Finite Group Databasebeta

← C31×#18635 #21475394

Subgroups — C31×#18635 #21475394

Order 7936.

1004 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3508C211yesmaximal, characteristic
2C31×#3209C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#3079C211yesmaximal, characteristic, complemented
5C31×#1641C214yesmaximal, complemented
6C31×#1640C214yesmaximal, complemented
7C31×#1545C211yesmaximal, characteristic, complemented
8C31×#1493C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C31×#559—21no—
11C31×#559C2212yes—
12C31×#550—21no—
13C31×#550C2212yes—
14C31×#546—22no—
15C31×#546—22no—
16C31×#540—22no—
17C31×#535—21no—
18C31×#533C2211yescharacteristic
19C31×#532—21no—
20C31×#532—21no—
21C2×C31×C4⋊Q8C2211yescharacteristic
22C2×C31×#127C2211yescharacteristic
23C31×#518—21no—
24C4×#41×C31C2211yescharacteristic
25C31×#494—21no—
26C31×#487—21no—
27C31×#486—22no—
28C31×#486C2214yes—
29C31×#477—21no—
30C31×#476—21no—
31C31×#475C2212yescomplemented
32C31×#474C2212yescomplemented
33C31×#462C2214yescomplemented
34C31×#451—22no—
35C31×#431C2211yescharacteristic
36C31×#428—21no—
37C31×#423C2211yescharacteristic
38C31×#417—21no—
39C2×C31×#102—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#331C2214yes—
42C31×Q8⋊C8C2212yescomplemented
43C31×Q8⋊C8C2214yescomplemented
44C31×D4⋊C8C2212yescomplemented
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31D411yescharacteristic
48C22×Q8×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31D412yes—
52C2×Q16×C31—24no—
53C2×M16×C31—21no—
54C22×C8×C31—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8D411yescharacteristic
59C31×C4⋊Q8D411yescharacteristic
60C31×C4⋊Q8C2312yes—
61C31×#127—22no—
62C31×#127—21no—
63C31×#127—21no—
64C31×#127D412yes—
65C31×#127C2312yes—
66C31×#125—21no—
67C31×#124—22no—
68C31×#124—22no—
69C31×#124—22no—
70C31×#123—24no—
71C31×#123—24no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8D412yes—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—24no—
78C31×#121—24no—
79C31×#121—22no—
80C31×#120—24no—
81C4×Q8×C31—21no—
82C4×Q8×C31D411yescharacteristic
83C4×Q8×C31C2312yes—
84C4×Q8×C31C2311yescharacteristic
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31D411yescharacteristic
94C4×D4×C31C2311yescharacteristic
95C31×#117—21no—
96C31×#117—21no—
97C31×#117D411yescharacteristic
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C22⋊C4×C31—24no—
101C2×C42×C31—21no—
102C2×C42×C31C2311yescharacteristic
103C31×#107—22no—
104C31×#106—22no—
105C31×C4⋊C8—21no—
106C31×C4⋊C8C2312yes—
107C31×C4⋊C8C2312yes—
108C31×#103—22no—
109C31×#103—24no—
110C31×#102—28no—
111C31×#102—22no—
112C31×C22⋊C8—21no—
113C4×C8×C31—21no—
114C4×C8×C31—21no—
115C4×C8×C31C2312yes—
116C24×C31—22no—
117#41×C31—22no—
118#41×C31—22no—
119#41×C31—84no—
120#41×C31—84no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—41no—
124C2×Q8×C31—82no—
125C2×Q8×C31C2×D411yescharacteristic
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—21no—
128C2×D4×C31—24no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—44no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—82no—
139C2×D4×C31C2×D411yescharacteristic
140C2×D4×C31C2×D414yes—
141C22×C4×C31—24no—
142C22×C4×C31—24no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—41no—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150Q16×C31—84no—
151M16×C31—42no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—42no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—41no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31C2×D412yes—
168C4⋊C4×C31C2×D411yescharacteristic
169C4⋊C4×C31C2×D411yescharacteristic
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—21no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—22no—
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D411yescharacteristic
185C42×C31C2411yescharacteristic
186#18635C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—24no—
188C23×C31—21no—
189C23×C31—44no—
190C23×C31—82no—
191C23×C31—82no—
192C23×C31#12011yescharacteristic
193Q8×C31—21no—
194Q8×C31—21no—
195Q8×C31—24no—
196Q8×C31—42no—
197Q8×C31—44no—
198Q8×C31—82no—
199D4×C31—21no—
200D4×C31—21no—
201D4×C31—44no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—44no—
206D4×C31—44no—
207D4×C31—44no—
208D4×C31—82no—
209D4×C31—84no—
210D4×C31—84no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—84no—
223C2×C4×C31—84no—
224C2×C4×C31—82no—
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#13512yes—
229C2×C4×C31#13712yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C8×C31—42no—
234C8×C31—42no—
235C8×C31—42no—
236#3508C2×C3111yescharacteristic
237#3209C2×C3111yescharacteristic, complemented
238C2×#486C2×C3111yescharacteristic, complemented
239#3079C2×C3111yescharacteristic, complemented
240#1641C2×C3114yescomplemented
241#1640C2×C3114yescomplemented
242#1545C2×C3111yescharacteristic, complemented
243#1493C2×C3111yescharacteristic, complemented
244#1476C2×C3111yescharacteristic, complemented
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—41no—
248C22×C31—44no—
249C22×C31—41no—
250C22×C31—44no—
251C22×C31—41no—
252C22×C31—44no—
253C22×C31—164no—
254C22×C31#45212yes—
255C22×C31C2×#12011yescharacteristic
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—41no—
259C4×C31—41no—
260C4×C31—42no—
261C4×C31—41no—
262C4×C31—42no—
263C4×C31—82no—
264C4×C31#45212yes—
265C4×C31#45512yes—
266C4×C31C2×#13511yescharacteristic
267C4×C31C2×#13711yescharacteristic
268C4×C31#57612yes—
269#559—21no—
270#559C22×C3112yes—
271#550—21no—
272#550C22×C3112yes—
273#546—22no—
274#546—22no—
275#540—22no—
276#535—21no—
277#533C22×C3111yescharacteristic
278#532—21no—
279#532—21no—
280C2×C4⋊Q8C22×C3111yescharacteristic
281C2×#127C22×C3111yescharacteristic
282#518—21no—
283C4×#41C22×C3111yescharacteristic
284#494—21no—
285#487—21no—
286#486—22no—
287#486C22×C3114yes—
288#477—21no—
289#476—21no—
290#475C22×C3112yescomplemented
291#474C22×C3112yescomplemented
292#462C22×C3114yescomplemented
293#451—22no—
294#431C22×C3111yescharacteristic
295#428—21no—
296#423C22×C3111yescharacteristic
297#417—21no—
298C2×#102—22no—
299C2×C4×C8C22×C3111yescharacteristic
300#331C22×C3114yes—
301Q8⋊C8C22×C3112yescomplemented
302Q8⋊C8C22×C3114yescomplemented
303D4⋊C8C22×C3112yescomplemented
304C2×C31—22no—
305C2×C31—81no—
306C2×C31—84no—
307C2×C31C2×#45211yescharacteristic
308C2×C31#300811yescharacteristic
309C2×C31C2×#45511yescharacteristic
310C2×#41—22no—
311C2×#41—22no—
312C2×#41D4×C3111yescharacteristic
313C22×Q8—21no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4D4×C3112yes—
317C2×Q16—24no—
318C2×M16—21no—
319C22×C8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8D4×C3111yescharacteristic
324C4⋊Q8D4×C3111yescharacteristic
325C4⋊Q8C23×C3112yes—
326#127—22no—
327#127—21no—
328#127—21no—
329#127D4×C3112yes—
330#127C23×C3112yes—
331#125—21no—
332#124—22no—
333#124—22no—
334#124—22no—
335#123—24no—
336#123—24no—
337C22⋊Q8—21no—
338C22⋊Q8D4×C3112yes—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—24no—
343#121—24no—
344#121—22no—
345#120—24no—
346C4×Q8—21no—
347C4×Q8D4×C3111yescharacteristic
348C4×Q8C23×C3111yescharacteristic
349C4×Q8C23×C3112yes—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4D4×C3111yescharacteristic
359C4×D4C23×C3111yescharacteristic
360#117—21no—
361#117—21no—
362#117D4×C3111yescharacteristic
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C42—21no—
367C2×C42C23×C3111yescharacteristic
368#107—22no—
369#106—22no—
370C4⋊C8—21no—
371C4⋊C8C23×C3112yes—
372C4⋊C8C23×C3112yes—
373#103—22no—
374#103—24no—
375#102—28no—
376#102—22no—
377C22⋊C8—21no—
378C4×C8—21no—
379C4×C8—21no—
380C4×C8C23×C3112yes—
381C31#1863511yescharacteristic, direct product factor, Sylow
382C24—22no—
383#41—22no—
384#41—22no—
385#41—84no—
386#41—84no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—41no—
390C2×Q8—82no—
391C2×Q8C2×D4×C3111yescharacteristic
392C2×Q8C2×D4×C3112yes—
393C2×D4—21no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×D4×C3111yescharacteristic
406C2×D4C2×D4×C3114yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416Q16—84no—
417M16—42no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—42no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4C2×D4×C3112yes—
434C4⋊C4C2×D4×C3111yescharacteristic
435C4⋊C4C2×D4×C3111yescharacteristic
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42C2×D4×C3111yescharacteristic
449C42C2×D4×C3111yescharacteristic
450C42C2×D4×C3111yescharacteristic
451C42C24×C3111yescharacteristic
452C23—24no—
453C23—21no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23C31×#12011yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—24no—
461Q8—42no—
462Q8—44no—
463Q8—82no—
464D4—21no—
465D4—21no—
466D4—44no—
467D4—42no—
468D4—44no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—82no—
474D4—84no—
475D4—84no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4C31×#12011yescharacteristic
491C2×C4C31×#12011yescharacteristic
492C2×C4C31×#12011yescharacteristic
493C2×C4C31×#13512yes—
494C2×C4C31×#13712yes—
495C2×C4C22×D4×C3111yescharacteristic
496C2×C4C22×D4×C3111yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C8—42no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—44no—
505C22—41no—
506C22—44no—
507C22—41no—
508C22—44no—
509C22—164no—
510C22C31×#45212yes—
511C22C2×C31×#12011yescharacteristic
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—82no—
520C4C31×#45212yes—
521C4C31×#45512yes—
522C4C2×C31×#13511yescharacteristic
523C4C2×C31×#13711yescharacteristic
524C4C31×#57612yes—
525C2—22no—
526C2—84no—
527C2—81no—
528C2C2×C31×#45211yescharacteristic
529C2C31×#300811yescharacteristic
530C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3508C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3209C2semidirect
3. C2×C31×#486C2semidirect
4. C31×#3079C2semidirect
5. C31×#1641C2semidirect
6. C31×#1640C2semidirect
7. C31×#1545C2semidirect
8. C31×#1493C2semidirect
9. C31×#1476C2semidirect
31. C31×#475C22semidirect
32. C31×#474C22semidirect
33. C31×#462C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×Q8⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
186. #18635C31direct (N × H)
237. #3209C2×C31semidirect
238. C2×#486C2×C31semidirect
239. #3079C2×C31semidirect
240. #1641C2×C31semidirect
241. #1640C2×C31semidirect
242. #1545C2×C31semidirect
243. #1493C2×C31semidirect
244. #1476C2×C31semidirect
290. #475C22×C31semidirect
291. #474C22×C31semidirect
292. #462C22×C31semidirect
301. Q8⋊C8C22×C31semidirect
302. Q8⋊C8C22×C31semidirect
303. D4⋊C8C22×C31semidirect
381. C31#18635direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35082
C31×#32092
C2×C31×#4862
C31×#30792
C31×#16412
C31×#16402
C31×#15452
C31×#14932
C31×#14762
#1863531