Finite Group Databasebeta

← C31×#25622 #21482381

Subgroups — C31×#25622 #21482381

Order 7936.

1294 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic, complemented
2C31×#3151C211yesmaximal, characteristic, complemented
3C31×#3138C211yesmaximal, characteristic, complemented
4C31×#2541C211yesmaximal, characteristic, complemented
5C31×#2002C214yesmaximal, complemented
6C31×#1986C211yesmaximal, characteristic, complemented
7C31×#1980C211yesmaximal, characteristic, complemented
8C31×#1883C214yesmaximal, complemented
9C4×C31×#100C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)C2214yes—
11C31×#559—22no—
12C31×#556—21no—
13C31×#555—21no—
14C31×#553—22no—
15C31×#550—21no—
16C31×#547—21no—
17C31×#538—22no—
18C31×#535C2212yes—
19C31×#533C2211yescharacteristic
20C31×#532C2211yescharacteristic
21C2×C31×#127C2212yes—
22C2×C31×#127C2211yescharacteristic
23C31×#525—22no—
24C2×C31×#121—22no—
25C31×#520C2212yes—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#496—22no—
28C31×#493—21no—
29C31×#492—21no—
30C31×#490—22no—
31C31×#486—21no—
32C31×#486—21no—
33C2×C31×#104C2214yes—
34C31×#418C2214yes—
35C2×C31×#100C2212yes—
36C4×M16×C31C2211yescharacteristic
37C4×M16×C31C2211yescharacteristic
38C31×#392—22no—
39C31×#390C2212yes—
40C4×C4⋊C4×C31C2211yescharacteristic
41C31×#343C2212yes—
42C31×#339C2214yes—
43C2×#41×C31—22no—
44C2×#41×C31C2312yes—
45C22×D4×C31—24no—
46C22×D4×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31C2312yes—
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31D412yes—
51C31×AGL(1,ℤ/8)—28no—
52C31×AGL(1,ℤ/8)—44no—
53C2×SD16×C31—24no—
54C2×D8×C31—24no—
55C2×M16×C31C2312yes—
56C2×M16×C31C2312yes—
57C31×C4⋊Q8D411yescharacteristic, complemented
58C31×C4⋊Q8D411yescharacteristic, complemented
59C31×C4⋊Q8D411yescharacteristic, complemented
60C31×#127—22no—
61C31×#127—21no—
62C31×#127—21no—
63C31×#127—21no—
64C31×#127—21no—
65C31×#127—44no—
66C31×#127D411yescharacteristic, complemented
67C31×#126—24no—
68C31×#125D412yescomplemented
69C31×#124—24no—
70C31×#124—22no—
71C31×#124D412yescomplemented
72C31×#123—24no—
73C31×#123—24no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—21no—
77C31×#121—22no—
78C31×#121—24no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—48no—
84C31×#120—24no—
85C4×Q8×C31—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×D4×C31—24no—
89C4×D4×C31—22no—
90C4×D4×C31—24no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C31×#117—22no—
98C31×#117—22no—
99C31×#117C2312yes—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31C2311yescharacteristic
103C2×C22⋊C4×C31—24no—
104C2×C22⋊C4×C31D412yes—
105C2×C42×C31C2312yes—
106C2×C42×C31C2311yescharacteristic
107C31×#104—44no—
108C31×#102—24no—
109C31×#102—24no—
110C31×#100—24no—
111C31×#97—21no—
112C31×#97—21no—
113C4×C8×C31—21no—
114C4×C8×C31—21no—
115C31×#95—21no—
116C24×C31—22no—
117C24×C31C2×D412yes—
118#41×C31—24no—
119#41×C31—24no—
120#41×C31—84no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31C2×D412yes—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31—24no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—24no—
135C2×D4×C31—22no—
136C2×D4×C31—24no—
137C2×D4×C31—44no—
138C2×D4×C31—48no—
139C2×D4×C31—48no—
140C2×D4×C31—41no—
141C2×D4×C31—41no—
142C2×D4×C31—41no—
143C2×D4×C31—41no—
144C2×D4×C31—82no—
145C2×D4×C31—84no—
146C2×D4×C31C2×D412yes—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—44no—
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2411yescharacteristic
157SD16×C31—48no—
158D8×C31—48no—
159M16×C31—22no—
160M16×C31—22no—
161M16×C31—22no—
162M16×C31#4114yescomplemented
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31C2×D412yes—
176C4⋊C4×C31C2×D412yes—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—48no—
186C42×C31—24no—
187C42×C31—22no—
188C42×C31—24no—
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191C42×C31C2×D411yescharacteristic
192C42×C31C2×D411yescharacteristic
193#25622C3111yesmaximal, characteristic, direct product factor, Sylow
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31—44no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31C22×D411yescharacteristic
206Q8×C31—24no—
207Q8×C31—42no—
208Q8×C31—81no—
209D4×C31—24no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—44no—
214D4×C31—42no—
215D4×C31—44no—
216D4×C31—42no—
217D4×C31—81no—
218D4×C31—82no—
219D4×C31—82no—
220D4×C31—82no—
221D4×C31—84no—
222D4×C31—84no—
223D4×C31—816no—
224D4×C31—82no—
225C2×C4×C31—22no—
226C2×C4×C31—24no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—44no—
235C2×C4×C31—44no—
236C2×C4×C31—42no—
237C2×C4×C31—82no—
238C2×C4×C31—84no—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C2×#4112yes—
246C2×C4×C31H(5,𝔽2)12yes—
247C8×C31—24no—
248C8×C31—42no—
249#3506C2×C3111yescharacteristic, complemented
250#3151C2×C3111yescharacteristic, complemented
251#3138C2×C3111yescharacteristic, complemented
252#2541C2×C3111yescharacteristic, complemented
253#2002C2×C3114yescomplemented
254#1986C2×C3111yescharacteristic, complemented
255#1980C2×C3111yescharacteristic, complemented
256#1883C2×C3114yescomplemented
257C4×#100C2×C3111yescharacteristic, complemented
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—44no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—82no—
268C22×C31—84no—
269C22×C31—84no—
270C22×C31—82no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31C2×#12011yescharacteristic
274C22×C31#53211yescharacteristic
275C22×C31#53511yescharacteristic
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—44no—
280C4×C31—42no—
281C4×C31—42no—
282C4×C31—42no—
283C4×C31—42no—
284C4×C31#45312yes—
285C4×C31#54612yes—
286C4×C31#55012yes—
287C2×AGL(1,ℤ/8)C22×C3114yes—
288#559—22no—
289#556—21no—
290#555—21no—
291#553—22no—
292#550—21no—
293#547—21no—
294#538—22no—
295#535C22×C3112yes—
296#533C22×C3111yescharacteristic
297#532C22×C3111yescharacteristic
298C2×#127C22×C3111yescharacteristic
299C2×#127C22×C3112yes—
300#525—22no—
301C2×#121—22no—
302#520C22×C3112yes—
303C2×C4×D4C22×C3111yescharacteristic
304#496—22no—
305#493—21no—
306#492—21no—
307#490—22no—
308#486—21no—
309#486—21no—
310C2×#104C22×C3114yes—
311#418C22×C3114yes—
312C2×#100C22×C3112yes—
313C4×M16C22×C3111yescharacteristic
314C4×M16C22×C3111yescharacteristic
315#392—22no—
316#390C22×C3112yes—
317C4×C4⋊C4C22×C3111yescharacteristic
318#343C22×C3112yes—
319#339C22×C3114yes—
320C2×C31—21no—
321C2×C31—21no—
322C2×C31—82no—
323C2×C31—82no—
324C2×C31—162no—
325C2×C31#243111yescharacteristic
326C2×C31C2×#45311yescharacteristic
327C2×C31#302111yescharacteristic
328C2×#41—22no—
329C2×#41C23×C3112yes—
330C22×D4—24no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4C23×C3112yes—
334C22×D4C23×C3111yescharacteristic
335C23×C4D4×C3112yes—
336AGL(1,ℤ/8)—28no—
337AGL(1,ℤ/8)—44no—
338C2×SD16—24no—
339C2×D8—24no—
340C2×M16C23×C3112yes—
341C2×M16C23×C3112yes—
342C4⋊Q8D4×C3111yescharacteristic, complemented
343C4⋊Q8D4×C3111yescharacteristic, complemented
344C4⋊Q8D4×C3111yescharacteristic, complemented
345#127—22no—
346#127—21no—
347#127—21no—
348#127—21no—
349#127—21no—
350#127—44no—
351#127D4×C3111yescharacteristic, complemented
352#126—24no—
353#125D4×C3112yescomplemented
354#124—24no—
355#124—22no—
356#124D4×C3112yescomplemented
357#123—24no—
358#123—24no—
359C22⋊Q8—22no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362#121—22no—
363#121—24no—
364#121—21no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—48no—
369#120—24no—
370C4×Q8—22no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—24no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382#117—22no—
383#117—22no—
384#117C23×C3112yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4C23×C3111yescharacteristic
388C2×C22⋊C4—24no—
389C2×C22⋊C4D4×C3112yes—
390C2×C42C23×C3112yes—
391C2×C42C23×C3111yescharacteristic
392#104—44no—
393#102—24no—
394#102—24no—
395#100—24no—
396#97—21no—
397#97—21no—
398C4×C8—21no—
399C4×C8—21no—
400#95—21no—
401C31#2562211yescharacteristic, direct product factor, Sylow
402C24—22no—
403C24C2×D4×C3112yes—
404#41—24no—
405#41—24no—
406#41—84no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×D4×C3112yes—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—82no—
431C2×D4—84no—
432C2×D4C2×D4×C3112yes—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3112yes—
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C24×C3111yescharacteristic
443SD16—48no—
444D8—48no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16#41×C3114yescomplemented
449C2×C8—22no—
450C2×C8—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4C2×D4×C3112yes—
462C4⋊C4C2×D4×C3112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—48no—
472C42—24no—
473C42—22no—
474C42—24no—
475C42C2×D4×C3111yescharacteristic
476C42C2×D4×C3111yescharacteristic
477C42C2×D4×C3111yescharacteristic
478C42C2×D4×C3111yescharacteristic
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C22×D4×C3111yescharacteristic
491Q8—24no—
492Q8—42no—
493Q8—81no—
494D4—24no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—42no—
501D4—44no—
502D4—81no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—84no—
508D4—816no—
509D4—82no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—84no—
524C2×C4C31×#12011yescharacteristic
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#12011yescharacteristic
528C2×C4C22×D4×C3111yescharacteristic
529C2×C4C22×D4×C3111yescharacteristic
530C2×C4C2×#41×C3112yes—
531C2×C4C31×H(5,𝔽2)12yes—
532C8—24no—
533C8—42no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C2×C31×#12011yescharacteristic
550C22C31×#53211yescharacteristic
551C22C31×#53511yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4C31×#45312yes—
561C4C31×#54612yes—
562C4C31×#55012yes—
563C2—21no—
564C2—21no—
565C2—82no—
566C2—82no—
567C2—162no—
568C2C31×#243111yescharacteristic
569C2C2×C31×#45311yescharacteristic
570C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3506C2semidirect
2. C31×#3151C2semidirect
3. C31×#3138C2semidirect
4. C31×#2541C2semidirect
5. C31×#2002C2semidirect
6. C31×#1986C2semidirect
7. C31×#1980C2semidirect
8. C31×#1883C2semidirect
9. C4×C31×#100C2semidirect
57. C31×C4⋊Q8D4semidirect
58. C31×C4⋊Q8D4semidirect
59. C31×C4⋊Q8D4semidirect
66. C31×#127D4semidirect
68. C31×#125D4semidirect
71. C31×#124D4semidirect
162. M16×C31#41semidirect
193. #25622C31direct (N × H)
249. #3506C2×C31semidirect
250. #3151C2×C31semidirect
251. #3138C2×C31semidirect
252. #2541C2×C31semidirect
253. #2002C2×C31semidirect
254. #1986C2×C31semidirect
255. #1980C2×C31semidirect
256. #1883C2×C31semidirect
257. C4×#100C2×C31semidirect
342. C4⋊Q8D4×C31semidirect
343. C4⋊Q8D4×C31semidirect
344. C4⋊Q8D4×C31semidirect
351. #127D4×C31semidirect
353. #125D4×C31semidirect
356. #124D4×C31semidirect
401. C31#25622direct (N × H)
448. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#31512
C31×#31382
C31×#25412
C31×#20022
C31×#19862
C31×#19802
C31×#18832
C4×C31×#1002
#2562231