Finite Group Databasebeta

← C31×#33110 #21489869

Subgroups — C31×#33110 #21489869

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#484C211yesmaximal, characteristic, complemented
2C2×C31×#481C211yesmaximal, characteristic
3C31×#2850C211yesmaximal, characteristic
4C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
5C31×#1983C212yesmaximal, complemented
6C31×#1949C212yesmaximal, complemented
7C31×#1927C212yesmaximal, complemented
8C31×#1925C212yesmaximal, complemented
9C31×#1874C211yesmaximal, characteristic
10C31×#1872C211yesmaximal, characteristic, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C2×C31×#124C2211yescharacteristic
13C2×C31×C22⋊Q8C2211yescharacteristic
14C2×C31×#121C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C42×C31—21no—
18C22×C42×C31C2211yescharacteristic
19C31×#484—24no—
20C31×#481—24no—
21C2×C31×#107C2212yes—
22C2×C31×#106C2212yes—
23C2×C31×C4⋊C8C2212yes—
24C2×C31×#103C2212yes—
25C2×C31×#103C2212yes—
26C2×C31×#102C2212yes—
27C2×C31×#102C2212yes—
28C2×C31×C22⋊C8C2211yescharacteristic
29C2×C31×C22⋊C8C2211yescharacteristic
30C31×#391C2211yescharacteristic
31C31×#390C2211yescharacteristic
32C31×#388—21no—
33C31×#388—21no—
34C31×#386—21no—
35C31×#386—21no—
36C31×#382C2211yescharacteristic
37C4×C4⋊C4×C31—21no—
38C4×C4⋊C4×C31—21no—
39C4×C4⋊C4×C31C2212yes—
40C4×C4⋊C4×C31C2212yes—
41C2×C31×#95—21no—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#336C2212yes—
45C22×Q8×C31C2311yescharacteristic
46C22×D4×C31—21no—
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C22×C8×C31C2312yes—
54C22×C8×C31C2312yes—
55C31×#124—24no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—22no—
58C31×#121—22no—
59C31×#121—22no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31D414yes—
66C2×C4⋊C4×C31C2311yescharacteristic
67C2×C4⋊C4×C31C2311yescharacteristic
68C2×C4⋊C4×C31C2312yes—
69C2×C4⋊C4×C31C2312yes—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C22⋊C4×C31—21no—
76C2×C42×C31—24no—
77C2×C42×C31—22no—
78C2×C42×C31—22no—
79C2×C42×C31—44no—
80C2×C42×C31D411yescharacteristic
81C2×C42×C31D411yescharacteristic
82C2×C42×C31C2311yescharacteristic
83C2×C42×C31C2311yescharacteristic
84C31×#107—24no—
85C31×#106—24no—
86C31×C4⋊C8—24no—
87C31×#103—24no—
88C31×#103—24no—
89C31×#102—24no—
90C31×#102—24no—
91C31×C22⋊C8—22no—
92C31×C22⋊C8—22no—
93C31×#95—22no—
94C31×#95—22no—
95C31×#95—21no—
96C31×#95—21no—
97C31×#95—21no—
98C31×#95—21no—
99C24×C31—21no—
100C24×C31C2×D411yescharacteristic
101C2×Q8×C31—21no—
102C2×Q8×C31—22no—
103C2×Q8×C31—21no—
104C2×Q8×C31—22no—
105C2×D4×C31—21no—
106C2×D4×C31—22no—
107C2×D4×C31—21no—
108C2×D4×C31—22no—
109C2×D4×C31—42no—
110C2×D4×C31—42no—
111C2×D4×C31—42no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—24no—
118C22×C4×C31—24no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—41no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D412yes—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31#4112yes—
138C22×C4×C31C2411yescharacteristic
139C2×C8×C31—22no—
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—24no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—42no—
154C4⋊C4×C31C2×D414yes—
155C4⋊C4×C31#4114yes—
156C22⋊C4×C31—42no—
157C22⋊C4×C31—42no—
158C22⋊C4×C31—42no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31—42no—
161C22⋊C4×C31—42no—
162C42×C31—22no—
163C42×C31—22no—
164C42×C31—22no—
165C42×C31—44no—
166C42×C31—44no—
167C42×C31#4114yes—
168#33110C3111yesmaximal, characteristic, direct product factor, Sylow
169C23×C31—21no—
170C23×C31—21no—
171C23×C31—21no—
172C23×C31—21no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31#12112yes—
181C23×C31#13512yes—
182C23×C31#13712yes—
183C23×C31C22×D411yescharacteristic
184Q8×C31—42no—
185Q8×C31—42no—
186D4×C31—42no—
187D4×C31—42no—
188D4×C31—84no—
189C2×C4×C31—28no—
190C2×C4×C31—21no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—21no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—21no—
202C2×C4×C31—24no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—44no—
206C2×C4×C31—48no—
207C2×C4×C31—48no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31#12112yes—
222C2×C4×C31#12312yes—
223C2×C4×C31#12312yes—
224C2×C4×C31AGL(1,ℤ/8)12yes—
225C2×C4×C31C22×D411yescharacteristic
226C2×C4×C31C2×#4111yescharacteristic
227C2×C4×C31C2×#4112yes—
228C2×C4×C31C2×#4112yes—
229C8×C31—42no—
230C8×C31—42no—
231C2×#484C2×C3111yescharacteristic, complemented
232C2×#481C2×C3111yescharacteristic
233#2850C2×C3111yescharacteristic
234C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
235#1983C2×C3112yescomplemented
236#1949C2×C3112yescomplemented
237#1927C2×C3112yescomplemented
238#1925C2×C3112yescomplemented
239#1874C2×C3111yescharacteristic
240#1872C2×C3111yescharacteristic, complemented
241#1797C2×C3111yescharacteristic, complemented
242C22×C31—22no—
243C22×C31—22no—
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—81no—
249C22×C31—81no—
250C22×C31—81no—
251C22×C31—81no—
252C22×C31—81no—
253C22×C31—81no—
254C22×C31—81no—
255C22×C31#46414yes—
256C22×C31C2×#12111yescharacteristic
257C22×C31C2×#12311yescharacteristic
258C22×C31#52511yescharacteristic
259C22×C31C2×#13511yescharacteristic
260C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C31C2×#13711yescharacteristic
262C22×C31#57511yescharacteristic
263C4×C31—22no—
264C4×C31—22no—
265C4×C31—22no—
266C4×C31—22no—
267C4×C31—41no—
268C4×C31—44no—
269C4×C31—44no—
270C4×C31—81no—
271C4×C31—81no—
272C4×C31—81no—
273C4×C31#52912yes—
274C4×C31#53212yes—
275C2×#124C22×C3111yescharacteristic
276C2×C22⋊Q8C22×C3111yescharacteristic
277C2×#121C22×C3111yescharacteristic
278C22×C4⋊C4C22×C3111yescharacteristic
279C22×C4⋊C4C22×C3111yescharacteristic
280C22×C42—21no—
281C22×C42C22×C3111yescharacteristic
282#484—24no—
283#481—24no—
284C2×#107C22×C3112yes—
285C2×#106C22×C3112yes—
286C2×C4⋊C8C22×C3112yes—
287C2×#103C22×C3112yes—
288C2×#103C22×C3112yes—
289C2×#102C22×C3112yes—
290C2×#102C22×C3112yes—
291C2×C22⋊C8C22×C3111yescharacteristic
292C2×C22⋊C8C22×C3111yescharacteristic
293#391C22×C3111yescharacteristic
294#390C22×C3111yescharacteristic
295#388—21no—
296#388—21no—
297#386—21no—
298#386—21no—
299#382C22×C3111yescharacteristic
300C4×C4⋊C4—21no—
301C4×C4⋊C4—21no—
302C4×C4⋊C4C22×C3112yes—
303C4×C4⋊C4C22×C3112yes—
304C2×#95—21no—
305#340C22×C3112yes—
306#340C22×C3112yes—
307#336C22×C3112yes—
308C2×C31—24no—
309C2×C31—161no—
310C2×C31#285411yescharacteristic
311C2×C31C2×#46411yescharacteristic
312C2×C31#303611yescharacteristic
313C2×C31#309111yescharacteristic
314C2×C31#309211yescharacteristic
315C2×C31#310511yescharacteristic
316C2×C31#311711yescharacteristic
317C22×Q8C23×C3111yescharacteristic
318C22×D4—21no—
319C22×D4C23×C3111yescharacteristic
320C23×C4—21no—
321C23×C4—21no—
322C23×C4D4×C3111yescharacteristic
323C23×C4D4×C3111yescharacteristic
324C23×C4C23×C3111yescharacteristic
325C22×C8C23×C3112yes—
326C22×C8C23×C3112yes—
327#124—24no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330#121—22no—
331#121—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4D4×C3114yes—
338C2×C4⋊C4C23×C3111yescharacteristic
339C2×C4⋊C4C23×C3111yescharacteristic
340C2×C4⋊C4C23×C3112yes—
341C2×C4⋊C4C23×C3112yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42D4×C3111yescharacteristic
353C2×C42D4×C3111yescharacteristic
354C2×C42C23×C3111yescharacteristic
355C2×C42C23×C3111yescharacteristic
356#107—24no—
357#106—24no—
358C4⋊C8—24no—
359#103—24no—
360#103—24no—
361#102—24no—
362#102—24no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371C31#3311011yescharacteristic, direct product factor, Sylow
372C24—21no—
373C24C2×D4×C3111yescharacteristic
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3112yes—
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4#41×C3112yes—
411C22×C4C24×C3111yescharacteristic
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4C2×D4×C3114yes—
428C4⋊C4#41×C3114yes—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—44no—
439C42—44no—
440C42#41×C3114yes—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23C31×#12112yes—
453C23C31×#13512yes—
454C23C31×#13712yes—
455C23C22×D4×C3111yescharacteristic
456Q8—42no—
457Q8—42no—
458D4—42no—
459D4—42no—
460D4—84no—
461C2×C4—28no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4C31×#12112yes—
494C2×C4C31×#12312yes—
495C2×C4C31×#12312yes—
496C2×C4C31×AGL(1,ℤ/8)12yes—
497C2×C4C22×D4×C3111yescharacteristic
498C2×C4C2×#41×C3111yescharacteristic
499C2×C4C2×#41×C3112yes—
500C2×C4C2×#41×C3112yes—
501C8—42no—
502C8—42no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22C31×#46414yes—
517C22C2×C31×#12111yescharacteristic
518C22C2×C31×#12311yescharacteristic
519C22C31×#52511yescharacteristic
520C22C2×C31×#13511yescharacteristic
521C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
522C22C2×C31×#13711yescharacteristic
523C22C31×#57511yescharacteristic
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—41no—
529C4—44no—
530C4—44no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4C31×#52912yes—
535C4C31×#53212yes—
536C2—24no—
537C2—161no—
538C2C31×#285411yescharacteristic
539C2C2×C31×#46411yescharacteristic
540C2C31×#303611yescharacteristic
541C2C31×#309111yescharacteristic
542C2C31×#309211yescharacteristic
543C2C31×#310511yescharacteristic
544C2C31×#311711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#484C2semidirect
4. C2×C4×C4⋊C4×C31C2semidirect
5. C31×#1983C2semidirect
6. C31×#1949C2semidirect
7. C31×#1927C2semidirect
8. C31×#1925C2semidirect
10. C31×#1872C2semidirect
11. C31×#1797C2semidirect
168. #33110C31direct (N × H)
231. C2×#484C2×C31semidirect
234. C2×C4×C4⋊C4C2×C31semidirect
235. #1983C2×C31semidirect
236. #1949C2×C31semidirect
237. #1927C2×C31semidirect
238. #1925C2×C31semidirect
240. #1872C2×C31semidirect
241. #1797C2×C31semidirect
371. C31#33110direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4842
C2×C31×#4812
C31×#28502
C2×C4×C4⋊C4×C312
C31×#19832
C31×#19492
C31×#19272
C31×#19252
C31×#18742
C31×#18722
C31×#17972
#3311031