Finite Group Databasebeta

← C31×#25866 #21482625

Subgroups — C31×#25866 #21482625

Order 7936.

1238 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic
2C2×C31×#489C211yesmaximal, characteristic, complemented
3C2×C31×#487C211yesmaximal, characteristic, complemented
4C31×#2628C211yesmaximal, characteristic, complemented
5C31×#2043C212yesmaximal, complemented
6C31×#2042C212yesmaximal, complemented
7C31×#1960C211yesmaximal, characteristic, complemented
8C31×#1958C211yesmaximal, characteristic, complemented
9C31×#1788C212yesmaximal
10C31×#1787C212yesmaximal, complemented
11C31×#1750C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C23×Q8×C31C2211yescharacteristic
14C31×#536—24no—
15C31×#536—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×C4⋊Q8C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#124C2211yescharacteristic
21C2×C31×#123—21no—
22C2×C31×#123—21no—
23C2×C31×C22⋊Q8—21no—
24C2×C31×C22⋊Q8—21no—
25C2×C31×#117C2211yescharacteristic
26C31×#489—24no—
27C31×#487—24no—
28C2×C31×#103C2212yes—
29C2×C31×#103C2212yes—
30C2×C31×#103C2212yes—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×C22⋊C8C2212yes—
37C2×C31×#97C2211yescharacteristic
38C2×C4×C8×C31C2211yescharacteristic
39C31×#400—21no—
40C31×#399—21no—
41C31×#395—21no—
42C31×#393—21no—
43C31×#389C2212yes—
44C31×#382C2211yescharacteristic
45C31×#380C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—24no—
50C2×#41×C31—42no—
51C22×Q8×C31—22no—
52C22×Q8×C31—22no—
53C22×Q8×C31—21no—
54C22×Q8×C31—21no—
55C22×Q8×C31—24no—
56C22×Q8×C31—42no—
57C22×Q8×C31C2311yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C22×Q8×C31C2311yescharacteristic
60C22×D4×C31—21no—
61C22×D4×C31C2311yescharacteristic
62C23×C4×C31—21no—
63C23×C4×C31—21no—
64C23×C4×C31C2311yescharacteristic
65C22×C8×C31C2312yes—
66C22×C8×C31C2312yes—
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—22no—
71C31×C4⋊Q8—22no—
72C31×C4⋊Q8D414yes—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124—22no—
76C31×#124D414yes—
77C31×#123—28no—
78C31×#123—28no—
79C31×C22⋊Q8—28no—
80C31×C22⋊Q8—28no—
81C31×#117—22no—
82C31×#117—22no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2312yes—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31D411yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C2×C42×C31C2311yescharacteristic
102C31×#103—24no—
103C31×#103—24no—
104C31×#103—24no—
105C31×#103—42no—
106C31×#103—42no—
107C31×#103—42no—
108C31×#102—24no—
109C31×#102—42no—
110C31×C22⋊C8—24no—
111C31×#97—22no—
112C4×C8×C31—22no—
113C31×#95—22no—
114C31×#95—21no—
115C31×#95—21no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119#41×C31—48no—
120#41×C31—48no—
121C2×Q8×C31—24no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—48no—
132C2×Q8×C31—48no—
133C2×Q8×C31—42no—
134C2×Q8×C31—42no—
135C2×Q8×C31—42no—
136C2×Q8×C31—82no—
137C2×Q8×C31C2×D412yes—
138C2×Q8×C31C2×D412yes—
139C2×Q8×C31C2×D412yes—
140C2×D4×C31—24no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31C2×D412yes—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—42no—
158C22×C4×C31—42no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31#4114yes—
167C22×C4×C31C2411yescharacteristic
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—24no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—42no—
185C4⋊C4×C31—82no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—82no—
193C42×C31—21no—
194C42×C31—21no—
195C42×C31C2×D412yes—
196C42×C31C2×D412yes—
197#25866C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—22no—
201C23×C31—22no—
202C23×C31—22no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31#12412yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—24no—
210Q8×C31—24no—
211Q8×C31—24no—
212Q8×C31—48no—
213Q8×C31—42no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—84no—
217D4×C31—24no—
218D4×C31—48no—
219D4×C31—42no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—48no—
234C2×C4×C31—48no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31#12012yes—
248C2×C4×C31#12012yes—
249C2×C4×C31#12412yes—
250C2×C4×C31C22×D411yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C2×#4112yes—
253C2×C4×C31H(5,𝔽2)12yes—
254C8×C31—42no—
255C8×C31—42no—
256C2×#536C2×C3111yescharacteristic
257C2×#489C2×C3111yescharacteristic, complemented
258C2×#487C2×C3111yescharacteristic, complemented
259#2628C2×C3111yescharacteristic, complemented
260#2043C2×C3112yescomplemented
261#2042C2×C3112yescomplemented
262#1960C2×C3111yescharacteristic, complemented
263#1958C2×C3111yescharacteristic, complemented
264#1788C2×C3112yes—
265#1787C2×C3112yescomplemented
266#1750C2×C3111yescharacteristic, complemented
267C22×C31—24no—
268C22×C31—22no—
269C22×C31—42no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31C2×#12011yescharacteristic
278C22×C31C2×#12411yescharacteristic
279C22×C31#53511yescharacteristic
280C22×C31#57712yes—
281C22×C31#57812yes—
282C4×C31—22no—
283C4×C31—22no—
284C4×C31—22no—
285C4×C31—42no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—41no—
290C4×C31—82no—
291C4×C31—82no—
292C4×C31#54612yes—
293C4×C31#54612yes—
294C22×#41C22×C3111yescharacteristic
295C23×Q8C22×C3111yescharacteristic
296#536—24no—
297#536—24no—
298C2×C4⋊Q8C22×C3111yescharacteristic
299C2×C4⋊Q8C22×C3111yescharacteristic
300C2×C4⋊Q8C22×C3111yescharacteristic
301C2×#124C22×C3111yescharacteristic
302C2×#124C22×C3111yescharacteristic
303C2×#123—21no—
304C2×#123—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×#117C22×C3111yescharacteristic
308#489—24no—
309#487—24no—
310C2×#103C22×C3112yes—
311C2×#103C22×C3112yes—
312C2×#103C22×C3112yes—
313C2×#103C22×C3112yes—
314C2×#103C22×C3112yes—
315C2×#103C22×C3112yes—
316C2×#102C22×C3112yes—
317C2×#102C22×C3112yes—
318C2×C22⋊C8C22×C3112yes—
319C2×#97C22×C3111yescharacteristic
320C2×C4×C8C22×C3111yescharacteristic
321#400—21no—
322#399—21no—
323#395—21no—
324#393—21no—
325#389C22×C3112yes—
326#382C22×C3111yescharacteristic
327#380C22×C3112yes—
328#336C22×C3112yes—
329C2×C31—42no—
330C2×C31—82no—
331C2×C31#243511yescharacteristic
332C2×C31#301011yescharacteristic
333C2×C31#301111yescharacteristic
334C2×C31#313712yes—
335C2×C31#322512yes—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—24no—
339C2×#41—42no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—21no—
343C22×Q8—21no—
344C22×Q8—24no—
345C22×Q8—42no—
346C22×Q8C23×C3111yescharacteristic
347C22×Q8C23×C3111yescharacteristic
348C22×Q8C23×C3111yescharacteristic
349C22×D4—21no—
350C22×D4C23×C3111yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4C23×C3111yescharacteristic
354C22×C8C23×C3112yes—
355C22×C8C23×C3112yes—
356C4⋊Q8—22no—
357C4⋊Q8—22no—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361C4⋊Q8D4×C3114yes—
362#124—22no—
363#124—22no—
364#124—22no—
365#124D4×C3114yes—
366#123—28no—
367#123—28no—
368C22⋊Q8—28no—
369C22⋊Q8—28no—
370#117—22no—
371#117—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C3111yescharacteristic
379C2×C4⋊C4D4×C3111yescharacteristic
380C2×C4⋊C4C23×C3112yes—
381C2×C4⋊C4C23×C3112yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4D4×C3111yescharacteristic
388C2×C22⋊C4D4×C3111yescharacteristic
389C2×C42C23×C3111yescharacteristic
390C2×C42C23×C3111yescharacteristic
391#103—24no—
392#103—24no—
393#103—24no—
394#103—42no—
395#103—42no—
396#103—42no—
397#102—24no—
398#102—42no—
399C22⋊C8—24no—
400#97—22no—
401C4×C8—22no—
402#95—22no—
403#95—21no—
404#95—21no—
405#95—22no—
406C31#2586611yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24C2×D4×C3111yescharacteristic
409#41—48no—
410#41—48no—
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8C2×D4×C3112yes—
428C2×Q8C2×D4×C3112yes—
429C2×Q8C2×D4×C3112yes—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C2×D4×C3112yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4#41×C3114yes—
457C22×C4C24×C3111yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—82no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—82no—
483C42—21no—
484C42—21no—
485C42C2×D4×C3112yes—
486C42C2×D4×C3112yes—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23C31×#12412yes—
497C23C22×D4×C3111yescharacteristic
498Q8—24no—
499Q8—24no—
500Q8—24no—
501Q8—48no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—84no—
506D4—24no—
507D4—48no—
508D4—42no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4C31×#12012yes—
537C2×C4C31×#12012yes—
538C2×C4C31×#12412yes—
539C2×C4C22×D4×C3111yescharacteristic
540C2×C4C22×D4×C3111yescharacteristic
541C2×C4C2×#41×C3112yes—
542C2×C4C31×H(5,𝔽2)12yes—
543C8—42no—
544C8—42no—
545C22—24no—
546C22—22no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22C2×C31×#12011yescharacteristic
556C22C2×C31×#12411yescharacteristic
557C22C31×#53511yescharacteristic
558C22C31×#57712yes—
559C22C31×#57812yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—82no—
569C4—82no—
570C4C31×#54612yes—
571C4C31×#54612yes—
572C2—42no—
573C2—82no—
574C2C31×#243511yescharacteristic
575C2C31×#301011yescharacteristic
576C2C31×#301111yescharacteristic
577C2C31×#313712yes—
578C2C31×#322512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#489C2semidirect
3. C2×C31×#487C2semidirect
4. C31×#2628C2semidirect
5. C31×#2043C2semidirect
6. C31×#2042C2semidirect
7. C31×#1960C2semidirect
8. C31×#1958C2semidirect
10. C31×#1787C2semidirect
11. C31×#1750C2semidirect
197. #25866C31direct (N × H)
257. C2×#489C2×C31semidirect
258. C2×#487C2×C31semidirect
259. #2628C2×C31semidirect
260. #2043C2×C31semidirect
261. #2042C2×C31semidirect
262. #1960C2×C31semidirect
263. #1958C2×C31semidirect
265. #1787C2×C31semidirect
266. #1750C2×C31semidirect
406. C31#25866direct (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×#4892
C2×C31×#4872
C31×#26282
C31×#20432
C31×#20422
C31×#19602
C31×#19582
C31×#17882
C31×#17872
C31×#17502
#2586631