Finite Group Databasebeta

← C31×#24752 #21481511

Subgroups — C31×#24752 #21481511

Order 7936.

992 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#467C212yesmaximal
2C2×C31×#462C211yesmaximal, characteristic, complemented
3C22×C31×C4⋊C8C211yesmaximal, characteristic
4C31×#2857C211yesmaximal, characteristic
5C4×C31×C22⋊Q8C211yesmaximal, characteristic
6C31×#2068C212yesmaximal, complemented
7C31×#2056C212yesmaximal, complemented
8C31×#1981C211yesmaximal, characteristic, complemented
9C31×#1947C211yesmaximal, characteristic, complemented
10C31×#1892C212yesmaximal
11C31×#1803C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2212yes—
13C23×C8×C31C2212yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C31C2211yescharacteristic
18C22×C42×C31C2211yescharacteristic
19C31×#467—28no—
20C31×#462—24no—
21C2×C31×#107C2212yes—
22C2×C31×C4⋊C8—24no—
23C2×C31×C4⋊C8C2212yes—
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C2×C31×#103C2212yes—
27C2×C31×#103C2212yes—
28C2×C31×#103C2212yes—
29C2×C31×#103C2212yes—
30C31×#391C2211yescharacteristic
31C31×#391C2212yes—
32C31×#389—21no—
33C31×#389—21no—
34C31×#389C2211yescharacteristic
35C31×#387—21no—
36C31×#385—21no—
37C31×#384C2211yescharacteristic
38C31×#380—21no—
39C31×#380—21no—
40C31×#380C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C22×Q8×C31C2312yes—
47C22×Q8×C31C2311yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31D411yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C2×Q16×C31—24no—
52C2×Q16×C31—24no—
53C2×Q16×C31—42no—
54C22×C8×C31—24no—
55C22×C8×C31—24no—
56C22×C8×C31—42no—
57C22×C8×C31C2312yes—
58C22×C8×C31C2312yes—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C31×C22⋊Q8D414yes—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31C2311yescharacteristic
76C2×C4⋊C4×C31C2311yescharacteristic
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—21no—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—21no—
85C2×C42×C31—21no—
86C2×C42×C31D411yescharacteristic
87C2×C42×C31D411yescharacteristic
88C2×C42×C31C2311yescharacteristic
89C2×C42×C31C2311yescharacteristic
90C31×#107—24no—
91C31×C4⋊C8—24no—
92C31×C4⋊C8—22no—
93C31×C4⋊C8D414yes—
94C31×#103—24no—
95C31×#103—24no—
96C31×#103—24no—
97C31×#103—42no—
98C31×#95—21no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—22no—
104C31×#95—22no—
105C24×C31C2×D411yescharacteristic
106C2×Q8×C31—21no—
107C2×Q8×C31—21no—
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—21no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—21no—
114C2×Q8×C31—22no—
115C2×Q8×C31—21no—
116C2×Q8×C31—22no—
117C2×Q8×C31C2×D412yes—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—41no—
133C22×C4×C31Q1614yescomplemented
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31#4112yes—
141C22×C4×C31C2411yescharacteristic
142Q16×C31—48no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C2×C8×C31—48no—
149C2×C8×C31C2×D414yes—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—44no—
162C4⋊C4×C31—42no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31C2×D412yes—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—44no—
168C22⋊C4×C31—42no—
169C42×C31—22no—
170C42×C31—22no—
171C42×C31—21no—
172C42×C31—21no—
173C42×C31—42no—
174C42×C31—42no—
175C42×C31C2×D412yes—
176C42×C31#4112yes—
177#24752C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—21no—
182C23×C31#12112yes—
183C23×C31C2×Q1612yes—
184C23×C31#13712yes—
185C23×C31C22×D411yescharacteristic
186Q8×C31—22no—
187Q8×C31—22no—
188Q8×C31—41no—
189Q8×C31—41no—
190Q8×C31—42no—
191Q8×C31—42no—
192Q8×C31—42no—
193Q8×C31—42no—
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—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—28no—
205C2×C4×C31—21no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—44no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—41no—
227C2×C4×C31#12112yes—
228C2×C4×C31#12112yes—
229C2×C4×C31#12112yes—
230C2×C4×C31C2×Q1612yes—
231C2×C4×C31C2×Q1612yes—
232C2×C4×C31#13512yes—
233C2×C4×C31C22×D411yescharacteristic
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C2×#4111yescharacteristic
236C2×C4×C31C2×#4111yescharacteristic
237C2×C4×C31H(5,𝔽2)11yescharacteristic
238C2×C4×C312−1+411yescharacteristic
239C8×C31—24no—
240C8×C31—42no—
241C2×#467C2×C3112yes—
242C2×#462C2×C3111yescharacteristic, complemented
243C22×C4⋊C8C2×C3111yescharacteristic
244#2857C2×C3111yescharacteristic
245C4×C22⋊Q8C2×C3111yescharacteristic
246#2068C2×C3112yescomplemented
247#2056C2×C3112yescomplemented
248#1981C2×C3111yescharacteristic, complemented
249#1947C2×C3111yescharacteristic, complemented
250#1892C2×C3112yes—
251#1803C2×C3111yescharacteristic, complemented
252C22×C31—22no—
253C22×C31—22no—
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31#46214yescomplemented
259C22×C31C2×#12111yescharacteristic
260C22×C31C2×#12111yescharacteristic
261C22×C31#53711yescharacteristic
262C22×C31C22×Q1611yescharacteristic
263C22×C31C2×#13511yescharacteristic
264C22×C31C2×#13711yescharacteristic
265C22×C31#57511yescharacteristic
266C4×C31—22no—
267C4×C31—22no—
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—41no—
272C4×C31—82no—
273C4×C31—81no—
274C4×C31—82no—
275C4×C31—81no—
276C4×C31#46714yescomplemented
277C4×C31D4211yescharacteristic
278C4×C31#54711yescharacteristic
279C4×C31#54811yescharacteristic
280C4×C31#55311yescharacteristic
281C22×Q16C22×C3112yes—
282C23×C8C22×C3112yes—
283C2×C4⋊Q8C22×C3111yescharacteristic
284C2×C22⋊Q8C22×C3112yes—
285C2×C22⋊Q8C22×C3111yescharacteristic
286C2×C4×Q8C22×C3111yescharacteristic
287C22×C42C22×C3111yescharacteristic
288#467—28no—
289#462—24no—
290C2×#107C22×C3112yes—
291C2×C4⋊C8—24no—
292C2×C4⋊C8C22×C3112yes—
293C2×C4⋊C8C22×C3111yescharacteristic
294C2×C4⋊C8C22×C3111yescharacteristic
295C2×#103C22×C3112yes—
296C2×#103C22×C3112yes—
297C2×#103C22×C3112yes—
298C2×#103C22×C3112yes—
299#391C22×C3111yescharacteristic
300#391C22×C3112yes—
301#389—21no—
302#389—21no—
303#389C22×C3111yescharacteristic
304#387—21no—
305#385—21no—
306#384C22×C3111yescharacteristic
307#380—21no—
308#380—21no—
309#380C22×C3111yescharacteristic
310C4×C4⋊C4—21no—
311C4×C4⋊C4C22×C3111yescharacteristic
312C4×C22⋊C4—21no—
313#340C22×C3112yes—
314#340C22×C3112yes—
315C2×C31—24no—
316C2×C31#241611yescharacteristic
317C2×C31C2×#46211yescharacteristic
318C2×C31#303511yescharacteristic
319C2×C31C2×#46711yescharacteristic
320C2×C31#305611yescharacteristic
321C2×C31#324811yescharacteristic
322C2×C31#324911yescharacteristic
323C22×Q8C23×C3112yes—
324C22×Q8C23×C3111yescharacteristic
325C23×C4D4×C3111yescharacteristic
326C23×C4D4×C3111yescharacteristic
327C23×C4C23×C3111yescharacteristic
328C2×Q16—24no—
329C2×Q16—24no—
330C2×Q16—42no—
331C22×C8—24no—
332C22×C8—24no—
333C22×C8—42no—
334C22×C8C23×C3112yes—
335C22×C8C23×C3112yes—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8D4×C3114yes—
342C4×Q8—22no—
343C4×Q8—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C4⋊C4C23×C3112yes—
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42D4×C3111yescharacteristic
364C2×C42D4×C3111yescharacteristic
365C2×C42C23×C3111yescharacteristic
366C2×C42C23×C3111yescharacteristic
367#107—24no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8D4×C3114yes—
371#103—24no—
372#103—24no—
373#103—24no—
374#103—42no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382C31#2475211yescharacteristic, direct product factor, Sylow
383C24C2×D4×C3111yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8C2×D4×C3112yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—41no—
411C22×C4Q16×C3114yescomplemented
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3112yes—
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4#41×C3112yes—
419C22×C4C24×C3111yescharacteristic
420Q16—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—48no—
427C2×C8C2×D4×C3114yes—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4C2×D4×C3112yes—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—21no—
451C42—42no—
452C42—42no—
453C42C2×D4×C3112yes—
454C42#41×C3112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23C31×#12112yes—
460C23C2×Q16×C3112yes—
461C23C31×#13712yes—
462C23C22×D4×C3111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4C31×#12112yes—
505C2×C4C31×#12112yes—
506C2×C4C31×#12112yes—
507C2×C4C2×Q16×C3112yes—
508C2×C4C2×Q16×C3112yes—
509C2×C4C31×#13512yes—
510C2×C4C22×D4×C3111yescharacteristic
511C2×C4C22×D4×C3111yescharacteristic
512C2×C4C2×#41×C3111yescharacteristic
513C2×C4C2×#41×C3111yescharacteristic
514C2×C4C31×H(5,𝔽2)11yescharacteristic
515C2×C4C31×2−1+411yescharacteristic
516C8—24no—
517C8—42no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22C31×#46214yescomplemented
525C22C2×C31×#12111yescharacteristic
526C22C2×C31×#12111yescharacteristic
527C22C31×#53711yescharacteristic
528C22C22×Q16×C3111yescharacteristic
529C22C2×C31×#13511yescharacteristic
530C22C2×C31×#13711yescharacteristic
531C22C31×#57511yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—41no—
538C4—82no—
539C4—81no—
540C4—82no—
541C4—81no—
542C4C31×#46714yescomplemented
543C4D42×C3111yescharacteristic
544C4C31×#54711yescharacteristic
545C4C31×#54811yescharacteristic
546C4C31×#55311yescharacteristic
547C2—24no—
548C2C31×#241611yescharacteristic
549C2C2×C31×#46211yescharacteristic
550C2C31×#303511yescharacteristic
551C2C2×C31×#46711yescharacteristic
552C2C31×#305611yescharacteristic
553C2C31×#324811yescharacteristic
554C2C31×#324911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
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×#462C2semidirect
6. C31×#2068C2semidirect
7. C31×#2056C2semidirect
8. C31×#1981C2semidirect
9. C31×#1947C2semidirect
11. C31×#1803C2semidirect
133. C22×C4×C31Q16semidirect
177. #24752C31direct (N × H)
242. C2×#462C2×C31semidirect
246. #2068C2×C31semidirect
247. #2056C2×C31semidirect
248. #1981C2×C31semidirect
249. #1947C2×C31semidirect
251. #1803C2×C31semidirect
258. C22×C31#462semidirect
276. C4×C31#467semidirect
382. C31#24752direct (N × H)
411. C22×C4Q16×C31semidirect
524. C22C31×#462semidirect
542. C4C31×#467semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4672
C2×C31×#4622
C22×C31×C4⋊C82
C31×#28572
C4×C31×C22⋊Q82
C31×#20682
C31×#20562
C31×#19812
C31×#19472
C31×#18922
C31×#18032
#2475231