Finite Group Databasebeta

← C31×#18362 #21475121

Subgroups — C31×#18362 #21475121

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic
2C31×#3239C212yesmaximal, complemented
3C31×#2628C211yesmaximal, characteristic, complemented
4C31×#1615C214yesmaximal, complemented
5C31×#1614C212yesmaximal, complemented
6C31×#1613C212yesmaximal
7C31×#1512C212yesmaximal, complemented
8C31×#1464C211yesmaximal, characteristic, complemented
9C31×#552—22no—
10C31×#552—22no—
11C31×#551—22no—
12C31×#551—22no—
13C31×#543C2211yescharacteristic
14C31×#539—21no—
15C31×#539C2212yes—
16C31×#534C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic, complemented
18C31×#529—21no—
19C31×#529C2212yes—
20C2×C31×#123—21no—
21C31×#518—21no—
22C2×C31×#117C2211yescharacteristic
23C31×#501—22no—
24C31×#489—22no—
25C31×#484C2214yescomplemented
26C31×#481C2214yes—
27C31×#475—22no—
28C31×#474—22no—
29C31×#432C2212yescomplemented
30C31×#400C2212yescomplemented
31C31×#399C2212yes—
32C31×#395—21no—
33C31×#393—21no—
34C31×#389—21no—
35C31×#382C2211yescharacteristic, complemented
36C31×#380—21no—
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C22×Q8×C31—21no—
42C22×D4×C31—21no—
43C23×C4×C31D411yescharacteristic
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8—22no—
46C31×#126—22no—
47C31×#126—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#125—21no—
51C31×#125—22no—
52C31×#125—22no—
53C31×#125D412yes—
54C31×#124—21no—
55C31×#124D412yes—
56C31×#123—22no—
57C31×#123—22no—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8C2312yes—
69C31×#121—21no—
70C31×#121C2312yes—
71C31×#120—22no—
72C4×Q8×C31—21no—
73C4×Q8×C31D412yes—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31—21no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C4×D4×C31D412yes—
80C31×#117—22no—
81C31×#117—22no—
82C31×#117—22no—
83C31×#117—22no—
84C31×#117—22no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31C2311yescharacteristic
100C2×C42×C31D411yescharacteristic
101C2×C42×C31C2311yescharacteristic
102C31×#107—24no—
103C31×#106—24no—
104C31×C4⋊C8—22no—
105C31×#103—24no—
106C31×#102—24no—
107C31×C22⋊C8C2314yes—
108C31×#97—22no—
109C31×#95—22no—
110C31×#95—21no—
111C31×#95—22no—
112C31×#95C2312yes—
113C24×C31—21no—
114C24×C31C2×D411yescharacteristic
115C2×Q8×C31—21no—
116C2×Q8×C31—42no—
117C2×Q8×C31—82no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31C2×D412yes—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—41no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2411yescharacteristic
142C2×C8×C31—24no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—42no—
157C4⋊C4×C31—41no—
158C4⋊C4×C31—41no—
159C4⋊C4×C31—82no—
160C4⋊C4×C31C2×D412yes—
161C4⋊C4×C31C2×D412yes—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
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—
174C22⋊C4×C31—82no—
175C42×C31—22no—
176C42×C31—21no—
177C42×C31—21no—
178C42×C31—22no—
179C42×C31C2×D412yes—
180C42×C31C2×D412yes—
181#18362C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31—21no—
183C23×C31—21no—
184C23×C31—22no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—81no—
189C23×C31—81no—
190C23×C31#13712yes—
191C23×C31C22×D411yescharacteristic
192Q8×C31—42no—
193Q8×C31—84no—
194Q8×C31—81no—
195D4×C31—42no—
196D4×C31—44no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—81no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—82no—
224C2×C4×C31—81no—
225C2×C4×C31—82no—
226C2×C4×C31—81no—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12012yes—
230C2×C4×C31C22×D412yes—
231C8×C31—44no—
232#3491C2×C3111yescharacteristic
233#3239C2×C3112yescomplemented
234#2628C2×C3111yescharacteristic, complemented
235#1615C2×C3114yescomplemented
236#1614C2×C3112yescomplemented
237#1613C2×C3112yes—
238#1512C2×C3112yescomplemented
239#1464C2×C3111yescharacteristic, complemented
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—22no—
243C22×C31—42no—
244C22×C31—41no—
245C22×C31—41no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—81no—
250C22×C31—162no—
251C22×C31C2×#12011yescharacteristic
252C22×C31C2×#13711yescharacteristic
253C22×C31#57511yescharacteristic
254C4×C31—21no—
255C4×C31—21no—
256C4×C31—22no—
257C4×C31—22no—
258C4×C31—41no—
259C4×C31—42no—
260C4×C31—41no—
261C4×C31—42no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—82no—
265C4×C31—82no—
266#552—22no—
267#552—22no—
268#551—22no—
269#551—22no—
270#543C22×C3111yescharacteristic
271#539—21no—
272#539C22×C3112yes—
273#534C22×C3111yescharacteristic
274C2×C4⋊Q8C22×C3111yescharacteristic, complemented
275#529—21no—
276#529C22×C3112yes—
277C2×#123—21no—
278#518—21no—
279C2×#117C22×C3111yescharacteristic
280#501—22no—
281#489—22no—
282#484C22×C3114yescomplemented
283#481C22×C3114yes—
284#475—22no—
285#474—22no—
286#432C22×C3112yescomplemented
287#400C22×C3112yescomplemented
288#399C22×C3112yes—
289#395—21no—
290#393—21no—
291#389—21no—
292#382C22×C3111yescharacteristic, complemented
293#380—21no—
294#328C22×C3114yescomplemented
295#327C22×C3114yes—
296#324C22×C3112yescomplemented
297#323C22×C3112yes—
298C2×C31—21no—
299C2×C31—21no—
300C2×C31—42no—
301C2×C31—82no—
302C2×C31#300711yescharacteristic
303C2×C31#302111yescharacteristic
304C2×C31#302811yescharacteristic
305C22×Q8—21no—
306C22×D4—21no—
307C23×C4D4×C3111yescharacteristic
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#126—22no—
311#126—22no—
312#126—22no—
313#126—22no—
314#125—21no—
315#125—22no—
316#125—22no—
317#125D4×C3112yes—
318#124—21no—
319#124D4×C3112yes—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—22no—
325#123—22no—
326#123—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—22no—
332C22⋊Q8C23×C3112yes—
333#121—21no—
334#121C23×C3112yes—
335#120—22no—
336C4×Q8—21no—
337C4×Q8D4×C3112yes—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4D4×C3112yes—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348#117—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4C23×C3112yes—
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4D4×C3111yescharacteristic
363C2×C22⋊C4C23×C3111yescharacteristic
364C2×C42D4×C3111yescharacteristic
365C2×C42C23×C3111yescharacteristic
366#107—24no—
367#106—24no—
368C4⋊C8—22no—
369#103—24no—
370#102—24no—
371C22⋊C8C23×C3114yes—
372#97—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95C23×C3112yes—
377C31#1836211yescharacteristic, direct product factor, Sylow
378C24—21no—
379C24C2×D4×C3111yescharacteristic
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—82no—
383C2×Q8C2×D4×C3112yes—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C2×D4×C3112yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C24×C3111yescharacteristic
407C2×C8—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—82no—
425C4⋊C4C2×D4×C3112yes—
426C4⋊C4C2×D4×C3112yes—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—82no—
440C42—22no—
441C42—21no—
442C42—21no—
443C42—22no—
444C42C2×D4×C3112yes—
445C42C2×D4×C3112yes—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—81no—
453C23—81no—
454C23C31×#13712yes—
455C23C22×D4×C3111yescharacteristic
456Q8—42no—
457Q8—84no—
458Q8—81no—
459D4—42no—
460D4—44no—
461D4—42no—
462D4—42no—
463D4—81no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4C31×#12011yescharacteristic
492C2×C4C31×#12011yescharacteristic
493C2×C4C31×#12012yes—
494C2×C4C22×D4×C3112yes—
495C8—44no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—162no—
507C22C2×C31×#12011yescharacteristic
508C22C2×C31×#13711yescharacteristic
509C22C31×#57511yescharacteristic
510C4—21no—
511C4—21no—
512C4—22no—
513C4—22no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—82no—
521C4—82no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—82no—
526C2C31×#300711yescharacteristic
527C2C31×#302111yescharacteristic
528C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#534C22
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×#3239C2semidirect
3. C31×#2628C2semidirect
4. C31×#1615C2semidirect
5. C31×#1614C2semidirect
7. C31×#1512C2semidirect
8. C31×#1464C2semidirect
17. C2×C31×C4⋊Q8C22semidirect
25. C31×#484C22semidirect
29. C31×#432C22semidirect
30. C31×#400C22semidirect
35. C31×#382C22semidirect
37. C31×#328C22semidirect
39. C31×#324C22semidirect
181. #18362C31direct (N × H)
233. #3239C2×C31semidirect
234. #2628C2×C31semidirect
235. #1615C2×C31semidirect
236. #1614C2×C31semidirect
238. #1512C2×C31semidirect
239. #1464C2×C31semidirect
274. C2×C4⋊Q8C22×C31semidirect
282. #484C22×C31semidirect
286. #432C22×C31semidirect
287. #400C22×C31semidirect
292. #382C22×C31semidirect
294. #328C22×C31semidirect
296. #324C22×C31semidirect
377. C31#18362direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#32392
C31×#26282
C31×#16152
C31×#16142
C31×#16132
C31×#15122
C31×#14642
#1836231