Finite Group Databasebeta

← C31×#18124 #21474883

Subgroups — C31×#18124 #21474883

Order 7936.

1012 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3458C211yesmaximal, characteristic, complemented
2C31×#3101C212yesmaximal, complemented
3C31×#2392C211yesmaximal, characteristic
4C31×#1611C212yesmaximal, complemented
5C31×#1608C212yesmaximal, complemented
6C31×#1599C212yesmaximal, complemented
7C31×#1596C212yesmaximal
8C31×#1503C212yesmaximal, complemented
9C31×#1458C211yesmaximal, characteristic, complemented
10C31×#544—22no—
11C31×#544—22no—
12C31×#544—21no—
13C31×#544C2212yes—
14C31×#543—21no—
15C31×#543C2212yes—
16C31×#533—24no—
17C31×#533—21no—
18C31×#533C2212yes—
19C2×C4×D4×C31—21no—
20C2×C4×D4×C31C2211yescharacteristic
21C2×C31×#117—21no—
22C2×C31×#117C2211yescharacteristic
23C31×#483C2214yes—
24C31×#476—22no—
25C31×#461—22no—
26C31×#450C2214yes—
27C4×SD16×C31—24no—
28C31×#431C2212yes—
29C31×#398C2211yescharacteristic
30C31×#397C2211yescharacteristic
31C31×#396—21no—
32C31×#394—21no—
33C31×#394C2211yescharacteristic
34C31×#392C2211yescharacteristic
35C4×C4⋊C4×C31C2211yescharacteristic
36C4×C22⋊C4×C31—21no—
37C31×#328C2214yes—
38C31×#328C2214yes—
39C31×#324C2212yes—
40C31×#323C2212yes—
41C22×D4×C31—21no—
42C22×D4×C31C2311yescharacteristic
43C23×C4×C31—21no—
44C23×C4×C31D411yescharacteristic
45C2×SD16×C31—24no—
46C31×C4⋊Q8—24no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—24no—
50C31×C4⋊Q8D412yescomplemented
51C31×C4⋊Q8D412yescomplemented
52C31×#125—24no—
53C31×#125—22no—
54C31×#125—24no—
55C31×#125—22no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—24no—
59C31×C22⋊Q8—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8C2314yes—
64C4×Q8×C31—22no—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31D414yescomplemented
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—44no—
71C31×#117—22no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117—22no—
75C31×#117—22no—
76C31×#117—22no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31D411yescharacteristic
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31D411yescharacteristic
91C2×C22⋊C4×C31C2311yescharacteristic
92C2×C42×C31—22no—
93C2×C42×C31—21no—
94C2×C42×C31D411yescharacteristic
95C2×C42×C31C2311yescharacteristic
96C31×#106—24no—
97C31×C4⋊C8—22no—
98C31×#103—24no—
99C31×#102—24no—
100C31×C22⋊C8C2314yes—
101C4×C8×C31—22no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95C2311yescharacteristic
105C31×#95C2311yescharacteristic
106C24×C31—21no—
107C24×C31C2×D411yescharacteristic
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31C2×D414yes—
112C2×D4×C31—22no—
113C2×D4×C31—42no—
114C2×D4×C31—42no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—41no—
126C22×C4×C31—41no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31SD1614yescomplemented
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2411yescharacteristic
136SD16×C31—48no—
137C2×C8×C31—24no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—42no—
153C4⋊C4×C31—42no—
154C4⋊C4×C31C2×D414yes—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31—42no—
161C22⋊C4×C31—42no—
162C22⋊C4×C31—42no—
163C22⋊C4×C31—42no—
164C42×C31—24no—
165C42×C31—21no—
166C42×C31—24no—
167C42×C31—21no—
168C42×C31—42no—
169C42×C31C2×D412yes—
170C42×C31C2×D412yes—
171#18124C3111yesmaximal, characteristic, direct product factor, Sylow
172C23×C31—21no—
173C23×C31—21no—
174C23×C31—22no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31C2×SD1612yes—
184C23×C31C22×D411yescharacteristic
185Q8×C31—44no—
186Q8×C31—82no—
187Q8×C31—82no—
188Q8×C31—82no—
189D4×C31—24no—
190D4×C31—42no—
191D4×C31—48no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—22no—
197C2×C4×C31—24no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—24no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—82no—
214C2×C4×C31—82no—
215C2×C4×C31—82no—
216C2×C4×C31—82no—
217C2×C4×C31#12012yes—
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12011yescharacteristic
220C2×C4×C31C2×SD1612yes—
221C2×C4×C31C2×SD1612yes—
222C2×C4×C31#13712yes—
223C2×C4×C31C22×D412yes—
224C8×C31—44no—
225#3458C2×C3111yescharacteristic, complemented
226#3101C2×C3112yescomplemented
227#2392C2×C3111yescharacteristic
228#1611C2×C3112yescomplemented
229#1608C2×C3112yescomplemented
230#1599C2×C3112yescomplemented
231#1596C2×C3112yes—
232#1503C2×C3112yescomplemented
233#1458C2×C3111yescharacteristic, complemented
234C22×C31—21no—
235C22×C31—21no—
236C22×C31—22no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—41no—
241C22×C31—41no—
242C22×C31—42no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31C2×#12011yescharacteristic
249C22×C31C22×SD1611yescharacteristic
250C22×C31C2×#13711yescharacteristic
251C4×C31—22no—
252C4×C31—22no—
253C4×C31—22no—
254C4×C31—42no—
255C4×C31—41no—
256C4×C31—44no—
257C4×C31—41no—
258C4×C31—44no—
259C4×C31—81no—
260C4×C31—81no—
261C4×C31#44812yes—
262C4×C31#44812yes—
263#544—22no—
264#544—22no—
265#544—21no—
266#544C22×C3112yes—
267#543—21no—
268#543C22×C3112yes—
269#533—24no—
270#533—21no—
271#533C22×C3112yes—
272C2×C4×D4—21no—
273C2×C4×D4C22×C3111yescharacteristic
274C2×#117—21no—
275C2×#117C22×C3111yescharacteristic
276#483C22×C3114yes—
277#476—22no—
278#461—22no—
279#450C22×C3114yes—
280C4×SD16—24no—
281#431C22×C3112yes—
282#398C22×C3111yescharacteristic
283#397C22×C3111yescharacteristic
284#396—21no—
285#394—21no—
286#394C22×C3111yescharacteristic
287#392C22×C3111yescharacteristic
288C4×C4⋊C4C22×C3111yescharacteristic
289C4×C22⋊C4—21no—
290#328C22×C3114yes—
291#328C22×C3114yes—
292#324C22×C3112yes—
293#323C22×C3112yes—
294C2×C31—21no—
295C2×C31—21no—
296C2×C31—42no—
297C2×C31—82no—
298C2×C31C2×#44811yescharacteristic
299C2×C31#301811yescharacteristic
300C2×C31#302511yescharacteristic
301C22×D4—21no—
302C22×D4C23×C3111yescharacteristic
303C23×C4—21no—
304C23×C4D4×C3111yescharacteristic
305C2×SD16—24no—
306C4⋊Q8—24no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—24no—
310C4⋊Q8D4×C3112yescomplemented
311C4⋊Q8D4×C3112yescomplemented
312#125—24no—
313#125—22no—
314#125—24no—
315#125—22no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8C23×C3114yes—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8D4×C3114yescomplemented
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—44no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4D4×C3111yescharacteristic
342C2×C4⋊C4C23×C3111yescharacteristic
343C2×C4⋊C4C23×C3111yescharacteristic
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4D4×C3111yescharacteristic
351C2×C22⋊C4C23×C3111yescharacteristic
352C2×C42—22no—
353C2×C42—21no—
354C2×C42D4×C3111yescharacteristic
355C2×C42C23×C3111yescharacteristic
356#106—24no—
357C4⋊C8—22no—
358#103—24no—
359#102—24no—
360C22⋊C8C23×C3114yes—
361C4×C8—22no—
362#95—21no—
363#95—21no—
364#95C23×C3111yescharacteristic
365#95C23×C3111yescharacteristic
366C31#1812411yescharacteristic, direct product factor, Sylow
367C24—21no—
368C24C2×D4×C3111yescharacteristic
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8C2×D4×C3114yes—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4SD16×C3114yescomplemented
391C22×C4C2×D4×C3111yescharacteristic
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3111yescharacteristic
394C22×C4C2×D4×C3111yescharacteristic
395C22×C4C2×D4×C3111yescharacteristic
396C22×C4C24×C3111yescharacteristic
397SD16—48no—
398C2×C8—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C2×D4×C3114yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—24no—
426C42—24no—
427C42—21no—
428C42—21no—
429C42—42no—
430C42C2×D4×C3112yes—
431C42C2×D4×C3112yes—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C2×SD16×C3112yes—
444C23C22×D4×C3111yescharacteristic
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449D4—24no—
450D4—42no—
451D4—48no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4C31×#12012yes—
478C2×C4C31×#12011yescharacteristic
479C2×C4C31×#12011yescharacteristic
480C2×C4C2×SD16×C3112yes—
481C2×C4C2×SD16×C3112yes—
482C2×C4C31×#13712yes—
483C2×C4C22×D4×C3112yes—
484C8—44no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C2×C31×#12011yescharacteristic
500C22C22×SD16×C3111yescharacteristic
501C22C2×C31×#13711yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—42no—
506C4—41no—
507C4—44no—
508C4—41no—
509C4—44no—
510C4—81no—
511C4—81no—
512C4C31×#44812yes—
513C4C31×#44812yes—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—82no—
518C2C2×C31×#44811yescharacteristic
519C2C31×#301811yescharacteristic
520C2C31×#302511yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
prime radicalC31×#392C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3458C2semidirect
2. C31×#3101C2semidirect
4. C31×#1611C2semidirect
5. C31×#1608C2semidirect
6. C31×#1599C2semidirect
8. C31×#1503C2semidirect
9. C31×#1458C2semidirect
50. C31×C4⋊Q8D4semidirect
51. C31×C4⋊Q8D4semidirect
67. C4×Q8×C31D4semidirect
129. C22×C4×C31SD16semidirect
171. #18124C31direct (N × H)
225. #3458C2×C31semidirect
226. #3101C2×C31semidirect
228. #1611C2×C31semidirect
229. #1608C2×C31semidirect
230. #1599C2×C31semidirect
232. #1503C2×C31semidirect
233. #1458C2×C31semidirect
310. C4⋊Q8D4×C31semidirect
311. C4⋊Q8D4×C31semidirect
327. C4×Q8D4×C31semidirect
366. C31#18124direct (N × H)
390. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34582
C31×#31012
C31×#23922
C31×#16112
C31×#16082
C31×#15992
C31×#15962
C31×#15032
C31×#14582
#1812431