Finite Group Databasebeta

← C31×#18352 #21475111

Subgroups — C31×#18352 #21475111

Order 7936.

996 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#3193C212yesmaximal, complemented
3C31×#2635C211yesmaximal, characteristic
4C31×#1617C212yesmaximal, complemented
5C31×#1614C212yesmaximal, complemented
6C31×#1599C212yesmaximal, complemented
7C31×#1596C212yesmaximal, complemented
8C31×#1511C212yesmaximal, complemented
9C31×#1465C211yesmaximal, characteristic, complemented
10C31×#554—24no—
11C31×#554—22no—
12D4×Q8×C31—22no—
13C31×#544C2212yes—
14C31×#543—21no—
15C31×#543C2212yes—
16C31×#533—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×#121C2211yescharacteristic
20C31×#518—21no—
21C2×C4×D4×C31C2211yescharacteristic
22C31×#484C2214yes—
23C31×#480—22no—
24C31×#468—24no—
25C31×#450C2214yes—
26C31×#448—22no—
27C31×#407C2212yes—
28C31×#400C2211yescharacteristic
29C31×#399C2211yescharacteristic
30C31×#394—21no—
31C31×#394C2211yescharacteristic
32C31×#392—21no—
33C31×#392C2211yescharacteristic
34C31×#388—21no—
35C31×#386—21no—
36C31×#381C2211yescharacteristic
37C31×#328C2214yes—
38C31×#328C2214yes—
39C31×#324C2212yes—
40C31×#323C2212yes—
41C22×Q8×C31—21no—
42C22×Q8×C31D412yescomplemented
43C22×D4×C31—21no—
44C22×D4×C31—21no—
45C22×D4×C31D411yescharacteristic
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31D411yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C2×SD16×C31—24no—
50C2×M16×C31—22no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—24no—
54C31×#125—22no—
55C31×#125—24no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—22no—
58C31×C22⋊Q8—24no—
59C31×C22⋊Q8—24no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8D414yescomplemented
67C31×C22⋊Q8C2314yes—
68C31×#121—22no—
69C31×#121—22no—
70C4×Q8×C31—22no—
71C4×Q8×C31—22no—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C31×#117—22no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31D412yescomplemented
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31D411yescharacteristic
93C2×C22⋊C4×C31C2311yescharacteristic
94C2×C42×C31D411yescharacteristic
95C31×#107—24no—
96C31×#103—24no—
97C31×#102—24no—
98C31×C22⋊C8—22no—
99C31×C22⋊C8C2314yes—
100C31×#95—21no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95C2311yescharacteristic
104C31×#95C2311yescharacteristic
105C24×C31—21no—
106C24×C31C2×D411yescharacteristic
107C24×C31C2×D411yescharacteristic
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31—82no—
112C2×Q8×C31—82no—
113C2×Q8×C31C2×D414yes—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31—81no—
121C2×D4×C31—81no—
122C2×D4×C31—81no—
123C2×D4×C31—81no—
124C2×D4×C31—81no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—81no—
135C22×C4×C31—81no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2411yescharacteristic
143SD16×C31—84no—
144M16×C31—82no—
145C2×C8×C31—24no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—81no—
158C4⋊C4×C31C2×D414yes—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—41no—
165C22⋊C4×C31—41no—
166C22⋊C4×C31—42no—
167C22⋊C4×C31—81no—
168C22⋊C4×C31—81no—
169C22⋊C4×C31—81no—
170C22⋊C4×C31—81no—
171C42×C31—24no—
172C42×C31—21no—
173C42×C31—21no—
174#18352C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31#12012yes—
191C23×C31#13712yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—44no—
194Q8×C31—44no—
195Q8×C31—82no—
196Q8×C31—82no—
197D4×C31—28no—
198D4×C31—44no—
199D4×C31—84no—
200D4×C31—84no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—21no—
207C2×C4×C31—42no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—81no—
224C2×C4×C31—82no—
225C2×C4×C31—82no—
226C2×C4×C31—82no—
227C2×C4×C31—82no—
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#13712yes—
231C2×C4×C31C22×D412yes—
232C8×C31—44no—
233#3501C2×C3111yescharacteristic, complemented
234#3193C2×C3112yescomplemented
235#2635C2×C3111yescharacteristic
236#1617C2×C3112yescomplemented
237#1614C2×C3112yescomplemented
238#1599C2×C3112yescomplemented
239#1596C2×C3112yescomplemented
240#1511C2×C3112yescomplemented
241#1465C2×C3111yescharacteristic, complemented
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—42no—
249C22×C31—41no—
250C22×C31—41no—
251C22×C31—81no—
252C22×C31—81no—
253C22×C31—81no—
254C22×C31—81no—
255C22×C31—81no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31—81no—
261C22×C31C2×#12011yescharacteristic
262C22×C31C2×#13711yescharacteristic
263C22×C31C2×#13711yescharacteristic
264C4×C31—22no—
265C4×C31—22no—
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—41no—
269C4×C31—44no—
270C4×C31—41no—
271C4×C31—44no—
272C4×C31—81no—
273C4×C31—81no—
274C4×C31—81no—
275#554—24no—
276#554—22no—
277D4×Q8—22no—
278#544C22×C3112yes—
279#543—21no—
280#543C22×C3112yes—
281#533—21no—
282C2×C22⋊Q8—21no—
283C2×C22⋊Q8C22×C3112yes—
284C2×#121C22×C3111yescharacteristic
285#518—21no—
286C2×C4×D4C22×C3111yescharacteristic
287#484C22×C3114yes—
288#480—22no—
289#468—24no—
290#450C22×C3114yes—
291#448—22no—
292#407C22×C3112yes—
293#400C22×C3111yescharacteristic
294#399C22×C3111yescharacteristic
295#394—21no—
296#394C22×C3111yescharacteristic
297#392—21no—
298#392C22×C3111yescharacteristic
299#388—21no—
300#386—21no—
301#381C22×C3111yescharacteristic
302#328C22×C3114yes—
303#328C22×C3114yes—
304#324C22×C3112yes—
305#323C22×C3112yes—
306C2×C31—21no—
307C2×C31—21no—
308C2×C31—42no—
309C2×C31—42no—
310C2×C31—161no—
311C2×C31#300411yescharacteristic
312C2×C31#302111yescharacteristic
313C2×C31#302611yescharacteristic
314C22×Q8—21no—
315C22×Q8D4×C3112yescomplemented
316C22×D4—21no—
317C22×D4—21no—
318C22×D4D4×C3111yescharacteristic
319C22×D4C23×C3111yescharacteristic
320C23×C4D4×C3111yescharacteristic
321C23×C4C23×C3111yescharacteristic
322C2×SD16—24no—
323C2×M16—22no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—24no—
327#125—22no—
328#125—24no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—24no—
339C22⋊Q8D4×C3114yescomplemented
340C22⋊Q8C23×C3114yes—
341#121—22no—
342#121—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—22no—
349#117—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4D4×C3112yescomplemented
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4D4×C3111yescharacteristic
366C2×C22⋊C4C23×C3111yescharacteristic
367C2×C42D4×C3111yescharacteristic
368#107—24no—
369#103—24no—
370#102—24no—
371C22⋊C8—22no—
372C22⋊C8C23×C3114yes—
373#95—21no—
374#95—21no—
375#95—21no—
376#95C23×C3111yescharacteristic
377#95C23×C3111yescharacteristic
378C31#1835211yescharacteristic, direct product factor, Sylow
379C24—21no—
380C24C2×D4×C3111yescharacteristic
381C24C2×D4×C3111yescharacteristic
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C2×D4×C3114yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—81no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—81no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3112yes—
415C22×C4C2×D4×C3112yes—
416C22×C4C24×C3111yescharacteristic
417SD16—84no—
418M16—82no—
419C2×C8—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—81no—
432C4⋊C4C2×D4×C3114yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—42no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—24no—
446C42—21no—
447C42—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C31×#12012yes—
464C23C31×#13712yes—
465C23C22×D4×C3111yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—82no—
469Q8—82no—
470D4—28no—
471D4—44no—
472D4—84no—
473D4—84no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4C31×#12011yescharacteristic
502C2×C4C31×#12011yescharacteristic
503C2×C4C31×#13712yes—
504C2×C4C22×D4×C3112yes—
505C8—44no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22C2×C31×#12011yescharacteristic
526C22C2×C31×#13711yescharacteristic
527C22C2×C31×#13711yescharacteristic
528C4—22no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—44no—
534C4—41no—
535C4—44no—
536C4—81no—
537C4—81no—
538C4—81no—
539C2—21no—
540C2—21no—
541C2—42no—
542C2—42no—
543C2—161no—
544C2C31×#300411yescharacteristic
545C2C31×#302111yescharacteristic
546C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2635C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3501C2semidirect
2. C31×#3193C2semidirect
4. C31×#1617C2semidirect
5. C31×#1614C2semidirect
6. C31×#1599C2semidirect
7. C31×#1596C2semidirect
8. C31×#1511C2semidirect
9. C31×#1465C2semidirect
42. C22×Q8×C31D4semidirect
66. C31×C22⋊Q8D4semidirect
81. C2×C4⋊C4×C31D4semidirect
174. #18352C31direct (N × H)
233. #3501C2×C31semidirect
234. #3193C2×C31semidirect
236. #1617C2×C31semidirect
237. #1614C2×C31semidirect
238. #1599C2×C31semidirect
239. #1596C2×C31semidirect
240. #1511C2×C31semidirect
241. #1465C2×C31semidirect
315. C22×Q8D4×C31semidirect
339. C22⋊Q8D4×C31semidirect
354. C2×C4⋊C4D4×C31semidirect
378. C31#18352direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#31932
C31×#26352
C31×#16172
C31×#16142
C31×#15992
C31×#15962
C31×#15112
C31×#14652
#1835231