Finite Group Databasebeta

← C31×#18440 #21475199

Subgroups — C31×#18440 #21475199

Order 7936.

904 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C31×#3226C211yesmaximal, characteristic, complemented
4C31×#2660C211yesmaximal, characteristic, complemented
5C31×#1610C214yesmaximal, complemented
6C31×#1609C214yesmaximal, complemented
7C31×#1505C211yesmaximal, characteristic, complemented
8C31×#1500C211yesmaximal, characteristic, complemented
9C31×#1470C211yesmaximal, characteristic, complemented
10C31×#563—21no—
11C31×#562—21no—
12C31×#554—21no—
13C31×#553—21no—
14C31×#553—21no—
15C31×#547—22no—
16C31×#547—21no—
17C31×#544—21no—
18C31×#540—21no—
19C31×#539—22no—
20C31×#539C2212yes—
21C31×#537C2212yes—
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C2×C31×#125C2211yescharacteristic
26C31×#501—21no—
27C31×#500—21no—
28C31×#489—21no—
29C31×#486—21no—
30C31×#483C2212yes—
31C31×#480C2214yes—
32C31×#480C2212yes—
33C31×#477—21no—
34C31×#476—21no—
35C31×#474—22no—
36C31×#432C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#398C2214yes—
39C31×#395—22no—
40C31×#384—21no—
41C31×#382—21no—
42C31×#376C2211yescharacteristic
43C31×#328C2212yes—
44C31×#327C2214yes—
45C31×#327C2212yes—
46C31×#324C2214yes—
47C2×#41×C31—21no—
48C2×#41×C31—22no—
49C22×Q8×C31—21no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8—22no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8D411yescharacteristic
57C31×#127D411yescharacteristic
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—21no—
61C31×#125—21no—
62C31×#125—22no—
63C31×#125—21no—
64C31×#125—21no—
65C31×#125—21no—
66C31×#125—21no—
67C31×#125D411yescharacteristic
68C31×#124—21no—
69C31×#124D411yescharacteristic
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—21no—
76C31×#123—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8C2311yescharacteristic
84C31×#121—21no—
85C31×#121—22no—
86C31×#121—21no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—21no—
90C31×#121—22no—
91C31×#121C2311yescharacteristic
92C31×#121C2312yes—
93C4×Q8×C31D411yescharacteristic
94C4×D4×C31—22no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—22no—
99C4×D4×C31—21no—
100C4×D4×C31D412yes—
101C4×D4×C31D411yescharacteristic
102C31×#117—21no—
103C31×#117—21no—
104C2×C4⋊C4×C31—22no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—24no—
108C2×C4⋊C4×C31D412yes—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C4⋊C4×C31D411yescharacteristic
111C2×C4⋊C4×C31C2312yes—
112C2×C4⋊C4×C31C2312yes—
113C2×C42×C31—21no—
114C2×C42×C31C2311yescharacteristic
115C31×#107—24no—
116C31×#107—22no—
117C31×#106—22no—
118C31×C4⋊C8—21no—
119C31×C4⋊C8—21no—
120C31×#103—22no—
121C31×#102—24no—
122C31×#102—22no—
123C31×C22⋊C8C2312yes—
124C31×C22⋊C8C2312yes—
125C31×#97—21no—
126C4×C8×C31—21no—
127C31×#95—21no—
128C31×#95—24no—
129C31×#95C2312yes—
130#41×C31—82no—
131#41×C31—84no—
132C2×Q8×C31—22no—
133C2×Q8×C31—21no—
134C2×Q8×C31—41no—
135C2×Q8×C31—41no—
136C2×Q8×C31—82no—
137C2×Q8×C31C2×D411yescharacteristic
138C2×D4×C31—21no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31C2×D411yescharacteristic
145C2×D4×C31C2×D412yes—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2411yescharacteristic
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—41no—
180C4⋊C4×C31—41no—
181C4⋊C4×C31—84no—
182C4⋊C4×C31C2×D411yescharacteristic
183C4⋊C4×C31C2×D412yes—
184C4⋊C4×C31C2×D411yescharacteristic
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—22no—
197C42×C31—24no—
198C42×C31—22no—
199C42×C31—21no—
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204#18440C3111yesmaximal, characteristic, direct product factor, Sylow
205C23×C31—21no—
206C23×C31—22no—
207C23×C31C22×D411yescharacteristic
208Q8×C31—41no—
209Q8×C31—81no—
210Q8×C31—82no—
211Q8×C31—82no—
212Q8×C31—82no—
213D4×C31—42no—
214D4×C31—41no—
215D4×C31—44no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—81no—
219D4×C31—82no—
220D4×C31—82no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—44no—
238C2×C4×C31—82no—
239C2×C4×C31—82no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C8×C31—42no—
249C8×C31—42no—
250#3559C2×C3111yescharacteristic
251#3238C2×C3111yescharacteristic, complemented
252#3226C2×C3111yescharacteristic, complemented
253#2660C2×C3111yescharacteristic, complemented
254#1610C2×C3114yescomplemented
255#1609C2×C3114yescomplemented
256#1505C2×C3111yescharacteristic, complemented
257#1500C2×C3111yescharacteristic, complemented
258#1470C2×C3111yescharacteristic, complemented
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—42no—
262C22×C31—41no—
263C22×C31—41no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—41no—
267C22×C31C2×#12011yescharacteristic
268C22×C31#57612yes—
269C4×C31—21no—
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—44no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—41no—
278C4×C31—41no—
279C4×C31—42no—
280C4×C31—41no—
281C4×C31—82no—
282C4×C31#45612yes—
283#563—21no—
284#562—21no—
285#554—21no—
286#553—21no—
287#553—21no—
288#547—22no—
289#547—21no—
290#544—21no—
291#540—21no—
292#539—22no—
293#539C22×C3112yes—
294#537C22×C3112yes—
295#532C22×C3111yescharacteristic
296C2×C4⋊Q8C22×C3111yescharacteristic
297#529C22×C3111yescharacteristic
298C2×#125C22×C3111yescharacteristic
299#501—21no—
300#500—21no—
301#489—21no—
302#486—21no—
303#483C22×C3112yes—
304#480C22×C3114yes—
305#480C22×C3112yes—
306#477—21no—
307#476—21no—
308#474—22no—
309#432C22×C3111yescharacteristic
310#431C22×C3111yescharacteristic
311#398C22×C3114yes—
312#395—22no—
313#384—21no—
314#382—21no—
315#376C22×C3111yescharacteristic
316#328C22×C3112yes—
317#327C22×C3114yes—
318#327C22×C3112yes—
319#324C22×C3114yes—
320C2×C31—22no—
321C2×C31—81no—
322C2×C31—82no—
323C2×C31#300811yescharacteristic
324C2×C31C2×#45611yescharacteristic
325C2×C31#302811yescharacteristic
326C2×#41—21no—
327C2×#41—22no—
328C22×Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—22no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8D4×C3111yescharacteristic
336#127D4×C3111yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125—21no—
341#125—22no—
342#125—21no—
343#125—21no—
344#125—21no—
345#125—21no—
346#125D4×C3111yescharacteristic
347#124—21no—
348#124D4×C3111yescharacteristic
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—21no—
355#123—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8C23×C3111yescharacteristic
363#121—21no—
364#121—22no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—21no—
369#121—22no—
370#121C23×C3111yescharacteristic
371#121C23×C3112yes—
372C4×Q8D4×C3111yescharacteristic
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4D4×C3111yescharacteristic
380C4×D4D4×C3112yes—
381#117—21no—
382#117—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4D4×C3111yescharacteristic
388C2×C4⋊C4D4×C3112yes—
389C2×C4⋊C4D4×C3111yescharacteristic
390C2×C4⋊C4C23×C3112yes—
391C2×C4⋊C4C23×C3112yes—
392C2×C42—21no—
393C2×C42C23×C3111yescharacteristic
394#107—24no—
395#107—22no—
396#106—22no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399#103—22no—
400#102—24no—
401#102—22no—
402C22⋊C8C23×C3112yes—
403C22⋊C8C23×C3112yes—
404#97—21no—
405C4×C8—21no—
406#95—21no—
407#95—24no—
408#95C23×C3112yes—
409C31#1844011yescharacteristic, direct product factor, Sylow
410#41—82no—
411#41—84no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—82no—
417C2×Q8C2×D4×C3111yescharacteristic
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×D4×C3111yescharacteristic
425C2×D4C2×D4×C3112yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3112yes—
437C22×C4C24×C3111yescharacteristic
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—84no—
462C4⋊C4C2×D4×C3111yescharacteristic
463C4⋊C4C2×D4×C3112yes—
464C4⋊C4C2×D4×C3111yescharacteristic
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C42—24no—
478C42—22no—
479C42—21no—
480C42C2×D4×C3111yescharacteristic
481C42C2×D4×C3111yescharacteristic
482C42C2×D4×C3111yescharacteristic
483C42C2×D4×C3111yescharacteristic
484C23—21no—
485C23—22no—
486C23C22×D4×C3111yescharacteristic
487Q8—41no—
488Q8—81no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492D4—42no—
493D4—41no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—81no—
498D4—82no—
499D4—82no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12011yescharacteristic
523C2×C4C31×#12011yescharacteristic
524C2×C4C31×#12011yescharacteristic
525C2×C4C22×D4×C3111yescharacteristic
526C2×C4C22×D4×C3111yescharacteristic
527C8—42no—
528C8—42no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22C2×C31×#12011yescharacteristic
538C22C31×#57612yes—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—82no—
552C4C31×#45612yes—
553C2—22no—
554C2—81no—
555C2—82no—
556C2C31×#300811yescharacteristic
557C2C2×C31×#45611yescharacteristic
558C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3559C2
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×#3226C2semidirect
4. C31×#2660C2semidirect
5. C31×#1610C2semidirect
6. C31×#1609C2semidirect
7. C31×#1505C2semidirect
8. C31×#1500C2semidirect
9. C31×#1470C2semidirect
204. #18440C31direct (N × H)
251. #3238C2×C31semidirect
252. #3226C2×C31semidirect
253. #2660C2×C31semidirect
254. #1610C2×C31semidirect
255. #1609C2×C31semidirect
256. #1505C2×C31semidirect
257. #1500C2×C31semidirect
258. #1470C2×C31semidirect
409. C31#18440direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#32382
C31×#32262
C31×#26602
C31×#16102
C31×#16092
C31×#15052
C31×#15002
C31×#14702
#1844031