Finite Group Databasebeta

← C31×#32988 #21489747

Subgroups — C31×#32988 #21489747

Order 7936.

1300 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic
2C2×C31×#485C211yesmaximal, characteristic, complemented
3C2×C31×#483C211yesmaximal, characteristic, complemented
4C31×#2616C211yesmaximal, characteristic
5C31×#1873C212yesmaximal, complemented
6C31×#1872C212yesmaximal, complemented
7C31×#1854C212yesmaximal, complemented
8C31×#1853C212yesmaximal, complemented
9C31×#1825C211yesmaximal, characteristic, complemented
10C31×#1824C211yesmaximal, characteristic, complemented
11C31×#1780C211yesmaximal, characteristic
12C22×#41×C31C2211yescharacteristic
13C2×C31×C22⋊Q8C2211yescharacteristic
14C2×C31×C22⋊Q8C2212yes—
15C31×#520—24no—
16C31×#520—24no—
17C2×C4×Q8×C31—21no—
18C2×C4×D4×C31—22no—
19C2×C4×D4×C31—21no—
20C2×C31×#117C2212yes—
21C2×C31×#117C2211yescharacteristic
22C22×C4⋊C4×C31C2212yes—
23C22×C4⋊C4×C31C2211yescharacteristic
24C31×#485—24no—
25C31×#483—24no—
26C2×C31×#107C2211yescharacteristic
27C2×C31×#106C2212yes—
28C2×C31×#106C2211yescharacteristic
29C2×C31×#103C2212yes—
30C2×C31×#103C2211yescharacteristic
31C2×C31×#103C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C2×C31×C22⋊C8C2211yescharacteristic
34C2×C31×C22⋊C8C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C31×#395—21no—
37C31×#394—21no—
38C31×#394—22no—
39C31×#380C2211yescharacteristic
40C31×#380C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31—24no—
48C2×#41×C31—24no—
49C22×Q8×C31D411yescharacteristic
50C22×Q8×C31C2311yescharacteristic
51C22×D4×C31D411yescharacteristic
52C22×D4×C31D412yes—
53C23×C4×C31—22no—
54C23×C4×C31—21no—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2312yes—
57C22×C8×C31C2312yes—
58C22×C8×C31C2311yescharacteristic
59C22×C8×C31C2311yescharacteristic
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C4×Q8×C31—28no—
65C4×D4×C31—216no—
66C4×D4×C31—28no—
67C31×#117—22no—
68C31×#117—24no—
69C31×#117—24no—
70C31×#117D414yes—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—24no—
75C2×C4⋊C4×C31—24no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31D414yes—
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2312yes—
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—22no—
90C2×C42×C31—22no—
91C2×C42×C31—21no—
92C31×#107—22no—
93C31×#106—24no—
94C31×#106—22no—
95C31×#103—24no—
96C31×#103—22no—
97C31×#103—22no—
98C31×C22⋊C8—22no—
99C31×C22⋊C8—22no—
100C31×C22⋊C8—24no—
101C31×C22⋊C8—24no—
102C31×#95—21no—
103C31×#95—22no—
104C24×C31C2×D411yescharacteristic
105C24×C31C2×D412yes—
106#41×C31—28no—
107#41×C31—28no—
108#41×C31—216no—
109C2×Q8×C31—22no—
110C2×Q8×C31—24no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×D4×C31—24no—
116C2×D4×C31—24no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—24no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—28no—
130C22×C4×C31—24no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—82no—
135C22×C4×C31—82no—
136C22×C4×C31—82no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31#4114yes—
142C22×C4×C31#4114yes—
143C22×C4×C31C2411yescharacteristic
144C2×C8×C31—22no—
145C2×C8×C31—21no—
146C2×C8×C31—21no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C4⋊C4×C31—28no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—44no—
162C4⋊C4×C31C2×D412yes—
163C4⋊C4×C31C2×D414yes—
164C4⋊C4×C31C2×D412yes—
165C22⋊C4×C31—28no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—42no—
168C22⋊C4×C31—44no—
169C22⋊C4×C31—81no—
170C22⋊C4×C31—82no—
171C42×C31—28no—
172C42×C31—24no—
173#32988C3111yesmaximal, characteristic, direct product factor, Sylow
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—21no—
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—21no—
184C23×C31#12012yes—
185C23×C31#12314yes—
186C23×C31C22×D411yescharacteristic
187Q8×C31—28no—
188Q8×C31—42no—
189Q8×C31—41no—
190Q8×C31—41no—
191D4×C31—28no—
192D4×C31—216no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—22no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—22no—
206C2×C4×C31—41no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—41no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—84no—
220C2×C4×C31—88no—
221C2×C4×C31#12012yes—
222C2×C4×C31#12314yes—
223C2×C4×C31C22×D411yescharacteristic
224C2×C4×C31C22×D411yescharacteristic
225C2×C4×C31C2×#4112yes—
226C2×C4×C31C2×#4112yes—
227C8×C31—42no—
228C8×C31—41no—
229C8×C31—41no—
230C2×#520C2×C3111yescharacteristic
231C2×#485C2×C3111yescharacteristic, complemented
232C2×#483C2×C3111yescharacteristic, complemented
233#2616C2×C3111yescharacteristic
234#1873C2×C3112yescomplemented
235#1872C2×C3112yescomplemented
236#1854C2×C3112yescomplemented
237#1853C2×C3112yescomplemented
238#1825C2×C3111yescharacteristic, complemented
239#1824C2×C3111yescharacteristic, complemented
240#1780C2×C3111yescharacteristic
241C22×C31—24no—
242C22×C31—24no—
243C22×C31—22no—
244C22×C31—24no—
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31C2×#12011yescharacteristic
254C22×C31C2×#12312yes—
255C22×C31#57712yes—
256C22×C31#57711yescharacteristic
257C22×C31#57811yescharacteristic
258C4×C31—22no—
259C4×C31—24no—
260C4×C31—44no—
261C4×C31—42no—
262C4×C31—42no—
263C4×C31—82no—
264C4×C31—81no—
265C4×C31—81no—
266C4×C31#52511yescharacteristic
267C4×C31#52512yes—
268C4×C31#52511yescharacteristic
269C22×#41C22×C3111yescharacteristic
270C2×C22⋊Q8C22×C3111yescharacteristic
271C2×C22⋊Q8C22×C3112yes—
272#520—24no—
273#520—24no—
274C2×C4×Q8—21no—
275C2×C4×D4—22no—
276C2×C4×D4—21no—
277C2×#117C22×C3112yes—
278C2×#117C22×C3111yescharacteristic
279C22×C4⋊C4C22×C3112yes—
280C22×C4⋊C4C22×C3111yescharacteristic
281#485—24no—
282#483—24no—
283C2×#107C22×C3111yescharacteristic
284C2×#106C22×C3112yes—
285C2×#106C22×C3111yescharacteristic
286C2×#103C22×C3112yes—
287C2×#103C22×C3111yescharacteristic
288C2×#103C22×C3111yescharacteristic
289C2×C22⋊C8C22×C3111yescharacteristic
290C2×C22⋊C8C22×C3111yescharacteristic
291C2×C22⋊C8C22×C3112yes—
292C2×C22⋊C8C22×C3112yes—
293#395—21no—
294#394—21no—
295#394—22no—
296#380C22×C3111yescharacteristic
297#380C22×C3112yes—
298#340C22×C3112yes—
299#340C22×C3112yes—
300#340C22×C3112yes—
301#340C22×C3112yes—
302C2×C31—44no—
303C2×C31—42no—
304C2×C31#284911yescharacteristic
305C2×C31#300911yescharacteristic
306C2×C31#301311yescharacteristic
307C2×C31#309712yes—
308C2×C31#309912yes—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—24no—
312C2×#41—24no—
313C22×Q8D4×C3111yescharacteristic
314C22×Q8C23×C3111yescharacteristic
315C22×D4D4×C3111yescharacteristic
316C22×D4D4×C3112yes—
317C23×C4—22no—
318C23×C4—21no—
319C23×C4C23×C3111yescharacteristic
320C23×C4C23×C3112yes—
321C22×C8C23×C3111yescharacteristic
322C22×C8C23×C3112yes—
323C22×C8C23×C3111yescharacteristic
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C4×Q8—28no—
329C4×D4—216no—
330C4×D4—28no—
331#117—22no—
332#117—24no—
333#117—24no—
334#117D4×C3114yes—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4D4×C3114yes—
343C2×C4⋊C4C23×C3112yes—
344C2×C4⋊C4C23×C3111yescharacteristic
345C2×C4⋊C4C23×C3111yescharacteristic
346C2×C4⋊C4C23×C3112yes—
347C2×C4⋊C4C23×C3111yescharacteristic
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C42—22no—
355C2×C42—21no—
356#107—22no—
357#106—24no—
358#106—22no—
359#103—24no—
360#103—22no—
361#103—22no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366#95—21no—
367#95—22no—
368C31#3298811yescharacteristic, direct product factor, Sylow
369C24C2×D4×C3111yescharacteristic
370C24C2×D4×C3112yes—
371#41—28no—
372#41—28no—
373#41—216no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3112yes—
404C22×C4C2×D4×C3112yes—
405C22×C4C2×D4×C3112yes—
406C22×C4#41×C3114yes—
407C22×C4#41×C3114yes—
408C22×C4C24×C3111yescharacteristic
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4C2×D4×C3112yes—
428C4⋊C4C2×D4×C3114yes—
429C4⋊C4C2×D4×C3112yes—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—81no—
435C22⋊C4—82no—
436C42—28no—
437C42—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—21no—
448C23C31×#12012yes—
449C23C31×#12314yes—
450C23C22×D4×C3111yescharacteristic
451Q8—28no—
452Q8—42no—
453Q8—41no—
454Q8—41no—
455D4—28no—
456D4—216no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—84no—
484C2×C4—88no—
485C2×C4C31×#12012yes—
486C2×C4C31×#12314yes—
487C2×C4C22×D4×C3111yescharacteristic
488C2×C4C22×D4×C3111yescharacteristic
489C2×C4C2×#41×C3112yes—
490C2×C4C2×#41×C3112yes—
491C8—42no—
492C8—41no—
493C8—41no—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22C2×C31×#12011yescharacteristic
507C22C2×C31×#12312yes—
508C22C31×#57712yes—
509C22C31×#57711yescharacteristic
510C22C31×#57811yescharacteristic
511C4—22no—
512C4—24no—
513C4—44no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—81no—
518C4—81no—
519C4C31×#52511yescharacteristic
520C4C31×#52512yes—
521C4C31×#52511yescharacteristic
522C2—44no—
523C2—42no—
524C2C31×#284911yescharacteristic
525C2C31×#300911yescharacteristic
526C2C31×#301311yescharacteristic
527C2C31×#309712yes—
528C2C31×#309912yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#485C2semidirect
3. C2×C31×#483C2semidirect
5. C31×#1873C2semidirect
6. C31×#1872C2semidirect
7. C31×#1854C2semidirect
8. C31×#1853C2semidirect
9. C31×#1825C2semidirect
10. C31×#1824C2semidirect
173. #32988C31direct (N × H)
231. C2×#485C2×C31semidirect
232. C2×#483C2×C31semidirect
234. #1873C2×C31semidirect
235. #1872C2×C31semidirect
236. #1854C2×C31semidirect
237. #1853C2×C31semidirect
238. #1825C2×C31semidirect
239. #1824C2×C31semidirect
368. C31#32988direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C2×C31×#4852
C2×C31×#4832
C31×#26162
C31×#18732
C31×#18722
C31×#18542
C31×#18532
C31×#18252
C31×#18242
C31×#17802
#3298831