Finite Group Databasebeta

← C31×#8444 #21465203

Subgroups — C31×#8444 #21465203

Order 7936.

1116 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×#550C211yesmaximal, characteristic
2C31×#1957C211yesmaximal, characteristic, complemented
3C31×#1803C211yesmaximal, characteristic, complemented
4C31×#1802C211yesmaximal, characteristic, complemented
5C2×C31×#331C211yesmaximal, characteristic, complemented
6C2×C31×Q8⋊C8C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#550—22no—
10C31×#550—22no—
11C31×#550—22no—
12C31×#550C414yescomplemented
13C2×C31×#127—21no—
14C2×C31×#127C2211yescharacteristic
15C2×C31×#121—22no—
16C2×C31×#121—21no—
17C2×C4×Q8×C31C2211yescharacteristic
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31C2211yescharacteristic
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×#103—21no—
23C2×C31×#102—21no—
24C2×C31×#102—22no—
25C2×C4×C8×C31C2211yescharacteristic
26C31×#384C2211yescharacteristic
27C31×#340—21no—
28C31×#340—21no—
29C31×#331—22no—
30C31×Q8⋊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×C31D411yescharacteristic
39C22×D4×C31—22no—
40C22×D4×C31—22no—
41C22×D4×C31—21no—
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C22×D4×C31D412yes—
45C23×C4×C31—22no—
46C23×C4×C31—21no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C22×C8×C31—21no—
50C31×#127—24no—
51C31×#127—24no—
52C31×#127—21no—
53C31×#127—21no—
54C31×#127C2×C412yes—
55C31×#127D412yes—
56C31×#121—24no—
57C31×#121—28no—
58C31×#121—28no—
59C31×#121—24no—
60C4×Q8×C31—21no—
61C4×Q8×C31—21no—
62C4×Q8×C31C2×C412yes—
63C4×Q8×C31D412yes—
64C4×D4×C31—24no—
65C4×D4×C31—24no—
66C4×D4×C31—21no—
67C4×D4×C31—21no—
68C4×D4×C31C2×C412yes—
69C4×D4×C31D412yes—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31D411yescharacteristic
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—21no—
78C2×C42×C31—21no—
79C2×C42×C31C2311yescharacteristic
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8—21no—
83C31×C4⋊C8—21no—
84C31×#103—42no—
85C31×#102—44no—
86C31×#102—42no—
87C4×C8×C31—21no—
88C4×C8×C31—21no—
89C24×C31—21no—
90C24×C31—22no—
91#41×C31—48no—
92#41×C31—48no—
93#41×C31—48no—
94#41×C31—48no—
95C2×Q8×C31—21no—
96C2×Q8×C31—21no—
97C2×Q8×C31—21no—
98C2×Q8×C31C22⋊C412yes—
99C2×Q8×C31D812yes—
100C2×Q8×C31SD1612yes—
101C2×D4×C31—28no—
102C2×D4×C31—28no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—21no—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—44no—
114C2×D4×C31—44no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C2×D4×C31—41no—
118C2×D4×C31—41no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31—41no—
123C2×D4×C31—41no—
124C2×D4×C31—41no—
125C2×D4×C31—82no—
126C2×D4×C31—82no—
127C2×D4×C31C22⋊C412yes—
128C2×D4×C31C22⋊C414yes—
129C22×C4×C31—21no—
130C22×C4×C31—28no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—44no—
136C22×C4×C31—44no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C2×C8×C31—22no—
146C2×C8×C31—41no—
147C2×C8×C31—41no—
148C2×C8×C31—41no—
149C2×C8×C31—41no—
150C2×C8×C31—41no—
151C2×C8×C31—41no—
152C2×C8×C31—41no—
153C2×C8×C31—41no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—41no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—41no—
163C4⋊C4×C31C22⋊C412yes—
164C4⋊C4×C31C22⋊C412yes—
165C22⋊C4×C31—28no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C42×C31—22no—
169C42×C31—22no—
170C42×C31C22×C411yescharacteristic
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173C42×C31C2×D411yescharacteristic
174#8444C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—24no—
176C23×C31—22no—
177C23×C31—42no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—41no—
189C23×C31#12011yescharacteristic
190Q8×C31—22no—
191Q8×C31—22no—
192Q8×C31—22no—
193Q8×C31#10214yescomplemented
194D4×C31—22no—
195D4×C31—22no—
196D4×C31—44no—
197D4×C31—416no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—41no—
204D4×C31—42no—
205D4×C31—42no—
206D4×C31—41no—
207D4×C31—42no—
208D4×C31—82no—
209D4×C31—88no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—416no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—81no—
227C2×C4×C31—81no—
228C2×C4×C31—81no—
229C2×C4×C31C2×C22⋊C411yescharacteristic
230C2×C4×C31C2×C22⋊C411yescharacteristic
231C2×C4×C31C2×C22⋊C411yescharacteristic
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31C2×D811yescharacteristic
236C2×C4×C31C2×SD1611yescharacteristic
237C2×C4×C31#13511yescharacteristic
238C2×C4×C31#13511yescharacteristic
239C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
240C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
241C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
242C2×C4×C31#13711yescharacteristic
243C8×C31—42no—
244C8×C31—81no—
245C8×C31—81no—
246C2×#550C2×C3111yescharacteristic
247#1957C2×C3111yescharacteristic, complemented
248#1803C2×C3111yescharacteristic, complemented
249#1802C2×C3111yescharacteristic, complemented
250C2×#331C2×C3111yescharacteristic, complemented
251C2×Q8⋊C8C2×C3111yescharacteristic, complemented
252C2×D4⋊C8C2×C3111yescharacteristic, complemented
253C22×C31—44no—
254C22×C31—42no—
255C22×C31—44no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—44no—
259C22×C31—81no—
260C22×C31—82no—
261C22×C31—81no—
262C22×C31—82no—
263C22×C31—81no—
264C22×C31—82no—
265C22×C31—81no—
266C22×C31—82no—
267C22×C31#37911yescharacteristic
268C22×C31#44711yescharacteristic
269C22×C31#44911yescharacteristic
270C22×C31#45011yescharacteristic
271C22×C31#45211yescharacteristic
272C22×C31#45311yescharacteristic
273C22×C31#45511yescharacteristic
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—24no—
277C4×C31—42no—
278C4×C31—42no—
279C4×C31—161no—
280C4×C31C2×#10211yescharacteristic
281C4×C31#41611yescharacteristic
282C4×C31#41711yescharacteristic
283C4×C31#41811yescharacteristic
284C4×C31#44711yescharacteristic
285C4×C31#44911yescharacteristic
286C4×C31#45011yescharacteristic
287C4×C31#45211yescharacteristic
288C22×#41—21no—
289#550—22no—
290#550—22no—
291#550—22no—
292#550C4×C3114yescomplemented
293C2×#127—21no—
294C2×#127C22×C3111yescharacteristic
295C2×#121—22no—
296C2×#121—21no—
297C2×C4×Q8C22×C3111yescharacteristic
298C2×C4×D4—21no—
299C2×C4×D4C22×C3111yescharacteristic
300C2×C4⋊C8C22×C3111yescharacteristic
301C2×C4⋊C8C22×C3111yescharacteristic
302C2×#103—21no—
303C2×#102—21no—
304C2×#102—22no—
305C2×C4×C8C22×C3111yescharacteristic
306#384C22×C3111yescharacteristic
307#340—21no—
308#340—21no—
309#331—22no—
310Q8⋊C8—22no—
311D4⋊C8—22no—
312C2×C31—84no—
313C2×C31—82no—
314C2×C31#162111yescharacteristic
315C2×C31#163111yescharacteristic
316C2×C31#164111yescharacteristic
317C2×C31#164211yescharacteristic
318C2×C31#178611yescharacteristic
319C2×C31#178811yescharacteristic
320C2×C31#179211yescharacteristic
321C2×#41—24no—
322C2×#41—24no—
323C2×#41—42no—
324C2×#41—42no—
325C2×#41—42no—
326C2×#41—42no—
327C22×Q8D4×C3111yescharacteristic
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4D4×C3111yescharacteristic
333C22×D4D4×C3112yes—
334C23×C4—22no—
335C23×C4—21no—
336C22×C8—21no—
337C22×C8—21no—
338C22×C8—21no—
339#127—24no—
340#127—24no—
341#127—21no—
342#127—21no—
343#127C2×C4×C3112yes—
344#127D4×C3112yes—
345#121—24no—
346#121—28no—
347#121—28no—
348#121—24no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C2×C4×C3112yes—
352C4×Q8D4×C3112yes—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4C2×C4×C3112yes—
358C4×D4D4×C3112yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4D4×C3111yescharacteristic
364C2×C4⋊C4D4×C3111yescharacteristic
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C42—21no—
368C2×C42C23×C3111yescharacteristic
369C4⋊C8—21no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373#103—42no—
374#102—44no—
375#102—42no—
376C4×C8—21no—
377C4×C8—21no—
378C31#844411yescharacteristic, direct product factor, Sylow
379C24—21no—
380C24—22no—
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8C22⋊C4×C3112yes—
389C2×Q8D8×C3112yes—
390C2×Q8SD16×C3112yes—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C22⋊C4×C3112yes—
418C2×D4C22⋊C4×C3114yes—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4C22⋊C4×C3112yes—
454C4⋊C4C22⋊C4×C3112yes—
455C22⋊C4—28no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C42—22no—
459C42—22no—
460C42C22×C4×C3111yescharacteristic
461C42C2×D4×C3111yescharacteristic
462C42C2×D4×C3111yescharacteristic
463C42C2×D4×C3111yescharacteristic
464C23—22no—
465C23—24no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23C31×#12011yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—22no—
482Q8C31×#10214yescomplemented
483D4—22no—
484D4—22no—
485D4—44no—
486D4—416no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—41no—
493D4—42no—
494D4—42no—
495D4—41no—
496D4—42no—
497D4—82no—
498D4—88no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—416no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4C2×C22⋊C4×C3111yescharacteristic
519C2×C4C2×C22⋊C4×C3111yescharacteristic
520C2×C4C2×C22⋊C4×C3111yescharacteristic
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12011yescharacteristic
523C2×C4C31×#12011yescharacteristic
524C2×C4C2×D8×C3111yescharacteristic
525C2×C4C2×SD16×C3111yescharacteristic
526C2×C4C31×#13511yescharacteristic
527C2×C4C31×#13511yescharacteristic
528C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
529C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
530C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
531C2×C4C31×#13711yescharacteristic
532C8—42no—
533C8—81no—
534C8—81no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—81no—
542C22—82no—
543C22—81no—
544C22—82no—
545C22—81no—
546C22—82no—
547C22—81no—
548C22—82no—
549C22C31×#37911yescharacteristic
550C22C31×#44711yescharacteristic
551C22C31×#44911yescharacteristic
552C22C31×#45011yescharacteristic
553C22C31×#45211yescharacteristic
554C22C31×#45311yescharacteristic
555C22C31×#45511yescharacteristic
556C4—24no—
557C4—21no—
558C4—21no—
559C4—42no—
560C4—42no—
561C4—161no—
562C4C2×C31×#10211yescharacteristic
563C4C31×#41611yescharacteristic
564C4C31×#41711yescharacteristic
565C4C31×#41811yescharacteristic
566C4C31×#44711yescharacteristic
567C4C31×#44911yescharacteristic
568C4C31×#45011yescharacteristic
569C4C31×#45211yescharacteristic
570C2—82no—
571C2—84no—
572C2C31×#162111yescharacteristic
573C2C31×#163111yescharacteristic
574C2C31×#164111yescharacteristic
575C2C31×#164211yescharacteristic
576C2C31×#178611yescharacteristic
577C2C31×#178811yescharacteristic
578C2C31×#179211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#550C2
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×#1957C2semidirect
3. C31×#1803C2semidirect
4. C31×#1802C2semidirect
5. C2×C31×#331C2semidirect
6. C2×C31×Q8⋊C8C2semidirect
7. C2×C31×D4⋊C8C2semidirect
12. C31×#550C4semidirect
174. #8444C31direct (N × H)
193. Q8×C31#102semidirect
247. #1957C2×C31semidirect
248. #1803C2×C31semidirect
249. #1802C2×C31semidirect
250. C2×#331C2×C31semidirect
251. C2×Q8⋊C8C2×C31semidirect
252. C2×D4⋊C8C2×C31semidirect
292. #550C4×C31semidirect
378. C31#8444direct (N × H)
482. Q8C31×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5502
C31×#19572
C31×#18032
C31×#18022
C2×C31×#3312
C2×C31×Q8⋊C82
C2×C31×D4⋊C82
#844431