Finite Group Databasebeta

← C31×#8427 #21465186

Subgroups — C31×#8427 #21465186

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3509C211yesmaximal, characteristic
2C31×#1988C211yesmaximal, characteristic, complemented
3C31×#1806C211yesmaximal, characteristic
4C31×#1550C211yesmaximal, characteristic, complemented
5C31×#1546C211yesmaximal, characteristic, complemented
6C31×#1503C211yesmaximal, characteristic, complemented
7C31×#1477C211yesmaximal, characteristic
8C31×#557—21no—
9C31×#557C412yescomplemented
10C31×#556—21no—
11C31×#556C412yescomplemented
12C31×#552—22no—
13C31×#552—22no—
14C31×#541—22no—
15C31×#536—21no—
16C31×#533—21no—
17C31×#533—21no—
18C31×#533C2211yescharacteristic
19C2×C31×C4⋊Q8C2211yescharacteristic
20C31×#519—21no—
21C4×#41×C31C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#104—21no—
26C4×C4⋊C4×C31C2211yescharacteristic
27C31×#340—22no—
28C31×#339—21no—
29C31×#333—21no—
30C31×#332—22no—
31C31×#332—21no—
32C31×#328—22no—
33C31×Q8⋊C8—21no—
34C31×D4⋊C8—21no—
35C2×#41×C31D411yescharacteristic
36C22×Q8×C31—21no—
37C2×M16×C31—21no—
38C22×C8×C31—21no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8C2×C412yes—
43C31×C4⋊Q8D411yescharacteristic
44C31×C4⋊Q8D412yes—
45C31×C4⋊Q8D411yescharacteristic
46C31×#126—24no—
47C31×#126—24no—
48C31×#125—22no—
49C31×#125—24no—
50C31×#125—22no—
51C31×#125—24no—
52C31×#125—21no—
53C31×#124—22no—
54C31×#123—24no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—22no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—21no—
60C31×C22⋊Q8D412yes—
61C4×Q8×C31—22no—
62C4×Q8×C31—22no—
63C4×Q8×C31—22no—
64C4×Q8×C31—22no—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31—21no—
68C4×Q8×C31C2×C411yescharacteristic
69C4×Q8×C31C2×C412yes—
70C4×Q8×C31D411yescharacteristic
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C4×D4×C31C2×C411yescharacteristic
74C4×D4×C31D411yescharacteristic
75C31×#117—24no—
76C31×#117—22no—
77C31×#117—21no—
78C31×#117—22no—
79C31×#117—21no—
80C31×#117D411yescharacteristic
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31D412yes—
84C2×C42×C31—21no—
85C2×C42×C31—21no—
86C2×C42×C31C2311yescharacteristic
87C31×C4⋊C8—21no—
88C31×C4⋊C8—21no—
89C31×C4⋊C8—21no—
90C31×C4⋊C8—21no—
91C31×C4⋊C8—21no—
92C31×#104—42no—
93C31×C22⋊C8—21no—
94C4×C8×C31—21no—
95C31×#95—21no—
96#41×C31—21no—
97#41×C31—21no—
98#41×C31—21no—
99#41×C31—21no—
100C2×Q8×C31—22no—
101C2×Q8×C31—21no—
102C2×Q8×C31—22no—
103C2×Q8×C31—41no—
104C2×Q8×C31—41no—
105C2×Q8×C31—41no—
106C2×Q8×C31—41no—
107C2×Q8×C31C22⋊C411yescharacteristic
108C2×Q8×C31C22⋊C412yes—
109C2×D4×C31—21no—
110C2×D4×C31C22⋊C411yescharacteristic
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—41no—
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2×D411yescharacteristic
119M16×C31—41no—
120M16×C31—41no—
121C2×C8×C31—42no—
122C2×C8×C31—41no—
123C2×C8×C31—41no—
124C2×C8×C31—41no—
125C4⋊C4×C31—24no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—24no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—41no—
141C4⋊C4×C31C22⋊C414yes—
142C4⋊C4×C31C22⋊C412yes—
143C4⋊C4×C31C22⋊C411yescharacteristic
144C4⋊C4×C31C22⋊C411yescharacteristic
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—21no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—21no—
150C22⋊C4×C31—21no—
151C42×C31—24no—
152C42×C31—24no—
153C42×C31—21no—
154C42×C31—22no—
155C42×C31—21no—
156C42×C31—21no—
157C42×C31—41no—
158C42×C31—41no—
159C42×C31C22×C411yescharacteristic
160C42×C31C2×D411yescharacteristic
161C42×C31C2×D411yescharacteristic
162C42×C31C2×D411yescharacteristic
163#8427C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—21no—
165C23×C31#12011yescharacteristic
166Q8×C31—21no—
167Q8×C31—21no—
168Q8×C31—42no—
169Q8×C31—42no—
170Q8×C31—42no—
171Q8×C31—42no—
172Q8×C31—42no—
173Q8×C31—41no—
174Q8×C31—41no—
175D4×C31—21no—
176D4×C31—21no—
177D4×C31—41no—
178D4×C31—41no—
179C2×C4×C31—21no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—21no—
187C2×C4×C31—41no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—42no—
196C2×C4×C31—81no—
197C2×C4×C31—81no—
198C2×C4×C31C2×C22⋊C411yescharacteristic
199C2×C4×C31C2×C22⋊C411yescharacteristic
200C2×C4×C31C2×C22⋊C411yescharacteristic
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31#12011yescharacteristic
203C2×C4×C31#12011yescharacteristic
204C2×C4×C31#13511yescharacteristic
205C2×C4×C31#13511yescharacteristic
206C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
207C2×C4×C31#13711yescharacteristic
208C8×C31—81no—
209C8×C31—81no—
210C8×C31—81no—
211#3509C2×C3111yescharacteristic
212#1988C2×C3111yescharacteristic, complemented
213#1806C2×C3111yescharacteristic
214#1550C2×C3111yescharacteristic, complemented
215#1546C2×C3111yescharacteristic, complemented
216#1503C2×C3111yescharacteristic, complemented
217#1477C2×C3111yescharacteristic
218C22×C31—21no—
219C22×C31—21no—
220C22×C31—41no—
221C22×C31—41no—
222C22×C31—41no—
223C22×C31#37911yescharacteristic
224C22×C31#44911yescharacteristic
225C22×C31#45211yescharacteristic
226C4×C31—21no—
227C4×C31—21no—
228C4×C31—21no—
229C4×C31—41no—
230C4×C31—42no—
231C4×C31—41no—
232C4×C31—42no—
233C4×C31—41no—
234C4×C31—44no—
235C4×C31—82no—
236C4×C31#41611yescharacteristic
237C4×C31#41711yescharacteristic
238C4×C31#44911yescharacteristic
239C4×C31#45211yescharacteristic
240C4×C31#45411yescharacteristic
241C4×C31#45611yescharacteristic
242#557—21no—
243#557C4×C3112yescomplemented
244#556—21no—
245#556C4×C3112yescomplemented
246#552—22no—
247#552—22no—
248#541—22no—
249#536—21no—
250#533—21no—
251#533—21no—
252#533C22×C3111yescharacteristic
253C2×C4⋊Q8C22×C3111yescharacteristic
254#519—21no—
255C4×#41C22×C3111yescharacteristic
256#431C22×C3111yescharacteristic
257#423C22×C3111yescharacteristic
258C2×C4⋊C8C22×C3111yescharacteristic
259C2×#104—21no—
260C4×C4⋊C4C22×C3111yescharacteristic
261#340—22no—
262#339—21no—
263#333—21no—
264#332—22no—
265#332—21no—
266#328—22no—
267Q8⋊C8—21no—
268D4⋊C8—21no—
269C2×C31—21no—
270C2×C31—21no—
271C2×C31—81no—
272C2×C31#178811yescharacteristic
273C2×C31#179311yescharacteristic
274C2×C31#179611yescharacteristic
275C2×#41D4×C3111yescharacteristic
276C22×Q8—21no—
277C2×M16—21no—
278C22×C8—21no—
279C4⋊Q8—22no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C2×C4×C3112yes—
283C4⋊Q8D4×C3111yescharacteristic
284C4⋊Q8D4×C3112yes—
285C4⋊Q8D4×C3111yescharacteristic
286#126—24no—
287#126—24no—
288#125—22no—
289#125—24no—
290#125—22no—
291#125—24no—
292#125—21no—
293#124—22no—
294#123—24no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—21no—
300C22⋊Q8D4×C3112yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—21no—
308C4×Q8C2×C4×C3111yescharacteristic
309C4×Q8C2×C4×C3112yes—
310C4×Q8D4×C3111yescharacteristic
311C4×D4—21no—
312C4×D4—21no—
313C4×D4C2×C4×C3111yescharacteristic
314C4×D4D4×C3111yescharacteristic
315#117—24no—
316#117—22no—
317#117—21no—
318#117—22no—
319#117—21no—
320#117D4×C3111yescharacteristic
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4D4×C3112yes—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42C23×C3111yescharacteristic
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332#104—42no—
333C22⋊C8—21no—
334C4×C8—21no—
335#95—21no—
336C31#842711yescharacteristic, direct product factor, Sylow
337#41—21no—
338#41—21no—
339#41—21no—
340#41—21no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8C22⋊C4×C3111yescharacteristic
349C2×Q8C22⋊C4×C3112yes—
350C2×D4—21no—
351C2×D4C22⋊C4×C3111yescharacteristic
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4C2×D4×C3111yescharacteristic
358C22×C4C2×D4×C3111yescharacteristic
359C22×C4C2×D4×C3111yescharacteristic
360M16—41no—
361M16—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4C22⋊C4×C3114yes—
383C4⋊C4C22⋊C4×C3112yes—
384C4⋊C4C22⋊C4×C3111yescharacteristic
385C4⋊C4C22⋊C4×C3111yescharacteristic
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—21no—
389C22⋊C4—22no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C42—24no—
393C42—24no—
394C42—21no—
395C42—22no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42C22×C4×C3111yescharacteristic
401C42C2×D4×C3111yescharacteristic
402C42C2×D4×C3111yescharacteristic
403C42C2×D4×C3111yescharacteristic
404C23—21no—
405C23C31×#12011yescharacteristic
406Q8—21no—
407Q8—21no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—41no—
414Q8—41no—
415D4—21no—
416D4—21no—
417D4—41no—
418D4—41no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4C2×C22⋊C4×C3111yescharacteristic
439C2×C4C2×C22⋊C4×C3111yescharacteristic
440C2×C4C2×C22⋊C4×C3111yescharacteristic
441C2×C4C31×#12011yescharacteristic
442C2×C4C31×#12011yescharacteristic
443C2×C4C31×#12011yescharacteristic
444C2×C4C31×#13511yescharacteristic
445C2×C4C31×#13511yescharacteristic
446C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
447C2×C4C31×#13711yescharacteristic
448C8—81no—
449C8—81no—
450C8—81no—
451C22—21no—
452C22—21no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22C31×#37911yescharacteristic
457C22C31×#44911yescharacteristic
458C22C31×#45211yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—42no—
464C4—41no—
465C4—44no—
466C4—41no—
467C4—42no—
468C4—82no—
469C4C31×#41611yescharacteristic
470C4C31×#41711yescharacteristic
471C4C31×#44911yescharacteristic
472C4C31×#45211yescharacteristic
473C4C31×#45411yescharacteristic
474C4C31×#45611yescharacteristic
475C2—21no—
476C2—21no—
477C2—81no—
478C2C31×#178811yescharacteristic
479C2C31×#179311yescharacteristic
480C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×#41×C31D4
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×#1988C2semidirect
4. C31×#1550C2semidirect
5. C31×#1546C2semidirect
6. C31×#1503C2semidirect
9. C31×#557C4semidirect
11. C31×#556C4semidirect
163. #8427C31direct (N × H)
212. #1988C2×C31semidirect
214. #1550C2×C31semidirect
215. #1546C2×C31semidirect
216. #1503C2×C31semidirect
243. #557C4×C31semidirect
245. #556C4×C31semidirect
336. C31#8427direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35092
C31×#19882
C31×#18062
C31×#15502
C31×#15462
C31×#15032
C31×#14772
#842731