Finite Group Databasebeta

← C31×#24608 #21481367

Subgroups — C31×#24608 #21481367

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#480C211yesmaximal, characteristic
2C2×C31×#432C211yesmaximal, characteristic, complemented
3C31×#2854C211yesmaximal, characteristic
4C2×C31×#389C211yesmaximal, characteristic, complemented
5C31×#2087C212yesmaximal, complemented
6C31×#2065C212yesmaximal, complemented
7C31×#1944C212yesmaximal, complemented
8C31×#1936C212yesmaximal, complemented
9C31×#1874C211yesmaximal, characteristic
10C31×#1853C211yesmaximal, characteristic, complemented
11C31×#1823C211yesmaximal, characteristic, complemented
12C2×C31×#123—21no—
13C2×C31×#121C2211yescharacteristic
14C22×C4⋊C4×C31—21no—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C42×C31C2211yescharacteristic
18C31×#480—24no—
19C31×#432—24no—
20C2×C31×#107C2212yes—
21C2×C31×#106C2212yes—
22C2×C31×C4⋊C8C2212yes—
23C2×C31×#102C2212yes—
24C2×C31×#102C2212yes—
25C2×C31×C22⋊C8C2211yescharacteristic
26C2×C31×C22⋊C8C2211yescharacteristic
27C2×C31×#97C2212yes—
28C31×#390C2211yescharacteristic
29C31×#390C2211yescharacteristic
30C31×#389—21no—
31C31×#389—21no—
32C31×#389C2212yes—
33C31×#389C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#389C2212yes—
36C31×#388—21no—
37C31×#388—21no—
38C31×#386—21no—
39C31×#380C2211yescharacteristic
40C2×C31×#95—21no—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×D4×C31—21no—
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31—21no—
48C23×C4×C31—21no—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C22×C8×C31C2312yes—
53C22×C8×C31C2312yes—
54C31×#123—44no—
55C31×#121—22no—
56C31×#121—22no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—44no—
67C2×C4⋊C4×C31C2312yes—
68C2×C4⋊C4×C31C2311yescharacteristic
69C2×C4⋊C4×C31C2311yescharacteristic
70C2×C4⋊C4×C31C2311yescharacteristic
71C2×C4⋊C4×C31C2312yes—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—21no—
77C2×C22⋊C4×C31—21no—
78C2×C42×C31—22no—
79C2×C42×C31—22no—
80C2×C42×C31D411yescharacteristic
81C2×C42×C31D411yescharacteristic
82C2×C42×C31C2311yescharacteristic
83C2×C42×C31C2311yescharacteristic
84C31×#107—24no—
85C31×#106—24no—
86C31×C4⋊C8—24no—
87C31×#102—24no—
88C31×#102—24no—
89C31×C22⋊C8—22no—
90C31×C22⋊C8D414yes—
91C31×#97—24no—
92C31×#95—22no—
93C31×#95—22no—
94C31×#95—21no—
95C31×#95—21no—
96C31×#95—21no—
97C24×C31—21no—
98C24×C31C2×D411yescharacteristic
99C2×D4×C31—21no—
100C2×D4×C31—22no—
101C2×D4×C31—21no—
102C2×D4×C31—22no—
103C2×D4×C31—42no—
104C2×D4×C31—42no—
105C2×D4×C31—42no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—24no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—41no—
120C22×C4×C31—42no—
121C22×C4×C31—42no—
122C22×C4×C31—42no—
123C22×C4×C31—42no—
124C22×C4×C31—42no—
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D412yes—
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31#4112yes—
132C22×C4×C31C2411yescharacteristic
133C2×C8×C31—22no—
134C2×C8×C31—22no—
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C2×C8×C31—22no—
138C2×C8×C31C2×D414yes—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—44no—
148C4⋊C4×C31—44no—
149C4⋊C4×C31—42no—
150C4⋊C4×C31—42no—
151C4⋊C4×C31—42no—
152C4⋊C4×C31#4114yes—
153C22⋊C4×C31—42no—
154C22⋊C4×C31—42no—
155C22⋊C4×C31—42no—
156C22⋊C4×C31—42no—
157C22⋊C4×C31—42no—
158C22⋊C4×C31—42no—
159C42×C31—22no—
160C42×C31—22no—
161C42×C31—21no—
162C42×C31—21no—
163C42×C31—21no—
164C42×C31—21no—
165#24608C3111yesmaximal, characteristic, direct product factor, Sylow
166C23×C31—21no—
167C23×C31—21no—
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—41no—
171C23×C31—41no—
172C23×C31—41no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31#12112yes—
178C23×C31AGL(1,ℤ/8)12yes—
179C23×C31AGL(1,ℤ/8)12yes—
180C23×C31C22×D411yescharacteristic
181D4×C31—42no—
182D4×C31—42no—
183D4×C31—84no—
184C2×C4×C31—21no—
185C2×C4×C31—21no—
186C2×C4×C31—21no—
187C2×C4×C31—21no—
188C2×C4×C31—24no—
189C2×C4×C31—24no—
190C2×C4×C31—21no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—24no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—44no—
201C2×C4×C31—48no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—44no—
205C2×C4×C31—41no—
206C2×C4×C31—42no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—84no—
218C2×C4×C31#12112yes—
219C2×C4×C31#13712yes—
220C2×C4×C31C22×D411yescharacteristic
221C2×C4×C31C2×#4112yes—
222C2×C4×C31C2×#4111yescharacteristic
223C2×C4×C312−1+412yes—
224C8×C31—24no—
225C8×C31—42no—
226C2×#480C2×C3111yescharacteristic
227C2×#432C2×C3111yescharacteristic, complemented
228#2854C2×C3111yescharacteristic
229C2×#389C2×C3111yescharacteristic, complemented
230#2087C2×C3112yescomplemented
231#2065C2×C3112yescomplemented
232#1944C2×C3112yescomplemented
233#1936C2×C3112yescomplemented
234#1874C2×C3111yescharacteristic
235#1853C2×C3111yescharacteristic, complemented
236#1823C2×C3111yescharacteristic, complemented
237C22×C31—22no—
238C22×C31—22no—
239C22×C31—22no—
240C22×C31—22no—
241C22×C31—22no—
242C22×C31—22no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31—81no—
249C22×C31—81no—
250C22×C31#46914yes—
251C22×C31C2×#12111yescharacteristic
252C22×C31#52511yescharacteristic
253C22×C31#53611yescharacteristic
254C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C31C2×#13711yescharacteristic
257C22×C31#57511yescharacteristic
258C4×C31—22no—
259C4×C31—22no—
260C4×C31—44no—
261C4×C31—41no—
262C4×C31—41no—
263C4×C31—41no—
264C4×C31—81no—
265C4×C31—81no—
266C4×C31—82no—
267C4×C31—81no—
268C4×C31#54712yes—
269C4×C31#55212yes—
270C2×#123—21no—
271C2×#121C22×C3111yescharacteristic
272C22×C4⋊C4—21no—
273C22×C4⋊C4C22×C3111yescharacteristic
274C22×C4⋊C4C22×C3111yescharacteristic
275C22×C42C22×C3111yescharacteristic
276#480—24no—
277#432—24no—
278C2×#107C22×C3112yes—
279C2×#106C22×C3112yes—
280C2×C4⋊C8C22×C3112yes—
281C2×#102C22×C3112yes—
282C2×#102C22×C3112yes—
283C2×C22⋊C8C22×C3111yescharacteristic
284C2×C22⋊C8C22×C3111yescharacteristic
285C2×#97C22×C3112yes—
286#390C22×C3111yescharacteristic
287#390C22×C3111yescharacteristic
288#389—21no—
289#389—21no—
290#389C22×C3112yes—
291#389C22×C3111yescharacteristic
292#389C22×C3111yescharacteristic
293#389C22×C3112yes—
294#388—21no—
295#388—21no—
296#386—21no—
297#380C22×C3111yescharacteristic
298C2×#95—21no—
299#340C22×C3112yes—
300#340C22×C3112yes—
301#340C22×C3112yes—
302#340C22×C3112yes—
303C2×C31—24no—
304C2×C31—161no—
305C2×C31#241311yescharacteristic
306C2×C31C2×#46911yescharacteristic
307C2×C31#305611yescharacteristic
308C2×C31#311211yescharacteristic
309C2×C31#311411yescharacteristic
310C2×C31#323411yescharacteristic
311C2×C31#323811yescharacteristic
312C22×D4—21no—
313C22×D4C23×C3111yescharacteristic
314C23×C4—21no—
315C23×C4—21no—
316C23×C4D4×C3111yescharacteristic
317C23×C4D4×C3111yescharacteristic
318C23×C4C23×C3111yescharacteristic
319C22×C8C23×C3112yes—
320C22×C8C23×C3112yes—
321#123—44no—
322#121—22no—
323#121—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C23×C3112yes—
335C2×C4⋊C4C23×C3111yescharacteristic
336C2×C4⋊C4C23×C3111yescharacteristic
337C2×C4⋊C4C23×C3111yescharacteristic
338C2×C4⋊C4C23×C3112yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42D4×C3111yescharacteristic
348C2×C42D4×C3111yescharacteristic
349C2×C42C23×C3111yescharacteristic
350C2×C42C23×C3111yescharacteristic
351#107—24no—
352#106—24no—
353C4⋊C8—24no—
354#102—24no—
355#102—24no—
356C22⋊C8—22no—
357C22⋊C8D4×C3114yes—
358#97—24no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—21no—
364C31#2460811yescharacteristic, direct product factor, Sylow
365C24—21no—
366C24C2×D4×C3111yescharacteristic
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4C2×D4×C3111yescharacteristic
394C22×C4C2×D4×C3111yescharacteristic
395C22×C4C2×D4×C3111yescharacteristic
396C22×C4C2×D4×C3111yescharacteristic
397C22×C4C2×D4×C3112yes—
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4#41×C3112yes—
400C22×C4C24×C3111yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C2×D4×C3114yes—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4#41×C3114yes—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23C31×#12112yes—
445C23C31×AGL(1,ℤ/8)12yes—
446C23C31×AGL(1,ℤ/8)12yes—
447C23C22×D4×C3111yescharacteristic
448D4—42no—
449D4—42no—
450D4—84no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—84no—
485C2×C4C31×#12112yes—
486C2×C4C31×#13712yes—
487C2×C4C22×D4×C3111yescharacteristic
488C2×C4C2×#41×C3112yes—
489C2×C4C2×#41×C3111yescharacteristic
490C2×C4C31×2−1+412yes—
491C8—24no—
492C8—42no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22C31×#46914yes—
507C22C2×C31×#12111yescharacteristic
508C22C31×#52511yescharacteristic
509C22C31×#53611yescharacteristic
510C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
511C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
512C22C2×C31×#13711yescharacteristic
513C22C31×#57511yescharacteristic
514C4—22no—
515C4—22no—
516C4—44no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4C31×#54712yes—
525C4C31×#55212yes—
526C2—24no—
527C2—161no—
528C2C31×#241311yescharacteristic
529C2C2×C31×#46911yescharacteristic
530C2C31×#305611yescharacteristic
531C2C31×#311211yescharacteristic
532C2C31×#311411yescharacteristic
533C2C31×#323411yescharacteristic
534C2C31×#323811yescharacteristic
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
2. C2×C31×#432C2semidirect
4. C2×C31×#389C2semidirect
5. C31×#2087C2semidirect
6. C31×#2065C2semidirect
7. C31×#1944C2semidirect
8. C31×#1936C2semidirect
10. C31×#1853C2semidirect
11. C31×#1823C2semidirect
165. #24608C31direct (N × H)
227. C2×#432C2×C31semidirect
229. C2×#389C2×C31semidirect
230. #2087C2×C31semidirect
231. #2065C2×C31semidirect
232. #1944C2×C31semidirect
233. #1936C2×C31semidirect
235. #1853C2×C31semidirect
236. #1823C2×C31semidirect
364. C31#24608direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4802
C2×C31×#4322
C31×#28542
C2×C31×#3892
C31×#20872
C31×#20652
C31×#19442
C31×#19362
C31×#18742
C31×#18532
C31×#18232
#2460831