Finite Group Databasebeta

← C31×#18646 #21475405

Subgroups — C31×#18646 #21475405

Order 7936.

1024 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic, complemented
2C31×#3205C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#3038C211yesmaximal, characteristic, complemented
5C31×#1628C214yesmaximal, complemented
6C31×#1625C214yesmaximal, complemented
7C31×#1549C211yesmaximal, characteristic, complemented
8C31×#1492C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11C31×#556C2212yescomplemented
12C31×#555—21no—
13C31×#554—21no—
14C31×#553—22no—
15C31×#548—22no—
16C31×#547C2212yescomplemented
17C31×#542—22no—
18C31×#537—21no—
19C31×#533—21no—
20C31×#532—21no—
21C31×#532C2211yescharacteristic
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#520—21no—
25C4×#41×C31C2211yescharacteristic
26C31×#493—21no—
27C31×#486—22no—
28C31×#486—21no—
29C31×#486C2214yes—
30C31×#460—21no—
31C31×#460C2212yes—
32C31×#459—21no—
33C31×#459C2212yes—
34C31×#459C2214yes—
35C31×#447—22no—
36C31×#431C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C31×#418—21no—
39C2×C31×#102—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#332C2214yescomplemented
42C31×Q8⋊C8C2212yescomplemented
43C31×D4⋊C8C2212yescomplemented
44C31×D4⋊C8C2214yescomplemented
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C2×#41×C31D411yescharacteristic
48C22×Q8×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C31×AGL(1,ℤ/8)—44no—
53C2×SD16×C31—22no—
54C2×D8×C31—22no—
55C2×D8×C31—24no—
56C2×M16×C31—21no—
57C22×C8×C31—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8D412yes—
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8C2312yes—
63C31×#127—21no—
64C31×#127—21no—
65C31×#127—21no—
66C31×#127D411yescharacteristic
67C31×#127C2312yes—
68C31×#126—24no—
69C31×#125—22no—
70C31×#124—24no—
71C31×#124—22no—
72C31×#124—21no—
73C31×#123—24no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—24no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121D412yes—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31D411yescharacteristic
87C4×Q8×C31C2311yescharacteristic
88C4×D4×C31—22no—
89C4×D4×C31—24no—
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31D411yescharacteristic
95C4×D4×C31C2311yescharacteristic
96C4×D4×C31C2312yes—
97C31×#117—22no—
98C31×#117—22no—
99C31×#117—21no—
100C31×#117D411yescharacteristic
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—22no—
103C2×C4⋊C4×C31D412yes—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C31×C4⋊C8—21no—
107C31×C4⋊C8C2312yes—
108C31×C4⋊C8C2312yes—
109C31×#102—28no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—24no—
113C31×C22⋊C8—21no—
114C4×C8×C31—21no—
115C4×C8×C31—21no—
116C4×C8×C31C2312yes—
117C24×C31—41no—
118#41×C31—22no—
119#41×C31—22no—
120#41×C31—42no—
121#41×C31—42no—
122#41×C31—84no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—82no—
127C2×Q8×C31C2×D411yescharacteristic
128C2×D4×C31—21no—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—44no—
137C2×D4×C31—41no—
138C2×D4×C31—41no—
139C2×D4×C31—41no—
140C2×D4×C31—42no—
141C2×D4×C31—42no—
142C2×D4×C31—82no—
143C2×D4×C31C2×D411yescharacteristic
144C2×D4×C31C2×D412yes—
145C22×C4×C31—24no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154SD16×C31—44no—
155D8×C31—44no—
156D8×C31—84no—
157M16×C31—42no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31—42no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31C2×D411yescharacteristic
171C4⋊C4×C31C2×D414yes—
172C4⋊C4×C31C2×D411yescharacteristic
173C4⋊C4×C31C2×D412yes—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—22no—
182C42×C31—24no—
183C42×C31—21no—
184C42×C31—21no—
185C42×C31—21no—
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188C42×C31C2×D411yescharacteristic
189C42×C31C2411yescharacteristic
190#18646C3111yesmaximal, characteristic, direct product factor, Sylow
191C23×C31—21no—
192C23×C31—22no—
193C23×C31—42no—
194C23×C31—82no—
195C23×C31—82no—
196C23×C31—82no—
197C23×C31#12011yescharacteristic
198Q8×C31—21no—
199Q8×C31—21no—
200Q8×C31—44no—
201Q8×C31—42no—
202Q8×C31—82no—
203D4×C31—21no—
204D4×C31—21no—
205D4×C31—24no—
206D4×C31—44no—
207D4×C31—42no—
208D4×C31—44no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—44no—
212D4×C31—42no—
213D4×C31—42no—
214D4×C31—82no—
215D4×C31—84no—
216D4×C31—82no—
217D4×C31—82no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—21no—
224C2×C4×C31—24no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—82no—
231C2×C4×C31—84no—
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31#13512yes—
236C2×C4×C31AGL(1,ℤ/8)12yes—
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C8×C31—42no—
241C8×C31—42no—
242C8×C31—42no—
243#3511C2×C3111yescharacteristic, complemented
244#3205C2×C3111yescharacteristic, complemented
245C2×#486C2×C3111yescharacteristic, complemented
246#3038C2×C3111yescharacteristic, complemented
247#1628C2×C3114yescomplemented
248#1625C2×C3114yescomplemented
249#1549C2×C3111yescharacteristic, complemented
250#1492C2×C3111yescharacteristic, complemented
251#1476C2×C3111yescharacteristic, complemented
252C22×C31—21no—
253C22×C31—21no—
254C22×C31—41no—
255C22×C31—42no—
256C22×C31—41no—
257C22×C31—42no—
258C22×C31—41no—
259C22×C31—42no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31—164no—
264C22×C31#44912yes—
265C22×C31C2×#12011yescharacteristic
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—41no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—44no—
272C4×C31—41no—
273C4×C31#44912yes—
274C4×C31#45412yes—
275C4×C31C2×#13511yescharacteristic
276C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C31#57612yes—
278C2×AGL(1,ℤ/8)—21no—
279#556C22×C3112yescomplemented
280#555—21no—
281#554—21no—
282#553—22no—
283#548—22no—
284#547C22×C3112yescomplemented
285#542—22no—
286#537—21no—
287#533—21no—
288#532—21no—
289#532C22×C3111yescharacteristic
290C2×C4⋊Q8C22×C3111yescharacteristic
291C2×#127C22×C3111yescharacteristic
292#520—21no—
293C4×#41C22×C3111yescharacteristic
294#493—21no—
295#486—22no—
296#486—21no—
297#486C22×C3114yes—
298#460—21no—
299#460C22×C3112yes—
300#459—21no—
301#459C22×C3112yes—
302#459C22×C3114yes—
303#447—22no—
304#431C22×C3111yescharacteristic
305#423C22×C3111yescharacteristic
306#418—21no—
307C2×#102—22no—
308C2×C4×C8C22×C3111yescharacteristic
309#332C22×C3114yescomplemented
310Q8⋊C8C22×C3112yescomplemented
311D4⋊C8C22×C3112yescomplemented
312D4⋊C8C22×C3114yescomplemented
313C2×C31—22no—
314C2×C31—81no—
315C2×C31—82no—
316C2×C31—162no—
317C2×C31C2×#44911yescharacteristic
318C2×C31#300811yescharacteristic
319C2×C31C2×#45411yescharacteristic
320C2×#41—22no—
321C2×#41—21no—
322C2×#41D4×C3111yescharacteristic
323C22×Q8—21no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327AGL(1,ℤ/8)—44no—
328C2×SD16—22no—
329C2×D8—22no—
330C2×D8—24no—
331C2×M16—21no—
332C22×C8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8D4×C3112yes—
336C4⋊Q8D4×C3111yescharacteristic
337C4⋊Q8C23×C3112yes—
338#127—21no—
339#127—21no—
340#127—21no—
341#127D4×C3111yescharacteristic
342#127C23×C3112yes—
343#126—24no—
344#125—22no—
345#124—24no—
346#124—22no—
347#124—21no—
348#123—24no—
349C22⋊Q8—22no—
350C22⋊Q8—24no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353#121—22no—
354#121—22no—
355#121—24no—
356#121—22no—
357#121—21no—
358#121D4×C3112yes—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8D4×C3111yescharacteristic
362C4×Q8C23×C3111yescharacteristic
363C4×D4—22no—
364C4×D4—24no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4D4×C3111yescharacteristic
370C4×D4C23×C3111yescharacteristic
371C4×D4C23×C3112yes—
372#117—22no—
373#117—22no—
374#117—21no—
375#117D4×C3111yescharacteristic
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C3112yes—
379C2×C42—21no—
380C2×C42C23×C3111yescharacteristic
381C4⋊C8—21no—
382C4⋊C8C23×C3112yes—
383C4⋊C8C23×C3112yes—
384#102—28no—
385#102—22no—
386#102—22no—
387#102—24no—
388C22⋊C8—21no—
389C4×C8—21no—
390C4×C8—21no—
391C4×C8C23×C3112yes—
392C31#1864611yescharacteristic, direct product factor, Sylow
393C24—41no—
394#41—22no—
395#41—22no—
396#41—42no—
397#41—42no—
398#41—84no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—82no—
403C2×Q8C2×D4×C3111yescharacteristic
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4C2×D4×C3111yescharacteristic
420C2×D4C2×D4×C3112yes—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430SD16—44no—
431D8—44no—
432D8—84no—
433M16—42no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—42no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4C2×D4×C3111yescharacteristic
447C4⋊C4C2×D4×C3114yes—
448C4⋊C4C2×D4×C3111yescharacteristic
449C4⋊C4C2×D4×C3112yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C42—24no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42C2×D4×C3111yescharacteristic
463C42C2×D4×C3111yescharacteristic
464C42C2×D4×C3111yescharacteristic
465C42C24×C3111yescharacteristic
466C23—21no—
467C23—22no—
468C23—42no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23C31×#12011yescharacteristic
473Q8—21no—
474Q8—21no—
475Q8—42no—
476Q8—44no—
477Q8—82no—
478D4—21no—
479D4—21no—
480D4—24no—
481D4—44no—
482D4—42no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—44no—
489D4—82no—
490D4—84no—
491D4—82no—
492D4—82no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4C31×#12011yescharacteristic
508C2×C4C31×#12011yescharacteristic
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#13512yes—
511C2×C4C31×AGL(1,ℤ/8)12yes—
512C2×C4C22×D4×C3111yescharacteristic
513C2×C4C22×D4×C3111yescharacteristic
514C2×C4C22×D4×C3111yescharacteristic
515C8—42no—
516C8—42no—
517C8—42no—
518C22—21no—
519C22—21no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—41no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—164no—
530C22C31×#44912yes—
531C22C2×C31×#12011yescharacteristic
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4C31×#44912yes—
540C4C31×#45412yes—
541C4C2×C31×#13511yescharacteristic
542C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
543C4C31×#57612yes—
544C2—22no—
545C2—81no—
546C2—82no—
547C2—162no—
548C2C2×C31×#44911yescharacteristic
549C2C31×#300811yescharacteristic
550C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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
1. C31×#3511C2semidirect
2. C31×#3205C2semidirect
3. C2×C31×#486C2semidirect
4. C31×#3038C2semidirect
5. C31×#1628C2semidirect
6. C31×#1625C2semidirect
7. C31×#1549C2semidirect
8. C31×#1492C2semidirect
9. C31×#1476C2semidirect
11. C31×#556C22semidirect
16. C31×#547C22semidirect
41. C31×#332C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×D4⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
190. #18646C31direct (N × H)
243. #3511C2×C31semidirect
244. #3205C2×C31semidirect
245. C2×#486C2×C31semidirect
246. #3038C2×C31semidirect
247. #1628C2×C31semidirect
248. #1625C2×C31semidirect
249. #1549C2×C31semidirect
250. #1492C2×C31semidirect
251. #1476C2×C31semidirect
279. #556C22×C31semidirect
284. #547C22×C31semidirect
309. #332C22×C31semidirect
310. Q8⋊C8C22×C31semidirect
311. D4⋊C8C22×C31semidirect
312. D4⋊C8C22×C31semidirect
392. C31#18646direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#32052
C2×C31×#4862
C31×#30382
C31×#16282
C31×#16252
C31×#15492
C31×#14922
C31×#14762
#1864631