Finite Group Databasebeta

← C31×#18122 #21474881

Subgroups — C31×#18122 #21474881

Order 7936.

1292 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3449C211yesmaximal, characteristic, complemented
2C31×#3100C212yesmaximal, complemented
3C31×#2392C211yesmaximal, characteristic, complemented
4C31×#1610C212yesmaximal, complemented
5C31×#1607C212yesmaximal, complemented
6C31×#1598C212yesmaximal, complemented
7C31×#1595C212yesmaximal, complemented
8C31×#1502C212yesmaximal, complemented
9C31×#1458C211yesmaximal, characteristic, complemented
10C22×#41×C31—21no—
11C31×#537—21no—
12C31×#537C2212yes—
13C31×#536—22no—
14C31×#535—22no—
15C31×#534—21no—
16C31×#534C2212yes—
17C31×#532—21no—
18C31×#532C2212yes—
19C31×#525—24no—
20C2×C4×D4×C31—21no—
21C2×C4×D4×C31C2211yescharacteristic, complemented
22C2×C31×#117C2211yescharacteristic
23C31×#480C2214yescomplemented
24C31×#474—22no—
25C31×#459—22no—
26C31×#447C2214yescomplemented
27C4×D8×C31—24no—
28C31×#431C2212yescomplemented
29C31×#398C2211yescharacteristic, complemented
30C31×#397C2211yescharacteristic
31C31×#396—21no—
32C31×#394—21no—
33C31×#394C2211yescharacteristic, complemented
34C31×#392C2211yescharacteristic
35C4×C4⋊C4×C31C2211yescharacteristic, complemented
36C4×C22⋊C4×C31—21no—
37C31×#327C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—44no—
46C2×#41×C31—44no—
47C22×Q8×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—22no—
50C22×D4×C31—21no—
51C22×D4×C31C2311yescharacteristic
52C23×C4×C31—22no—
53C23×C4×C31—21no—
54C23×C4×C31D411yescharacteristic
55C2×D8×C31—24no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8D412yescomplemented
58C31×#127—21no—
59C31×#127D412yescomplemented
60C31×#124—22no—
61C31×#124—22no—
62C31×#123—24no—
63C31×#123—24no—
64C31×C22⋊Q8—24no—
65C31×C22⋊Q8—24no—
66C31×#121—24no—
67C31×#121—24no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121C2314yes—
72C31×#120—24no—
73C31×#120—24no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C4×D4×C31—44no—
80C4×D4×C31D414yescomplemented
81C31×#117—22no—
82C31×#117—22no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31D411yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31D411yescharacteristic
94C2×C22⋊C4×C31C2311yescharacteristic
95C2×C42×C31—21no—
96C2×C42×C31D411yescharacteristic
97C2×C42×C31C2311yescharacteristic
98C31×#107—24no—
99C31×C4⋊C8—22no—
100C31×#102—24no—
101C31×#102—24no—
102C31×C22⋊C8C2314yes—
103C4×C8×C31—22no—
104C31×#95—21no—
105C31×#95—21no—
106C31×#95C2311yescharacteristic
107C31×#95C2311yescharacteristic
108C24×C31—22no—
109C24×C31—21no—
110C24×C31C2×D411yescharacteristic
111#41×C31—44no—
112#41×C31—48no—
113#41×C31—48no—
114#41×C31—44no—
115#41×C31—44no—
116#41×C31—44no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—44no—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—24no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—42no—
129C2×D4×C31—44no—
130C2×D4×C31—42no—
131C2×D4×C31—42no—
132C2×D4×C31—44no—
133C2×D4×C31—42no—
134C2×D4×C31—44no—
135C2×D4×C31C2×D414yes—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—22no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—24no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—48no—
146C22×C4×C31—41no—
147C22×C4×C31—41no—
148C22×C4×C31—42no—
149C22×C4×C31—42no—
150C22×C4×C31D814yescomplemented
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2411yescharacteristic
157D8×C31—48no—
158C2×C8×C31—24no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31C2×D414yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—42no—
175C22⋊C4×C31—42no—
176C22⋊C4×C31—42no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—42no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—42no—
182C42×C31C2×D412yes—
183C42×C31C2×D412yes—
184#18122C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—22no—
188C23×C31—24no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—44no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—82no—
199C23×C31—82no—
200C23×C31C2×D812yes—
201C23×C31C22×D411yescharacteristic
202Q8×C31—44no—
203Q8×C31—44no—
204D4×C31—24no—
205D4×C31—44no—
206D4×C31—42no—
207D4×C31—44no—
208D4×C31—48no—
209D4×C31—48no—
210D4×C31—44no—
211D4×C31—44no—
212D4×C31—44no—
213D4×C31—82no—
214D4×C31—82no—
215D4×C31—82no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—24no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—48no—
227C2×C4×C31—44no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—48no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—82no—
240C2×C4×C31—82no—
241C2×C4×C31#12012yes—
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31C2×D812yes—
245C2×C4×C31C2×D812yes—
246C2×C4×C31AGL(1,ℤ/8)12yes—
247C2×C4×C31C22×D412yes—
248C8×C31—44no—
249#3449C2×C3111yescharacteristic, complemented
250#3100C2×C3112yescomplemented
251#2392C2×C3111yescharacteristic, complemented
252#1610C2×C3112yescomplemented
253#1607C2×C3112yescomplemented
254#1598C2×C3112yescomplemented
255#1595C2×C3112yescomplemented
256#1502C2×C3112yescomplemented
257#1458C2×C3111yescharacteristic, complemented
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—22no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—41no—
265C22×C31—41no—
266C22×C31—44no—
267C22×C31—44no—
268C22×C31—42no—
269C22×C31—44no—
270C22×C31—81no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—88no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31C2×#12011yescharacteristic
277C22×C31C22×D811yescharacteristic
278C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
279C4×C31—22no—
280C4×C31—22no—
281C4×C31—22no—
282C4×C31—42no—
283C4×C31—44no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—81no—
287C4×C31—81no—
288C4×C31#44712yes—
289C4×C31#44712yes—
290C22×#41—21no—
291#537—21no—
292#537C22×C3112yes—
293#536—22no—
294#535—22no—
295#534—21no—
296#534C22×C3112yes—
297#532—21no—
298#532C22×C3112yes—
299#525—24no—
300C2×C4×D4—21no—
301C2×C4×D4C22×C3111yescharacteristic, complemented
302C2×#117C22×C3111yescharacteristic
303#480C22×C3114yescomplemented
304#474—22no—
305#459—22no—
306#447C22×C3114yescomplemented
307C4×D8—24no—
308#431C22×C3112yescomplemented
309#398C22×C3111yescharacteristic, complemented
310#397C22×C3111yescharacteristic
311#396—21no—
312#394—21no—
313#394C22×C3111yescharacteristic, complemented
314#392C22×C3111yescharacteristic
315C4×C4⋊C4C22×C3111yescharacteristic, complemented
316C4×C22⋊C4—21no—
317#327C22×C3114yescomplemented
318#327C22×C3114yescomplemented
319#324C22×C3112yescomplemented
320#323C22×C3112yes—
321C2×C31—21no—
322C2×C31—21no—
323C2×C31—42no—
324C2×C31—82no—
325C2×C31—84no—
326C2×C31C2×#44711yescharacteristic
327C2×C31#301811yescharacteristic
328C2×C31#302511yescharacteristic
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—44no—
334C2×#41—44no—
335C22×Q8—21no—
336C22×D4—21no—
337C22×D4—22no—
338C22×D4—21no—
339C22×D4C23×C3111yescharacteristic
340C23×C4—22no—
341C23×C4—21no—
342C23×C4D4×C3111yescharacteristic
343C2×D8—24no—
344C4⋊Q8—21no—
345C4⋊Q8D4×C3112yescomplemented
346#127—21no—
347#127D4×C3112yescomplemented
348#124—22no—
349#124—22no—
350#123—24no—
351#123—24no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354#121—24no—
355#121—24no—
356#121—22no—
357#121—22no—
358#121—22no—
359#121C23×C3114yes—
360#120—24no—
361#120—24no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—44no—
368C4×D4D4×C3114yescomplemented
369#117—22no—
370#117—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4D4×C3111yescharacteristic
374C2×C4⋊C4C23×C3111yescharacteristic
375C2×C4⋊C4C23×C3111yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4D4×C3111yescharacteristic
382C2×C22⋊C4C23×C3111yescharacteristic
383C2×C42—21no—
384C2×C42D4×C3111yescharacteristic
385C2×C42C23×C3111yescharacteristic
386#107—24no—
387C4⋊C8—22no—
388#102—24no—
389#102—24no—
390C22⋊C8C23×C3114yes—
391C4×C8—22no—
392#95—21no—
393#95—21no—
394#95C23×C3111yescharacteristic
395#95C23×C3111yescharacteristic
396C31#1812211yescharacteristic, direct product factor, Sylow
397C24—22no—
398C24—21no—
399C24C2×D4×C3111yescharacteristic
400#41—44no—
401#41—48no—
402#41—48no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4C2×D4×C3114yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—48no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4D8×C3114yescomplemented
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C24×C3111yescharacteristic
446D8—48no—
447C2×C8—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4C2×D4×C3114yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C42—21no—
469C42—21no—
470C42—42no—
471C42C2×D4×C3112yes—
472C42C2×D4×C3112yes—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—24no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—82no—
487C23—82no—
488C23C2×D8×C3112yes—
489C23C22×D4×C3111yescharacteristic
490Q8—44no—
491Q8—44no—
492D4—24no—
493D4—44no—
494D4—42no—
495D4—44no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4C31×#12012yes—
530C2×C4C31×#12011yescharacteristic
531C2×C4C31×#12011yescharacteristic
532C2×C4C2×D8×C3112yes—
533C2×C4C2×D8×C3112yes—
534C2×C4C31×AGL(1,ℤ/8)12yes—
535C2×C4C22×D4×C3112yes—
536C8—44no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—44no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—88no—
553C22—81no—
554C22—81no—
555C22C2×C31×#12011yescharacteristic
556C22C22×D8×C3111yescharacteristic
557C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—44no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4C31×#44712yes—
568C4C31×#44712yes—
569C2—21no—
570C2—21no—
571C2—42no—
572C2—82no—
573C2—84no—
574C2C2×C31×#44711yescharacteristic
575C2C31×#301811yescharacteristic
576C2C31×#302511yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3449C2semidirect
2. C31×#3100C2semidirect
3. C31×#2392C2semidirect
4. C31×#1610C2semidirect
5. C31×#1607C2semidirect
6. C31×#1598C2semidirect
7. C31×#1595C2semidirect
8. C31×#1502C2semidirect
9. C31×#1458C2semidirect
21. C2×C4×D4×C31C22semidirect
23. C31×#480C22semidirect
26. C31×#447C22semidirect
28. C31×#431C22semidirect
29. C31×#398C22semidirect
33. C31×#394C22semidirect
35. C4×C4⋊C4×C31C22semidirect
37. C31×#327C22semidirect
38. C31×#327C22semidirect
39. C31×#324C22semidirect
57. C31×C4⋊Q8D4semidirect
59. C31×#127D4semidirect
80. C4×D4×C31D4semidirect
150. C22×C4×C31D8semidirect
184. #18122C31direct (N × H)
249. #3449C2×C31semidirect
250. #3100C2×C31semidirect
251. #2392C2×C31semidirect
252. #1610C2×C31semidirect
253. #1607C2×C31semidirect
254. #1598C2×C31semidirect
255. #1595C2×C31semidirect
256. #1502C2×C31semidirect
257. #1458C2×C31semidirect
301. C2×C4×D4C22×C31semidirect
303. #480C22×C31semidirect
306. #447C22×C31semidirect
308. #431C22×C31semidirect
309. #398C22×C31semidirect
313. #394C22×C31semidirect
315. C4×C4⋊C4C22×C31semidirect
317. #327C22×C31semidirect
318. #327C22×C31semidirect
319. #324C22×C31semidirect
345. C4⋊Q8D4×C31semidirect
347. #127D4×C31semidirect
368. C4×D4D4×C31semidirect
396. C31#18122direct (N × H)
439. C22×C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34492
C31×#31002
C31×#23922
C31×#16102
C31×#16072
C31×#15982
C31×#15952
C31×#15022
C31×#14582
#1812231