Finite Group Databasebeta

← C31×#18530 #21475289

Subgroups — C31×#18530 #21475289

Order 7936.

962 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C31×#3139C211yesmaximal, characteristic, complemented
3C31×#3113C211yesmaximal, characteristic, complemented
4C31×#3110C211yesmaximal, characteristic
5C31×#1644C214yesmaximal, complemented
6C31×#1643C214yesmaximal, complemented
7C31×#1558C211yesmaximal, characteristic, complemented
8C31×#1491C211yesmaximal, characteristic, complemented
9C31×#1490C211yesmaximal, characteristic
10C31×#554—22no—
11C31×#554—22no—
12C31×#554—22no—
13C31×#554C2214yes—
14C31×#533C2211yescharacteristic
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#125—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8—22no—
19C2×C4×Q8×C31C2211yescharacteristic
20C2×C4×D4×C31—21no—
21C2×C4×D4×C31C2211yescharacteristic
22C22×C4⋊C4×C31—21no—
23C31×#491—22no—
24C31×#487C2212yescomplemented
25C31×#487C2212yescomplemented
26C31×#484—21no—
27C31×#483—21no—
28C31×#475C2212yescomplemented
29C31×#474C2212yes—
30C31×#462C2212yescomplemented
31C31×#461C2212yes—
32C31×#451—21no—
33C31×#450—21no—
34C31×#441—22no—
35C31×#440—22no—
36C31×#432C2211yescharacteristic
37C31×#432C2211yescharacteristic
38C31×#419—22no—
39C4×M16×C31C2211yescharacteristic
40C31×#333C2214yescomplemented
41C31×Q8⋊C8C2214yescomplemented
42C31×D4⋊C8C2214yes—
43C22×Q8×C31D412yes—
44C22×Q8×C31D411yescharacteristic
45C22×D4×C31D411yescharacteristic
46C23×C4×C31—22no—
47C23×C4×C31—21no—
48C2×Q16×C31—22no—
49C2×SD16×C31—22no—
50C2×M16×C31—21no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8D412yes—
54C31×C4⋊Q8C2311yescharacteristic
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2312yes—
57C31×#125—24no—
58C31×#125—24no—
59C31×#125—21no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—28no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—28no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8—21no—
66C4×Q8×C31—21no—
67C4×Q8×C31—21no—
68C4×Q8×C31—21no—
69C4×Q8×C31—21no—
70C4×Q8×C31D412yes—
71C4×Q8×C31C2312yes—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31D412yes—
77C4×D4×C31C2312yes—
78C31×#117—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—24no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—24no—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—21no—
90C2×C42×C31—21no—
91C2×C42×C31C2311yescharacteristic
92C31×#107—22no—
93C31×#106—22no—
94C31×C4⋊C8C2312yes—
95C31×C4⋊C8C2312yes—
96C31×#103—24no—
97C31×#103—24no—
98C31×#103—22no—
99C31×#103—22no—
100C31×#103—22no—
101C31×#102—22no—
102C31×C22⋊C8—21no—
103C31×C22⋊C8—21no—
104C31×#97—21no—
105C31×#97—21no—
106C31×#97—21no—
107C4×C8×C31C2312yes—
108C24×C31—21no—
109C2×Q8×C31—21no—
110C2×Q8×C31—22no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—42no—
114C2×Q8×C31—44no—
115C2×Q8×C31C2×D412yes—
116C2×Q8×C31C2×D414yes—
117C2×D4×C31—24no—
118C2×D4×C31—21no—
119C2×D4×C31C2×D412yes—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—28no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—44no—
128C22×C4×C31—84no—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132Q16×C31—44no—
133SD16×C31—44no—
134M16×C31—42no—
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C2×C8×C31—22no—
138C4⋊C4×C31—28no—
139C4⋊C4×C31—28no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—41no—
153C4⋊C4×C31—41no—
154C4⋊C4×C31C2×D412yes—
155C4⋊C4×C31C2×D412yes—
156C22⋊C4×C31—28no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—41no—
160C42×C31—22no—
161C42×C31—22no—
162C42×C31—41no—
163C42×C31C2×D411yescharacteristic
164C42×C31C2×D411yescharacteristic
165C42×C31C2×D411yescharacteristic
166C42×C31C2411yescharacteristic
167#18530C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—24no—
169C23×C31—22no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31#12011yescharacteristic
173Q8×C31—22no—
174Q8×C31—22no—
175Q8×C31—44no—
176Q8×C31—42no—
177Q8×C31—42no—
178Q8×C31—42no—
179Q8×C31—82no—
180D4×C31—24no—
181D4×C31—22no—
182D4×C31#13714yes—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—24no—
190C2×C4×C31—44no—
191C2×C4×C31—42no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—44no—
195C2×C4×C31—44no—
196C2×C4×C31—41no—
197C2×C4×C31—41no—
198C2×C4×C31—88no—
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31#12011yescharacteristic
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31C22×D411yescharacteristic
203C2×C4×C31C22×D411yescharacteristic
204C2×C4×C31C22×D411yescharacteristic
205C8×C31—42no—
206C8×C31—42no—
207C8×C31—42no—
208C2×#554C2×C3111yescharacteristic
209#3139C2×C3111yescharacteristic, complemented
210#3113C2×C3111yescharacteristic, complemented
211#3110C2×C3111yescharacteristic
212#1644C2×C3114yescomplemented
213#1643C2×C3114yescomplemented
214#1558C2×C3111yescharacteristic, complemented
215#1491C2×C3111yescharacteristic, complemented
216#1490C2×C3111yescharacteristic
217C22×C31—21no—
218C22×C31—21no—
219C22×C31—24no—
220C22×C31—48no—
221C22×C31—42no—
222C22×C31—42no—
223C22×C31#45512yes—
224C22×C31C2×#12011yescharacteristic
225C4×C31—21no—
226C4×C31—21no—
227C4×C31—21no—
228C4×C31—21no—
229C4×C31—42no—
230C4×C31—42no—
231C4×C31—42no—
232C4×C31—44no—
233C4×C31—81no—
234C4×C31—81no—
235C4×C31C2×#13712yes—
236C4×C31#57512yes—
237#554—22no—
238#554—22no—
239#554—22no—
240#554C22×C3114yes—
241#533C22×C3111yescharacteristic
242C2×C4⋊Q8C22×C3111yescharacteristic
243C2×#125—21no—
244C2×C22⋊Q8—21no—
245C2×C22⋊Q8—22no—
246C2×C4×Q8C22×C3111yescharacteristic
247C2×C4×D4—21no—
248C2×C4×D4C22×C3111yescharacteristic
249C22×C4⋊C4—21no—
250#491—22no—
251#487C22×C3112yescomplemented
252#487C22×C3112yescomplemented
253#484—21no—
254#483—21no—
255#475C22×C3112yescomplemented
256#474C22×C3112yes—
257#462C22×C3112yescomplemented
258#461C22×C3112yes—
259#451—21no—
260#450—21no—
261#441—22no—
262#440—22no—
263#432C22×C3111yescharacteristic
264#432C22×C3111yescharacteristic
265#419—22no—
266C4×M16C22×C3111yescharacteristic
267#333C22×C3114yescomplemented
268Q8⋊C8C22×C3114yescomplemented
269D4⋊C8C22×C3114yes—
270C2×C31—22no—
271C2×C31—44no—
272C2×C31#300411yescharacteristic
273C2×C31#300511yescharacteristic
274C2×C31C2×#45511yescharacteristic
275C22×Q8D4×C3111yescharacteristic
276C22×Q8D4×C3112yes—
277C22×D4D4×C3111yescharacteristic
278C23×C4—22no—
279C23×C4—21no—
280C2×Q16—22no—
281C2×SD16—22no—
282C2×M16—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8D4×C3112yes—
286C4⋊Q8C23×C3111yescharacteristic
287C4⋊Q8C23×C3111yescharacteristic
288C4⋊Q8C23×C3112yes—
289#125—24no—
290#125—24no—
291#125—21no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—28no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8D4×C3112yes—
303C4×Q8C23×C3112yes—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4D4×C3112yes—
309C4×D4C23×C3112yes—
310#117—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4D4×C3111yescharacteristic
319C2×C4⋊C4D4×C3111yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—21no—
323C2×C42C23×C3111yescharacteristic
324#107—22no—
325#106—22no—
326C4⋊C8C23×C3112yes—
327C4⋊C8C23×C3112yes—
328#103—24no—
329#103—24no—
330#103—22no—
331#103—22no—
332#103—22no—
333#102—22no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336#97—21no—
337#97—21no—
338#97—21no—
339C4×C8C23×C3112yes—
340C31#1853011yescharacteristic, direct product factor, Sylow
341C24—21no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8C2×D4×C3112yes—
349C2×Q8C2×D4×C3114yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4C2×D4×C3112yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4C2×D4×C3111yescharacteristic
363C22×C4C2×D4×C3111yescharacteristic
364C22×C4C2×D4×C3111yescharacteristic
365Q16—44no—
366SD16—44no—
367M16—42no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4C2×D4×C3112yes—
388C4⋊C4C2×D4×C3112yes—
389C22⋊C4—28no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—41no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42C2×D4×C3111yescharacteristic
397C42C2×D4×C3111yescharacteristic
398C42C2×D4×C3111yescharacteristic
399C42C24×C3111yescharacteristic
400C23—24no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23C31×#12011yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—82no—
412D4—24no—
413D4—22no—
414D4C31×#13714yes—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—88no—
431C2×C4C31×#12011yescharacteristic
432C2×C4C31×#12011yescharacteristic
433C2×C4C31×#12011yescharacteristic
434C2×C4C22×D4×C3111yescharacteristic
435C2×C4C22×D4×C3111yescharacteristic
436C2×C4C22×D4×C3111yescharacteristic
437C8—42no—
438C8—42no—
439C8—42no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22C31×#45512yes—
447C22C2×C31×#12011yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—44no—
456C4—81no—
457C4—81no—
458C4C2×C31×#13712yes—
459C4C31×#57512yes—
460C2—22no—
461C2—44no—
462C2C31×#300411yescharacteristic
463C2C31×#300511yescharacteristic
464C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3139C2semidirect
3. C31×#3113C2semidirect
5. C31×#1644C2semidirect
6. C31×#1643C2semidirect
7. C31×#1558C2semidirect
8. C31×#1491C2semidirect
24. C31×#487C22semidirect
25. C31×#487C22semidirect
28. C31×#475C22semidirect
30. C31×#462C22semidirect
40. C31×#333C22semidirect
41. C31×Q8⋊C8C22semidirect
167. #18530C31direct (N × H)
209. #3139C2×C31semidirect
210. #3113C2×C31semidirect
212. #1644C2×C31semidirect
213. #1643C2×C31semidirect
214. #1558C2×C31semidirect
215. #1491C2×C31semidirect
251. #487C22×C31semidirect
252. #487C22×C31semidirect
255. #475C22×C31semidirect
257. #462C22×C31semidirect
267. #333C22×C31semidirect
268. Q8⋊C8C22×C31semidirect
340. C31#18530direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C31×#31392
C31×#31132
C31×#31102
C31×#16442
C31×#16432
C31×#15582
C31×#14912
C31×#14902
#1853031