Finite Group Databasebeta

← C31×#18141 #21474900

Subgroups — C31×#18141 #21474900

Order 7936.

1004 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic
2C31×#3092C212yesmaximal, complemented
3C31×#2487C211yesmaximal, characteristic, complemented
4C31×#1615C214yesmaximal, complemented
5C31×#1614C212yesmaximal, complemented
6C31×#1613C212yesmaximal, complemented
7C31×#1513C212yesmaximal, complemented
8C31×#1465C211yesmaximal, characteristic, complemented
9C31×#543C2211yescharacteristic
10C31×#539—22no—
11C31×#539C2212yes—
12C31×#538—22no—
13C31×#538—21no—
14C31×#534—21no—
15C31×#534C2211yescharacteristic
16C31×#529—22no—
17C31×#529—21no—
18C31×#525—22no—
19C31×#525C2212yes—
20C2×C4×D4×C31—21no—
21C2×C4×D4×C31C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic, complemented
23C31×#484C2214yescomplemented
24C31×#483—22no—
25C31×#481C2214yes—
26C31×#480—22no—
27C31×#428—22no—
28C31×#417—22no—
29C31×#407C2212yescomplemented
30C31×#400C2212yescomplemented
31C31×#399C2212yes—
32C31×#389—21no—
33C31×#386—21no—
34C31×#385—21no—
35C31×#380—21no—
36C31×#380C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C2×#41×C31D412yes—
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C23×C4×C31—21no—
45C23×C4×C31—21no—
46C23×C4×C31D411yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×M16×C31—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#125—21no—
53C31×#125—22no—
54C31×#124—22no—
55C31×#124—21no—
56C31×#123—22no—
57C31×#123—22no—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8D412yes—
67C31×C22⋊Q8C2312yes—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—21no—
72C31×#121D412yes—
73C31×#121C2312yes—
74C31×#120—22no—
75C31×#120—22no—
76C31×#120—22no—
77C4×Q8×C31—21no—
78C4×D4×C31—22no—
79C4×D4×C31—22no—
80C4×D4×C31—21no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C31×#117—22no—
87C31×#117D412yes—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31—22no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31C2312yes—
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31D411yescharacteristic
101C2×C22⋊C4×C31C2311yescharacteristic
102C2×C42×C31—21no—
103C2×C42×C31D411yescharacteristic
104C31×#107—24no—
105C31×#106—24no—
106C31×#103—24no—
107C31×#102—24no—
108C31×C22⋊C8—22no—
109C31×C22⋊C8C2314yes—
110C31×#95—22no—
111C31×#95—22no—
112C31×#95C2312yes—
113C24×C31—21no—
114C24×C31C2×D411yescharacteristic
115C24×C31C2×D411yescharacteristic
116#41×C31—44no—
117C2×Q8×C31—21no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—41no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31C2×D412yes—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—82no—
143C22×C4×C31—82no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2411yescharacteristic
151M16×C31—44no—
152C2×C8×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31—41no—
165C4⋊C4×C31—41no—
166C4⋊C4×C31—82no—
167C4⋊C4×C31C2×D412yes—
168C4⋊C4×C31C2×D412yes—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—82no—
183C42×C31—22no—
184C42×C31—22no—
185C42×C31—21no—
186C42×C31—21no—
187#18141C3111yesmaximal, characteristic, direct product factor, Sylow
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—22no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—42no—
198C23×C31—81no—
199C23×C31—81no—
200C23×C31#12012yes—
201C23×C31#13712yes—
202C23×C31C22×D411yescharacteristic
203Q8×C31—42no—
204Q8×C31—81no—
205D4×C31—42no—
206D4×C31—44no—
207D4×C31—44no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—82no—
211D4×C31—82no—
212D4×C31—81no—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—21no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—44no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—81no—
237C2×C4×C31—81no—
238C2×C4×C31—82no—
239C2×C4×C31—81no—
240C2×C4×C31—84no—
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31C22×D412yes—
244C8×C31—44no—
245#3459C2×C3111yescharacteristic
246#3092C2×C3112yescomplemented
247#2487C2×C3111yescharacteristic, complemented
248#1615C2×C3114yescomplemented
249#1614C2×C3112yescomplemented
250#1613C2×C3112yescomplemented
251#1513C2×C3112yescomplemented
252#1465C2×C3111yescharacteristic, complemented
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—22no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—41no—
263C22×C31—41no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—81no—
268C22×C31—84no—
269C22×C31C2×#12011yescharacteristic
270C22×C31C2×#13711yescharacteristic
271C22×C31#57511yescharacteristic
272C4×C31—22no—
273C4×C31—21no—
274C4×C31—41no—
275C4×C31—42no—
276C4×C31—41no—
277C4×C31—42no—
278C4×C31—41no—
279C4×C31—42no—
280C4×C31—82no—
281C4×C31—82no—
282C4×C31#45812yes—
283#543C22×C3111yescharacteristic
284#539—22no—
285#539C22×C3112yes—
286#538—22no—
287#538—21no—
288#534—21no—
289#534C22×C3111yescharacteristic
290#529—22no—
291#529—21no—
292#525—22no—
293#525C22×C3112yes—
294C2×C4×D4—21no—
295C2×C4×D4C22×C3111yescharacteristic
296C22×C4⋊C4C22×C3111yescharacteristic, complemented
297#484C22×C3114yescomplemented
298#483—22no—
299#481C22×C3114yes—
300#480—22no—
301#428—22no—
302#417—22no—
303#407C22×C3112yescomplemented
304#400C22×C3112yescomplemented
305#399C22×C3112yes—
306#389—21no—
307#386—21no—
308#385—21no—
309#380—21no—
310#380C22×C3111yescharacteristic, complemented
311#328C22×C3114yescomplemented
312#327C22×C3114yes—
313#324C22×C3112yescomplemented
314#323C22×C3112yes—
315C2×C31—21no—
316C2×C31—21no—
317C2×C31—42no—
318C2×C31—42no—
319C2×C31—82no—
320C2×C31#300711yescharacteristic
321C2×C31#302111yescharacteristic
322C2×C31C2×#45811yescharacteristic
323C2×#41D4×C3112yes—
324C22×D4—21no—
325C22×D4D4×C3111yescharacteristic
326C23×C4—21no—
327C23×C4—21no—
328C23×C4D4×C3111yescharacteristic
329C23×C4C23×C3111yescharacteristic
330C2×M16—22no—
331#126—22no—
332#126—22no—
333#126—22no—
334#125—21no—
335#125—22no—
336#124—22no—
337#124—21no—
338#123—22no—
339#123—22no—
340#123—22no—
341#123—22no—
342#123—22no—
343#123—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—21no—
347C22⋊Q8—22no—
348C22⋊Q8D4×C3112yes—
349C22⋊Q8C23×C3112yes—
350#121—22no—
351#121—22no—
352#121—22no—
353#121—21no—
354#121D4×C3112yes—
355#121C23×C3112yes—
356#120—22no—
357#120—22no—
358#120—22no—
359C4×Q8—21no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—21no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368#117—22no—
369#117D4×C3112yes—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C23×C3112yes—
377C2×C4⋊C4C23×C3111yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4D4×C3111yescharacteristic
383C2×C22⋊C4C23×C3111yescharacteristic
384C2×C42—21no—
385C2×C42D4×C3111yescharacteristic
386#107—24no—
387#106—24no—
388#103—24no—
389#102—24no—
390C22⋊C8—22no—
391C22⋊C8C23×C3114yes—
392#95—22no—
393#95—22no—
394#95C23×C3112yes—
395C31#1814111yescharacteristic, direct product factor, Sylow
396C24—21no—
397C24C2×D4×C3111yescharacteristic
398C24C2×D4×C3111yescharacteristic
399#41—44no—
400C2×Q8—21no—
401C2×Q8C2×D4×C3112yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4C2×D4×C3112yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3112yes—
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3112yes—
433C22×C4C24×C3111yescharacteristic
434M16—44no—
435C2×C8—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—42no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—82no—
450C4⋊C4C2×D4×C3112yes—
451C4⋊C4C2×D4×C3112yes—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—42no—
465C22⋊C4—82no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—81no—
481C23—81no—
482C23C31×#12012yes—
483C23C31×#13712yes—
484C23C22×D4×C3111yescharacteristic
485Q8—42no—
486Q8—81no—
487D4—42no—
488D4—44no—
489D4—44no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—82no—
494D4—81no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4C31×#12011yescharacteristic
524C2×C4C31×#12011yescharacteristic
525C2×C4C22×D4×C3112yes—
526C8—44no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—84no—
543C22C2×C31×#12011yescharacteristic
544C22C2×C31×#13711yescharacteristic
545C22C31×#57511yescharacteristic
546C4—22no—
547C4—21no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—41no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4C31×#45812yes—
557C2—21no—
558C2—21no—
559C2—42no—
560C2—42no—
561C2—82no—
562C2C31×#300711yescharacteristic
563C2C31×#302111yescharacteristic
564C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3092C2semidirect
3. C31×#2487C2semidirect
4. C31×#1615C2semidirect
5. C31×#1614C2semidirect
6. C31×#1613C2semidirect
7. C31×#1513C2semidirect
8. C31×#1465C2semidirect
22. C22×C4⋊C4×C31C22semidirect
23. C31×#484C22semidirect
29. C31×#407C22semidirect
30. C31×#400C22semidirect
36. C31×#380C22semidirect
37. C31×#328C22semidirect
39. C31×#324C22semidirect
187. #18141C31direct (N × H)
246. #3092C2×C31semidirect
247. #2487C2×C31semidirect
248. #1615C2×C31semidirect
249. #1614C2×C31semidirect
250. #1613C2×C31semidirect
251. #1513C2×C31semidirect
252. #1465C2×C31semidirect
296. C22×C4⋊C4C22×C31semidirect
297. #484C22×C31semidirect
303. #407C22×C31semidirect
304. #400C22×C31semidirect
310. #380C22×C31semidirect
311. #328C22×C31semidirect
313. #324C22×C31semidirect
395. C31#18141direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34592
C31×#30922
C31×#24872
C31×#16152
C31×#16142
C31×#16132
C31×#15132
C31×#14652
#1814131