Finite Group Databasebeta

← C31×#24179 #21480938

Subgroups — C31×#24179 #21480938

Order 7936.

1038 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic
2C31×#3078C211yesmaximal, characteristic, complemented
3C31×#3037C211yesmaximal, characteristic
4C4×C31×C4⋊Q8C211yesmaximal, characteristic, complemented
5C31×#2038C212yesmaximal, complemented
6C31×#2005C212yesmaximal, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1962C211yesmaximal, characteristic
9C31×#1882C212yesmaximal, complemented
10C31×#1859C212yesmaximal, complemented
11C31×#1801C211yesmaximal, characteristic, complemented
12C2×C31×#137C2212yes—
13C31×#559—21no—
14C31×#557—21no—
15C31×#556—21no—
16C31×#556—22no—
17C31×#554—21no—
18C31×#552—22no—
19C31×#539—22no—
20C31×#536C2212yes—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#529—22no—
25C31×#520C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#479—22no—
29C31×#477—21no—
30C31×#475—21no—
31C31×#464—22no—
32C31×#462—21no—
33C31×#460—21no—
34C31×#428C2212yes—
35C31×#423C2211yescharacteristic
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C2×C31×#104C2212yes—
39C31×#419C2212yes—
40C31×#417C2212yes—
41C2×C31×#101C2212yes—
42C31×#391C2211yescharacteristic
43C31×#391C2211yescharacteristic
44C31×#389—22no—
45C31×#384C2211yescharacteristic
46C4×C4⋊C4×C31—22no—
47C43×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31C2312yes—
52C22×Q8×C31C2311yescharacteristic
53C22×Q8×C31C2311yescharacteristic
54C31×#137—24no—
55C31×#137—42no—
56C2×Q16×C31—22no—
57C2×SD16×C31—22no—
58C2×M16×C31C2312yes—
59C2×M16×C31C2312yes—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8D411yescharacteristic
66C31×#127D411yescharacteristic
67C31×#126—24no—
68C31×#126—24no—
69C31×#125—22no—
70C31×#125—24no—
71C31×#125D412yes—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124D412yes—
75C31×#123—24no—
76C31×#123—24no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×#121—21no—
83C31×#121—21no—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—24no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C31×#117—24no—
102C31×#117—22no—
103C31×#117C2312yes—
104C2×C4⋊C4×C31—22no—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—22no—
108C2×C4⋊C4×C31D412yes—
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C4⋊C4×C31C2311yescharacteristic
111C2×C42×C31—21no—
112C2×C42×C31—21no—
113C2×C42×C31—21no—
114C2×C42×C31D412yes—
115C2×C42×C31C2311yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×C4⋊C8—21no—
124C31×#104—24no—
125C31×#104—42no—
126C31×#103—22no—
127C31×#103—22no—
128C31×#103—22no—
129C31×#102—22no—
130C31×#101—42no—
131C31×#95—22no—
132C31×#95—22no—
133#41×C31—22no—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31—21no—
147C2×Q8×C31—21no—
148C2×Q8×C31—21no—
149C2×Q8×C31—21no—
150C2×Q8×C31C2×D412yes—
151C2×Q8×C31#4112yes—
152C2×D4×C31—22no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31C2×D412yes—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—24no—
160C22×C4×C31—22no—
161C22×C4×C31—21no—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2411yescharacteristic
167Q16×C31—44no—
168SD16×C31—44no—
169M16×C31—22no—
170M16×C31—22no—
171M16×C31—22no—
172M16×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—28no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—44no—
192C4⋊C4×C31—41no—
193C4⋊C4×C31—41no—
194C4⋊C4×C31C2×D412yes—
195C4⋊C4×C31C2×D412yes—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C42×C31—24no—
203C42×C31—24no—
204C42×C31—22no—
205C42×C31—22no—
206C42×C31—22no—
207C42×C31—21no—
208C42×C31—21no—
209C42×C31—21no—
210C42×C31—42no—
211C42×C31—41no—
212C42×C31—41no—
213C42×C31C2×D411yescharacteristic
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31#4112yes—
218#24179C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—22no—
220C23×C31C22×D411yescharacteristic
221Q8×C31—22no—
222Q8×C31—22no—
223Q8×C31—22no—
224Q8×C31—22no—
225Q8×C31—41no—
226Q8×C31—41no—
227Q8×C31—41no—
228Q8×C31—41no—
229Q8×C31—41no—
230Q8×C31—41no—
231Q8×C31—42no—
232Q8×C31—81no—
233D4×C31—22no—
234D4×C31—22no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—81no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—41no—
262C2×C4×C31—84no—
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C2×#4111yescharacteristic
270C2×C4×C31C2×#4111yescharacteristic
271C2×C4×C31H(5,𝔽2)11yescharacteristic
272C2×C4×C312−1+411yescharacteristic
273C8×C31—42no—
274C8×C31—42no—
275#3507C2×C3111yescharacteristic
276#3078C2×C3111yescharacteristic, complemented
277#3037C2×C3111yescharacteristic
278C4×C4⋊Q8C2×C3111yescharacteristic, complemented
279#2038C2×C3112yescomplemented
280#2005C2×C3112yescomplemented
281#1994C2×C3111yescharacteristic, complemented
282#1962C2×C3111yescharacteristic
283#1882C2×C3112yescomplemented
284#1859C2×C3112yescomplemented
285#1801C2×C3111yescharacteristic, complemented
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31C2×#12011yescharacteristic
292C22×C31#52511yescharacteristic
293C22×C31#53711yescharacteristic
294C4×C31—24no—
295C4×C31—21no—
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—42no—
299C4×C31—42no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—42no—
303C4×C31—42no—
304C4×C31—42no—
305C4×C31—81no—
306C4×C31#45612yes—
307C4×C31#54611yescharacteristic
308C4×C31#54711yescharacteristic
309C4×C31#54811yescharacteristic
310C4×C31#55011yescharacteristic
311C2×#137C22×C3112yes—
312#559—21no—
313#557—21no—
314#556—21no—
315#556—22no—
316#554—21no—
317#552—22no—
318#539—22no—
319#536C22×C3112yes—
320#533C22×C3111yescharacteristic
321#532C22×C3111yescharacteristic
322C2×C4⋊Q8C22×C3111yescharacteristic
323#529—22no—
324#520C22×C3112yes—
325C2×C4×Q8C22×C3111yescharacteristic
326C2×C4×Q8C22×C3111yescharacteristic
327#479—22no—
328#477—21no—
329#475—21no—
330#464—22no—
331#462—21no—
332#460—21no—
333#428C22×C3112yes—
334#423C22×C3111yescharacteristic
335#423C22×C3111yescharacteristic
336C2×#104C22×C3112yes—
337C2×#104C22×C3112yes—
338#419C22×C3112yes—
339#417C22×C3112yes—
340C2×#101C22×C3112yes—
341#391C22×C3111yescharacteristic
342#391C22×C3111yescharacteristic
343#389—22no—
344#384C22×C3111yescharacteristic
345C4×C4⋊C4—22no—
346C43C22×C3111yescharacteristic
347#343C22×C3112yes—
348#339C22×C3112yes—
349#339C22×C3112yes—
350C2×C31—21no—
351C2×C31—21no—
352C2×C31—82no—
353C2×C31#240711yescharacteristic
354C2×C31C2×#45611yescharacteristic
355C2×C31#301811yescharacteristic
356C2×#41C23×C3112yes—
357C22×Q8C23×C3111yescharacteristic
358C22×Q8C23×C3111yescharacteristic
359#137—24no—
360#137—42no—
361C2×Q16—22no—
362C2×SD16—22no—
363C2×M16C23×C3112yes—
364C2×M16C23×C3112yes—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368C4⋊Q8D4×C3111yescharacteristic
369C4⋊Q8D4×C3111yescharacteristic
370C4⋊Q8D4×C3111yescharacteristic
371#127D4×C3111yescharacteristic
372#126—24no—
373#126—24no—
374#125—22no—
375#125—24no—
376#125D4×C3112yes—
377#124—22no—
378#124—22no—
379#124D4×C3112yes—
380#123—24no—
381#123—24no—
382C22⋊Q8—24no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387#121—21no—
388#121—21no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—24no—
404C4×D4—21no—
405C4×D4—21no—
406#117—24no—
407#117—22no—
408#117C23×C3112yes—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4D4×C3112yes—
414C2×C4⋊C4C23×C3111yescharacteristic
415C2×C4⋊C4C23×C3111yescharacteristic
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42D4×C3112yes—
420C2×C42C23×C3111yescharacteristic
421C2×C42C23×C3111yescharacteristic
422C2×C42C23×C3111yescharacteristic
423#107—22no—
424#106—22no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428C4⋊C8—21no—
429#104—24no—
430#104—42no—
431#103—22no—
432#103—22no—
433#103—22no—
434#102—22no—
435#101—42no—
436#95—22no—
437#95—22no—
438C31#2417911yescharacteristic, direct product factor, Sylow
439#41—22no—
440#41—22no—
441#41—22no—
442#41—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C2×D4×C3112yes—
457C2×Q8#41×C3112yes—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×D4×C3112yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4C2×D4×C3112yes—
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3112yes—
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C24×C3111yescharacteristic
473Q16—44no—
474SD16—44no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479C2×C8—22no—
480C2×C8—22no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—44no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C2×D4×C3112yes—
501C4⋊C4C2×D4×C3112yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—41no—
518C42—41no—
519C42C2×D4×C3111yescharacteristic
520C42C2×D4×C3111yescharacteristic
521C42C2×D4×C3111yescharacteristic
522C42C2×D4×C3111yescharacteristic
523C42#41×C3112yes—
524C23—22no—
525C23C22×D4×C3111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8—22no—
530Q8—42no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—81no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—81no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4C31×#12011yescharacteristic
569C2×C4C31×#12011yescharacteristic
570C2×C4C31×#12011yescharacteristic
571C2×C4C31×#12011yescharacteristic
572C2×C4C22×D4×C3111yescharacteristic
573C2×C4C22×D4×C3111yescharacteristic
574C2×C4C2×#41×C3111yescharacteristic
575C2×C4C2×#41×C3111yescharacteristic
576C2×C4C31×H(5,𝔽2)11yescharacteristic
577C2×C4C31×2−1+411yescharacteristic
578C8—42no—
579C8—42no—
580C22—21no—
581C22—21no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C2×C31×#12011yescharacteristic
586C22C31×#52511yescharacteristic
587C22C31×#53711yescharacteristic
588C4—24no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4C31×#45612yes—
601C4C31×#54611yescharacteristic
602C4C31×#54711yescharacteristic
603C4C31×#54811yescharacteristic
604C4C31×#55011yescharacteristic
605C2—21no—
606C2—21no—
607C2—82no—
608C2C31×#240711yescharacteristic
609C2C2×C31×#45611yescharacteristic
610C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3078C2semidirect
4. C4×C31×C4⋊Q8C2semidirect
5. C31×#2038C2semidirect
6. C31×#2005C2semidirect
7. C31×#1994C2semidirect
9. C31×#1882C2semidirect
10. C31×#1859C2semidirect
11. C31×#1801C2semidirect
218. #24179C31direct (N × H)
276. #3078C2×C31semidirect
278. C4×C4⋊Q8C2×C31semidirect
279. #2038C2×C31semidirect
280. #2005C2×C31semidirect
281. #1994C2×C31semidirect
283. #1882C2×C31semidirect
284. #1859C2×C31semidirect
285. #1801C2×C31semidirect
438. C31#24179direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#30782
C31×#30372
C4×C31×C4⋊Q82
C31×#20382
C31×#20052
C31×#19942
C31×#19622
C31×#18822
C31×#18592
C31×#18012
#2417931