Finite Group Databasebeta

← C31×#8446 #21465205

Subgroups — C31×#8446 #21465205

Order 7936.

1048 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C31×#1990C211yesmaximal, characteristic, complemented
3C31×#1802C211yesmaximal, characteristic, complemented
4C31×#1802C211yesmaximal, characteristic, complemented
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#547—22no—
10C31×#547—22no—
11C31×#547—22no—
12C31×#547C414yescomplemented
13C2×C31×C4⋊Q8C2211yescharacteristic
14C2×C31×#123—22no—
15C2×C31×C22⋊Q8—21no—
16C2×C31×#121—21no—
17C2×C4×D4×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C22×C4⋊C4×C31—21no—
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×#102—21no—
23C2×C31×#102—21no—
24C2×C4×C8×C31C2211yescharacteristic
25C31×#384C2211yescharacteristic
26C31×#340—21no—
27C31×#340—21no—
28C31×#340—22no—
29C31×#332—22no—
30C31×D4⋊C8—22no—
31C31×D4⋊C8—22no—
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—42no—
35C2×#41×C31—42no—
36C2×#41×C31—42no—
37C2×#41×C31—42no—
38C22×Q8×C31—21no—
39C22×D4×C31—22no—
40C22×D4×C31D411yescharacteristic
41C22×D4×C31D411yescharacteristic
42C23×C4×C31—22no—
43C23×C4×C31—21no—
44C23×C4×C31—21no—
45C22×C8×C31—21no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8C2×C412yes—
51C31×C4⋊Q8D412yes—
52C31×#123—28no—
53C31×#123—28no—
54C31×C22⋊Q8—24no—
55C31×C22⋊Q8—24no—
56C31×#121—24no—
57C31×#121—24no—
58C4×D4×C31—21no—
59C4×D4×C31—21no—
60C4×D4×C31—21no—
61C4×D4×C31—21no—
62C4×D4×C31C2×C412yes—
63C4×D4×C31C2×C412yes—
64C4×D4×C31D412yes—
65C4×D4×C31D412yes—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—24no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—24no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31D411yescharacteristic
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31D412yes—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×#102—42no—
85C31×#102—42no—
86C4×C8×C31—21no—
87C4×C8×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90#41×C31—48no—
91#41×C31—48no—
92#41×C31—48no—
93#41×C31—48no—
94C2×Q8×C31—22no—
95C2×Q8×C31—22no—
96C2×Q8×C31—41no—
97C2×Q8×C31—41no—
98C2×Q8×C31—41no—
99C2×Q8×C31—41no—
100C2×D4×C31—28no—
101C2×D4×C31—21no—
102C2×D4×C31—21no—
103C2×D4×C31—21no—
104C2×D4×C31—21no—
105C2×D4×C31—21no—
106C2×D4×C31—21no—
107C2×D4×C31—21no—
108C2×D4×C31—21no—
109C2×D4×C31—44no—
110C2×D4×C31—44no—
111C2×D4×C31C22⋊C412yes—
112C2×D4×C31C22⋊C412yes—
113C2×D4×C31D812yes—
114C2×D4×C31SD1612yes—
115C22×C4×C31—28no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—44no—
124C22×C4×C31—44no—
125C22×C4×C31—41no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—41no—
130C22×C4×C31—41no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31—41no—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C2×C8×C31—22no—
138C2×C8×C31—41no—
139C2×C8×C31—41no—
140C2×C8×C31—41no—
141C2×C8×C31—41no—
142C2×C8×C31—41no—
143C2×C8×C31—41no—
144C2×C8×C31—41no—
145C2×C8×C31—41no—
146C4⋊C4×C31—28no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—42no—
153C4⋊C4×C31—41no—
154C4⋊C4×C31—41no—
155C4⋊C4×C31—41no—
156C4⋊C4×C31—41no—
157C4⋊C4×C31C22⋊C412yes—
158C4⋊C4×C31C22⋊C412yes—
159C4⋊C4×C31C22⋊C414yes—
160C22⋊C4×C31—28no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C42×C31C22×C411yescharacteristic
166C42×C31C2×D411yescharacteristic
167C42×C31C2×D411yescharacteristic
168C42×C31C2×D411yescharacteristic
169#8446C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31#12011yescharacteristic
185Q8×C31—42no—
186Q8×C31—42no—
187Q8×C31—42no—
188Q8×C31—42no—
189D4×C31—22no—
190D4×C31—22no—
191D4×C31—22no—
192D4×C31—22no—
193D4×C31—22no—
194D4×C31—416no—
195D4×C31—41no—
196D4×C31—41no—
197D4×C31#10214yescomplemented
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—416no—
204C2×C4×C31—41no—
205C2×C4×C31—42no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—42no—
209C2×C4×C31—41no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—81no—
219C2×C4×C31—81no—
220C2×C4×C31—81no—
221C2×C4×C31C2×C22⋊C411yescharacteristic
222C2×C4×C31C2×C22⋊C411yescharacteristic
223C2×C4×C31C2×C22⋊C411yescharacteristic
224C2×C4×C31#12011yescharacteristic
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31C2×D811yescharacteristic
228C2×C4×C31C2×SD1611yescharacteristic
229C2×C4×C31#13511yescharacteristic
230C2×C4×C31#13511yescharacteristic
231C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
232C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
233C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
234C2×C4×C31#13711yescharacteristic
235C8×C31—42no—
236C8×C31—81no—
237C8×C31—81no—
238C2×#547C2×C3111yescharacteristic
239#1990C2×C3111yescharacteristic, complemented
240#1802C2×C3111yescharacteristic, complemented
241#1802C2×C3111yescharacteristic, complemented
242C2×#332C2×C3111yescharacteristic, complemented
243C2×D4⋊C8C2×C3111yescharacteristic, complemented
244C2×D4⋊C8C2×C3111yescharacteristic, complemented
245C22×C31—24no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—81no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31—81no—
259C22×C31#37911yescharacteristic
260C22×C31#44711yescharacteristic
261C22×C31#44911yescharacteristic
262C22×C31#45011yescharacteristic
263C22×C31#45211yescharacteristic
264C22×C31#45411yescharacteristic
265C22×C31#45611yescharacteristic
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—42no—
269C4×C31—44no—
270C4×C31—42no—
271C4×C31—161no—
272C4×C31C2×#10211yescharacteristic
273C4×C31#41611yescharacteristic
274C4×C31#41711yescharacteristic
275C4×C31#41811yescharacteristic
276C4×C31#44711yescharacteristic
277C4×C31#44911yescharacteristic
278C4×C31#45011yescharacteristic
279C4×C31#45211yescharacteristic
280C22×#41—21no—
281#547—22no—
282#547—22no—
283#547—22no—
284#547C4×C3114yescomplemented
285C2×C4⋊Q8C22×C3111yescharacteristic
286C2×#123—22no—
287C2×C22⋊Q8—21no—
288C2×#121—21no—
289C2×C4×D4C22×C3111yescharacteristic
290C2×C4×D4C22×C3111yescharacteristic
291C22×C4⋊C4—21no—
292C2×C4⋊C8C22×C3111yescharacteristic
293C2×C4⋊C8C22×C3111yescharacteristic
294C2×#102—21no—
295C2×#102—21no—
296C2×C4×C8C22×C3111yescharacteristic
297#384C22×C3111yescharacteristic
298#340—21no—
299#340—21no—
300#340—22no—
301#332—22no—
302D4⋊C8—22no—
303D4⋊C8—22no—
304C2×C31—44no—
305C2×C31—82no—
306C2×C31#162511yescharacteristic
307C2×C31#163511yescharacteristic
308C2×C31#164711yescharacteristic
309C2×C31#165411yescharacteristic
310C2×C31#178611yescharacteristic
311C2×C31#178811yescharacteristic
312C2×C31#179311yescharacteristic
313C2×#41—24no—
314C2×#41—24no—
315C2×#41—42no—
316C2×#41—42no—
317C2×#41—42no—
318C2×#41—42no—
319C22×Q8—21no—
320C22×D4—22no—
321C22×D4D4×C3111yescharacteristic
322C22×D4D4×C3111yescharacteristic
323C23×C4—22no—
324C23×C4—21no—
325C23×C4—21no—
326C22×C8—21no—
327C22×C8—21no—
328C22×C8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4⋊Q8C2×C4×C3112yes—
332C4⋊Q8D4×C3112yes—
333#123—28no—
334#123—28no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337#121—24no—
338#121—24no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4C2×C4×C3112yes—
344C4×D4C2×C4×C3112yes—
345C4×D4D4×C3112yes—
346C4×D4D4×C3112yes—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4D4×C3112yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42C23×C3111yescharacteristic
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365#102—42no—
366#102—42no—
367C4×C8—21no—
368C4×C8—21no—
369C31#844611yescharacteristic, direct product factor, Sylow
370C24—21no—
371C24—21no—
372#41—48no—
373#41—48no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×D4—28no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4C22⋊C4×C3112yes—
394C2×D4C22⋊C4×C3112yes—
395C2×D4D8×C3112yes—
396C2×D4SD16×C3112yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3111yescharacteristic
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4C22⋊C4×C3112yes—
440C4⋊C4C22⋊C4×C3112yes—
441C4⋊C4C22⋊C4×C3114yes—
442C22⋊C4—28no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C42C22×C4×C3111yescharacteristic
448C42C2×D4×C3111yescharacteristic
449C42C2×D4×C3111yescharacteristic
450C42C2×D4×C3111yescharacteristic
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23C31×#12011yescharacteristic
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478D4C31×#10214yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—416no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C2×C22⋊C4×C3111yescharacteristic
503C2×C4C2×C22⋊C4×C3111yescharacteristic
504C2×C4C2×C22⋊C4×C3111yescharacteristic
505C2×C4C31×#12011yescharacteristic
506C2×C4C31×#12011yescharacteristic
507C2×C4C31×#12011yescharacteristic
508C2×C4C2×D8×C3111yescharacteristic
509C2×C4C2×SD16×C3111yescharacteristic
510C2×C4C31×#13511yescharacteristic
511C2×C4C31×#13511yescharacteristic
512C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
514C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C31×#13711yescharacteristic
516C8—42no—
517C8—81no—
518C8—81no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22C31×#37911yescharacteristic
534C22C31×#44711yescharacteristic
535C22C31×#44911yescharacteristic
536C22C31×#45011yescharacteristic
537C22C31×#45211yescharacteristic
538C22C31×#45411yescharacteristic
539C22C31×#45611yescharacteristic
540C4—21no—
541C4—21no—
542C4—42no—
543C4—44no—
544C4—42no—
545C4—161no—
546C4C2×C31×#10211yescharacteristic
547C4C31×#41611yescharacteristic
548C4C31×#41711yescharacteristic
549C4C31×#41811yescharacteristic
550C4C31×#44711yescharacteristic
551C4C31×#44911yescharacteristic
552C4C31×#45011yescharacteristic
553C4C31×#45211yescharacteristic
554C2—44no—
555C2—82no—
556C2C31×#162511yescharacteristic
557C2C31×#163511yescharacteristic
558C2C31×#164711yescharacteristic
559C2C31×#165411yescharacteristic
560C2C31×#178611yescharacteristic
561C2C31×#178811yescharacteristic
562C2C31×#179311yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#547C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1990C2semidirect
3. C31×#1802C2semidirect
4. C31×#1802C2semidirect
5. C2×C31×#332C2semidirect
6. C2×C31×D4⋊C8C2semidirect
7. C2×C31×D4⋊C8C2semidirect
12. C31×#547C4semidirect
169. #8446C31direct (N × H)
197. D4×C31#102semidirect
239. #1990C2×C31semidirect
240. #1802C2×C31semidirect
241. #1802C2×C31semidirect
242. C2×#332C2×C31semidirect
243. C2×D4⋊C8C2×C31semidirect
244. C2×D4⋊C8C2×C31semidirect
284. #547C4×C31semidirect
369. C31#8446direct (N × H)
478. D4C31×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C31×#19902
C31×#18022
C2×C31×#3322
C2×C31×D4⋊C82
#844631