Finite Group Databasebeta

← C31×#18344 #21475103

Subgroups — C31×#18344 #21475103

Order 7936.

924 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
1C31×#3461C211yesmaximal, characteristic
2C31×#3238C212yesmaximal, complemented
3C31×#2546C211yesmaximal, characteristic, complemented
4C31×#1611C212yesmaximal, complemented
5C31×#1610C212yesmaximal
6C31×#1609C214yesmaximal, complemented
7C31×#1509C212yesmaximal, complemented
8C31×#1467C211yesmaximal, characteristic, complemented
9C31×#544C2211yescharacteristic
10C31×#542—22no—
11C31×#541—22no—
12C31×#539—21no—
13C31×#539C2212yes—
14C31×#537C2211yescharacteristic
15C31×#536—21no—
16C31×#533—22no—
17C2×C31×C4⋊Q8—21no—
18C2×C31×C4⋊Q8C2211yescharacteristic
19C31×#529—22no—
20C31×#529—21no—
21C31×#529C2212yes—
22C4×#41×C31—21no—
23C2×C4×Q8×C31C2211yescharacteristic
24C31×#501—22no—
25C31×#489—22no—
26C31×#483C2214yes—
27C31×#480C2214yes—
28C31×#477—22no—
29C31×#476—22no—
30C31×#432C2212yes—
31C31×#398C2212yes—
32C31×#398C2212yes—
33C31×#391—21no—
34C31×#389—21no—
35C31×#389—21no—
36C4×C4⋊C4×C31C2211yescharacteristic
37C31×#328C2214yes—
38C31×#327C2214yes—
39C31×#324C2212yes—
40C31×#324C2212yes—
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8—22no—
46C31×C4⋊Q8—22no—
47C31×C4⋊Q8—22no—
48C31×C4⋊Q8—44no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#125—21no—
54C31×#125—22no—
55C31×#125—22no—
56C31×#125—22no—
57C31×#125D412yes—
58C31×#124—22no—
59C31×#124—21no—
60C31×#124D412yes—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8C2312yes—
72C31×#121—22no—
73C31×#121—21no—
74C31×#121C2312yes—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31—22no—
79C4×Q8×C31—22no—
80C4×Q8×C31D412yes—
81C4×D4×C31—21no—
82C4×D4×C31—21no—
83C4×D4×C31—21no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31D412yes—
87C31×#117—22no—
88C31×#117—22no—
89C31×#117—21no—
90C31×#117—22no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31D411yescharacteristic
96C2×C4⋊C4×C31C2312yes—
97C2×C4⋊C4×C31C2311yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C42×C31—22no—
100C2×C42×C31—21no—
101C2×C42×C31D411yescharacteristic
102C2×C42×C31D411yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×#107—24no—
105C31×#106—24no—
106C31×C4⋊C8—22no—
107C31×#103—24no—
108C31×#102—24no—
109C31×C22⋊C8C2314yes—
110C31×#97—22no—
111C31×#95—22no—
112C31×#95—22no—
113C31×#95C2312yes—
114#41×C31—42no—
115#41×C31—42no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31—41no—
121C2×Q8×C31—42no—
122C2×Q8×C31—42no—
123C2×Q8×C31—42no—
124C2×Q8×C31C2×D412yes—
125C2×D4×C31—21no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31C2×D412yes—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2411yescharacteristic
141C2×C8×C31—24no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—44no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—44no—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31C2×D412yes—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C42×C31—22no—
175C42×C31—22no—
176C42×C31—21no—
177C42×C31—22no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—42no—
182C42×C31C2×D412yes—
183C42×C31C2×D412yes—
184#18344C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31C22×D411yescharacteristic
187Q8×C31—44no—
188Q8×C31—42no—
189Q8×C31—42no—
190Q8×C31—84no—
191Q8×C31—81no—
192D4×C31—42no—
193D4×C31—42no—
194D4×C31—42no—
195D4×C31—42no—
196D4×C31—81no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—21no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—22no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—41no—
212C2×C4×C31—44no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—44no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—82no—
222C2×C4×C31#12011yescharacteristic
223C2×C4×C31#12012yes—
224C2×C4×C31#12011yescharacteristic
225C2×C4×C31AGL(1,ℤ/8)12yes—
226C2×C4×C31C22×D412yes—
227C8×C31—44no—
228#3461C2×C3111yescharacteristic
229#3238C2×C3112yescomplemented
230#2546C2×C3111yescharacteristic, complemented
231#1611C2×C3112yescomplemented
232#1610C2×C3112yes—
233#1609C2×C3114yescomplemented
234#1509C2×C3112yescomplemented
235#1467C2×C3111yescharacteristic, complemented
236C22×C31—21no—
237C22×C31—21no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31C2×#12011yescharacteristic
242C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C31#57511yescharacteristic
244C4×C31—22no—
245C4×C31—22no—
246C4×C31—21no—
247C4×C31—22no—
248C4×C31—44no—
249C4×C31—41no—
250C4×C31—42no—
251C4×C31—42no—
252C4×C31—41no—
253C4×C31—41no—
254C4×C31—42no—
255C4×C31—82no—
256C4×C31#45612yes—
257#544C22×C3111yescharacteristic
258#542—22no—
259#541—22no—
260#539—21no—
261#539C22×C3112yes—
262#537C22×C3111yescharacteristic
263#536—21no—
264#533—22no—
265C2×C4⋊Q8—21no—
266C2×C4⋊Q8C22×C3111yescharacteristic
267#529—22no—
268#529—21no—
269#529C22×C3112yes—
270C4×#41—21no—
271C2×C4×Q8C22×C3111yescharacteristic
272#501—22no—
273#489—22no—
274#483C22×C3114yes—
275#480C22×C3114yes—
276#477—22no—
277#476—22no—
278#432C22×C3112yes—
279#398C22×C3112yes—
280#398C22×C3112yes—
281#391—21no—
282#389—21no—
283#389—21no—
284C4×C4⋊C4C22×C3111yescharacteristic
285#328C22×C3114yes—
286#327C22×C3114yes—
287#324C22×C3112yes—
288#324C22×C3112yes—
289C2×C31—21no—
290C2×C31—21no—
291C2×C31—82no—
292C2×C31#300611yescharacteristic
293C2×C31C2×#45611yescharacteristic
294C2×C31#302811yescharacteristic
295C2×#41—21no—
296C22×Q8—21no—
297C22×Q8D4×C3111yescharacteristic
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—44no—
303#126—22no—
304#126—22no—
305#126—22no—
306#126—22no—
307#125—21no—
308#125—22no—
309#125—22no—
310#125—22no—
311#125D4×C3112yes—
312#124—22no—
313#124—21no—
314#124D4×C3112yes—
315#123—22no—
316#123—22no—
317#123—22no—
318#123—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—22no—
325C22⋊Q8C23×C3112yes—
326#121—22no—
327#121—21no—
328#121C23×C3112yes—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8D4×C3112yes—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4D4×C3112yes—
341#117—22no—
342#117—22no—
343#117—21no—
344#117—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D4×C3111yescharacteristic
350C2×C4⋊C4C23×C3112yes—
351C2×C4⋊C4C23×C3111yescharacteristic
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C42—22no—
354C2×C42—21no—
355C2×C42D4×C3111yescharacteristic
356C2×C42D4×C3111yescharacteristic
357C2×C42C23×C3111yescharacteristic
358#107—24no—
359#106—24no—
360C4⋊C8—22no—
361#103—24no—
362#102—24no—
363C22⋊C8C23×C3114yes—
364#97—22no—
365#95—22no—
366#95—22no—
367#95C23×C3112yes—
368C31#1834411yescharacteristic, direct product factor, Sylow
369#41—42no—
370#41—42no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8C2×D4×C3112yes—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4C2×D4×C3112yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4C2×D4×C3111yescharacteristic
390C22×C4C2×D4×C3111yescharacteristic
391C22×C4C2×D4×C3111yescharacteristic
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3111yescharacteristic
394C22×C4C2×D4×C3111yescharacteristic
395C22×C4C24×C3111yescharacteristic
396C2×C8—24no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—44no—
418C4⋊C4C2×D4×C3112yes—
419C4⋊C4C2×D4×C3112yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C42—22no—
430C42—22no—
431C42—21no—
432C42—22no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—42no—
437C42C2×D4×C3112yes—
438C42C2×D4×C3112yes—
439C23—22no—
440C23C22×D4×C3111yescharacteristic
441Q8—44no—
442Q8—42no—
443Q8—42no—
444Q8—84no—
445Q8—81no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—81no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4C31×#12011yescharacteristic
477C2×C4C31×#12012yes—
478C2×C4C31×#12011yescharacteristic
479C2×C4C31×AGL(1,ℤ/8)12yes—
480C2×C4C22×D4×C3112yes—
481C8—44no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22C2×C31×#12011yescharacteristic
488C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
489C22C31×#57511yescharacteristic
490C4—22no—
491C4—22no—
492C4—21no—
493C4—22no—
494C4—44no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—41no—
499C4—41no—
500C4—42no—
501C4—82no—
502C4C31×#45612yes—
503C2—21no—
504C2—21no—
505C2—82no—
506C2C31×#300611yescharacteristic
507C2C2×C31×#45611yescharacteristic
508C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#537C22
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×#3238C2semidirect
3. C31×#2546C2semidirect
4. C31×#1611C2semidirect
6. C31×#1609C2semidirect
7. C31×#1509C2semidirect
8. C31×#1467C2semidirect
184. #18344C31direct (N × H)
229. #3238C2×C31semidirect
230. #2546C2×C31semidirect
231. #1611C2×C31semidirect
233. #1609C2×C31semidirect
234. #1509C2×C31semidirect
235. #1467C2×C31semidirect
368. C31#18344direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34612
C31×#32382
C31×#25462
C31×#16112
C31×#16102
C31×#16092
C31×#15092
C31×#14672
#1834431