Finite Group Databasebeta

← C31×#18342 #21475101

Subgroups — C31×#18342 #21475101

Order 7936.

998 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3459C211yesmaximal, characteristic
2C31×#3238C212yesmaximal, complemented
3C31×#2547C211yesmaximal, characteristic, complemented
4C31×#1612C214yesmaximal, complemented
5C31×#1608C212yesmaximal, complemented
6C31×#1607C212yesmaximal, complemented
7C31×#1509C212yesmaximal, complemented
8C31×#1466C211yesmaximal, characteristic, complemented
9C31×#543C2211yescharacteristic
10C31×#539—22no—
11C31×#539—21no—
12C31×#538—22no—
13C31×#538C2212yes—
14C31×#534—21no—
15C31×#534C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic, complemented
17C31×#529—22no—
18C31×#529C2212yes—
19C2×C31×#124—21no—
20C31×#525—21no—
21C31×#525—22no—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#501—22no—
25C31×#489—22no—
26C31×#483C2214yescomplemented
27C31×#480C2214yes—
28C31×#477—22no—
29C31×#476—22no—
30C31×#432C2212yescomplemented
31C31×#397C2212yes—
32C31×#397C2212yescomplemented
33C31×#391—21no—
34C31×#380—21no—
35C31×#380—21no—
36C4×C4⋊C4×C31C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#323C2212yes—
40C31×#323C2212yescomplemented
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31D411yescharacteristic
45C23×C4×C31—21no—
46C23×C4×C31D411yescharacteristic
47C23×C4×C31D411yescharacteristic
48C31×C4⋊Q8—22no—
49C31×C4⋊Q8—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#125—22no—
54C31×#125D412yes—
55C31×#124—22no—
56C31×#124—44no—
57C31×#124D412yes—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8C2312yes—
70C31×#121—22no—
71C31×#121—21no—
72C31×#121—22no—
73C31×#121—22no—
74C31×#121—21no—
75C31×#121C2312yes—
76C31×#120—22no—
77C31×#120—22no—
78C31×#120—22no—
79C4×Q8×C31D412yes—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31D412yes—
88C31×#117—21no—
89C31×#117—22no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31D411yescharacteristic
94C2×C4⋊C4×C31C2312yes—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31C2311yescharacteristic
100C2×C22⋊C4×C31C2311yescharacteristic
101C2×C42×C31—21no—
102C2×C42×C31—21no—
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—
114C24×C31—21no—
115C24×C31C2×D411yescharacteristic
116C24×C31C2×D411yescharacteristic
117#41×C31—82no—
118C2×Q8×C31—21no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31—42no—
122C2×Q8×C31C2×D412yes—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—41no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31C2×D412yes—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—82no—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2411yescharacteristic
153C2×C8×C31—24no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—41no—
166C4⋊C4×C31—41no—
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—44no—
182C22⋊C4×C31—44no—
183C42×C31—22no—
184C42×C31—22no—
185C42×C31—42no—
186C42×C31C2×D412yes—
187C42×C31C2×D412yes—
188#18342C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—22no—
193C23×C31—21no—
194C23×C31—21no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—42no—
200C23×C31—81no—
201C23×C31—81no—
202C23×C31AGL(1,ℤ/8)12yes—
203C23×C31C22×D411yescharacteristic
204Q8×C31—42no—
205Q8×C31—84no—
206Q8×C31—81no—
207D4×C31—44no—
208D4×C31—42no—
209D4×C31—44no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—82no—
213D4×C31—81no—
214D4×C31—82no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
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—81no—
239C2×C4×C31—82no—
240C2×C4×C31—81no—
241C2×C4×C31#12012yes—
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31C22×D412yes—
245C8×C31—44no—
246#3459C2×C3111yescharacteristic
247#3238C2×C3112yescomplemented
248#2547C2×C3111yescharacteristic, complemented
249#1612C2×C3114yescomplemented
250#1608C2×C3112yescomplemented
251#1607C2×C3112yescomplemented
252#1509C2×C3112yescomplemented
253#1466C2×C3111yescharacteristic, complemented
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—22no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—41no—
260C22×C31—41no—
261C22×C31—41no—
262C22×C31—42no—
263C22×C31—41no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—41no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31—84no—
270C22×C31C2×#12011yescharacteristic
271C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C31#57511yescharacteristic
273C4×C31—22no—
274C4×C31—21no—
275C4×C31—22no—
276C4×C31—42no—
277C4×C31—41no—
278C4×C31—41no—
279C4×C31—42no—
280C4×C31—42no—
281C4×C31—82no—
282C4×C31—82no—
283C4×C31#45512yes—
284#543C22×C3111yescharacteristic
285#539—22no—
286#539—21no—
287#538—22no—
288#538C22×C3112yes—
289#534—21no—
290#534C22×C3111yescharacteristic
291C2×C4⋊Q8C22×C3111yescharacteristic, complemented
292#529—22no—
293#529C22×C3112yes—
294C2×#124—21no—
295#525—21no—
296#525—22no—
297C2×C4×D4—21no—
298C2×C4×D4C22×C3111yescharacteristic
299#501—22no—
300#489—22no—
301#483C22×C3114yescomplemented
302#480C22×C3114yes—
303#477—22no—
304#476—22no—
305#432C22×C3112yescomplemented
306#397C22×C3112yes—
307#397C22×C3112yescomplemented
308#391—21no—
309#380—21no—
310#380—21no—
311C4×C4⋊C4C22×C3111yescharacteristic, complemented
312#328C22×C3114yescomplemented
313#327C22×C3114yes—
314#323C22×C3112yes—
315#323C22×C3112yescomplemented
316C2×C31—21no—
317C2×C31—21no—
318C2×C31—42no—
319C2×C31—81no—
320C2×C31—82no—
321C2×C31#300611yescharacteristic
322C2×C31C2×#45511yescharacteristic
323C2×C31#302811yescharacteristic
324C2×#41—21no—
325C22×Q8—21no—
326C22×D4—21no—
327C22×D4D4×C3111yescharacteristic
328C23×C4—21no—
329C23×C4D4×C3111yescharacteristic
330C23×C4D4×C3111yescharacteristic
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333#126—22no—
334#126—22no—
335#126—22no—
336#125—22no—
337#125D4×C3112yes—
338#124—22no—
339#124—44no—
340#124D4×C3112yes—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8C23×C3112yes—
353#121—22no—
354#121—21no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121C23×C3112yes—
359#120—22no—
360#120—22no—
361#120—22no—
362C4×Q8D4×C3112yes—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4D4×C3112yes—
371#117—21no—
372#117—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4D4×C3111yescharacteristic
377C2×C4⋊C4C23×C3112yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C23×C3111yescharacteristic
383C2×C22⋊C4C23×C3111yescharacteristic
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C23×C3111yescharacteristic
387#107—24no—
388#106—24no—
389C4⋊C8—22no—
390#103—24no—
391#102—24no—
392C22⋊C8C23×C3114yes—
393#97—22no—
394#95—22no—
395#95—22no—
396#95C23×C3112yes—
397C31#1834211yescharacteristic, direct product factor, Sylow
398C24—21no—
399C24C2×D4×C3111yescharacteristic
400C24C2×D4×C3111yescharacteristic
401#41—82no—
402C2×Q8—21no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8C2×D4×C3112yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—41no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×D4×C3112yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C24×C3111yescharacteristic
437C2×C8—24no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4C2×D4×C3112yes—
452C4⋊C4C2×D4×C3112yes—
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—22no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42C2×D4×C3112yes—
471C42C2×D4×C3112yes—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23C31×AGL(1,ℤ/8)12yes—
486C23C22×D4×C3111yescharacteristic
487Q8—42no—
488Q8—84no—
489Q8—81no—
490D4—44no—
491D4—42no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—82no—
496D4—81no—
497D4—82no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4C31×#12012yes—
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C22×D4×C3112yes—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—81no—
543C22—81no—
544C22—84no—
545C22C2×C31×#12011yescharacteristic
546C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
547C22C31×#57511yescharacteristic
548C4—22no—
549C4—21no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4C31×#45512yes—
559C2—21no—
560C2—21no—
561C2—42no—
562C2—81no—
563C2—82no—
564C2C31×#300611yescharacteristic
565C2C2×C31×#45511yescharacteristic
566C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3459C2
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×#2547C2semidirect
4. C31×#1612C2semidirect
5. C31×#1608C2semidirect
6. C31×#1607C2semidirect
7. C31×#1509C2semidirect
8. C31×#1466C2semidirect
16. C2×C31×C4⋊Q8C22semidirect
26. C31×#483C22semidirect
30. C31×#432C22semidirect
32. C31×#397C22semidirect
36. C4×C4⋊C4×C31C22semidirect
37. C31×#328C22semidirect
40. C31×#323C22semidirect
188. #18342C31direct (N × H)
247. #3238C2×C31semidirect
248. #2547C2×C31semidirect
249. #1612C2×C31semidirect
250. #1608C2×C31semidirect
251. #1607C2×C31semidirect
252. #1509C2×C31semidirect
253. #1466C2×C31semidirect
291. C2×C4⋊Q8C22×C31semidirect
301. #483C22×C31semidirect
305. #432C22×C31semidirect
307. #397C22×C31semidirect
311. C4×C4⋊C4C22×C31semidirect
312. #328C22×C31semidirect
315. #323C22×C31semidirect
397. C31#18342direct (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×#32382
C31×#25472
C31×#16122
C31×#16082
C31×#16072
C31×#15092
C31×#14662
#1834231