Finite Group Databasebeta

← C31×#24016 #21480775

Subgroups — C31×#24016 #21480775

Order 7936.

1276 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic
2C2×C31×#485C212yesmaximal, complemented
3C31×#2311C211yesmaximal, characteristic
4C31×#2036C212yesmaximal, complemented
5C31×#2035C212yesmaximal, complemented
6C31×#1872C212yesmaximal, complemented
7C31×#1854C212yesmaximal, complemented
8C31×#1787C212yesmaximal, complemented
9C31×#1780C211yesmaximal, characteristic
10C22×#41×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C31×#536—28no—
13C2×C31×C4⋊Q8—21no—
14C2×C31×#124—21no—
15C2×C31×#123—21no—
16C2×C31×#123—21no—
17C2×C31×C22⋊Q8C2212yes—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×#117C2211yescharacteristic
20C2×C31×#117C2211yescharacteristic
21C22×C4⋊C4×C31C2211yescharacteristic
22C31×#485—28no—
23C2×C31×#107C2212yes—
24C2×C31×#106C2212yes—
25C2×C31×#103C2212yes—
26C2×C31×#103C2212yes—
27C2×C31×#103C2212yes—
28C2×C31×#103C2212yes—
29C2×C31×C22⋊C8C2212yes—
30C2×C31×C22⋊C8C2212yes—
31C2×C31×C22⋊C8C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C31×#391—21no—
34C31×#390—21no—
35C31×#386—21no—
36C31×#386—21no—
37C31×#380C2212yes—
38C31×#380C2212yes—
39C31×#340C2212yes—
40C31×#340C2212yes—
41C2×#41×C31—24no—
42C2×#41×C31—24no—
43C2×#41×C31—41no—
44C2×#41×C31—41no—
45C22×Q8×C31—24no—
46C22×Q8×C31—21no—
47C22×Q8×C31—24no—
48C22×Q8×C31—41no—
49C22×Q8×C31—41no—
50C22×Q8×C31D411yescharacteristic
51C22×Q8×C31C2312yes—
52C22×D4×C31D411yescharacteristic
53C22×D4×C31D411yescharacteristic
54C22×D4×C31D411yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31—21no—
57C23×C4×C31C2311yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C22×C8×C31C2312yes—
61C22×C8×C31C2312yes—
62C31×C4⋊Q8—28no—
63C31×#124—28no—
64C31×#123—28no—
65C31×#123—28no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8D418yes—
70C31×#117—24no—
71C31×#117—24no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—41no—
76C2×C4⋊C4×C31—41no—
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2312yes—
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C42×C31—21no—
87C2×C42×C31—21no—
88C31×#107—24no—
89C31×#106—24no—
90C31×#103—24no—
91C31×#103—24no—
92C31×#103—42no—
93C31×#103—42no—
94C31×C22⋊C8—24no—
95C31×C22⋊C8—24no—
96C31×C22⋊C8—41no—
97C31×C22⋊C8—41no—
98C31×#95—22no—
99C31×#95—22no—
100C24×C31C2×D411yescharacteristic
101C24×C31C2×D411yescharacteristic
102C24×C31C2×D411yescharacteristic
103#41×C31—48no—
104#41×C31—48no—
105C2×Q8×C31—24no—
106C2×Q8×C31—22no—
107C2×Q8×C31—22no—
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—41no—
114C2×Q8×C31—48no—
115C2×Q8×C31—48no—
116C2×Q8×C31—42no—
117C2×Q8×C31—42no—
118C2×Q8×C31C2×D414yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—41no—
126C2×D4×C31—41no—
127C2×D4×C31—41no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—82no—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D414yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31#4114yes—
152C22×C4×C31C2411yescharacteristic
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C4⋊C4×C31—28no—
160C4⋊C4×C31—28no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31C2×D414yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—28no—
170C22⋊C4×C31—28no—
171C22⋊C4×C31—42no—
172C22⋊C4×C31—82no—
173C22⋊C4×C31—82no—
174C42×C31—24no—
175C42×C31—42no—
176#24016C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31#12012yes—
191C23×C31#12312yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—24no—
194Q8×C31—24no—
195Q8×C31—44no—
196Q8×C31—48no—
197Q8×C31—42no—
198Q8×C31—42no—
199D4×C31—44no—
200D4×C31—44no—
201D4×C31—44no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—48no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—84no—
229C2×C4×C31#12012yes—
230C2×C4×C31#12312yes—
231C2×C4×C31C22×D412yes—
232C2×C4×C31C2×#4112yes—
233C2×C4×C31H(5,𝔽2)11yescharacteristic
234C2×C4×C31H(5,𝔽2)11yescharacteristic
235C8×C31—42no—
236C8×C31—42no—
237C2×#536C2×C3111yescharacteristic
238C2×#485C2×C3112yescomplemented
239#2311C2×C3111yescharacteristic
240#2036C2×C3112yescomplemented
241#2035C2×C3112yescomplemented
242#1872C2×C3112yescomplemented
243#1854C2×C3112yescomplemented
244#1787C2×C3112yescomplemented
245#1780C2×C3111yescharacteristic
246C22×C31—24no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—24no—
250C22×C31—42no—
251C22×C31—41no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31—41no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31—41no—
260C22×C31—42no—
261C22×C31C2×#12011yescharacteristic
262C22×C31C2×#12311yescharacteristic
263C22×C31#53411yescharacteristic
264C22×C31#57712yes—
265C22×C31#57812yes—
266C4×C31—22no—
267C4×C31—22no—
268C4×C31—41no—
269C4×C31—44no—
270C4×C31—44no—
271C4×C31—82no—
272C4×C31—82no—
273C4×C31#54612yes—
274C4×C31#54612yes—
275C22×#41C22×C3111yescharacteristic
276C23×Q8C22×C3111yescharacteristic
277#536—28no—
278C2×C4⋊Q8—21no—
279C2×#124—21no—
280C2×#123—21no—
281C2×#123—21no—
282C2×C22⋊Q8C22×C3112yes—
283C2×C22⋊Q8C22×C3112yes—
284C2×#117C22×C3111yescharacteristic
285C2×#117C22×C3111yescharacteristic
286C22×C4⋊C4C22×C3111yescharacteristic
287#485—28no—
288C2×#107C22×C3112yes—
289C2×#106C22×C3112yes—
290C2×#103C22×C3112yes—
291C2×#103C22×C3112yes—
292C2×#103C22×C3112yes—
293C2×#103C22×C3112yes—
294C2×C22⋊C8C22×C3112yes—
295C2×C22⋊C8C22×C3112yes—
296C2×C22⋊C8C22×C3111yescharacteristic
297C2×C22⋊C8C22×C3111yescharacteristic
298#391—21no—
299#390—21no—
300#386—21no—
301#386—21no—
302#380C22×C3112yes—
303#380C22×C3112yes—
304#340C22×C3112yes—
305#340C22×C3112yes—
306C2×C31—42no—
307C2×C31—42no—
308C2×C31—81no—
309C2×C31#240511yescharacteristic
310C2×C31#301312yes—
311C2×C31#309712yes—
312C2×C31#320011yescharacteristic
313C2×C31#320211yescharacteristic
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—41no—
317C2×#41—41no—
318C22×Q8—24no—
319C22×Q8—21no—
320C22×Q8—24no—
321C22×Q8—41no—
322C22×Q8—41no—
323C22×Q8D4×C3111yescharacteristic
324C22×Q8C23×C3112yes—
325C22×D4D4×C3111yescharacteristic
326C22×D4D4×C3111yescharacteristic
327C22×D4D4×C3111yescharacteristic
328C23×C4—21no—
329C23×C4—21no—
330C23×C4C23×C3111yescharacteristic
331C23×C4C23×C3111yescharacteristic
332C23×C4C23×C3111yescharacteristic
333C22×C8C23×C3112yes—
334C22×C8C23×C3112yes—
335C4⋊Q8—28no—
336#124—28no—
337#123—28no—
338#123—28no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8D4×C3118yes—
343#117—24no—
344#117—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4C23×C3112yes—
351C2×C4⋊C4C23×C3112yes—
352C2×C4⋊C4C23×C3112yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C42—21no—
360C2×C42—21no—
361#107—24no—
362#106—24no—
363#103—24no—
364#103—24no—
365#103—42no—
366#103—42no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—41no—
370C22⋊C8—41no—
371#95—22no—
372#95—22no—
373C31#2401611yescharacteristic, direct product factor, Sylow
374C24C2×D4×C3111yescharacteristic
375C24C2×D4×C3111yescharacteristic
376C24C2×D4×C3111yescharacteristic
377#41—48no—
378#41—48no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—41no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8C2×D4×C3114yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C2×D4×C3114yes—
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4#41×C3114yes—
426C22×C4C24×C3111yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4C2×D4×C3114yes—
442C22⋊C4—24no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—42no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C42—24no—
449C42—42no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23C31×#12012yes—
464C23C31×#12312yes—
465C23C22×D4×C3111yescharacteristic
466Q8—24no—
467Q8—24no—
468Q8—44no—
469Q8—48no—
470Q8—42no—
471Q8—42no—
472D4—44no—
473D4—44no—
474D4—44no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4C31×#12012yes—
503C2×C4C31×#12312yes—
504C2×C4C22×D4×C3112yes—
505C2×C4C2×#41×C3112yes—
506C2×C4C31×H(5,𝔽2)11yescharacteristic
507C2×C4C31×H(5,𝔽2)11yescharacteristic
508C8—42no—
509C8—42no—
510C22—24no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22C2×C31×#12011yescharacteristic
526C22C2×C31×#12311yescharacteristic
527C22C31×#53411yescharacteristic
528C22C31×#57712yes—
529C22C31×#57812yes—
530C4—22no—
531C4—22no—
532C4—41no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4C31×#54612yes—
538C4C31×#54612yes—
539C2—42no—
540C2—42no—
541C2—81no—
542C2C31×#240511yescharacteristic
543C2C31×#301312yes—
544C2C31×#309712yes—
545C2C31×#320011yescharacteristic
546C2C31×#320211yescharacteristic
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
4. C31×#2036C2semidirect
5. C31×#2035C2semidirect
6. C31×#1872C2semidirect
7. C31×#1854C2semidirect
8. C31×#1787C2semidirect
176. #24016C31direct (N × H)
238. C2×#485C2×C31semidirect
240. #2036C2×C31semidirect
241. #2035C2×C31semidirect
242. #1872C2×C31semidirect
243. #1854C2×C31semidirect
244. #1787C2×C31semidirect
373. C31#24016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C2×C31×#4852
C31×#23112
C31×#20362
C31×#20352
C31×#18722
C31×#18542
C31×#17872
C31×#17802
#2401631