Finite Group Databasebeta

← C31×#61825 #21518584

Subgroups — C31×#61825 #21518584

Order 7936.

1990 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3497C211yesmaximal, characteristic, complemented
2C2×C31×#554C211yesmaximal, characteristic, complemented
3C31×#3384C218yesmaximal, complemented
4C31×#3366C218yesmaximal, complemented
5C31×#3234C214yesmaximal, complemented
6C31×#3227C212yesmaximal, complemented
7C31×#3109C214yesmaximal, complemented
8C31×#3100C212yesmaximal, complemented
9C31×#2983C211yesmaximal, characteristic, complemented
10C2×C31×2−1+4—21no—
11C31×#554—24no—
12C31×#554C2218yes—
13C31×#550C2214yescomplemented
14D4×Q8×C31C2214yescomplemented
15C31×#548—24no—
16C31×#537—21no—
17C31×#537—22no—
18C31×#532C2212yes—
19C31×#532C2214yes—
20C2×C31×C4⋊Q8C2211yescharacteristic
21C2×C31×#125C2212yes—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8—22no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C2×C4×D4×C31C2212yes—
27C22×C4⋊C4×C31—21no—
28C31×#501C2218yescomplemented
29C31×#498C2214yescomplemented
30C31×#490C2218yescomplemented
31C31×#486C2214yescomplemented
32C31×#481C2218yes—
33C31×#480C2214yes—
34C31×#478C22116yescomplemented
35C31×#476C2218yes—
36C31×#476C2218yescomplemented
37C31×#474C2214yes—
38C31×#459C2214yes—
39C31×#459C2218yes—
40C31×#447C2214yes—
41C31×#446C2218yescomplemented
42C31×#442C22116yes—
43C4×D8×C31C2218yes—
44C31×#432C2214yes—
45C31×#431C2212yes—
46C31×2−1+4—48no—
47C2×#41×C31—24no—
48C2×#41×C31—24no—
49C2×#41×C31—22no—
50C22×Q8×C31—24no—
51C22×Q8×C31—21no—
52C22×Q8×C31D411yescharacteristic
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31—22no—
56C2×D8×C31C2318yes—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8C2312yes—
59C31×C4⋊Q8C2314yes—
60C31×C4⋊Q8C2314yes—
61C31×#127C2312yes—
62C31×#127C2314yes—
63C31×#125—24no—
64C31×#125C2318yes—
65C31×#124—24no—
66C31×#124—22no—
67C31×C22⋊Q8—28no—
68C31×C22⋊Q8—216no—
69C31×C22⋊Q8—24no—
70C31×C22⋊Q8—28no—
71C31×#121—22no—
72C31×#121—24no—
73C31×#121—24no—
74C31×#121—28no—
75C31×#121C2314yes—
76C4×Q8×C31D414yes—
77C4×Q8×C31C2314yes—
78C4×D4×C31—24no—
79C4×D4×C31—22no—
80C4×D4×C31—24no—
81C4×D4×C31—22no—
82C4×D4×C31C2314yes—
83C4×D4×C31C2314yes—
84C4×D4×C31C2318yes—
85C4×D4×C31C2318yes—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—28no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D412yes—
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C4⋊C4×C31C2312yes—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2312yes—
98C31×#107C2318yes—
99C31×#106C23116yes—
100C31×C4⋊C8C2314yes—
101C31×C4⋊C8C2318yes—
102C31×#102C2318yes—
103C31×#102C2318yes—
104C31×#102C23116yes—
105C31×C22⋊C8C2314yes—
106C31×#97C2318yes—
107C4×C8×C31C2314yes—
108C24×C31—21no—
109#41×C31—416no—
110#41×C31—416no—
111#41×C31—48no—
112C2×Q8×C31—24no—
113C2×Q8×C31—24no—
114C2×Q8×C31—22no—
115C2×Q8×C31—24no—
116C2×Q8×C31—416no—
117C2×Q8×C31—82no—
118C2×Q8×C31C2×D414yes—
119C2×D4×C31—24no—
120C2×D4×C31—24no—
121C2×D4×C31—21no—
122C2×D4×C31—24no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31C2414yes—
127C2×D4×C31C2412yes—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—24no—
131C22×C4×C31—28no—
132C22×C4×C31—24no—
133C22×C4×C31—28no—
134C22×C4×C31—44no—
135C22×C4×C31—84no—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2411yescharacteristic
141D8×C31—216no—
142C2×C8×C31C2418yes—
143C4⋊C4×C31—28no—
144C4⋊C4×C31—24no—
145C4⋊C4×C31—28no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31C2×D414yes—
151C4⋊C4×C31C2×D418yes—
152C4⋊C4×C31C2414yes—
153C4⋊C4×C31C2412yes—
154C4⋊C4×C31C2414yes—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—28no—
157C22⋊C4×C31—24no—
158C22⋊C4×C31—24no—
159C22⋊C4×C31—28no—
160C42×C31C2×D414yes—
161C42×C31C2×D412yes—
162C42×C31C2412yes—
163C42×C31C2414yes—
164#61825C3111yesmaximal, characteristic, direct product factor, Sylow
165C23×C31—24no—
166C23×C31—22no—
167C23×C31—24no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31C22×D411yescharacteristic
171Q8×C31—28no—
172Q8×C31—48no—
173Q8×C31—44no—
174Q8×C31—84no—
175D4×C31—24no—
176D4×C31—22no—
177D4×C31—28no—
178D4×C31—48no—
179D4×C31—48no—
180D4×C31—44no—
181D4×C31—84no—
182D4×C31—82no—
183D4×C312−1+414yescomplemented
184C2×C4×C31—24no—
185C2×C4×C31—22no—
186C2×C4×C31—24no—
187C2×C4×C31—22no—
188C2×C4×C31—24no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—416no—
192C2×C4×C31—44no—
193C2×C4×C31—42no—
194C2×C4×C31—48no—
195C2×C4×C31—48no—
196C2×C4×C31—88no—
197C2×C4×C31AGL(1,ℤ/8)14yes—
198C2×C4×C31C22×D411yescharacteristic
199C2×C4×C31C22×D412yes—
200C2×C4×C31C22×D414yes—
201C2×C4×C31C22×D414yes—
202C2×C4×C31C22×D412yes—
203C2×C4×C31C2511yescharacteristic
204C8×C31—28no—
205#3497C2×C3111yescharacteristic, complemented
206C2×#554C2×C3111yescharacteristic, complemented
207#3384C2×C3118yescomplemented
208#3366C2×C3118yescomplemented
209#3234C2×C3114yescomplemented
210#3227C2×C3112yescomplemented
211#3109C2×C3114yescomplemented
212#3100C2×C3112yescomplemented
213#2983C2×C3111yescharacteristic, complemented
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—24no—
217C22×C31—48no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—44no—
221C22×C31—44no—
222C22×C31—44no—
223C22×C31C2×AGL(1,ℤ/8)12yes—
224C22×C31C23×D411yescharacteristic
225C4×C31—24no—
226C4×C31—22no—
227C4×C31—21no—
228C4×C31—24no—
229C4×C31—44no—
230C4×C31—42no—
231C4×C31—44no—
232C4×C31C2×AGL(1,ℤ/8)14yes—
233C4×C31C2×2−1+412yes—
234C2×2−1+4—21no—
235#554—24no—
236#554C22×C3118yes—
237#550C22×C3114yescomplemented
238D4×Q8C22×C3114yescomplemented
239#548—24no—
240#537—21no—
241#537—22no—
242#532C22×C3112yes—
243#532C22×C3114yes—
244C2×C4⋊Q8C22×C3111yescharacteristic
245C2×#125C22×C3112yes—
246C2×C22⋊Q8—21no—
247C2×C22⋊Q8—22no—
248C2×C4×Q8C22×C3111yescharacteristic
249C2×C4×D4C22×C3111yescharacteristic
250C2×C4×D4C22×C3112yes—
251C22×C4⋊C4—21no—
252#501C22×C3118yescomplemented
253#498C22×C3114yescomplemented
254#490C22×C3118yescomplemented
255#486C22×C3114yescomplemented
256#481C22×C3118yes—
257#480C22×C3114yes—
258#478C22×C31116yescomplemented
259#476C22×C3118yes—
260#476C22×C3118yescomplemented
261#474C22×C3114yes—
262#459C22×C3114yes—
263#459C22×C3118yes—
264#447C22×C3114yes—
265#446C22×C3118yescomplemented
266#442C22×C31116yes—
267C4×D8C22×C3118yes—
268#432C22×C3114yes—
269#431C22×C3112yes—
270C2×C31—22no—
271C2×C31—44no—
272C2×C31—84no—
273C2×C31#348611yescharacteristic
274C2×C31C22×AGL(1,ℤ/8)11yescharacteristic
275C2×C31#358511yescharacteristic
2762−1+4—48no—
277C2×#41—24no—
278C2×#41—24no—
279C2×#41—22no—
280C22×Q8—24no—
281C22×Q8—21no—
282C22×Q8D4×C3111yescharacteristic
283C22×D4C23×C3111yescharacteristic
284C23×C4—21no—
285C23×C4—22no—
286C2×D8C23×C3118yes—
287C4⋊Q8—22no—
288C4⋊Q8C23×C3114yes—
289C4⋊Q8C23×C3112yes—
290C4⋊Q8C23×C3114yes—
291#127C23×C3112yes—
292#127C23×C3114yes—
293#125—24no—
294#125C23×C3118yes—
295#124—24no—
296#124—22no—
297C22⋊Q8—28no—
298C22⋊Q8—216no—
299C22⋊Q8—24no—
300C22⋊Q8—28no—
301#121—22no—
302#121—24no—
303#121—24no—
304#121—28no—
305#121C23×C3114yes—
306C4×Q8D4×C3114yes—
307C4×Q8C23×C3114yes—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—24no—
311C4×D4—22no—
312C4×D4C23×C3114yes—
313C4×D4C23×C3118yes—
314C4×D4C23×C3114yes—
315C4×D4C23×C3118yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4D4×C3111yescharacteristic
321C2×C4⋊C4D4×C3112yes—
322C2×C4⋊C4C23×C3111yescharacteristic
323C2×C4⋊C4C23×C3112yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C42C23×C3112yes—
327C2×C42C23×C3111yescharacteristic
328#107C23×C3118yes—
329#106C23×C31116yes—
330C4⋊C8C23×C3114yes—
331C4⋊C8C23×C3118yes—
332#102C23×C3118yes—
333#102C23×C3118yes—
334#102C23×C31116yes—
335C22⋊C8C23×C3114yes—
336#97C23×C3118yes—
337C4×C8C23×C3114yes—
338C31#6182511yescharacteristic, direct product factor, Sylow
339C24—21no—
340#41—416no—
341#41—416no—
342#41—48no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—416no—
348C2×Q8—82no—
349C2×Q8C2×D4×C3114yes—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4C24×C3114yes—
358C2×D4C24×C3112yes—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—44no—
366C22×C4—84no—
367C22×C4C2×D4×C3112yes—
368C22×C4C2×D4×C3111yescharacteristic
369C22×C4C2×D4×C3112yes—
370C22×C4C2×D4×C3111yescharacteristic
371C22×C4C24×C3111yescharacteristic
372D8—216no—
373C2×C8C24×C3118yes—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4C2×D4×C3114yes—
382C4⋊C4C2×D4×C3118yes—
383C4⋊C4C24×C3114yes—
384C4⋊C4C24×C3112yes—
385C4⋊C4C24×C3114yes—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—28no—
391C42C2×D4×C3114yes—
392C42C2×D4×C3112yes—
393C42C24×C3112yes—
394C42C24×C3114yes—
395C23—24no—
396C23—22no—
397C23—24no—
398C23—42no—
399C23—42no—
400C23C22×D4×C3111yescharacteristic
401Q8—28no—
402Q8—48no—
403Q8—44no—
404Q8—84no—
405D4—24no—
406D4—22no—
407D4—28no—
408D4—48no—
409D4—48no—
410D4—44no—
411D4—84no—
412D4—82no—
413D4C31×2−1+414yescomplemented
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—416no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—88no—
427C2×C4C31×AGL(1,ℤ/8)14yes—
428C2×C4C22×D4×C3111yescharacteristic
429C2×C4C22×D4×C3112yes—
430C2×C4C22×D4×C3114yes—
431C2×C4C22×D4×C3114yes—
432C2×C4C22×D4×C3112yes—
433C2×C4C25×C3111yescharacteristic
434C8—28no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—48no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22C2×C31×AGL(1,ℤ/8)12yes—
445C22C23×D4×C3111yescharacteristic
446C4—21no—
447C4—24no—
448C4—24no—
449C4—22no—
450C4—44no—
451C4—42no—
452C4—44no—
453C4C2×C31×AGL(1,ℤ/8)14yes—
454C4C2×C31×2−1+412yes—
455C2—22no—
456C2—44no—
457C2—84no—
458C2C31×#348611yescharacteristic
459C2C22×C31×AGL(1,ℤ/8)11yescharacteristic
460C2C31×#358511yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3497C2semidirect
2. C2×C31×#554C2semidirect
3. C31×#3384C2semidirect
4. C31×#3366C2semidirect
5. C31×#3234C2semidirect
6. C31×#3227C2semidirect
7. C31×#3109C2semidirect
8. C31×#3100C2semidirect
9. C31×#2983C2semidirect
13. C31×#550C22semidirect
14. D4×Q8×C31C22semidirect
28. C31×#501C22semidirect
29. C31×#498C22semidirect
30. C31×#490C22semidirect
31. C31×#486C22semidirect
34. C31×#478C22semidirect
36. C31×#476C22semidirect
41. C31×#446C22semidirect
164. #61825C31direct (N × H)
183. D4×C312−1+4semidirect
205. #3497C2×C31semidirect
206. C2×#554C2×C31semidirect
207. #3384C2×C31semidirect
208. #3366C2×C31semidirect
209. #3234C2×C31semidirect
210. #3227C2×C31semidirect
211. #3109C2×C31semidirect
212. #3100C2×C31semidirect
213. #2983C2×C31semidirect
237. #550C22×C31semidirect
238. D4×Q8C22×C31semidirect
252. #501C22×C31semidirect
253. #498C22×C31semidirect
254. #490C22×C31semidirect
255. #486C22×C31semidirect
258. #478C22×C31semidirect
260. #476C22×C31semidirect
265. #446C22×C31semidirect
338. C31#61825direct (N × H)
413. D4C31×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C31×#34972
C2×C31×#5542
C31×#33842
C31×#33662
C31×#32342
C31×#32272
C31×#31092
C31×#31002
C31×#29832
#6182531