Finite Group Databasebeta

← C31×#25621 #21482380

Subgroups — C31×#25621 #21482380

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3152C211yesmaximal, characteristic, complemented
3C31×#3139C211yesmaximal, characteristic, complemented
4C31×#2547C211yesmaximal, characteristic, complemented
5C31×#2004C214yesmaximal, complemented
6C31×#1986C211yesmaximal, characteristic, complemented
7C31×#1980C211yesmaximal, characteristic, complemented
8C31×#1882C214yesmaximal, complemented
9C4×C31×#100C211yesmaximal, characteristic, complemented
10C2×C31×#137C2214yes—
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×C4⋊Q8C2211yescharacteristic
22C2×C31×#124C2212yes—
23C31×#525—22no—
24C31×#520C2212yes—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#497—22no—
27C31×#494—21no—
28C31×#492—21no—
29C31×#491—22no—
30C31×#487—21no—
31C31×#487—21no—
32C2×C31×#104C2214yes—
33C31×#419C2214yes—
34C2×C31×#100C2212yes—
35C4×M16×C31C2211yescharacteristic
36C4×M16×C31C2211yescharacteristic
37C31×#397—22no—
38C31×#391C2212yes—
39C31×#380—22no—
40C4×C4⋊C4×C31C2211yescharacteristic
41C31×#343C2212yes—
42C31×#339C2214yes—
43C2×#41×C31—22no—
44C2×#41×C31C2312yes—
45C22×Q8×C31C2312yes—
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31D412yes—
48C31×#137—28no—
49C31×#137—44no—
50C2×Q16×C31—24no—
51C2×SD16×C31—24no—
52C2×M16×C31C2312yes—
53C2×M16×C31C2312yes—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8D411yescharacteristic
59C31×C4⋊Q8D411yescharacteristic
60C31×C4⋊Q8D411yescharacteristic
61C31×#127—22no—
62C31×#127D411yescharacteristic
63C31×#126—24no—
64C31×#125D412yes—
65C31×#124—24no—
66C31×#124—22no—
67C31×#124—44no—
68C31×#124D412yes—
69C31×#123—24no—
70C31×#123—24no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—21no—
74C31×#121—22no—
75C31×#121—24no—
76C31×#121—21no—
77C31×#121—21no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#120—24no—
81C4×Q8×C31—22no—
82C4×Q8×C31—21no—
83C4×Q8×C31—21no—
84C4×D4×C31—24no—
85C4×D4×C31—22no—
86C4×D4×C31—24no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117C2312yes—
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C22⋊C4×C31—24no—
102C2×C22⋊C4×C31D412yes—
103C2×C42×C31C2312yes—
104C2×C42×C31C2311yescharacteristic
105C31×#104—44no—
106C31×#103—24no—
107C31×#103—24no—
108C31×#100—24no—
109C31×#97—21no—
110C31×#97—21no—
111C4×C8×C31—21no—
112C4×C8×C31—21no—
113C31×#95—21no—
114C31×#95—24no—
115C24×C31C2×D412yes—
116#41×C31—24no—
117#41×C31—24no—
118#41×C31—84no—
119C2×Q8×C31—21no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—24no—
124C2×Q8×C31—24no—
125C2×Q8×C31C2×D412yes—
126C2×D4×C31—22no—
127C2×D4×C31—24no—
128C2×D4×C31—24no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31C2×D412yes—
136C22×C4×C31—24no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—44no—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2411yescharacteristic
147Q16×C31—48no—
148SD16×C31—48no—
149M16×C31—22no—
150M16×C31—22no—
151M16×C31—22no—
152M16×C31#4114yescomplemented
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—41no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31—41no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—48no—
180C42×C31—24no—
181C42×C31—22no—
182C42×C31—24no—
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187#25621C3111yesmaximal, characteristic, direct product factor, Sylow
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31C22×D411yescharacteristic
195Q8×C31—24no—
196Q8×C31—42no—
197Q8×C31—44no—
198Q8×C31—44no—
199Q8×C31—81no—
200D4×C31—24no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—42no—
206D4×C31—81no—
207D4×C31—84no—
208D4×C31—84no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—41no—
215C2×C4×C31—44no—
216C2×C4×C31—44no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—82no—
225C2×C4×C31—84no—
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C2×#4112yes—
233C2×C4×C31H(5,𝔽2)12yes—
234C8×C31—24no—
235C8×C31—42no—
236#3506C2×C3111yescharacteristic
237#3152C2×C3111yescharacteristic, complemented
238#3139C2×C3111yescharacteristic, complemented
239#2547C2×C3111yescharacteristic, complemented
240#2004C2×C3114yescomplemented
241#1986C2×C3111yescharacteristic, complemented
242#1980C2×C3111yescharacteristic, complemented
243#1882C2×C3114yescomplemented
244C4×#100C2×C3111yescharacteristic, complemented
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—44no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—82no—
255C22×C31C2×#12011yescharacteristic
256C22×C31#53211yescharacteristic
257C22×C31#53511yescharacteristic
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—42no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—82no—
266C4×C31—82no—
267C4×C31#45512yes—
268C4×C31#54612yes—
269C4×C31#55012yes—
270C2×#137C22×C3114yes—
271#559—22no—
272#556—21no—
273#555—21no—
274#553—22no—
275#550—21no—
276#547—21no—
277#538—22no—
278#535C22×C3112yes—
279#533C22×C3111yescharacteristic
280#532C22×C3111yescharacteristic
281C2×C4⋊Q8C22×C3111yescharacteristic
282C2×#124C22×C3112yes—
283#525—22no—
284#520C22×C3112yes—
285C2×C4×D4C22×C3111yescharacteristic
286#497—22no—
287#494—21no—
288#492—21no—
289#491—22no—
290#487—21no—
291#487—21no—
292C2×#104C22×C3114yes—
293#419C22×C3114yes—
294C2×#100C22×C3112yes—
295C4×M16C22×C3111yescharacteristic
296C4×M16C22×C3111yescharacteristic
297#397—22no—
298#391C22×C3112yes—
299#380—22no—
300C4×C4⋊C4C22×C3111yescharacteristic
301#343C22×C3112yes—
302#339C22×C3114yes—
303C2×C31—21no—
304C2×C31—21no—
305C2×C31—82no—
306C2×C31—82no—
307C2×C31#243111yescharacteristic
308C2×C31C2×#45511yescharacteristic
309C2×C31#302211yescharacteristic
310C2×#41—22no—
311C2×#41C23×C3112yes—
312C22×Q8C23×C3112yes—
313C22×D4C23×C3111yescharacteristic
314C23×C4D4×C3112yes—
315#137—28no—
316#137—44no—
317C2×Q16—24no—
318C2×SD16—24no—
319C2×M16C23×C3112yes—
320C2×M16C23×C3112yes—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8D4×C3111yescharacteristic
326C4⋊Q8D4×C3111yescharacteristic
327C4⋊Q8D4×C3111yescharacteristic
328#127—22no—
329#127D4×C3111yescharacteristic
330#126—24no—
331#125D4×C3112yes—
332#124—24no—
333#124—22no—
334#124—44no—
335#124D4×C3112yes—
336#123—24no—
337#123—24no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341#121—22no—
342#121—24no—
343#121—21no—
344#121—21no—
345#121—22no—
346#121—22no—
347#120—24no—
348C4×Q8—22no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×D4—24no—
352C4×D4—22no—
353C4×D4—24no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360#117—22no—
361#117—22no—
362#117C23×C3112yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4C23×C3111yescharacteristic
368C2×C22⋊C4—24no—
369C2×C22⋊C4D4×C3112yes—
370C2×C42C23×C3112yes—
371C2×C42C23×C3111yescharacteristic
372#104—44no—
373#103—24no—
374#103—24no—
375#100—24no—
376#97—21no—
377#97—21no—
378C4×C8—21no—
379C4×C8—21no—
380#95—21no—
381#95—24no—
382C31#2562111yescharacteristic, direct product factor, Sylow
383C24C2×D4×C3112yes—
384#41—24no—
385#41—24no—
386#41—84no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8C2×D4×C3112yes—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4C2×D4×C3112yes—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C24×C3111yescharacteristic
415Q16—48no—
416SD16—48no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16#41×C3114yescomplemented
421C2×C8—22no—
422C2×C8—22no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4C2×D4×C3112yes—
438C4⋊C4C2×D4×C3112yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—48no—
448C42—24no—
449C42—22no—
450C42—24no—
451C42C2×D4×C3111yescharacteristic
452C42C2×D4×C3111yescharacteristic
453C42C2×D4×C3111yescharacteristic
454C42C2×D4×C3111yescharacteristic
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23C22×D4×C3111yescharacteristic
462Q8—24no—
463Q8—42no—
464Q8—44no—
465Q8—44no—
466Q8—81no—
467D4—24no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—81no—
474D4—84no—
475D4—84no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4C31×#12011yescharacteristic
494C2×C4C31×#12011yescharacteristic
495C2×C4C31×#12011yescharacteristic
496C2×C4C31×#12011yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C2×C4C22×D4×C3111yescharacteristic
499C2×C4C2×#41×C3112yes—
500C2×C4C31×H(5,𝔽2)12yes—
501C8—24no—
502C8—42no—
503C22—21no—
504C22—21no—
505C22—44no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22C2×C31×#12011yescharacteristic
514C22C31×#53211yescharacteristic
515C22C31×#53511yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—82no—
525C4C31×#45512yes—
526C4C31×#54612yes—
527C4C31×#55012yes—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—82no—
532C2C31×#243111yescharacteristic
533C2C2×C31×#45511yescharacteristic
534C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3506C2
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×#3152C2semidirect
3. C31×#3139C2semidirect
4. C31×#2547C2semidirect
5. C31×#2004C2semidirect
6. C31×#1986C2semidirect
7. C31×#1980C2semidirect
8. C31×#1882C2semidirect
9. C4×C31×#100C2semidirect
152. M16×C31#41semidirect
187. #25621C31direct (N × H)
237. #3152C2×C31semidirect
238. #3139C2×C31semidirect
239. #2547C2×C31semidirect
240. #2004C2×C31semidirect
241. #1986C2×C31semidirect
242. #1980C2×C31semidirect
243. #1882C2×C31semidirect
244. C4×#100C2×C31semidirect
382. C31#25621direct (N × H)
420. 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×#31522
C31×#31392
C31×#25472
C31×#20042
C31×#19862
C31×#19802
C31×#18822
C4×C31×#1002
#2562131