Finite Group Databasebeta

← C31×#18536 #21475295

Subgroups — C31×#18536 #21475295

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#556C211yesmaximal, characteristic, complemented
2C31×#3141C211yesmaximal, characteristic, complemented
3C31×#3112C211yesmaximal, characteristic, complemented
4C31×#3109C211yesmaximal, characteristic
5C31×#1638C214yesmaximal, complemented
6C31×#1628C214yesmaximal, complemented
7C31×#1556C211yesmaximal, characteristic, complemented
8C31×#1491C211yesmaximal, characteristic, complemented
9C31×#1490C211yesmaximal, characteristic, complemented
10C31×#556—22no—
11C31×#556—22no—
12C31×#556—22no—
13C31×#556C2214yes—
14C31×#532C2211yescharacteristic
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#126—22no—
17C2×C31×#124—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C4×Q8×C31—21no—
20C2×C4×Q8×C31C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C2×C31×#117—21no—
23C31×#489—22no—
24C31×#487C2212yescomplemented
25C31×#486C2212yescomplemented
26C31×#481—21no—
27C31×#480—21no—
28C31×#477C2212yescomplemented
29C31×#476C2212yes—
30C31×#460C2212yescomplemented
31C31×#459C2212yes—
32C31×#448—21no—
33C31×#447—21no—
34C31×#442—22no—
35C31×#440—22no—
36C31×#432C2211yescharacteristic
37C31×#432C2211yescharacteristic
38C31×#417—22no—
39C4×M16×C31C2211yescharacteristic
40C31×#332C2214yescomplemented
41C31×Q8⋊C8C2214yescomplemented
42C31×D4⋊C8C2214yes—
43C2×#41×C31—21no—
44C22×Q8×C31—21no—
45C22×Q8×C31D411yescharacteristic
46C22×D4×C31D411yescharacteristic
47C23×C4×C31—21no—
48C2×SD16×C31—22no—
49C2×D8×C31—22no—
50C2×M16×C31—21no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8D412yes—
54C31×C4⋊Q8C2311yescharacteristic
55C31×C4⋊Q8C2312yes—
56C31×#127C2311yescharacteristic
57C31×#126—28no—
58C31×#126—28no—
59C31×#124—24no—
60C31×#124—24no—
61C31×#124—21no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C31×#121—21no—
65C31×#121—21no—
66C4×Q8×C31—24no—
67C4×Q8×C31—24no—
68C4×Q8×C31—21no—
69C4×Q8×C31—21no—
70C4×Q8×C31D412yes—
71C4×Q8×C31C2312yes—
72C4×D4×C31—21no—
73C4×D4×C31—21no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31D412yes—
77C4×D4×C31C2312yes—
78C31×#117—24no—
79C31×#117—24no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31D412yes—
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31D411yescharacteristic
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—21no—
89C2×C42×C31—22no—
90C2×C42×C31—21no—
91C2×C42×C31C2311yescharacteristic
92C31×#107—22no—
93C31×#106—22no—
94C31×C4⋊C8C2312yes—
95C31×C4⋊C8C2312yes—
96C31×#103—22no—
97C31×#103—24no—
98C31×#102—22no—
99C31×#102—24no—
100C31×#102—22no—
101C31×#102—22no—
102C31×C22⋊C8—21no—
103C31×C22⋊C8—21no—
104C31×#97—21no—
105C31×#97—21no—
106C31×#97—21no—
107C4×C8×C31C2312yes—
108C24×C31—21no—
109#41×C31—44no—
110C2×Q8×C31—21no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—21no—
114C2×Q8×C31—42no—
115C2×Q8×C31—44no—
116C2×Q8×C31C2×D412yes—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—42no—
122C2×D4×C31—41no—
123C2×D4×C31C2×D412yes—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—44no—
131C22×C4×C31—41no—
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135SD16×C31—44no—
136D8×C31—44no—
137M16×C31—42no—
138C2×C8×C31—22no—
139C2×C8×C31—22no—
140C2×C8×C31—22no—
141C4⋊C4×C31—28no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31C2×D414yes—
152C4⋊C4×C31C2×D412yes—
153C4⋊C4×C31C2×D412yes—
154C22⋊C4×C31—28no—
155C22⋊C4×C31—28no—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—41no—
159C22⋊C4×C31—41no—
160C42×C31—28no—
161C42×C31—22no—
162C42×C31—22no—
163C42×C31C2×D411yescharacteristic
164C42×C31C2×D411yescharacteristic
165C42×C31C2×D411yescharacteristic
166C42×C31C2411yescharacteristic
167#18536C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—22no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—41no—
173C23×C31#12011yescharacteristic
174Q8×C31—22no—
175Q8×C31—22no—
176Q8×C31—42no—
177Q8×C31—44no—
178Q8×C31—42no—
179Q8×C31—44no—
180D4×C31—22no—
181D4×C31—22no—
182D4×C31—42no—
183D4×C31—42no—
184D4×C31—42no—
185D4×C31—42no—
186D4×C31—82no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—24no—
194C2×C4×C31—44no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—44no—
198C2×C4×C31—42no—
199C2×C4×C31—44no—
200C2×C4×C31—41no—
201C2×C4×C31—82no—
202C2×C4×C31#12011yescharacteristic
203C2×C4×C31#12011yescharacteristic
204C2×C4×C31#12011yescharacteristic
205C2×C4×C31C22×D411yescharacteristic
206C2×C4×C31C22×D411yescharacteristic
207C2×C4×C31C22×D411yescharacteristic
208C8×C31—42no—
209C8×C31—42no—
210C8×C31—42no—
211C2×#556C2×C3111yescharacteristic, complemented
212#3141C2×C3111yescharacteristic, complemented
213#3112C2×C3111yescharacteristic, complemented
214#3109C2×C3111yescharacteristic
215#1638C2×C3114yescomplemented
216#1628C2×C3114yescomplemented
217#1556C2×C3111yescharacteristic, complemented
218#1491C2×C3111yescharacteristic, complemented
219#1490C2×C3111yescharacteristic, complemented
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—42no—
223C22×C31—42no—
224C22×C31—42no—
225C22×C31—84no—
226C22×C31—81no—
227C22×C31—81no—
228C22×C31—81no—
229C22×C31#45412yes—
230C22×C31C2×#12011yescharacteristic
231C4×C31—21no—
232C4×C31—21no—
233C4×C31—21no—
234C4×C31—21no—
235C4×C31—42no—
236C4×C31—42no—
237C4×C31—44no—
238C4×C31—42no—
239C4×C31—81no—
240C4×C31#57512yes—
241C4×C31#57512yes—
242#556—22no—
243#556—22no—
244#556—22no—
245#556C22×C3114yes—
246#532C22×C3111yescharacteristic
247C2×C4⋊Q8C22×C3111yescharacteristic
248C2×#126—22no—
249C2×#124—21no—
250C2×C22⋊Q8—21no—
251C2×C4×Q8—21no—
252C2×C4×Q8C22×C3111yescharacteristic
253C2×C4×D4C22×C3111yescharacteristic
254C2×#117—21no—
255#489—22no—
256#487C22×C3112yescomplemented
257#486C22×C3112yescomplemented
258#481—21no—
259#480—21no—
260#477C22×C3112yescomplemented
261#476C22×C3112yes—
262#460C22×C3112yescomplemented
263#459C22×C3112yes—
264#448—21no—
265#447—21no—
266#442—22no—
267#440—22no—
268#432C22×C3111yescharacteristic
269#432C22×C3111yescharacteristic
270#417—22no—
271C4×M16C22×C3111yescharacteristic
272#332C22×C3114yescomplemented
273Q8⋊C8C22×C3114yescomplemented
274D4⋊C8C22×C3114yes—
275C2×C31—22no—
276C2×C31—82no—
277C2×C31—161no—
278C2×C31#300511yescharacteristic
279C2×C31#300511yescharacteristic
280C2×C31C2×#45411yescharacteristic
281C2×#41—21no—
282C22×Q8—21no—
283C22×Q8D4×C3111yescharacteristic
284C22×D4D4×C3111yescharacteristic
285C23×C4—21no—
286C2×SD16—22no—
287C2×D8—22no—
288C2×M16—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8D4×C3112yes—
292C4⋊Q8C23×C3111yescharacteristic
293C4⋊Q8C23×C3112yes—
294#127C23×C3111yescharacteristic
295#126—28no—
296#126—28no—
297#124—24no—
298#124—24no—
299#124—21no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302#121—21no—
303#121—21no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8D4×C3112yes—
309C4×Q8C23×C3112yes—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4D4×C3112yes—
315C4×D4C23×C3112yes—
316#117—24no—
317#117—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4D4×C3111yescharacteristic
322C2×C4⋊C4D4×C3111yescharacteristic
323C2×C4⋊C4D4×C3112yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42C23×C3111yescharacteristic
330#107—22no—
331#106—22no—
332C4⋊C8C23×C3112yes—
333C4⋊C8C23×C3112yes—
334#103—22no—
335#103—24no—
336#102—22no—
337#102—24no—
338#102—22no—
339#102—22no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342#97—21no—
343#97—21no—
344#97—21no—
345C4×C8C23×C3112yes—
346C31#1853611yescharacteristic, direct product factor, Sylow
347C24—21no—
348#41—44no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—44no—
355C2×Q8C2×D4×C3112yes—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4C2×D4×C3112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4C2×D4×C3111yescharacteristic
372C22×C4C2×D4×C3111yescharacteristic
373C22×C4C2×D4×C3111yescharacteristic
374SD16—44no—
375D8—44no—
376M16—42no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4C2×D4×C3114yes—
391C4⋊C4C2×D4×C3112yes—
392C4⋊C4C2×D4×C3112yes—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42C2×D4×C3111yescharacteristic
403C42C2×D4×C3111yescharacteristic
404C42C2×D4×C3111yescharacteristic
405C42C24×C3111yescharacteristic
406C23—22no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23C31×#12011yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—42no—
417Q8—44no—
418D4—22no—
419D4—22no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—82no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—41no—
439C2×C4—82no—
440C2×C4C31×#12011yescharacteristic
441C2×C4C31×#12011yescharacteristic
442C2×C4C31×#12011yescharacteristic
443C2×C4C22×D4×C3111yescharacteristic
444C2×C4C22×D4×C3111yescharacteristic
445C2×C4C22×D4×C3111yescharacteristic
446C8—42no—
447C8—42no—
448C8—42no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—84no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22C31×#45412yes—
459C22C2×C31×#12011yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—42no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—81no—
469C4C31×#57512yes—
470C4C31×#57512yes—
471C2—22no—
472C2—82no—
473C2—161no—
474C2C31×#300511yescharacteristic
475C2C31×#300511yescharacteristic
476C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3109C2
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. C2×C31×#556C2semidirect
2. C31×#3141C2semidirect
3. C31×#3112C2semidirect
5. C31×#1638C2semidirect
6. C31×#1628C2semidirect
7. C31×#1556C2semidirect
8. C31×#1491C2semidirect
9. C31×#1490C2semidirect
24. C31×#487C22semidirect
25. C31×#486C22semidirect
28. C31×#477C22semidirect
30. C31×#460C22semidirect
40. C31×#332C22semidirect
41. C31×Q8⋊C8C22semidirect
167. #18536C31direct (N × H)
211. C2×#556C2×C31semidirect
212. #3141C2×C31semidirect
213. #3112C2×C31semidirect
215. #1638C2×C31semidirect
216. #1628C2×C31semidirect
217. #1556C2×C31semidirect
218. #1491C2×C31semidirect
219. #1490C2×C31semidirect
256. #487C22×C31semidirect
257. #486C22×C31semidirect
260. #477C22×C31semidirect
262. #460C22×C31semidirect
272. #332C22×C31semidirect
273. Q8⋊C8C22×C31semidirect
346. C31#18536direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5562
C31×#31412
C31×#31122
C31×#31092
C31×#16382
C31×#16282
C31×#15562
C31×#14912
C31×#14902
#1853631