Finite Group Databasebeta

← C31×#9922 #21466681

Subgroups — C31×#9922 #21466681

Order 7936.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#450C211yesmaximal, characteristic, complemented
2C31×#2626C211yesmaximal, characteristic, complemented
3C2×C31×#392C211yesmaximal, characteristic
4C31×#1786C211yesmaximal, characteristic, complemented
5C2×C31×#328C211yesmaximal, characteristic, complemented
6C2×C31×#323C211yesmaximal, characteristic, complemented
7C31×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C31—21no—
9C23×D4×C31C2211yescharacteristic
10C22×SD16×C31—21no—
11C2×C31×C22⋊Q8C2211yescharacteristic
12C22×C22⋊C4×C31—21no—
13C22×C22⋊C4×C31C2211yescharacteristic
14C31×#450—22no—
15C31×#450—22no—
16C2×C31×#102—21no—
17C2×C31×#102—22no—
18C2×C31×C22⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8C2211yescharacteristic
20C31×#399—21no—
21C31×#396—22no—
22C31×#393—21no—
23C31×#392—22no—
24C31×#392—22no—
25C31×#385—21no—
26C31×#380C2211yescharacteristic
27C31×#379—21no—
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C31×#323—22no—
31C25×C31—21no—
32C25×C31D411yescharacteristic
33C22×Q8×C31D411yescharacteristic, complemented
34C22×D4×C31—24no—
35C22×D4×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—21no—
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31—44no—
41C22×D4×C31—44no—
42C22×D4×C31—44no—
43C22×D4×C31—42no—
44C22×D4×C31—42no—
45C22×D4×C31D412yes—
46C23×C4×C31—21no—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C2×SD16×C31—24no—
50C2×SD16×C31—44no—
51C22×C8×C31—21no—
52C22×C8×C31—21no—
53C31×C22⋊Q8—21no—
54C31×C22⋊Q8—21no—
55C31×C22⋊Q8—21no—
56C31×C22⋊Q8—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31D411yescharacteristic, complemented
60C2×C4⋊C4×C31D412yescomplemented
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—44no—
73C31×#102—24no—
74C31×#102—44no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8D412yescomplemented
79C31×#95—22no—
80C31×#95—21no—
81C31×#95—21no—
82C31×#95D412yes—
83C24×C31—24no—
84C24×C31—21no—
85C24×C31—22no—
86C24×C31—21no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—42no—
92C24×C31—42no—
93C24×C31—42no—
94C24×C31—42no—
95C24×C31—42no—
96C24×C31—41no—
97C24×C31—41no—
98C24×C31—41no—
99C24×C31—41no—
100C24×C31—41no—
101C24×C31—41no—
102C24×C31SD1612yescomplemented
103C24×C31C2×D411yescharacteristic
104C2×Q8×C31—21no—
105C2×Q8×C31—21no—
106C2×Q8×C31—42no—
107C2×D4×C31—24no—
108C2×D4×C31—24no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31—48no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—44no—
118C2×D4×C31—48no—
119C2×D4×C31—48no—
120C2×D4×C31—416no—
121C2×D4×C31—416no—
122C2×D4×C31—41no—
123C2×D4×C31—41no—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—41no—
135C22×C4×C31—41no—
136C22×C4×C31—42no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31#4111yescharacteristic
143SD16×C31—48no—
144C2×C8×C31—22no—
145C2×C8×C31—42no—
146C2×C8×C31—42no—
147C2×C8×C31—41no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—42no—
151C4⋊C4×C31—41no—
152C4⋊C4×C31—41no—
153C4⋊C4×C31—82no—
154C22⋊C4×C31—48no—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—41no—
157C22⋊C4×C31—41no—
158C22⋊C4×C31—41no—
159C22⋊C4×C31—41no—
160C22⋊C4×C31—41no—
161C22⋊C4×C31—41no—
162C22⋊C4×C31—41no—
163C22⋊C4×C31—82no—
164C22⋊C4×C31—82no—
165#9922C3111yesmaximal, characteristic, direct product factor, Sylow
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—24no—
169C23×C31—22no—
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—44no—
173C23×C31—42no—
174C23×C31—44no—
175C23×C31—44no—
176C23×C31—44no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—44no—
182C23×C31—44no—
183C23×C31—44no—
184C23×C31—41no—
185C23×C31—42no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—44no—
193C23×C31—41no—
194C23×C31—82no—
195C23×C31—82no—
196C23×C31—81no—
197C23×C31—81no—
198C23×C31—82no—
199C23×C31—82no—
200C23×C31—82no—
201C23×C31—82no—
202C23×C31—82no—
203C23×C31—81no—
204C23×C31—81no—
205C23×C31—82no—
206C23×C31—81no—
207C23×C31—81no—
208C23×C31#12011yescharacteristic
209C23×C31#12011yescharacteristic
210C23×C31#12111yescharacteristic
211C23×C31C2×SD1611yescharacteristic
212C23×C31#13511yescharacteristic
213C23×C31AGL(1,ℤ/8)11yescharacteristic
214C23×C31#13711yescharacteristic
215Q8×C31—42no—
216Q8×C31—81no—
217D4×C31—24no—
218D4×C31—24no—
219D4×C31—42no—
220D4×C31—44no—
221D4×C31—42no—
222D4×C31—44no—
223D4×C31—432no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—24no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—44no—
232C2×C4×C31—44no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—81no—
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#12111yescharacteristic
248C2×C4×C31#12111yescharacteristic
249C2×C4×C31#12711yescharacteristic
250C8×C31—42no—
251C8×C31—81no—
252C2×#450C2×C3111yescharacteristic, complemented
253#2626C2×C3111yescharacteristic, complemented
254C2×#392C2×C3111yescharacteristic
255#1786C2×C3111yescharacteristic, complemented
256C2×#328C2×C3111yescharacteristic, complemented
257C2×#323C2×C3111yescharacteristic, complemented
258#1294C2×C3111yescharacteristic, complemented
259C22×C31—22no—
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—21no—
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—44no—
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—44no—
271C22×C31—42no—
272C22×C31—44no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—84no—
278C22×C31—82no—
279C22×C31—84no—
280C22×C31—84no—
281C22×C31—82no—
282C22×C31—84no—
283C22×C31—82no—
284C22×C31—81no—
285C22×C31—82no—
286C22×C31—81no—
287C22×C31#39211yescharacteristic
288C22×C31#44811yescharacteristic
289C22×C31#44911yescharacteristic
290C22×C31#45011yescharacteristic
291C22×C31#45211yescharacteristic
292C22×C31#45312yes—
293C22×C31B(4,𝔽2)12yes—
294C22×C31#46511yescharacteristic
295C22×C31#46811yescharacteristic
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—44no—
301C4×C31—82no—
302C4×C31—81no—
303C4×C31—81no—
304C4×C31—81no—
305C23×D4—21no—
306C23×D4C22×C3111yescharacteristic
307C22×SD16—21no—
308C2×C22⋊Q8C22×C3111yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4C22×C3111yescharacteristic
311#450—22no—
312#450—22no—
313C2×#102—21no—
314C2×#102—22no—
315C2×C22⋊C8C22×C3111yescharacteristic
316C2×C22⋊C8C22×C3111yescharacteristic
317#399—21no—
318#396—22no—
319#393—21no—
320#392—22no—
321#392—22no—
322#385—21no—
323#380C22×C3111yescharacteristic
324#379—21no—
325C2×#95C22×C3111yescharacteristic
326#328—22no—
327#323—22no—
328C2×C31—21no—
329C2×C31—21no—
330C2×C31—21no—
331C2×C31—21no—
332C2×C31—82no—
333C2×C31—84no—
334C2×C31#159612yes—
335C2×C31#160012yes—
336C2×C31#200011yescharacteristic
337C2×C31#200211yescharacteristic
338C2×C31#202111yescharacteristic
339C25—21no—
340C25D4×C3111yescharacteristic
341C22×Q8D4×C3111yescharacteristic, complemented
342C22×D4—24no—
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—44no—
349C22×D4—44no—
350C22×D4—44no—
351C22×D4—42no—
352C22×D4—42no—
353C22×D4D4×C3112yes—
354C23×C4—21no—
355C23×C4D4×C3111yescharacteristic
356C23×C4C23×C3111yescharacteristic
357C2×SD16—24no—
358C2×SD16—44no—
359C22×C8—21no—
360C22×C8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4D4×C3111yescharacteristic, complemented
368C2×C4⋊C4D4×C3112yescomplemented
369C2×C22⋊C4—24no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—44no—
381#102—24no—
382#102—44no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8D4×C3112yescomplemented
387#95—22no—
388#95—21no—
389#95—21no—
390#95D4×C3112yes—
391C31#992211yescharacteristic, direct product factor, Sylow
392C24—24no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24SD16×C3112yescomplemented
412C24C2×D4×C3111yescharacteristic
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—41no—
432C2×D4—41no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4#41×C3111yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—41no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—82no—
463C22⋊C4—48no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—82no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23C31×#12011yescharacteristic
517C23C31×#12011yescharacteristic
518C23C31×#12111yescharacteristic
519C23C2×SD16×C3111yescharacteristic
520C23C31×#13511yescharacteristic
521C23C31×AGL(1,ℤ/8)11yescharacteristic
522C23C31×#13711yescharacteristic
523Q8—42no—
524Q8—81no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—44no—
531D4—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4C31×#12011yescharacteristic
555C2×C4C31×#12111yescharacteristic
556C2×C4C31×#12111yescharacteristic
557C2×C4C31×#12711yescharacteristic
558C8—42no—
559C8—81no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—84no—
581C22—84no—
582C22—82no—
583C22—84no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—81no—
588C22C31×#39211yescharacteristic
589C22C31×#44811yescharacteristic
590C22C31×#44911yescharacteristic
591C22C31×#45011yescharacteristic
592C22C31×#45211yescharacteristic
593C22C31×#45312yes—
594C22C31×B(4,𝔽2)12yes—
595C22C31×#46511yescharacteristic
596C22C31×#46811yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—82no—
603C4—81no—
604C4—81no—
605C4—81no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2C31×#159612yes—
613C2C31×#160012yes—
614C2C31×#200011yescharacteristic
615C2C31×#200211yescharacteristic
616C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#392C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#450C2semidirect
2. C31×#2626C2semidirect
4. C31×#1786C2semidirect
5. C2×C31×#328C2semidirect
6. C2×C31×#323C2semidirect
7. C31×#1294C2semidirect
33. C22×Q8×C31D4semidirect
59. C2×C4⋊C4×C31D4semidirect
60. C2×C4⋊C4×C31D4semidirect
78. C31×C22⋊C8D4semidirect
102. C24×C31SD16semidirect
165. #9922C31direct (N × H)
252. C2×#450C2×C31semidirect
253. #2626C2×C31semidirect
255. #1786C2×C31semidirect
256. C2×#328C2×C31semidirect
257. C2×#323C2×C31semidirect
258. #1294C2×C31semidirect
341. C22×Q8D4×C31semidirect
367. C2×C4⋊C4D4×C31semidirect
368. C2×C4⋊C4D4×C31semidirect
386. C22⋊C8D4×C31semidirect
391. C31#9922direct (N × H)
411. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4502
C31×#26262
C2×C31×#3922
C31×#17862
C2×C31×#3282
C2×C31×#3232
C31×#12942
#992231