Finite Group Databasebeta

← C31×#18309 #21475068

Subgroups — C31×#18309 #21475068

Order 7936.

1168 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3449C211yesmaximal, characteristic
2C31×#3227C212yesmaximal, complemented
3C31×#2628C211yesmaximal, characteristic, complemented
4C31×#1616C214yesmaximal, complemented
5C31×#1613C214yesmaximal, complemented
6C31×#1517C212yesmaximal, complemented
7C31×#1458C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#537—21no—
10C31×#537C2212yes—
11C31×#536—22no—
12C31×#535—22no—
13C31×#534—21no—
14C31×#534C2212yes—
15C31×#532—21no—
16C31×#532C2212yes—
17C2×C31×C4⋊Q8C2211yescharacteristic, complemented
18C31×#525—24no—
19C2×C31×#117C2211yescharacteristic
20C31×#498—22no—
21C31×#486—22no—
22C31×#481C2218yescomplemented
23C31×#476—24no—
24C31×#431C2212yescomplemented
25C31×#400C2212yescomplemented
26C31×#399C2212yes—
27C31×#395—21no—
28C31×#393—21no—
29C31×#389—21no—
30C31×#382C2211yescharacteristic, complemented
31C31×#380—21no—
32C31×#327C2218yescomplemented
33C31×#324C2212yescomplemented
34C31×#323C2212yes—
35C2×#41×C31—22no—
36C2×#41×C31—22no—
37C2×#41×C31—22no—
38C2×#41×C31—22no—
39C2×#41×C31—44no—
40C2×#41×C31—44no—
41C22×Q8×C31—21no—
42C22×Q8×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31—22no—
45C23×C4×C31—22no—
46C23×C4×C31D411yescharacteristic
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—22no—
49C31×C4⋊Q8—22no—
50C31×C4⋊Q8D412yes—
51C31×#127—21no—
52C31×#127D412yes—
53C31×#124—22no—
54C31×#124—22no—
55C31×#123—24no—
56C31×#123—24no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—24no—
59C31×#121—24no—
60C31×#121—24no—
61C31×#121—22no—
62C31×#121—22no—
63C31×#121—22no—
64C31×#121C2314yes—
65C31×#120—24no—
66C31×#120—24no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31D414yes—
71C31×#117—22no—
72C31×#117—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31D411yescharacteristic
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31D411yescharacteristic
81C2×C22⋊C4×C31C2311yescharacteristic
82C2×C42×C31D411yescharacteristic
83C2×C42×C31C2311yescharacteristic
84C31×#106—28no—
85C31×C4⋊C8—22no—
86C31×#102—28no—
87C31×C22⋊C8C2314yes—
88C4×C8×C31—22no—
89C31×#95—22no—
90C31×#95—21no—
91C31×#95—22no—
92C31×#95C2312yes—
93C24×C31—22no—
94C24×C31C2×D411yescharacteristic
95#41×C31—44no—
96#41×C31—48no—
97#41×C31—48no—
98#41×C31—44no—
99#41×C31—44no—
100#41×C31—44no—
101C2×Q8×C31—22no—
102C2×Q8×C31—22no—
103C2×Q8×C31—44no—
104C2×Q8×C31—42no—
105C2×Q8×C31—82no—
106C2×D4×C31—24no—
107C2×D4×C31—22no—
108C2×D4×C31—24no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—22no—
112C2×D4×C31—22no—
113C2×D4×C31—44no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31C2×D414yes—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—24no—
121C22×C4×C31—22no—
122C22×C4×C31—24no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—48no—
126C22×C4×C31SD1614yes—
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2411yescharacteristic
133C2×C8×C31—24no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—41no—
141C4⋊C4×C31—41no—
142C4⋊C4×C31—42no—
143C4⋊C4×C31—82no—
144C4⋊C4×C31C2×D414yes—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—24no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—82no—
152C42×C31—21no—
153C42×C31—21no—
154C42×C31C2×D412yes—
155C42×C31C2×D412yes—
156#18309C3111yesmaximal, characteristic, direct product factor, Sylow
157C23×C31—21no—
158C23×C31—21no—
159C23×C31—24no—
160C23×C31—41no—
161C23×C31—41no—
162C23×C31—44no—
163C23×C31—82no—
164C23×C31—82no—
165C23×C31C2×SD1612yes—
166C23×C31C22×D411yescharacteristic
167Q8×C31—44no—
168Q8×C31—44no—
169Q8×C31—84no—
170D4×C31—44no—
171D4×C31—44no—
172D4×C31—48no—
173D4×C31—44no—
174D4×C31—44no—
175D4×C31—44no—
176D4×C31—82no—
177D4×C31—82no—
178D4×C31—82no—
179C2×C4×C31—21no—
180C2×C4×C31—21no—
181C2×C4×C31—24no—
182C2×C4×C31—24no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—24no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—48no—
189C2×C4×C31—44no—
190C2×C4×C31—48no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31#12012yes—
200C2×C4×C31#12011yescharacteristic
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31C2×SD1612yes—
203C2×C4×C31C2×SD1612yes—
204C2×C4×C31AGL(1,ℤ/8)12yes—
205C2×C4×C31C22×D412yes—
206C8×C31—44no—
207#3449C2×C3111yescharacteristic
208#3227C2×C3112yescomplemented
209#2628C2×C3111yescharacteristic, complemented
210#1616C2×C3114yescomplemented
211#1613C2×C3114yescomplemented
212#1517C2×C3112yescomplemented
213#1458C2×C3111yescharacteristic, complemented
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—22no—
217C22×C31—42no—
218C22×C31—41no—
219C22×C31—41no—
220C22×C31—44no—
221C22×C31—44no—
222C22×C31—44no—
223C22×C31—81no—
224C22×C31—88no—
225C22×C31C2×#12011yescharacteristic
226C22×C31C22×SD1611yescharacteristic
227C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
228C4×C31—22no—
229C4×C31—22no—
230C4×C31—22no—
231C4×C31—44no—
232C4×C31—41no—
233C4×C31—41no—
234C4×C31—82no—
235C4×C31—82no—
236C4×C31#45012yes—
237C4×C31#45012yes—
238C22×#41—21no—
239#537—21no—
240#537C22×C3112yes—
241#536—22no—
242#535—22no—
243#534—21no—
244#534C22×C3112yes—
245#532—21no—
246#532C22×C3112yes—
247C2×C4⋊Q8C22×C3111yescharacteristic, complemented
248#525—24no—
249C2×#117C22×C3111yescharacteristic
250#498—22no—
251#486—22no—
252#481C22×C3118yescomplemented
253#476—24no—
254#431C22×C3112yescomplemented
255#400C22×C3112yescomplemented
256#399C22×C3112yes—
257#395—21no—
258#393—21no—
259#389—21no—
260#382C22×C3111yescharacteristic, complemented
261#380—21no—
262#327C22×C3118yescomplemented
263#324C22×C3112yescomplemented
264#323C22×C3112yes—
265C2×C31—21no—
266C2×C31—21no—
267C2×C31—42no—
268C2×C31—84no—
269C2×C31C2×#45011yescharacteristic
270C2×C31#302111yescharacteristic
271C2×C31#302811yescharacteristic
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—44no—
277C2×#41—44no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×D4—21no—
281C22×D4—22no—
282C23×C4—22no—
283C23×C4D4×C3111yescharacteristic
284C4⋊Q8—21no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8D4×C3112yes—
288#127—21no—
289#127D4×C3112yes—
290#124—22no—
291#124—22no—
292#123—24no—
293#123—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296#121—24no—
297#121—24no—
298#121—22no—
299#121—22no—
300#121—22no—
301#121C23×C3114yes—
302#120—24no—
303#120—24no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4D4×C3114yes—
308#117—22no—
309#117—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4D4×C3111yescharacteristic
314C2×C4⋊C4C23×C3112yes—
315C2×C4⋊C4C23×C3111yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4D4×C3111yescharacteristic
318C2×C22⋊C4C23×C3111yescharacteristic
319C2×C42D4×C3111yescharacteristic
320C2×C42C23×C3111yescharacteristic
321#106—28no—
322C4⋊C8—22no—
323#102—28no—
324C22⋊C8C23×C3114yes—
325C4×C8—22no—
326#95—22no—
327#95—21no—
328#95—22no—
329#95C23×C3112yes—
330C31#1830911yescharacteristic, direct product factor, Sylow
331C24—22no—
332C24C2×D4×C3111yescharacteristic
333#41—44no—
334#41—48no—
335#41—48no—
336#41—44no—
337#41—44no—
338#41—44no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—44no—
342C2×Q8—42no—
343C2×Q8—82no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4C2×D4×C3114yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—48no—
364C22×C4SD16×C3114yes—
365C22×C4C2×D4×C3111yescharacteristic
366C22×C4C2×D4×C3111yescharacteristic
367C22×C4C2×D4×C3111yescharacteristic
368C22×C4C2×D4×C3111yescharacteristic
369C22×C4C2×D4×C3111yescharacteristic
370C22×C4C24×C3111yescharacteristic
371C2×C8—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—82no—
382C4⋊C4C2×D4×C3114yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—82no—
390C42—21no—
391C42—21no—
392C42C2×D4×C3112yes—
393C42C2×D4×C3112yes—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—44no—
400C23—82no—
401C23—82no—
402C23C2×SD16×C3112yes—
403C23C22×D4×C3111yescharacteristic
404Q8—44no—
405Q8—44no—
406Q8—84no—
407D4—44no—
408D4—44no—
409D4—48no—
410D4—44no—
411D4—44no—
412D4—44no—
413D4—82no—
414D4—82no—
415D4—82no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4C31×#12012yes—
437C2×C4C31×#12011yescharacteristic
438C2×C4C31×#12011yescharacteristic
439C2×C4C2×SD16×C3112yes—
440C2×C4C2×SD16×C3112yes—
441C2×C4C31×AGL(1,ℤ/8)12yes—
442C2×C4C22×D4×C3112yes—
443C8—44no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—81no—
454C22—88no—
455C22C2×C31×#12011yescharacteristic
456C22C22×SD16×C3111yescharacteristic
457C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—44no—
462C4—41no—
463C4—41no—
464C4—82no—
465C4—82no—
466C4C31×#45012yes—
467C4C31×#45012yes—
468C2—21no—
469C2—21no—
470C2—42no—
471C2—84no—
472C2C2×C31×#45011yescharacteristic
473C2C31×#302111yescharacteristic
474C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3449C2
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
2. C31×#3227C2semidirect
3. C31×#2628C2semidirect
4. C31×#1616C2semidirect
5. C31×#1613C2semidirect
6. C31×#1517C2semidirect
7. C31×#1458C2semidirect
17. C2×C31×C4⋊Q8C22semidirect
22. C31×#481C22semidirect
24. C31×#431C22semidirect
25. C31×#400C22semidirect
30. C31×#382C22semidirect
32. C31×#327C22semidirect
33. C31×#324C22semidirect
156. #18309C31direct (N × H)
208. #3227C2×C31semidirect
209. #2628C2×C31semidirect
210. #1616C2×C31semidirect
211. #1613C2×C31semidirect
212. #1517C2×C31semidirect
213. #1458C2×C31semidirect
247. C2×C4⋊Q8C22×C31semidirect
252. #481C22×C31semidirect
254. #431C22×C31semidirect
255. #400C22×C31semidirect
260. #382C22×C31semidirect
262. #327C22×C31semidirect
263. #324C22×C31semidirect
330. C31#18309direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34492
C31×#32272
C31×#26282
C31×#16162
C31×#16132
C31×#15172
C31×#14582
#1830931