Finite Group Databasebeta

← C31×#60520 #21517279

Subgroups — C31×#60520 #21517279

Order 7936.

2170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C31×#3428C211yesmaximal, characteristic, complemented
3C31×#3099C218yesmaximal, complemented
4C31×#3098C218yesmaximal, complemented
5C31×#3095C214yesmaximal, complemented
6C31×#3094C214yesmaximal, complemented
7C2×C31×#428C212yesmaximal, complemented
8C31×#2896C212yesmaximal, complemented
9C31×#2879C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C22×#41×C31C2211yescharacteristic
12C22×M16×C31C2212yes—
13C31×#537—24no—
14C31×#537C2218yes—
15C2×C31×C22⋊Q8—21no—
16C2×C31×C22⋊Q8C2212yes—
17C2×C31×#121—21no—
18C2×C31×#121—22no—
19C2×C31×#121C2212yes—
20C31×#520C2214yescomplemented
21C31×#520C2214yescomplemented
22C31×#519—21no—
23C31×#518—21no—
24C31×#518—22no—
25C4×#41×C31—24no—
26C2×C31×#117C2214yes—
27C2×C31×#117C2212yes—
28C22×C4⋊C4×C31C2211yescharacteristic
29C31×#485C22116yescomplemented
30C31×#484C2218yescomplemented
31C31×#483C2218yescomplemented
32C31×#482C22116yescomplemented
33C31×#481C2218yescomplemented
34C31×#480C2218yescomplemented
35C31×#428C2218yes—
36C31×#428C2218yescomplemented
37C2×C31×#107C2214yescomplemented
38C2×C31×#106C2214yescomplemented
39C31×#419C2214yes—
40C31×#418C2214yes—
41C31×#417C2214yescomplemented
42C31×#417C2214yescomplemented
43C31×#416C2218yescomplemented
44C31×#408C2218yescomplemented
45C31×#407C2214yes—
46C2×#41×C31—22no—
47C2×#41×C31—28no—
48C2×#41×C31—28no—
49C2×#41×C31D414yes—
50C2×#41×C31C2314yes—
51C2×#41×C31C2314yes—
52C22×Q8×C31D411yescharacteristic
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31—21no—
55C22×D4×C31—22no—
56C22×D4×C31D412yes—
57C22×D4×C31D411yescharacteristic
58C22×D4×C31C2311yescharacteristic
59C23×C4×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31C2312yes—
62C23×C4×C31C2311yescharacteristic
63C2×M16×C31—28no—
64C2×M16×C31C2314yes—
65C2×M16×C31C2314yes—
66C22×C8×C31C2314yes—
67C31×C22⋊Q8—28no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8C2318yes—
70C31×#121—28no—
71C31×#121—216no—
72C31×#121—24no—
73C31×#121C2318yes—
74C4×Q8×C31—24no—
75C4×Q8×C31—24no—
76C4×D4×C31—28no—
77C4×D4×C31—24no—
78C4×D4×C31—28no—
79C4×D4×C31—24no—
80C31×#117—24no—
81C31×#117—22no—
82C31×#117—24no—
83C31×#117—22no—
84C31×#117C2314yes—
85C31×#117C2314yes—
86C31×#117C2318yes—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2314yes—
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2314yes—
95C2×C4⋊C4×C31C2314yes—
96C2×C22⋊C4×C31—24no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31—22no—
100C2×C42×C31—24no—
101C2×C42×C31—22no—
102C31×#107C23116yes—
103C31×#106C23116yes—
104C31×#103C2318yes—
105C31×#103C2318yes—
106C31×#102C2318yes—
107C31×#102C2318yes—
108C31×C22⋊C8C2318yes—
109C31×C22⋊C8C2318yes—
110C24×C31—21no—
111C24×C31C2×D412yes—
112C24×C31C2×D411yescharacteristic
113#41×C31—216no—
114#41×C31—28no—
115#41×C31—28no—
116#41×C31—28no—
117#41×C31—28no—
118#41×C31—48no—
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—24no—
122C2×Q8×C31C2×D414yes—
123C2×Q8×C31C2412yes—
124C2×D4×C31—28no—
125C2×D4×C31—24no—
126C2×D4×C31—28no—
127C2×D4×C31—24no—
128C2×D4×C31—21no—
129C2×D4×C31—24no—
130C2×D4×C31—44no—
131C2×D4×C31—84no—
132C2×D4×C31C2×D418yes—
133C2×D4×C31C2×D414yes—
134C2×D4×C31C2412yes—
135C22×C4×C31—28no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—24no—
143C22×C4×C31—24no—
144C22×C4×C31—44no—
145C22×C4×C31—82no—
146C22×C4×C31C2×D414yes—
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31#4118yes—
151C22×C4×C31C2414yes—
152C22×C4×C31C2412yes—
153C22×C4×C31C2411yescharacteristic
154M16×C31—28no—
155M16×C31—28no—
156C2×C8×C31—28no—
157C2×C8×C31C2418yes—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—28no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31C2414yes—
166C4⋊C4×C31C2414yes—
167C4⋊C4×C31C2414yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—28no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—28no—
172C22⋊C4×C31—28no—
173C42×C31—24no—
174C42×C31—28no—
175C42×C31—24no—
176#60520C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—24no—
178C23×C31—22no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31#12314yes—
187C23×C31C22×D414yes—
188C23×C31C22×D411yescharacteristic
189C23×C31C22×D412yes—
190Q8×C31—28no—
191Q8×C31—24no—
192Q8×C31—42no—
193D4×C31—216no—
194D4×C31—28no—
195D4×C31—24no—
196D4×C31—48no—
197D4×C31—42no—
198D4×C31—88no—
199C2×C4×C31—24no—
200C2×C4×C31—28no—
201C2×C4×C31—28no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—22no—
206C2×C4×C31—44no—
207C2×C4×C31—42no—
208C2×C4×C31—48no—
209C2×C4×C31—82no—
210C2×C4×C31—84no—
211C2×C4×C31—82no—
212C2×C4×C31—84no—
213C2×C4×C31—84no—
214C2×C4×C31#12314yes—
215C2×C4×C31#12318yes—
216C2×C4×C31C22×D414yes—
217C2×C4×C31C22×D411yescharacteristic
218C2×C4×C31C22×D412yes—
219C2×C4×C31C2×#4114yes—
220C2×C4×C31C2×#4114yes—
221C2×C4×C31C2×#4114yes—
222C2×C4×C31C2511yescharacteristic
223C8×C31—28no—
224C2×#537C2×C3111yescharacteristic
225#3428C2×C3111yescharacteristic, complemented
226#3099C2×C3118yescomplemented
227#3098C2×C3118yescomplemented
228#3095C2×C3114yescomplemented
229#3094C2×C3114yescomplemented
230C2×#428C2×C3112yescomplemented
231#2896C2×C3112yescomplemented
232#2879C2×C3111yescharacteristic, complemented
233C22×C31—21no—
234C22×C31—22no—
235C22×C31—22no—
236C22×C31—21no—
237C22×C31—24no—
238C22×C31—24no—
239C22×C31—24no—
240C22×C31—44no—
241C22×C31—42no—
242C22×C31—42no—
243C22×C31—42no—
244C22×C31—84no—
245C22×C31—84no—
246C22×C31C2×#12312yes—
247C22×C31C2×#12314yes—
248C22×C31C23×D411yescharacteristic
249C4×C31—24no—
250C4×C31—44no—
251C4×C31—44no—
252C4×C31—44no—
253C4×C31—42no—
254C4×C31C2×#12312yes—
255C4×C31C2×#12314yes—
256C4×C31C22×#4112yes—
257C22×#41C22×C3111yescharacteristic
258C22×#41C22×C3111yescharacteristic
259C22×M16C22×C3112yes—
260#537—24no—
261#537C22×C3118yes—
262C2×C22⋊Q8—21no—
263C2×C22⋊Q8C22×C3112yes—
264C2×#121—21no—
265C2×#121—22no—
266C2×#121C22×C3112yes—
267#520C22×C3114yescomplemented
268#520C22×C3114yescomplemented
269#519—21no—
270#518—21no—
271#518—22no—
272C4×#41—24no—
273C2×#117C22×C3114yes—
274C2×#117C22×C3112yes—
275C22×C4⋊C4C22×C3111yescharacteristic
276#485C22×C31116yescomplemented
277#484C22×C3118yescomplemented
278#483C22×C3118yescomplemented
279#482C22×C31116yescomplemented
280#481C22×C3118yescomplemented
281#480C22×C3118yescomplemented
282#428C22×C3118yes—
283#428C22×C3118yescomplemented
284C2×#107C22×C3114yescomplemented
285C2×#106C22×C3114yescomplemented
286#419C22×C3114yes—
287#418C22×C3114yes—
288#417C22×C3114yescomplemented
289#417C22×C3114yescomplemented
290#416C22×C3118yescomplemented
291#408C22×C3118yescomplemented
292#407C22×C3114yes—
293C2×C31—22no—
294C2×C31—24no—
295C2×C31—44no—
296C2×C31—82no—
297C2×C31C22×#12311yescharacteristic
298C2×C31#358511yescharacteristic
299C2×C31#358611yescharacteristic
300C2×#41—22no—
301C2×#41—28no—
302C2×#41—28no—
303C2×#41D4×C3114yes—
304C2×#41C23×C3114yes—
305C2×#41C23×C3114yes—
306C22×Q8D4×C3111yescharacteristic
307C22×Q8C23×C3111yescharacteristic
308C22×D4—21no—
309C22×D4—22no—
310C22×D4D4×C3112yes—
311C22×D4D4×C3111yescharacteristic
312C22×D4C23×C3111yescharacteristic
313C23×C4—21no—
314C23×C4—21no—
315C23×C4C23×C3112yes—
316C23×C4C23×C3111yescharacteristic
317C2×M16—28no—
318C2×M16C23×C3114yes—
319C2×M16C23×C3114yes—
320C22×C8C23×C3114yes—
321C22⋊Q8—28no—
322C22⋊Q8—24no—
323C22⋊Q8C23×C3118yes—
324#121—28no—
325#121—216no—
326#121—24no—
327#121C23×C3118yes—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×D4—28no—
331C4×D4—24no—
332C4×D4—28no—
333C4×D4—24no—
334#117—24no—
335#117—22no—
336#117—24no—
337#117—22no—
338#117C23×C3114yes—
339#117C23×C3114yes—
340#117C23×C3118yes—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C23×C3112yes—
345C2×C4⋊C4C23×C3114yes—
346C2×C4⋊C4C23×C3112yes—
347C2×C4⋊C4C23×C3112yes—
348C2×C4⋊C4C23×C3114yes—
349C2×C4⋊C4C23×C3114yes—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—24no—
355C2×C42—22no—
356#107C23×C31116yes—
357#106C23×C31116yes—
358#103C23×C3118yes—
359#103C23×C3118yes—
360#102C23×C3118yes—
361#102C23×C3118yes—
362C22⋊C8C23×C3118yes—
363C22⋊C8C23×C3118yes—
364C31#6052011yescharacteristic, direct product factor, Sylow
365C24—21no—
366C24C2×D4×C3112yes—
367C24C2×D4×C3111yescharacteristic
368#41—216no—
369#41—28no—
370#41—28no—
371#41—28no—
372#41—28no—
373#41—48no—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—24no—
377C2×Q8C2×D4×C3114yes—
378C2×Q8C24×C3112yes—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—21no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—84no—
387C2×D4C2×D4×C3118yes—
388C2×D4C2×D4×C3114yes—
389C2×D4C24×C3112yes—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—82no—
401C22×C4C2×D4×C3114yes—
402C22×C4C2×D4×C3112yes—
403C22×C4C2×D4×C3112yes—
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4#41×C3118yes—
406C22×C4C24×C3114yes—
407C22×C4C24×C3112yes—
408C22×C4C24×C3111yescharacteristic
409M16—28no—
410M16—28no—
411C2×C8—28no—
412C2×C8C24×C3118yes—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4C24×C3114yes—
421C4⋊C4C24×C3114yes—
422C4⋊C4C24×C3114yes—
423C22⋊C4—24no—
424C22⋊C4—28no—
425C22⋊C4—24no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C42—24no—
429C42—28no—
430C42—24no—
431C23—24no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23C31×#12314yes—
441C23C22×D4×C3114yes—
442C23C22×D4×C3111yescharacteristic
443C23C22×D4×C3112yes—
444Q8—28no—
445Q8—24no—
446Q8—42no—
447D4—216no—
448D4—28no—
449D4—24no—
450D4—48no—
451D4—42no—
452D4—88no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4C31×#12314yes—
469C2×C4C31×#12318yes—
470C2×C4C22×D4×C3114yes—
471C2×C4C22×D4×C3111yescharacteristic
472C2×C4C22×D4×C3112yes—
473C2×C4C2×#41×C3114yes—
474C2×C4C2×#41×C3114yes—
475C2×C4C2×#41×C3114yes—
476C2×C4C25×C3111yescharacteristic
477C8—28no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—21no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—84no—
490C22—84no—
491C22C2×C31×#12312yes—
492C22C2×C31×#12314yes—
493C22C23×D4×C3111yescharacteristic
494C4—24no—
495C4—42no—
496C4—44no—
497C4—44no—
498C4—44no—
499C4C2×C31×#12312yes—
500C4C2×C31×#12314yes—
501C4C22×#41×C3112yes—
502C2—22no—
503C2—24no—
504C2—44no—
505C2—82no—
506C2C22×C31×#12311yescharacteristic
507C2C31×#358511yescharacteristic
508C2C31×#358611yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC2×C31×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3428C2semidirect
3. C31×#3099C2semidirect
4. C31×#3098C2semidirect
5. C31×#3095C2semidirect
6. C31×#3094C2semidirect
7. C2×C31×#428C2semidirect
8. C31×#2896C2semidirect
9. C31×#2879C2semidirect
20. C31×#520C22semidirect
21. C31×#520C22semidirect
29. C31×#485C22semidirect
30. C31×#484C22semidirect
31. C31×#483C22semidirect
32. C31×#482C22semidirect
33. C31×#481C22semidirect
34. C31×#480C22semidirect
36. C31×#428C22semidirect
37. C2×C31×#107C22semidirect
38. C2×C31×#106C22semidirect
41. C31×#417C22semidirect
42. C31×#417C22semidirect
43. C31×#416C22semidirect
44. C31×#408C22semidirect
176. #60520C31direct (N × H)
225. #3428C2×C31semidirect
226. #3099C2×C31semidirect
227. #3098C2×C31semidirect
228. #3095C2×C31semidirect
229. #3094C2×C31semidirect
230. C2×#428C2×C31semidirect
231. #2896C2×C31semidirect
232. #2879C2×C31semidirect
267. #520C22×C31semidirect
268. #520C22×C31semidirect
276. #485C22×C31semidirect
277. #484C22×C31semidirect
278. #483C22×C31semidirect
279. #482C22×C31semidirect
280. #481C22×C31semidirect
281. #480C22×C31semidirect
283. #428C22×C31semidirect
284. C2×#107C22×C31semidirect
285. C2×#106C22×C31semidirect
288. #417C22×C31semidirect
289. #417C22×C31semidirect
290. #416C22×C31semidirect
291. #408C22×C31semidirect
364. C31#60520direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C31×#34282
C31×#30992
C31×#30982
C31×#30952
C31×#30942
C2×C31×#4282
C31×#28962
C31×#28792
#6052031