Finite Group Databasebeta

← C31×#24098 #21480857

Subgroups — C31×#24098 #21480857

Order 7936.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C22×C31×#107C211yesmaximal, characteristic, complemented
4C31×#2487C211yesmaximal, characteristic
5C31×#2057C212yesmaximal, complemented
6C31×#2036C212yesmaximal, complemented
7C31×#1892C211yesmaximal, characteristic, complemented
8C31×#1869C211yesmaximal, characteristic
9C31×#1863C212yesmaximal, complemented
10C31×#1852C212yesmaximal, complemented
11C31×#1824C211yesmaximal, characteristic, complemented
12C23×C8×C31C2211yescharacteristic
13C31×#554—24no—
14C31×#554—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#125—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C22×C4⋊C4×C31C2211yescharacteristic
25C22×C4⋊C4×C31C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#484—24no—
28C2×C31×#107—24no—
29C2×C31×#107C2212yes—
30C2×C31×#107C2212yes—
31C2×C31×#107C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×C22⋊C8C2211yescharacteristic
35C2×C31×C22⋊C8C2212yes—
36C31×#400—21no—
37C31×#399—21no—
38C31×#389C2212yes—
39C31×#386—21no—
40C31×#385—21no—
41C31×#380C2211yescharacteristic
42C31×#380C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31D411yescharacteristic
50C23×C4×C31—21no—
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C22×C8×C31—22no—
56C22×C8×C31—24no—
57C22×C8×C31C2312yes—
58C22×C8×C31C2312yes—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×#125—28no—
62C31×C22⋊Q8—28no—
63C31×C22⋊Q8—28no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8D414yes—
68C4×Q8×C31—22no—
69C4×Q8×C31—22no—
70C4×D4×C31—28no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D414yes—
88C2×C4⋊C4×C31C2312yes—
89C2×C4⋊C4×C31C2312yes—
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#107—24no—
103C31×#107—24no—
104C31×#103—24no—
105C31×#103—24no—
106C31×C22⋊C8—22no—
107C31×C22⋊C8—24no—
108C31×#95—21no—
109C31×#95—22no—
110C31×#95—22no—
111C24×C31C2×D411yescharacteristic
112C24×C31C2×D411yescharacteristic
113C2×Q8×C31—24no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—82no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31Q1614yes—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—24no—
129C22×C4×C31—24no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—42no—
137C22×C4×C31—82no—
138C22×C4×C31—82no—
139C22×C4×C31—82no—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2411yescharacteristic
147C2×C8×C31—28no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31#4114yes—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31—82no—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—82no—
179C42×C31—24no—
180C42×C31—21no—
181C42×C31—21no—
182#24098C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—21no—
186C23×C31—22no—
187C23×C31—21no—
188C23×C31—22no—
189C23×C31#12012yes—
190C23×C31C2×Q1612yes—
191C23×C31C2×Q1612yes—
192C23×C31#13712yes—
193C23×C31C22×D411yescharacteristic
194Q8×C31—24no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—84no—
198D4×C31—28no—
199C2×C4×C31—21no—
200C2×C4×C31—22no—
201C2×C4×C31—21no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—22no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
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—84no—
227C2×C4×C31—84no—
228C2×C4×C31—84no—
229C2×C4×C31#12012yes—
230C2×C4×C31C2×Q1612yes—
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C2×C4×C31C2×#4112yes—
234C2×C4×C312−1+412yes—
235C8×C31—24no—
236C8×C31—42no—
237C2×#554C2×C3111yescharacteristic
238C2×#484C2×C3111yescharacteristic, complemented
239C22×#107C2×C3111yescharacteristic, complemented
240#2487C2×C3111yescharacteristic
241#2057C2×C3112yescomplemented
242#2036C2×C3112yescomplemented
243#1892C2×C3111yescharacteristic, complemented
244#1869C2×C3111yescharacteristic
245#1863C2×C3112yescomplemented
246#1852C2×C3112yescomplemented
247#1824C2×C3111yescharacteristic, complemented
248C22×C31—24no—
249C22×C31—24no—
250C22×C31—22no—
251C22×C31—22no—
252C22×C31—24no—
253C22×C31—22no—
254C22×C31—42no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31#45114yes—
258C22×C31C2×#12011yescharacteristic
259C22×C31#54712yes—
260C22×C31C22×Q1611yescharacteristic
261C22×C31C2×#13711yescharacteristic
262C22×C31#57612yes—
263C4×C31—22no—
264C4×C31—22no—
265C4×C31—41no—
266C4×C31—42no—
267C4×C31—41no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—82no—
272C4×C31—82no—
273C4×C31#52511yescharacteristic
274C4×C31#53611yescharacteristic
275C4×C31#54712yes—
276C23×C8C22×C3111yescharacteristic
277#554—24no—
278#554—24no—
279C2×C4⋊Q8C22×C3111yescharacteristic
280C2×#125—21no—
281C2×C22⋊Q8—21no—
282C2×C22⋊Q8—21no—
283C2×C22⋊Q8C22×C3111yescharacteristic
284C2×C22⋊Q8C22×C3111yescharacteristic
285C2×C4×Q8C22×C3111yescharacteristic
286C2×C4×D4—21no—
287C2×C4×D4C22×C3111yescharacteristic
288C22×C4⋊C4C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290C22×C4⋊C4C22×C3111yescharacteristic
291#484—24no—
292C2×#107—24no—
293C2×#107C22×C3112yes—
294C2×#107C22×C3112yes—
295C2×#107C22×C3112yes—
296C2×#103C22×C3112yes—
297C2×#103C22×C3112yes—
298C2×C22⋊C8C22×C3111yescharacteristic
299C2×C22⋊C8C22×C3112yes—
300#400—21no—
301#399—21no—
302#389C22×C3112yes—
303#386—21no—
304#385—21no—
305#380C22×C3111yescharacteristic
306#380C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309#340C22×C3112yes—
310#336C22×C3112yes—
311C2×C31—24no—
312C2×C31—42no—
313C2×C31#240611yescharacteristic
314C2×C31C2×#45111yescharacteristic
315C2×C31#300811yescharacteristic
316C2×C31#333812yes—
317C2×C31#334612yes—
318C22×Q8—21no—
319C22×Q8C23×C3111yescharacteristic
320C22×D4D4×C3111yescharacteristic
321C23×C4—21no—
322C23×C4—21no—
323C23×C4—21no—
324C23×C4C23×C3111yescharacteristic
325C23×C4C23×C3111yescharacteristic
326C22×C8—22no—
327C22×C8—24no—
328C22×C8C23×C3112yes—
329C22×C8C23×C3112yes—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332#125—28no—
333C22⋊Q8—28no—
334C22⋊Q8—28no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8D4×C3114yes—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×D4—28no—
342C4×D4—22no—
343C4×D4—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—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4D4×C3111yescharacteristic
358C2×C4⋊C4D4×C3114yes—
359C2×C4⋊C4C23×C3112yes—
360C2×C4⋊C4C23×C3112yes—
361C2×C4⋊C4C23×C3111yescharacteristic
362C2×C4⋊C4C23×C3111yescharacteristic
363C2×C4⋊C4C23×C3111yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4D4×C3111yescharacteristic
369C2×C22⋊C4D4×C3111yescharacteristic
370C2×C42—21no—
371C2×C42C23×C3111yescharacteristic
372#107—24no—
373#107—24no—
374#107—24no—
375#103—24no—
376#103—24no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379#95—22no—
380#95—21no—
381#95—22no—
382C31#2409811yescharacteristic, direct product factor, Sylow
383C24C2×D4×C3111yescharacteristic
384C24C2×D4×C3111yescharacteristic
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—82no—
390C2×Q8C2×D4×C3112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4Q16×C3114yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3112yes—
416C22×C4C2×D4×C3112yes—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C24×C3111yescharacteristic
419C2×C8—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#41×C3114yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4C2×D4×C3112yes—
443C4⋊C4C2×D4×C3112yes—
444C4⋊C4C2×D4×C3112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—42no—
450C22⋊C4—82no—
451C42—24no—
452C42—21no—
453C42—21no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23C31×#12012yes—
461C23C2×Q16×C3112yes—
462C23C2×Q16×C3112yes—
463C23C31×#13712yes—
464C23C22×D4×C3111yescharacteristic
465Q8—24no—
466Q8—41no—
467Q8—41no—
468Q8—84no—
469D4—28no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4C31×#12012yes—
501C2×C4C2×Q16×C3112yes—
502C2×C4C22×D4×C3111yescharacteristic
503C2×C4C22×D4×C3111yescharacteristic
504C2×C4C2×#41×C3112yes—
505C2×C4C31×2−1+412yes—
506C8—24no—
507C8—42no—
508C22—24no—
509C22—24no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—22no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22C31×#45114yes—
518C22C2×C31×#12011yescharacteristic
519C22C31×#54712yes—
520C22C22×Q16×C3111yescharacteristic
521C22C2×C31×#13711yescharacteristic
522C22C31×#57612yes—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4C31×#52511yescharacteristic
534C4C31×#53611yescharacteristic
535C4C31×#54712yes—
536C2—24no—
537C2—42no—
538C2C31×#240611yescharacteristic
539C2C2×C31×#45111yescharacteristic
540C2C31×#300811yescharacteristic
541C2C31×#333812yes—
542C2C31×#334612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#484C2semidirect
3. C22×C31×#107C2semidirect
5. C31×#2057C2semidirect
6. C31×#2036C2semidirect
7. C31×#1892C2semidirect
9. C31×#1863C2semidirect
10. C31×#1852C2semidirect
11. C31×#1824C2semidirect
182. #24098C31direct (N × H)
238. C2×#484C2×C31semidirect
239. C22×#107C2×C31semidirect
241. #2057C2×C31semidirect
242. #2036C2×C31semidirect
243. #1892C2×C31semidirect
245. #1863C2×C31semidirect
246. #1852C2×C31semidirect
247. #1824C2×C31semidirect
382. C31#24098direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4842
C22×C31×#1072
C31×#24872
C31×#20572
C31×#20362
C31×#18922
C31×#18692
C31×#18632
C31×#18522
C31×#18242
#2409831