Finite Group Databasebeta

← C31×#18165 #21474924

Subgroups — C31×#18165 #21474924

Order 7936.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic
2C31×#3004C212yesmaximal, complemented
3C2×C31×#395C211yesmaximal, characteristic, complemented
4C31×#1605C212yesmaximal, complemented
5C31×#1604C212yesmaximal, complemented
6C31×#1603C214yesmaximal, complemented
7C31×#1504C212yesmaximal, complemented
8C31×#1463C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C2×C31×#137—22no—
11C31×#554—22no—
12C31×#553—22no—
13C31×#552—22no—
14C31×#547—22no—
15C31×#544C2211yescharacteristic
16C31×#539—21no—
17C31×#539C2212yes—
18C31×#537C2211yescharacteristic
19C31×#525—21no—
20C31×#525—21no—
21C31×#525C2212yes—
22C31×#520—21no—
23C22×C4⋊C4×C31—21no—
24C22×C4⋊C4×C31C2211yescharacteristic
25C31×#451—22no—
26C31×#451C2214yes—
27C31×#448—22no—
28C31×#448C2214yes—
29C31×#419—22no—
30C31×#407C2212yes—
31C31×#395—21no—
32C31×#395—21no—
33C31×#395C2212yes—
34C31×#395C2212yes—
35C2×C31×#95—21no—
36C2×C31×#95C2211yescharacteristic
37C31×#328C2214yes—
38C31×#327C2214yes—
39C31×#324C2212yes—
40C31×#324C2212yes—
41C2×#41×C31—21no—
42C2×#41×C31—21no—
43C2×#41×C31D412yes—
44C22×Q8×C31—21no—
45C22×Q8×C31—22no—
46C22×Q8×C31—22no—
47C22×Q8×C31—44no—
48C22×Q8×C31C2312yes—
49C23×C4×C31—21no—
50C23×C4×C31—21no—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C31×#137—48no—
55C2×Q16×C31—24no—
56C2×SD16×C31—24no—
57C2×M16×C31—22no—
58C31×C4⋊Q8—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#125—22no—
62C31×#125—22no—
63C31×#125—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×#123—22no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8D412yes—
77C31×C22⋊Q8C2312yes—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121D412yes—
84C31×#121C2312yes—
85C31×#120—22no—
86C4×Q8×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C31×#117—22no—
95C31×#117—21no—
96C31×#117—21no—
97C31×#117D412yes—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31—22no—
103C2×C4⋊C4×C31—44no—
104C2×C4⋊C4×C31D411yescharacteristic
105C2×C4⋊C4×C31D411yescharacteristic
106C2×C4⋊C4×C31C2311yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C31×#103—24no—
109C31×#103—24no—
110C31×C22⋊C8—22no—
111C31×C22⋊C8C2314yes—
112C31×#95—21no—
113C31×#95—22no—
114C31×#95—22no—
115C31×#95C2312yes—
116C24×C31C2×D411yescharacteristic
117#41×C31—42no—
118#41×C31—44no—
119#41×C31—42no—
120#41×C31—82no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31—28no—
124C2×Q8×C31—24no—
125C2×Q8×C31—44no—
126C2×Q8×C31—44no—
127C2×Q8×C31—44no—
128C2×Q8×C31—48no—
129C2×Q8×C31—42no—
130C2×Q8×C31C2×D412yes—
131C2×D4×C31—21no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—21no—
136C2×D4×C31C2×D412yes—
137C22×C4×C31—22no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—41no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31—41no—
155C22×C4×C31—41no—
156C22×C4×C31—42no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2411yescharacteristic
165Q16×C31—48no—
166SD16×C31—48no—
167M16×C31—44no—
168C2×C8×C31—24no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—44no—
184C4⋊C4×C31—44no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D412yes—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C42×C31—22no—
197C42×C31—22no—
198C42×C31—22no—
199#18165C3111yesmaximal, characteristic, direct product factor, Sylow
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—22no—
205C23×C31#12011yescharacteristic
206C23×C31#12011yescharacteristic
207C23×C31C22×D411yescharacteristic
208Q8×C31—28no—
209Q8×C31—42no—
210Q8×C31—44no—
211Q8×C31—48no—
212Q8×C31—42no—
213Q8×C31—81no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—44no—
219D4×C31—44no—
220D4×C31—81no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—42no—
241C2×C4×C31—48no—
242C2×C4×C31—42no—
243C2×C4×C31—44no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—44no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—82no—
251C2×C4×C31—81no—
252C2×C4×C31—82no—
253C2×C4×C31#12012yes—
254C2×C4×C31#13712yescomplemented
255C2×C4×C31C22×D412yes—
256C8×C31—44no—
257#3492C2×C3111yescharacteristic
258#3004C2×C3112yescomplemented
259C2×#395C2×C3111yescharacteristic, complemented
260#1605C2×C3112yescomplemented
261#1604C2×C3112yescomplemented
262#1603C2×C3114yescomplemented
263#1504C2×C3112yescomplemented
264#1463C2×C3111yescharacteristic, complemented
265C22×C31—24no—
266C22×C31—24no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—21no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31#45612yes—
277C22×C31B(4,𝔽2)12yes—
278C22×C31C2×#12011yescharacteristic
279C22×C31C2×#13711yescharacteristic
280C22×C31#57511yescharacteristic
281C4×C31—22no—
282C4×C31—22no—
283C4×C31—22no—
284C4×C31—44no—
285C4×C31—41no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—44no—
289C4×C31—41no—
290C4×C31—41no—
291C4×C31—42no—
292C23×Q8C22×C3111yescharacteristic
293C2×#137—22no—
294#554—22no—
295#553—22no—
296#552—22no—
297#547—22no—
298#544C22×C3111yescharacteristic
299#539—21no—
300#539C22×C3112yes—
301#537C22×C3111yescharacteristic
302#525—21no—
303#525—21no—
304#525C22×C3112yes—
305#520—21no—
306C22×C4⋊C4—21no—
307C22×C4⋊C4C22×C3111yescharacteristic
308#451—22no—
309#451C22×C3114yes—
310#448—22no—
311#448C22×C3114yes—
312#419—22no—
313#407C22×C3112yes—
314#395—21no—
315#395—21no—
316#395C22×C3112yes—
317#395C22×C3112yes—
318C2×#95—21no—
319C2×#95C22×C3111yescharacteristic
320#328C22×C3114yes—
321#327C22×C3114yes—
322#324C22×C3112yes—
323#324C22×C3112yes—
324C2×C31—21no—
325C2×C31—21no—
326C2×C31—22no—
327C2×C31—22no—
328C2×C31—82no—
329C2×C31#300711yescharacteristic
330C2×C31C2×#45611yescharacteristic
331C2×C31C2×B(4,𝔽2)11yescharacteristic
332C2×#41—21no—
333C2×#41—21no—
334C2×#41D4×C3112yes—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—44no—
339C22×Q8C23×C3112yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4D4×C3111yescharacteristic
343C23×C4D4×C3111yescharacteristic
344C23×C4C23×C3111yescharacteristic
345#137—48no—
346C2×Q16—24no—
347C2×SD16—24no—
348C2×M16—22no—
349C4⋊Q8—22no—
350#126—22no—
351#126—22no—
352#125—22no—
353#125—22no—
354#125—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8D4×C3112yes—
368C22⋊Q8C23×C3112yes—
369#121—21no—
370#121—21no—
371#121—21no—
372#121—22no—
373#121—21no—
374#121D4×C3112yes—
375#121C23×C3112yes—
376#120—22no—
377C4×Q8—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385#117—22no—
386#117—21no—
387#117—21no—
388#117D4×C3112yes—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—44no—
395C2×C4⋊C4D4×C3111yescharacteristic
396C2×C4⋊C4D4×C3111yescharacteristic
397C2×C4⋊C4C23×C3111yescharacteristic
398C2×C4⋊C4C23×C3111yescharacteristic
399#103—24no—
400#103—24no—
401C22⋊C8—22no—
402C22⋊C8C23×C3114yes—
403#95—22no—
404#95—22no—
405#95—21no—
406#95C23×C3112yes—
407C31#1816511yescharacteristic, direct product factor, Sylow
408C24C2×D4×C3111yescharacteristic
409#41—42no—
410#41—44no—
411#41—42no—
412#41—82no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—28no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8C2×D4×C3112yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4C2×D4×C3112yes—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3112yes—
456C22×C4C24×C3111yescharacteristic
457Q16—48no—
458SD16—48no—
459M16—44no—
460C2×C8—24no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4C2×D4×C3112yes—
478C4⋊C4C2×D4×C3112yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23C31×#12011yescharacteristic
497C23C31×#12011yescharacteristic
498C23C22×D4×C3111yescharacteristic
499Q8—28no—
500Q8—42no—
501Q8—44no—
502Q8—48no—
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—42no—
507D4—44no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—81no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4C31×#12012yes—
545C2×C4C31×#13712yescomplemented
546C2×C4C22×D4×C3112yes—
547C8—44no—
548C22—24no—
549C22—24no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22C31×#45612yes—
560C22C31×B(4,𝔽2)12yes—
561C22C2×C31×#12011yescharacteristic
562C22C2×C31×#13711yescharacteristic
563C22C31×#57511yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—41no—
573C4—41no—
574C4—42no—
575C2—21no—
576C2—22no—
577C2—22no—
578C2—21no—
579C2—82no—
580C2C31×#300711yescharacteristic
581C2C2×C31×#45611yescharacteristic
582C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3492C2
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×#3004C2semidirect
3. C2×C31×#395C2semidirect
4. C31×#1605C2semidirect
5. C31×#1604C2semidirect
6. C31×#1603C2semidirect
7. C31×#1504C2semidirect
8. C31×#1463C2semidirect
199. #18165C31direct (N × H)
254. C2×C4×C31#137semidirect
258. #3004C2×C31semidirect
259. C2×#395C2×C31semidirect
260. #1605C2×C31semidirect
261. #1604C2×C31semidirect
262. #1603C2×C31semidirect
263. #1504C2×C31semidirect
264. #1463C2×C31semidirect
407. C31#18165direct (N × H)
545. C2×C4C31×#137semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30042
C2×C31×#3952
C31×#16052
C31×#16042
C31×#16032
C31×#15042
C31×#14632
#1816531