Finite Group Databasebeta

← C31×#25946 #21482705

Subgroups — C31×#25946 #21482705

Order 7936.

1176 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic, complemented
2C2×C31×#490C211yesmaximal, characteristic, complemented
3C2×C31×#459C211yesmaximal, characteristic
4C31×#2548C211yesmaximal, characteristic, complemented
5C31×#2025C212yesmaximal, complemented
6C31×#2023C212yesmaximal, complemented
7C31×#1989C211yesmaximal, characteristic, complemented
8C31×#1985C211yesmaximal, characteristic, complemented
9C31×#1879C212yesmaximal, complemented
10C31×#1879C212yesmaximal, complemented
11C31×#1762C211yesmaximal, characteristic, complemented
12C22×D8×C31C2212yes—
13C31×#554—24no—
14C31×#554—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#127C2211yescharacteristic
17C2×C31×#125C2211yescharacteristic
18C2×C31×#125C2211yescharacteristic
19C2×C31×#124—21no—
20C2×C31×C22⋊Q8—21no—
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C2×C4×D4×C31C2211yescharacteristic
27C22×C4⋊C4×C31—21no—
28C31×#490—24no—
29C31×#459—24no—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#102C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#97C2211yescharacteristic
36C31×#394—21no—
37C31×#394—21no—
38C31×#390—21no—
39C31×#390C2212yes—
40C31×#390C2212yes—
41C4×C4⋊C4×C31C2211yescharacteristic
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31D411yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31C2312yes—
52C22×D4×C31C2311yescharacteristic
53C23×C4×C31—21no—
54C23×C4×C31—21no—
55C23×C4×C31—21no—
56C2×D8×C31—24no—
57C2×D8×C31—24no—
58C2×D8×C31—24no—
59C22×C8×C31C2312yes—
60C22×C8×C31C2312yes—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×#127—22no—
64C31×#127—22no—
65C31×#125—22no—
66C31×#125—22no—
67C31×#125—22no—
68C31×#125D414yes—
69C31×#124—44no—
70C31×C22⋊Q8—28no—
71C31×C22⋊Q8—28no—
72C31×C22⋊Q8—28no—
73C4×Q8×C31—22no—
74C4×Q8×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C4×D4×C31—22no—
80C4×D4×C31D414yes—
81C2×C4⋊C4×C31—28no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31—22no—
101C2×C42×C31—21no—
102C2×C42×C31C2311yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C2×C42×C31C2311yescharacteristic
105C31×C4⋊C8—22no—
106C31×#102—24no—
107C31×#102—24no—
108C31×#102—24no—
109C31×#102—24no—
110C31×#97—22no—
111C31×#95—21no—
112C24×C31—21no—
113C24×C31—22no—
114C2×Q8×C31—24no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—41no—
118C2×Q8×C31—82no—
119C2×D4×C31—21no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31—82no—
132C2×D4×C31C2×D412yes—
133C2×D4×C31#4114yes—
134C22×C4×C31—24no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—42no—
142C22×C4×C31—42no—
143C22×C4×C31—82no—
144C22×C4×C31—82no—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2411yescharacteristic
152D8×C31—48no—
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—24no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31C2×D412yes—
173C4⋊C4×C31C2×D412yes—
174C4⋊C4×C31C2×D412yes—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—44no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—82no—
181C22⋊C4×C31—82no—
182C42×C31—21no—
183C42×C31—21no—
184C42×C31—21no—
185C42×C31—21no—
186C42×C31—42no—
187C42×C31C2×D412yes—
188C42×C31C2×D412yes—
189#25946C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—22no—
191C23×C31—24no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31C22×D411yescharacteristic
203Q8×C31—44no—
204Q8×C31—84no—
205D4×C31—22no—
206D4×C31—24no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—84no—
212D4×C31—84no—
213D4×C31#12414yescomplemented
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—48no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—84no—
238C2×C4×C31—84no—
239C2×C4×C31#12012yes—
240C2×C4×C31#12012yes—
241C2×C4×C31AGL(1,ℤ/8)12yes—
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C2×#4112yes—
245C2×C4×C312−1+412yes—
246C8×C31—42no—
247C8×C31—42no—
248C2×#554C2×C3111yescharacteristic, complemented
249C2×#490C2×C3111yescharacteristic, complemented
250C2×#459C2×C3111yescharacteristic
251#2548C2×C3111yescharacteristic, complemented
252#2025C2×C3112yescomplemented
253#2023C2×C3112yescomplemented
254#1989C2×C3111yescharacteristic, complemented
255#1985C2×C3111yescharacteristic, complemented
256#1879C2×C3112yescomplemented
257#1879C2×C3112yescomplemented
258#1762C2×C3111yescharacteristic, complemented
259C22×C31—24no—
260C22×C31—44no—
261C22×C31—44no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—82no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—82no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31C2×#12011yescharacteristic
272C22×C31#54812yes—
273C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C31#57511yescharacteristic
275C22×C31#57612yes—
276C4×C31—22no—
277C4×C31—22no—
278C4×C31—22no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—41no—
282C4×C31—41no—
283C4×C31—42no—
284C4×C31—42no—
285C4×C31—82no—
286C4×C31C2×#12411yescharacteristic
287C4×C31#53611yescharacteristic
288C4×C31#54812yes—
289C22×D8C22×C3112yes—
290#554—24no—
291#554—24no—
292C2×C4⋊Q8C22×C3111yescharacteristic
293C2×#127C22×C3111yescharacteristic
294C2×#125C22×C3111yescharacteristic
295C2×#125C22×C3111yescharacteristic
296C2×#124—21no—
297C2×C22⋊Q8—21no—
298C2×C22⋊Q8—21no—
299C2×C22⋊Q8—21no—
300C2×C4×Q8C22×C3111yescharacteristic
301C2×C4×D4C22×C3111yescharacteristic
302C2×C4×D4C22×C3111yescharacteristic
303C2×C4×D4C22×C3111yescharacteristic
304C22×C4⋊C4—21no—
305#490—24no—
306#459—24no—
307C2×C4⋊C8C22×C3111yescharacteristic
308C2×#102C22×C3112yes—
309C2×#102C22×C3112yes—
310C2×#102C22×C3112yes—
311C2×#102C22×C3112yes—
312C2×#97C22×C3111yescharacteristic
313#394—21no—
314#394—21no—
315#390—21no—
316#390C22×C3112yes—
317#390C22×C3112yes—
318C4×C4⋊C4C22×C3111yescharacteristic
319#340C22×C3112yes—
320#340C22×C3112yes—
321#340C22×C3112yes—
322#336C22×C3112yes—
323#336C22×C3112yes—
324C2×C31—44no—
325C2×C31—162no—
326C2×C31#243611yescharacteristic
327C2×C31#300611yescharacteristic
328C2×C31#300811yescharacteristic
329C2×C31#336612yes—
330C2×C31#337012yes—
331C22×Q8—21no—
332C22×Q8D4×C3111yescharacteristic
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C23×C3112yes—
336C22×D4C23×C3111yescharacteristic
337C23×C4—21no—
338C23×C4—21no—
339C23×C4—21no—
340C2×D8—24no—
341C2×D8—24no—
342C2×D8—24no—
343C22×C8C23×C3112yes—
344C22×C8C23×C3112yes—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347#127—22no—
348#127—22no—
349#125—22no—
350#125—22no—
351#125—22no—
352#125D4×C3114yes—
353#124—44no—
354C22⋊Q8—28no—
355C22⋊Q8—28no—
356C22⋊Q8—28no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4D4×C3114yes—
365C2×C4⋊C4—28no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4D4×C3111yescharacteristic
372C2×C4⋊C4D4×C3111yescharacteristic
373C2×C4⋊C4D4×C3111yescharacteristic
374C2×C4⋊C4C23×C3112yes—
375C2×C4⋊C4C23×C3111yescharacteristic
376C2×C4⋊C4C23×C3111yescharacteristic
377C2×C4⋊C4C23×C3111yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C42—21no—
386C2×C42C23×C3111yescharacteristic
387C2×C42C23×C3111yescharacteristic
388C2×C42C23×C3111yescharacteristic
389C4⋊C8—22no—
390#102—24no—
391#102—24no—
392#102—24no—
393#102—24no—
394#97—22no—
395#95—21no—
396C31#2594611yescharacteristic, direct product factor, Sylow
397C24—21no—
398C24—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—41no—
403C2×Q8—82no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—82no—
417C2×D4C2×D4×C3112yes—
418C2×D4#41×C3114yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C24×C3111yescharacteristic
437D8—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4C2×D4×C3112yes—
458C4⋊C4C2×D4×C3112yes—
459C4⋊C4C2×D4×C3112yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42C2×D4×C3112yes—
473C42C2×D4×C3112yes—
474C23—22no—
475C23—24no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C22×D4×C3111yescharacteristic
487Q8—44no—
488Q8—84no—
489D4—22no—
490D4—24no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—84no—
496D4—84no—
497D4C31×#12414yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4C31×#12012yes—
524C2×C4C31×#12012yes—
525C2×C4C31×AGL(1,ℤ/8)12yes—
526C2×C4C22×D4×C3111yescharacteristic
527C2×C4C22×D4×C3111yescharacteristic
528C2×C4C2×#41×C3112yes—
529C2×C4C31×2−1+412yes—
530C8—42no—
531C8—42no—
532C22—24no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×C31×#12011yescharacteristic
545C22C31×#54812yes—
546C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
547C22C31×#57511yescharacteristic
548C22C31×#57612yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4C2×C31×#12411yescharacteristic
560C4C31×#53611yescharacteristic
561C4C31×#54812yes—
562C2—44no—
563C2—162no—
564C2C31×#243611yescharacteristic
565C2C31×#300611yescharacteristic
566C2C31×#300811yescharacteristic
567C2C31×#336612yes—
568C2C31×#337012yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#554C2semidirect
2. C2×C31×#490C2semidirect
4. C31×#2548C2semidirect
5. C31×#2025C2semidirect
6. C31×#2023C2semidirect
7. C31×#1989C2semidirect
8. C31×#1985C2semidirect
9. C31×#1879C2semidirect
10. C31×#1879C2semidirect
11. C31×#1762C2semidirect
189. #25946C31direct (N × H)
213. D4×C31#124semidirect
248. C2×#554C2×C31semidirect
249. C2×#490C2×C31semidirect
251. #2548C2×C31semidirect
252. #2025C2×C31semidirect
253. #2023C2×C31semidirect
254. #1989C2×C31semidirect
255. #1985C2×C31semidirect
256. #1879C2×C31semidirect
257. #1879C2×C31semidirect
258. #1762C2×C31semidirect
396. C31#25946direct (N × H)
497. D4C31×#124semidirect

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×#4902
C2×C31×#4592
C31×#25482
C31×#20252
C31×#20232
C31×#19892
C31×#19852
C31×#18792
C31×#17622
#2594631