Finite Group Databasebeta

← C31×#18412 #21475171

Subgroups — C31×#18412 #21475171

Order 7936.

994 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3229C211yesmaximal, characteristic, complemented
3C31×#3226C211yesmaximal, characteristic, complemented
4C31×#2547C211yesmaximal, characteristic, complemented
5C31×#1612C218yesmaximal, complemented
6C31×#1503C211yesmaximal, characteristic, complemented
7C31×#1502C211yesmaximal, characteristic, complemented
8C31×#1460C211yesmaximal, characteristic, complemented
9C31×#559—22no—
10C31×#556—21no—
11C31×#555—21no—
12C31×#553—22no—
13C31×#550—21no—
14C31×#547—21no—
15C31×#538C2214yes—
16C31×#535—21no—
17C31×#533C2211yescharacteristic
18C31×#532C2211yescharacteristic
19C2×C31×C4⋊Q8C2211yescharacteristic
20C2×C31×#124—21no—
21C31×#525—22no—
22C31×#520—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#500—21no—
25C31×#498—21no—
26C31×#487—21no—
27C31×#486—21no—
28C31×#483C2214yes—
29C31×#480C2214yes—
30C31×#475—22no—
31C31×#474—22no—
32C31×#431C2211yescharacteristic
33C31×#431C2211yescharacteristic
34C31×#397C2214yes—
35C31×#391—21no—
36C31×#380—22no—
37C4×C4⋊C4×C31C2211yescharacteristic
38C31×#328C2214yes—
39C31×#327C2214yes—
40C31×#323C2214yes—
41C2×#41×C31—22no—
42C2×#41×C31—21no—
43C22×Q8×C31—21no—
44C22×D4×C31D411yescharacteristic
45C23×C4×C31D412yes—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8D411yescharacteristic
51C31×C4⋊Q8D411yescharacteristic
52C31×C4⋊Q8D411yescharacteristic
53C31×#127—22no—
54C31×#127D411yescharacteristic
55C31×#126—24no—
56C31×#125—21no—
57C31×#124—22no—
58C31×#124—24no—
59C31×#124—21no—
60C31×#124—44no—
61C31×#123—24no—
62C31×#123—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8C2312yes—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—24no—
70C31×#121—21no—
71C31×#121C2312yes—
72C31×#120—24no—
73C4×Q8×C31—22no—
74C4×Q8×C31—21no—
75C4×Q8×C31D412yes—
76C4×D4×C31—22no—
77C4×D4×C31—24no—
78C4×D4×C31—24no—
79C4×D4×C31—21no—
80C4×D4×C31—21no—
81C4×D4×C31—21no—
82C4×D4×C31—21no—
83C4×D4×C31—21no—
84C4×D4×C31D412yes—
85C31×#117—22no—
86C31×#117—22no—
87C31×#117—21no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C22⋊C4×C31—24no—
94C2×C22⋊C4×C31C2312yes—
95C2×C42×C31—21no—
96C2×C42×C31C2311yescharacteristic
97C31×#107—24no—
98C31×#106—24no—
99C31×C4⋊C8—21no—
100C31×C4⋊C8—21no—
101C31×#103—24no—
102C31×#102—24no—
103C31×C22⋊C8C2312yes—
104C31×C22⋊C8C2312yes—
105C4×C8×C31—21no—
106C4×C8×C31—21no—
107C31×#95—24no—
108C31×#95C2312yes—
109C24×C31C2×D412yes—
110#41×C31—42no—
111#41×C31—42no—
112#41×C31—84no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—41no—
116C2×Q8×C31—41no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C2×Q8×C31C2×D412yes—
120C2×D4×C31—22no—
121C2×D4×C31—24no—
122C2×D4×C31—24no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31C2×D412yes—
130C22×C4×C31—24no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—44no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2411yescharacteristic
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—42no—
155C4⋊C4×C31—41no—
156C4⋊C4×C31—41no—
157C4⋊C4×C31C2×D412yes—
158C4⋊C4×C31C2×D412yes—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—24no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—48no—
168C42×C31—24no—
169C42×C31—22no—
170C42×C31—42no—
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173C42×C31C2×D411yescharacteristic
174C42×C31C2×D411yescharacteristic
175#18412C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—22no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31C22×D411yescharacteristic
183Q8×C31—42no—
184Q8×C31—42no—
185Q8×C31—42no—
186Q8×C31—82no—
187Q8×C31—82no—
188D4×C31—42no—
189D4×C31—42no—
190D4×C31—42no—
191D4×C31—42no—
192D4×C31—42no—
193D4×C31—42no—
194D4×C31—42no—
195D4×C31—84no—
196D4×C31—84no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—41no—
203C2×C4×C31—44no—
204C2×C4×C31—44no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—82no—
213C2×C4×C31—84no—
214C2×C4×C31#12011yescharacteristic
215C2×C4×C31#12011yescharacteristic
216C2×C4×C31#12011yescharacteristic
217C2×C4×C31#12011yescharacteristic
218C2×C4×C31#13512yes—
219C2×C4×C31AGL(1,ℤ/8)12yes—
220C2×C4×C31C22×D411yescharacteristic
221C2×C4×C31C22×D411yescharacteristic
222C8×C31—42no—
223C8×C31—42no—
224#3506C2×C3111yescharacteristic
225#3229C2×C3111yescharacteristic, complemented
226#3226C2×C3111yescharacteristic, complemented
227#2547C2×C3111yescharacteristic, complemented
228#1612C2×C3118yescomplemented
229#1503C2×C3111yescharacteristic, complemented
230#1502C2×C3111yescharacteristic, complemented
231#1460C2×C3111yescharacteristic, complemented
232C22×C31—21no—
233C22×C31—21no—
234C22×C31—44no—
235C22×C31—42no—
236C22×C31—42no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—82no—
242C22×C31C2×#12011yescharacteristic
243C22×C31C2×#13511yescharacteristic
244C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
245C4×C31—21no—
246C4×C31—21no—
247C4×C31—21no—
248C4×C31—42no—
249C4×C31—42no—
250C4×C31—42no—
251C4×C31—42no—
252C4×C31—82no—
253C4×C31—82no—
254C4×C31#44912yes—
255C4×C31#44912yes—
256C4×C31#45512yes—
257#559—22no—
258#556—21no—
259#555—21no—
260#553—22no—
261#550—21no—
262#547—21no—
263#538C22×C3114yes—
264#535—21no—
265#533C22×C3111yescharacteristic
266#532C22×C3111yescharacteristic
267C2×C4⋊Q8C22×C3111yescharacteristic
268C2×#124—21no—
269#525—22no—
270#520—21no—
271C2×C4×D4C22×C3111yescharacteristic
272#500—21no—
273#498—21no—
274#487—21no—
275#486—21no—
276#483C22×C3114yes—
277#480C22×C3114yes—
278#475—22no—
279#474—22no—
280#431C22×C3111yescharacteristic
281#431C22×C3111yescharacteristic
282#397C22×C3114yes—
283#391—21no—
284#380—22no—
285C4×C4⋊C4C22×C3111yescharacteristic
286#328C22×C3114yes—
287#327C22×C3114yes—
288#323C22×C3114yes—
289C2×C31—21no—
290C2×C31—21no—
291C2×C31—82no—
292C2×C31—82no—
293C2×C31C2×#44911yescharacteristic
294C2×C31C2×#45511yescharacteristic
295C2×C31#302811yescharacteristic
296C2×#41—22no—
297C2×#41—21no—
298C22×Q8—21no—
299C22×D4D4×C3111yescharacteristic
300C23×C4D4×C3112yes—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8D4×C3111yescharacteristic
306C4⋊Q8D4×C3111yescharacteristic
307C4⋊Q8D4×C3111yescharacteristic
308#127—22no—
309#127D4×C3111yescharacteristic
310#126—24no—
311#125—21no—
312#124—22no—
313#124—24no—
314#124—21no—
315#124—44no—
316#123—24no—
317#123—24no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8C23×C3112yes—
321#121—22no—
322#121—22no—
323#121—22no—
324#121—24no—
325#121—21no—
326#121C23×C3112yes—
327#120—24no—
328C4×Q8—22no—
329C4×Q8—21no—
330C4×Q8D4×C3112yes—
331C4×D4—22no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4D4×C3112yes—
340#117—22no—
341#117—22no—
342#117—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4D4×C3111yescharacteristic
347C2×C4⋊C4C23×C3112yes—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C23×C3112yes—
350C2×C42—21no—
351C2×C42C23×C3111yescharacteristic
352#107—24no—
353#106—24no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356#103—24no—
357#102—24no—
358C22⋊C8C23×C3112yes—
359C22⋊C8C23×C3112yes—
360C4×C8—21no—
361C4×C8—21no—
362#95—24no—
363#95C23×C3112yes—
364C31#1841211yescharacteristic, direct product factor, Sylow
365C24C2×D4×C3112yes—
366#41—42no—
367#41—42no—
368#41—84no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8C2×D4×C3112yes—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4C2×D4×C3112yes—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—44no—
393C22×C4C2×D4×C3111yescharacteristic
394C22×C4C2×D4×C3112yes—
395C22×C4C2×D4×C3111yescharacteristic
396C22×C4C24×C3111yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×D4×C3112yes—
414C4⋊C4C2×D4×C3112yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—48no—
424C42—24no—
425C42—22no—
426C42—42no—
427C42C2×D4×C3111yescharacteristic
428C42C2×D4×C3111yescharacteristic
429C42C2×D4×C3111yescharacteristic
430C42C2×D4×C3111yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23C22×D4×C3111yescharacteristic
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—82no—
442Q8—82no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4C31×#12011yescharacteristic
470C2×C4C31×#12011yescharacteristic
471C2×C4C31×#12011yescharacteristic
472C2×C4C31×#12011yescharacteristic
473C2×C4C31×#13512yes—
474C2×C4C31×AGL(1,ℤ/8)12yes—
475C2×C4C22×D4×C3111yescharacteristic
476C2×C4C22×D4×C3111yescharacteristic
477C8—42no—
478C8—42no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22C2×C31×#12011yescharacteristic
490C22C2×C31×#13511yescharacteristic
491C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—82no—
501C4C31×#44912yes—
502C4C31×#44912yes—
503C4C31×#45512yes—
504C2—21no—
505C2—21no—
506C2—82no—
507C2—82no—
508C2C2×C31×#44911yescharacteristic
509C2C2×C31×#45511yescharacteristic
510C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3506C2
LCSG ▷ C22×C4 ▷ 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×#3229C2semidirect
3. C31×#3226C2semidirect
4. C31×#2547C2semidirect
5. C31×#1612C2semidirect
6. C31×#1503C2semidirect
7. C31×#1502C2semidirect
8. C31×#1460C2semidirect
175. #18412C31direct (N × H)
225. #3229C2×C31semidirect
226. #3226C2×C31semidirect
227. #2547C2×C31semidirect
228. #1612C2×C31semidirect
229. #1503C2×C31semidirect
230. #1502C2×C31semidirect
231. #1460C2×C31semidirect
364. C31#18412direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#32292
C31×#32262
C31×#25472
C31×#16122
C31×#15032
C31×#15022
C31×#14602
#1841231