2312yes—
47C22×Q8×C31—22no—
48C22×Q8×C31—21no—
49C22×Q8×C31—21no—
50C22×Q8×C31—21no—
51C22×Q8×C31—21no—
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31—22no—
55C22×Q8×C31D411yescharacteristic
56C22×Q8×C31D412yes—
57C22×Q8×C31C2312yes—
58C22×D4×C31—21no—
59C22×D4×C31D411yescharacteristic
60C23×C4×C31—21no—
61C23×C4×C31D412yes—
62C23×C4×C31C2311yescharacteristic
63C2×M16×C31—21no—
64C2×M16×C31—22no—
65C2×M16×C31—21no—
66C2×M16×C31—22no—
67C2×M16×C31—22no—
68C2×M16×C31D414yes—
69C2×M16×C31C2312yes—
70C2×M16×C31C2312yes—
71C22×C8×C31—21no—
72C22×C8×C31—21no—
73C31×C4⋊Q8—24no—
74C31×C4⋊Q8—24no—
75C31×#124—24no—
76C31×#124—24no—
77C31×#123—28no—
78C31×#123—28no—
79C31×C22⋊Q8—28no—
80C31×C22⋊Q8—28no—
81C31×#117—21no—
82C31×#117—21no—
83C31×#117—22no—
84C31×#117—22no—
85C31×#117—21no—
86C31×#117—21no—
87C31×#117—21no—
88C31×#117—21no—
89C31×#117D412yes—
90C31×#117C2312yes—
91C2×C4⋊C4×C31—24no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31D412yes—
94C2×C4⋊C4×C31C2312yes—
95C2×C22⋊C4×C31—24no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31C2312yes—
102C31×#107—24no—
103C31×#106—24no—
104C31×#104—28no—
105C31×#103—24no—
106C31×#102—24no—
107C31×#101—28no—
108C31×#95—28no—
109C24×C31—21no—
110C24×C31C2×D411yescharacteristic
111#41×C31—24no—
112#41×C31—24no—
113#41×C31—22no—
114#41×C31—22no—
115#41×C31—22no—
116#41×C31—22no—
117#41×C31—42no—
118#41×C31—42no—
119#41×C31—42no—
120#41×C31—42no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—24no—
125C2×Q8×C31—24no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31—22no—
134C2×Q8×C31—42no—
135C2×Q8×C31—42no—
136C2×Q8×C31—42no—
137C2×Q8×C31—42no—
138C2×Q8×C31—42no—
139C2×Q8×C31—42no—
140C2×Q8×C31—42no—
141C2×Q8×C31—42no—
142C2×Q8×C31C2×D412yes—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—42no—
149C2×D4×C31—42no—
150C2×D4×C31C2×D412yes—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—24no—
156C22×C4×C31—24no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—42no—
162C22×C4×C31—42no—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D412yes—
170C22×C4×C31#4112yes—
171C22×C4×C31#4112yes—
172C22×C4×C31#4112yes—
173C22×C4×C31C2411yescharacteristic
174M16×C31—24no—
175M16×C31—22no—
176M16×C31—22no—
177M16×C31—22no—
178M16×C31—42no—
179M16×C31C2×D414yes—
180C2×C8×C31—24no—
181C2×C8×C31—22no—
182C2×C8×C31—22no—
183C2×C8×C31—42no—
184C4⋊C4×C31—216no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—42no—
189C4⋊C4×C31C2×D414yes—
190C22⋊C4×C31—216no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—41no—
194C22⋊C4×C31—41no—
195C22⋊C4×C31—42no—
196C42×C31—22no—
197C42×C31—22no—
198C42×C31—22no—
199C42×C31—22no—
200C42×C31—41no—
201C42×C31—41no—
202#23494C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—22no—
204C23×C31—42no—
205C23×C31—44no—
206C23×C31—42no—
207C23×C31#12011yescharacteristic
208C23×C31#12011yescharacteristic
209C23×C31#12112yes—
210C23×C31#12312yes—
211C23×C31C22×D411yescharacteristic
212Q8×C31—22no—
213Q8×C31—22no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—42no—
217Q8×C31—42no—
218Q8×C31—44no—
219Q8×C31—44no—
220Q8×C31—42no—
221D4×C31—22no—
222D4×C31—22no—
223D4×C31—42no—
224D4×C31—42no—
225D4×C31—44no—
226D4×C31—42no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—44no—
233C2×C4×C31—44no—
234C2×C4×C31—44no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—44no—
239C2×C4×C31—44no—
240C2×C4×C31—44no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31#12112yes—
246C2×C4×C31#12112yes—
247C2×C4×C31#12112yes—
248C2×C4×C31#12112yes—
249C2×C4×C31#12112yes—
250C2×C4×C31#12312yes—
251C2×C4×C31#12312yes—
252C2×C4×C31#12312yes—
253C2×C4×C31#12412yes—
254C2×C4×C31#12412yes—
255C2×C4×C31C22×D411yescharacteristic
256C2×C4×C31C22×D411yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C2×#4111yescharacteristic
259C2×C4×C31C2×#4111yescharacteristic
260C2×C4×C31C2×#4111yescharacteristic
261C8×C31—24no—
262C8×C31—42no—
263C2×#536C2×C3111yescharacteristic
264#2911C2×C3111yescharacteristic, complemented
265C2×#421C2×C3111yescharacteristic
266#2896C2×C3111yescharacteristic
267C22×#101C2×C3111yescharacteristic, complemented
268#2310C2×C3111yescharacteristic, complemented
269#2038C2×C3118yescomplemented
270#1733C2×C3111yescharacteristic, complemented
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—42no—
277C22×C31—44no—
278C22×C31—84no—
279C22×C31#39412yes—
280C22×C31#39412yes—
281C22×C31C2×#12011yescharacteristic
282C22×C31C2×#12111yescharacteristic
283C22×C31C2×#12311yescharacteristic
284C4×C31—44no—
285C4×C31—42no—
286C4×C31—44no—
287C4×C31—82no—
288C4×C31—82no—
289C4×C31#39412yes—
290C4×C31#39412yes—
291C4×C31C2×#12111yescharacteristic
292C4×C31C2×#12111yescharacteristic
293C4×C31C2×#12311yescharacteristic
294C4×C31C2×#12411yescharacteristic
295C22×#41C22×C3111yescharacteristic
296C23×Q8C22×C3111yescharacteristic
297C22×M16C22×C3111yescharacteristic
298C22×M16C22×C3111yescharacteristic
299#536—22no—
300#536—22no—
301#536—22no—
302#536C22×C3114yes—
303C2×C4⋊Q8—21no—
304C2×#124—21no—
305C2×#123—22no—
306C2×C22⋊Q8—22no—
307C2×#117C22×C3111yescharacteristic
308C2×#117C22×C3111yescharacteristic
309C2×#117C22×C3111yescharacteristic
310#428—22no—
311#428C22×C3114yescomplemented
312#427—22no—
313#421—24no—
314C2×#104C22×C3114yes—
315#419—21no—
316#418—21no—
317#417C22×C3114yes—
318#416—22no—
319C2×#101—24no—
320C2×#101C22×C3114yescomplemented
321#391—22no—
322#391C22×C3114yescomplemented
323#381—22no—
324#339C22×C3114yescomplemented
325C2×C31—21no—
326C2×C31—22no—
327C2×C31—21no—
328C2×C31—22no—
329C2×C31—82no—
330C2×C31C2×#39411yescharacteristic
331C2×C31#302012yes—
332C2×#41—21no—
333C2×#41—21no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41D4×C3112yes—
339C2×#41C23×C3112yes—
340C22×Q8—21no—
341C22×Q8—21no—
342C22×Q8—21no—
343C22×Q8—21no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—22no—
348C22×Q8D4×C3111yescharacteristic
349C22×Q8D4×C3112yes—
350C22×Q8C23×C3112yes—
351C22×D4—21no—
352C22×D4D4×C3111yescharacteristic
353C23×C4—21no—
354C23×C4D4×C3112yes—
355C23×C4C23×C3111yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—21no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16D4×C3114yes—
362C2×M16C23×C3112yes—
363C2×M16C23×C3112yes—
364C22×C8—21no—
365C22×C8—21no—
366C4⋊Q8—24no—
367C4⋊Q8—24no—
368#124—24no—
369#124—24no—
370#123—28no—
371#123—28no—
372C22⋊Q8—28no—
373C22⋊Q8—28no—
374#117—21no—
375#117—21no—
376#117—22no—
377#117—22no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117D4×C3112yes—
383#117C23×C3112yes—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4D4×C3112yes—
387C2×C4⋊C4C23×C3112yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C23×C3112yes—
395#107—24no—
396#106—24no—
397#104—28no—
398#103—24no—
399#102—24no—
400#101—28no—
401#95—28no—
402C31#2349411yescharacteristic, direct product factor, Sylow
403C24—21no—
404C24C2×D4×C3111yescharacteristic
405#41—24no—
406#41—24no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×D4×C3112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×D4×C3112yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C3114yes—
458C22×C4C2×D4×C3111yescharacteristic
459C22×C4C2×D4×C3111yescharacteristic
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3112yes—
464C22×C4#41×C3112yes—
465C22×C4#41×C3112yes—
466C22×C4#41×C3112yes—
467C22×C4C24×C3111yescharacteristic
468M16—24no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—42no—
473M16C2×D4×C3114yes—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—42no—
478C4⋊C4—216no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4C2×D4×C3114yes—
484C22⋊C4—216no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C23—22no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23C31×#12011yescharacteristic
501C23C31×#12011yescharacteristic
502C23C31×#12112yes—
503C23C31×#12312yes—
504C23C22×D4×C3111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—44no—
513Q8—42no—
514D4—22no—
515D4—22no—
516D4—42no—
517D4—42no—
518D4—44no—
519D4—42no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4C31×#12011yescharacteristic
537C2×C4C31×#12011yescharacteristic
538C2×C4C31×#12112yes—
539C2×C4C31×#12112yes—
540C2×C4C31×#12112yes—
541C2×C4C31×#12112yes—
542C2×C4C31×#12112yes—
543C2×C4C31×#12312yes—
544C2×C4C31×#12312yes—
545C2×C4C31×#12312yes—
546C2×C4C31×#12412yes—
547C2×C4C31×#12412yes—
548C2×C4C22×D4×C3111yescharacteristic
549C2×C4C22×D4×C3111yescharacteristic
550C2×C4C22×D4×C3111yescharacteristic
551C2×C4C2×#41×C3111yescharacteristic
552C2×C4C2×#41×C3111yescharacteristic
553C2×C4C2×#41×C3111yescharacteristic
554C8—24no—
555C8—42no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—84no—
564C22C31×#39412yes—
565C22C31×#39412yes—
566C22C2×C31×#12011yescharacteristic
567C22C2×C31×#12111yescharacteristic
568C22C2×C31×#12311yescharacteristic
569C4—44no—
570C4—42no—
571C4—44no—
572C4—82no—
573C4—82no—
574C4C31×#39412yes—
575C4C31×#39412yes—
576C4C2×C31×#12111yescharacteristic
577C4C2×C31×#12111yescharacteristic
578C4C2×C31×#12311yescharacteristic
579C4C2×C31×#12411yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2C2×C31×#39411yescharacteristic
586C2C31×#302012yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2911C2semidirect
5. C22×C31×#101C2semidirect
6. C31×#2310C2semidirect
7. C31×#2038C2semidirect
8. C31×#1733C2semidirect
25. C31×#428C22semidirect
34. C2×C31×#101C22semidirect
36. C31×#391C22semidirect
38. C31×#339C22semidirect
202. #23494C31direct (N × H)
264. #2911C2×C31semidirect
267. C22×#101C2×C31semidirect
268. #2310C2×C31semidirect
269. #2038C2×C31semidirect
270. #1733C2×C31semidirect
311. #428C22×C31semidirect
320. C2×#101C22×C31semidirect
322. #391C22×C31semidirect
324. #339C22×C31semidirect
402. C31#23494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C31×#29112
C2×C31×#4212
C31×#28962
C22×C31×#1012
C31×#23102
C31×#20382
C31×#17332
#2349431