Finite Group Databasebeta

← C31×#18318 #21475077

Subgroups — C31×#18318 #21475077

Order 7936.

1172 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#543C211yesmaximal, characteristic, complemented
2C31×#3193C212yesmaximal, complemented
3C31×#2539C211yesmaximal, characteristic
4C31×#1614C214yesmaximal, complemented
5C31×#1596C214yesmaximal, complemented
6C31×#1508C212yesmaximal, complemented
7C31×#1462C211yesmaximal, characteristic, complemented
8C31×#543—24no—
9C31×#543—22no—
10C31×#543C2214yes—
11C2×C31×#123—21no—
12C2×C31×C22⋊Q8—22no—
13C2×C31×C22⋊Q8—21no—
14C2×C31×C22⋊Q8—22no—
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×#121C2211yescharacteristic
17C2×C31×#120—21no—
18C22×C22⋊C4×C31—21no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C31×#484C2214yescomplemented
21C31×#480—22no—
22C31×#468—24no—
23C31×#450C2214yescomplemented
24C31×#448—22no—
25C31×#407C2212yes—
26C31×#399C2212yes—
27C31×#392C2212yes—
28C31×#390—22no—
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#328C2218yescomplemented
31C31×#323C2214yes—
32C25×C31D411yescharacteristic
33C22×Q8×C31—21no—
34C22×Q8×C31D412yescomplemented
35C22×D4×C31—21no—
36C22×D4×C31—22no—
37C22×D4×C31C2311yescharacteristic
38C23×C4×C31—21no—
39C23×C4×C31—21no—
40C23×C4×C31D411yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C2×SD16×C31—24no—
43C2×M16×C31—22no—
44C31×#123—44no—
45C31×C22⋊Q8—28no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8—24no—
48C31×C22⋊Q8—28no—
49C31×C22⋊Q8—24no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—28no—
53C31×C22⋊Q8D414yescomplemented
54C31×C22⋊Q8C2314yes—
55C31×#121—22no—
56C31×#121—22no—
57C31×#120—44no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31D412yescomplemented
63C2×C4⋊C4×C31C2311yescharacteristic
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31D412yes—
75C2×C22⋊C4×C31C2312yes—
76C2×C42×C31—21no—
77C31×#107—24no—
78C31×#103—24no—
79C31×#102—24no—
80C31×C22⋊C8—22no—
81C31×C22⋊C8C2314yes—
82C31×#95C2312yes—
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—24no—
86C24×C31—21no—
87C24×C31—42no—
88C24×C31—81no—
89C24×C31C2×D411yescharacteristic
90C24×C31C2×D412yes—
91C2×Q8×C31—24no—
92C2×Q8×C31—22no—
93C2×Q8×C31—82no—
94C2×Q8×C31—82no—
95C2×Q8×C31C2×D414yes—
96C2×D4×C31—22no—
97C2×D4×C31—44no—
98C2×D4×C31—42no—
99C2×D4×C31—42no—
100C2×D4×C31—41no—
101C2×D4×C31—41no—
102C2×D4×C31—44no—
103C2×D4×C31—81no—
104C2×D4×C31—82no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—24no—
110C22×C4×C31—24no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—42no—
114C22×C4×C31—81no—
115C22×C4×C31—82no—
116C22×C4×C31—82no—
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D412yes—
120C22×C4×C31C2×D412yes—
121C22×C4×C31C2411yescharacteristic
122SD16×C31—84no—
123M16×C31—82no—
124C2×C8×C31—24no—
125C4⋊C4×C31—24no—
126C4⋊C4×C31—28no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—44no—
132C4⋊C4×C31C2×D414yes—
133C22⋊C4×C31—28no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—24no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—44no—
140C22⋊C4×C31—44no—
141C22⋊C4×C31—42no—
142C42×C31—81no—
143#18318C3111yesmaximal, characteristic, direct product factor, Sylow
144C23×C31—24no—
145C23×C31—24no—
146C23×C31—22no—
147C23×C31—22no—
148C23×C31—22no—
149C23×C31—22no—
150C23×C31—41no—
151C23×C31—42no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—42no—
157C23×C31—42no—
158C23×C31—41no—
159C23×C31—42no—
160C23×C31—81no—
161C23×C31—81no—
162C23×C31—88no—
163C23×C31#12011yescharacteristic
164C23×C31#12011yescharacteristic
165C23×C31#13714yes—
166C23×C31C22×D411yescharacteristic
167Q8×C31—44no—
168Q8×C31—82no—
169Q8×C31—84no—
170D4×C31—44no—
171D4×C31—88no—
172D4×C31—82no—
173D4×C31—82no—
174C2×C4×C31—24no—
175C2×C4×C31—24no—
176C2×C4×C31—22no—
177C2×C4×C31—24no—
178C2×C4×C31—41no—
179C2×C4×C31—42no—
180C2×C4×C31—42no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—44no—
184C2×C4×C31—42no—
185C2×C4×C31—42no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—81no—
192C2×C4×C31—84no—
193C2×C4×C31—84no—
194C2×C4×C31#12012yes—
195C2×C4×C31C22×D412yes—
196C8×C31—44no—
197C2×#543C2×C3111yescharacteristic, complemented
198#3193C2×C3112yescomplemented
199#2539C2×C3111yescharacteristic
200#1614C2×C3114yescomplemented
201#1596C2×C3114yescomplemented
202#1508C2×C3112yescomplemented
203#1462C2×C3111yescharacteristic, complemented
204C22×C31—21no—
205C22×C31—21no—
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—24no—
210C22×C31—21no—
211C22×C31—21no—
212C22×C31—44no—
213C22×C31—44no—
214C22×C31—48no—
215C22×C31—42no—
216C22×C31—42no—
217C22×C31—82no—
218C22×C31—82no—
219C22×C31—82no—
220C22×C31—84no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31#45312yes—
225C22×C31C2×#12011yescharacteristic
226C22×C31C2×#13712yes—
227C4×C31—22no—
228C4×C31—22no—
229C4×C31—44no—
230C4×C31—42no—
231C4×C31—44no—
232C4×C31—44no—
233C4×C31—81no—
234#543—24no—
235#543—22no—
236#543C22×C3114yes—
237C2×#123—21no—
238C2×C22⋊Q8—22no—
239C2×C22⋊Q8—21no—
240C2×C22⋊Q8—22no—
241C2×C22⋊Q8C22×C3112yes—
242C2×#121C22×C3111yescharacteristic
243C2×#120—21no—
244C22×C22⋊C4—21no—
245C22×C22⋊C4C22×C3111yescharacteristic
246#484C22×C3114yescomplemented
247#480—22no—
248#468—24no—
249#450C22×C3114yescomplemented
250#448—22no—
251#407C22×C3112yes—
252#399C22×C3112yes—
253#392C22×C3112yes—
254#390—22no—
255C4×C22⋊C4C22×C3111yescharacteristic
256#328C22×C3118yescomplemented
257#323C22×C3114yes—
258C2×C31—22no—
259C2×C31—22no—
260C2×C31—41no—
261C2×C31—44no—
262C2×C31—161no—
263C2×C31#300411yescharacteristic
264C2×C31C2×#45311yescharacteristic
265C2×C31#302611yescharacteristic
266C25D4×C3111yescharacteristic
267C22×Q8—21no—
268C22×Q8D4×C3112yescomplemented
269C22×D4—21no—
270C22×D4—22no—
271C22×D4C23×C3111yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4D4×C3111yescharacteristic
275C23×C4C23×C3111yescharacteristic
276C2×SD16—24no—
277C2×M16—22no—
278#123—44no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—24no—
282C22⋊Q8—28no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—28no—
287C22⋊Q8D4×C3114yescomplemented
288C22⋊Q8C23×C3114yes—
289#121—22no—
290#121—22no—
291#120—44no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4D4×C3112yescomplemented
297C2×C4⋊C4C23×C3111yescharacteristic
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4D4×C3112yes—
309C2×C22⋊C4C23×C3112yes—
310C2×C42—21no—
311#107—24no—
312#103—24no—
313#102—24no—
314C22⋊C8—22no—
315C22⋊C8C23×C3114yes—
316#95C23×C3112yes—
317C31#1831811yescharacteristic, direct product factor, Sylow
318C24—21no—
319C24—21no—
320C24—24no—
321C24—21no—
322C24—42no—
323C24—81no—
324C24C2×D4×C3111yescharacteristic
325C24C2×D4×C3112yes—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8C2×D4×C3114yes—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—44no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C2×D4×C3112yes—
353C22×C4C2×D4×C3111yescharacteristic
354C22×C4C2×D4×C3112yes—
355C22×C4C2×D4×C3112yes—
356C22×C4C24×C3111yescharacteristic
357SD16—84no—
358M16—82no—
359C2×C8—24no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—44no—
367C4⋊C4C2×D4×C3114yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C42—81no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—81no—
395C23—81no—
396C23—88no—
397C23C31×#12011yescharacteristic
398C23C31×#12011yescharacteristic
399C23C31×#13714yes—
400C23C22×D4×C3111yescharacteristic
401Q8—44no—
402Q8—82no—
403Q8—84no—
404D4—44no—
405D4—88no—
406D4—82no—
407D4—82no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4C31×#12012yes—
429C2×C4C22×D4×C3112yes—
430C8—44no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—84no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C31×#45312yes—
452C22C2×C31×#12011yescharacteristic
453C22C2×C31×#13712yes—
454C4—22no—
455C4—22no—
456C4—44no—
457C4—42no—
458C4—44no—
459C4—44no—
460C4—81no—
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—161no—
466C2C31×#300411yescharacteristic
467C2C2×C31×#45311yescharacteristic
468C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2539C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#543C2semidirect
2. C31×#3193C2semidirect
4. C31×#1614C2semidirect
5. C31×#1596C2semidirect
6. C31×#1508C2semidirect
7. C31×#1462C2semidirect
20. C31×#484C22semidirect
23. C31×#450C22semidirect
30. C31×#328C22semidirect
34. C22×Q8×C31D4semidirect
53. C31×C22⋊Q8D4semidirect
62. C2×C4⋊C4×C31D4semidirect
143. #18318C31direct (N × H)
197. C2×#543C2×C31semidirect
198. #3193C2×C31semidirect
200. #1614C2×C31semidirect
201. #1596C2×C31semidirect
202. #1508C2×C31semidirect
203. #1462C2×C31semidirect
246. #484C22×C31semidirect
249. #450C22×C31semidirect
256. #328C22×C31semidirect
268. C22×Q8D4×C31semidirect
287. C22⋊Q8D4×C31semidirect
296. C2×C4⋊C4D4×C31semidirect
317. C31#18318direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C31×#31932
C31×#25392
C31×#16142
C31×#15962
C31×#15082
C31×#14622
#1831831