Finite Group Databasebeta

← C31×#18182 #21474941

Subgroups — C31×#18182 #21474941

Order 7936.

958 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic, complemented
2C31×#3110C211yesmaximal, characteristic, complemented
3C31×#3109C211yesmaximal, characteristic
4C31×#2398C211yesmaximal, characteristic, complemented
5C31×#1615C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1506C211yesmaximal, characteristic, complemented
8C31×#1505C211yesmaximal, characteristic, complemented
9C31×#1467C211yesmaximal, characteristic, complemented
10C31×#559—21no—
11C31×#557—21no—
12C31×#556—21no—
13C31×#556—22no—
14C31×#554—21no—
15C31×#552—22no—
16C31×#539C2214yes—
17C31×#536—21no—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C31×#529—22no—
21C31×#520—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C31×#484C2212yes—
25C31×#481C2212yes—
26C31×#476—21no—
27C31×#474—21no—
28C31×#461—21no—
29C31×#459—21no—
30C31×#450C2212yes—
31C31×#447C2212yes—
32C31×#442—22no—
33C31×#440—22no—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#400C2212yes—
37C31×#397—21no—
38C31×#394C2212yes—
39C31×#393—21no—
40C4×C22⋊C4×C31—22no—
41C31×#376C2211yescharacteristic
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#324C2214yes—
45C2×#41×C31—21no—
46C22×Q8×C31D411yescharacteristic
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31—21no—
49C2×SD16×C31—22no—
50C2×D8×C31—22no—
51C31×C4⋊Q8—22no—
52C31×C4⋊Q8D411yescharacteristic
53C31×C4⋊Q8D411yescharacteristic
54C31×C4⋊Q8D411yescharacteristic
55C31×#127D411yescharacteristic
56C31×#126—24no—
57C31×#126—24no—
58C31×#125—22no—
59C31×#125—24no—
60C31×#125—21no—
61C31×#124—22no—
62C31×#124—22no—
63C31×#124—21no—
64C31×#123—24no—
65C31×#123—24no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8C2312yes—
71C31×#121—21no—
72C31×#121C2312yes—
73C4×Q8×C31—22no—
74C4×Q8×C31—22no—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31—21no—
79C4×Q8×C31—21no—
80C4×Q8×C31D412yes—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—24no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31D412yes—
90C31×#117—22no—
91C31×#117—24no—
92C31×#117—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C4⋊C4×C31C2312yes—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31—22no—
100C2×C42×C31—21no—
101C2×C42×C31D412yes—
102C2×C42×C31C2311yescharacteristic
103C31×#107—22no—
104C31×#106—22no—
105C31×C4⋊C8—21no—
106C31×C4⋊C8—21no—
107C31×#103—22no—
108C31×#102—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×C22⋊C8C2312yes—
112C31×C22⋊C8C2312yes—
113C31×#97—21no—
114C31×#97—21no—
115C31×#95—22no—
116C31×#95—21no—
117C31×#95—22no—
118C31×#95C2312yes—
119C24×C31—21no—
120#41×C31—44no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—42no—
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—22no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—42no—
133C2×D4×C31C2×D412yes—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—44no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2411yescharacteristic
146SD16×C31—44no—
147D8×C31—44no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D412yes—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—44no—
175C22⋊C4×C31—41no—
176C22⋊C4×C31—41no—
177C42×C31—24no—
178C42×C31—22no—
179C42×C31—22no—
180C42×C31—22no—
181C42×C31—42no—
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D411yescharacteristic
185C42×C31C2×D411yescharacteristic
186#18182C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—42no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—42no—
193C23×C31C22×D411yescharacteristic
194Q8×C31—42no—
195Q8×C31—42no—
196Q8×C31—42no—
197Q8×C31—42no—
198Q8×C31—42no—
199D4×C31—22no—
200D4×C31—22no—
201D4×C31—42no—
202D4×C31—41no—
203D4×C31—41no—
204D4×C31—42no—
205D4×C31—44no—
206D4×C31—42no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—41no—
217C2×C4×C31—44no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—41no—
223C2×C4×C31—42no—
224C2×C4×C31—82no—
225C2×C4×C31—84no—
226C2×C4×C31—82no—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C8×C31—42no—
234C8×C31—42no—
235#3507C2×C3111yescharacteristic, complemented
236#3110C2×C3111yescharacteristic, complemented
237#3109C2×C3111yescharacteristic
238#2398C2×C3111yescharacteristic, complemented
239#1615C2×C3114yescomplemented
240#1597C2×C3114yescomplemented
241#1506C2×C3111yescharacteristic, complemented
242#1505C2×C3111yescharacteristic, complemented
243#1467C2×C3111yescharacteristic, complemented
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—44no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—82no—
252C22×C31—81no—
253C22×C31—81no—
254C22×C31C2×#12011yescharacteristic
255C22×C31#57512yes—
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—42no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—82no—
267C4×C31—81no—
268C4×C31#45412yes—
269#559—21no—
270#557—21no—
271#556—21no—
272#556—22no—
273#554—21no—
274#552—22no—
275#539C22×C3114yes—
276#536—21no—
277#533C22×C3111yescharacteristic
278#532C22×C3111yescharacteristic
279#529—22no—
280#520—21no—
281C2×C4×Q8C22×C3111yescharacteristic
282C2×C4×D4C22×C3111yescharacteristic
283#484C22×C3112yes—
284#481C22×C3112yes—
285#476—21no—
286#474—21no—
287#461—21no—
288#459—21no—
289#450C22×C3112yes—
290#447C22×C3112yes—
291#442—22no—
292#440—22no—
293#432C22×C3111yescharacteristic
294#432C22×C3111yescharacteristic
295#400C22×C3112yes—
296#397—21no—
297#394C22×C3112yes—
298#393—21no—
299C4×C22⋊C4—22no—
300#376C22×C3111yescharacteristic
301#328C22×C3114yes—
302#327C22×C3114yes—
303#324C22×C3114yes—
304C2×C31—22no—
305C2×C31—82no—
306C2×C31—82no—
307C2×C31#300511yescharacteristic
308C2×C31C2×#45411yescharacteristic
309C2×C31#302511yescharacteristic
310C2×#41—21no—
311C22×Q8D4×C3111yescharacteristic
312C22×D4C23×C3111yescharacteristic
313C23×C4—21no—
314C2×SD16—22no—
315C2×D8—22no—
316C4⋊Q8—22no—
317C4⋊Q8D4×C3111yescharacteristic
318C4⋊Q8D4×C3111yescharacteristic
319C4⋊Q8D4×C3111yescharacteristic
320#127D4×C3111yescharacteristic
321#126—24no—
322#126—24no—
323#125—22no—
324#125—24no—
325#125—21no—
326#124—22no—
327#124—22no—
328#124—21no—
329#123—24no—
330#123—24no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8C23×C3112yes—
336#121—21no—
337#121C23×C3112yes—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—21no—
345C4×Q8D4×C3112yes—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—24no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4D4×C3112yes—
355#117—22no—
356#117—24no—
357#117—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4D4×C3111yescharacteristic
360C2×C4⋊C4C23×C3111yescharacteristic
361C2×C4⋊C4C23×C3112yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—21no—
366C2×C42D4×C3112yes—
367C2×C42C23×C3111yescharacteristic
368#107—22no—
369#106—22no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372#103—22no—
373#102—22no—
374#102—22no—
375#102—22no—
376C22⋊C8C23×C3112yes—
377C22⋊C8C23×C3112yes—
378#97—21no—
379#97—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95C23×C3112yes—
384C31#1818211yescharacteristic, direct product factor, Sylow
385C24—21no—
386#41—44no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C3112yes—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4C2×D4×C3112yes—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—44no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4C2×D4×C3112yes—
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3112yes—
411C22×C4C24×C3111yescharacteristic
412SD16—44no—
413D8—44no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4C2×D4×C3112yes—
432C4⋊C4C2×D4×C3112yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42C2×D4×C3111yescharacteristic
449C42C2×D4×C3111yescharacteristic
450C42C2×D4×C3111yescharacteristic
451C42C2×D4×C3111yescharacteristic
452C23—22no—
453C23—22no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23C22×D4×C3111yescharacteristic
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464D4—22no—
465D4—22no—
466D4—42no—
467D4—41no—
468D4—41no—
469D4—42no—
470D4—44no—
471D4—42no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4C31×#12011yescharacteristic
493C2×C4C31×#12011yescharacteristic
494C2×C4C31×#12011yescharacteristic
495C2×C4C31×#12011yescharacteristic
496C2×C4C22×D4×C3111yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C8—42no—
499C8—42no—
500C22—21no—
501C22—21no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22C2×C31×#12011yescharacteristic
511C22C31×#57512yes—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—81no—
524C4C31×#45412yes—
525C2—22no—
526C2—82no—
527C2—82no—
528C2C31×#300511yescharacteristic
529C2C2×C31×#45411yescharacteristic
530C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3109C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3507C2semidirect
2. C31×#3110C2semidirect
4. C31×#2398C2semidirect
5. C31×#1615C2semidirect
6. C31×#1597C2semidirect
7. C31×#1506C2semidirect
8. C31×#1505C2semidirect
9. C31×#1467C2semidirect
186. #18182C31direct (N × H)
235. #3507C2×C31semidirect
236. #3110C2×C31semidirect
238. #2398C2×C31semidirect
239. #1615C2×C31semidirect
240. #1597C2×C31semidirect
241. #1506C2×C31semidirect
242. #1505C2×C31semidirect
243. #1467C2×C31semidirect
384. C31#18182direct (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×#31102
C31×#31092
C31×#23982
C31×#16152
C31×#15972
C31×#15062
C31×#15052
C31×#14672
#1818231