Finite Group Databasebeta

← C31×#18196 #21474955

Subgroups — C31×#18196 #21474955

Order 7936.

1370 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C2×C31×#450C212yesmaximal, complemented
3C31×#2408C211yesmaximal, characteristic
4C31×#1599C214yesmaximal, complemented
5C31×#1596C214yesmaximal, complemented
6C31×#1520C212yesmaximal, complemented
7C31×#1459C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×SD16×C31—22no—
10C31×#554—24no—
11C31×#554—22no—
12D4×Q8×C31—22no—
13C31×#544C2212yescomplemented
14C31×#543—21no—
15C31×#543C2212yescomplemented
16C31×#533—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×#121—21no—
20C2×C31×#120—21no—
21C31×#518—21no—
22C2×C4×D4×C31C2211yescharacteristic
23C31×#450—24no—
24C31×#450C2218yes—
25C2×C31×#102—22no—
26C2×C31×C22⋊C8C2212yes—
27C31×#394C2212yes—
28C31×#392C2212yes—
29C31×#390—21no—
30C31×#380C2211yescharacteristic
31C31×#379—21no—
32C31×#328C2218yescomplemented
33C31×#324C2212yescomplemented
34C31×#323C2212yescomplemented
35C25×C31—21no—
36C22×Q8×C31—21no—
37C22×Q8×C31D412yescomplemented
38C22×D4×C31—22no—
39C22×D4×C31—22no—
40C22×D4×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—44no—
43C22×D4×C31D411yescharacteristic
44C22×D4×C31C2312yes—
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C2×SD16×C31—28no—
48C2×SD16×C31—48no—
49C22×C8×C31—22no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—24no—
53C31×#125—22no—
54C31×#125—24no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—24no—
59C31×C22⋊Q8—24no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—24no—
65C31×C22⋊Q8D414yescomplemented
66C31×C22⋊Q8C2314yes—
67C31×#121—44no—
68C31×#120—44no—
69C4×Q8×C31—22no—
70C4×Q8×C31—22no—
71C4×D4×C31—24no—
72C4×D4×C31—24no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C31×#117—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31D412yescomplemented
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31C2311yescharacteristic
90C2×C42×C31D411yescharacteristic
91C31×#102—28no—
92C31×C22⋊C8—22no—
93C31×C22⋊C8C2314yes—
94C31×#95C2312yes—
95C24×C31—22no—
96C24×C31—44no—
97C24×C31—42no—
98C24×C31—42no—
99C24×C31—42no—
100C24×C31—82no—
101C24×C31C2×D411yescharacteristic
102C24×C31C2×D411yescharacteristic
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×Q8×C31—22no—
106C2×Q8×C31—44no—
107C2×Q8×C31—82no—
108C2×Q8×C31C2×D414yes—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—28no—
112C2×D4×C31—24no—
113C2×D4×C31—24no—
114C2×D4×C31—42no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—48no—
118C2×D4×C31—44no—
119C2×D4×C31—44no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31—82no—
123C2×D4×C31SD1614yescomplemented
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D412yes—
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2411yescharacteristic
140SD16×C31—416no—
141C2×C8×C31—24no—
142C2×C8×C31—44no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—24no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—42no—
154C4⋊C4×C31C2×D414yes—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—24no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—44no—
161C22⋊C4×C31—42no—
162C22⋊C4×C31—44no—
163C22⋊C4×C31—82no—
164C42×C31—24no—
165C42×C31—21no—
166C42×C31—21no—
167#18196C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—24no—
171C23×C31—41no—
172C23×C31—41no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—42no—
176C23×C31—48no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—84no—
183C23×C31—88no—
184C23×C31—88no—
185C23×C31—84no—
186C23×C31#12012yes—
187C23×C31C2×SD1612yes—
188C23×C31C2×SD1612yescomplemented
189C23×C31#13712yes—
190C23×C31C22×D411yescharacteristic
191Q8×C31—44no—
192Q8×C31—44no—
193Q8×C31—82no—
194Q8×C31—82no—
195D4×C31—28no—
196D4×C31—28no—
197D4×C31—44no—
198D4×C31—44no—
199D4×C31—44no—
200D4×C31—88no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—21no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—21no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—81no—
218C2×C4×C31—82no—
219C2×C4×C31—82no—
220C2×C4×C31—82no—
221C2×C4×C31—82no—
222C2×C4×C31#12011yescharacteristic
223C2×C4×C31#12011yescharacteristic
224C2×C4×C31C2×SD1612yescomplemented
225C2×C4×C31C22×D412yes—
226C8×C31—44no—
227#3501C2×C3111yescharacteristic, complemented
228C2×#450C2×C3112yescomplemented
229#2408C2×C3111yescharacteristic
230#1599C2×C3114yescomplemented
231#1596C2×C3114yescomplemented
232#1520C2×C3112yescomplemented
233#1459C2×C3111yescharacteristic, complemented
234C22×C31—21no—
235C22×C31—22no—
236C22×C31—21no—
237C22×C31—22no—
238C22×C31—22no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—42no—
242C22×C31—44no—
243C22×C31—44no—
244C22×C31—41no—
245C22×C31—41no—
246C22×C31—44no—
247C22×C31—81no—
248C22×C31—82no—
249C22×C31—82no—
250C22×C31—816no—
251C22×C31—82no—
252C22×C31—82no—
253C22×C31#44814yes—
254C22×C31C2×#12011yescharacteristic
255C22×C31C22×SD1611yescharacteristic
256C22×C31C2×#13711yescharacteristic
257C4×C31—22no—
258C4×C31—22no—
259C4×C31—21no—
260C4×C31—41no—
261C4×C31—44no—
262C4×C31—41no—
263C4×C31—44no—
264C4×C31—82no—
265C4×C31B(4,𝔽2)12yes—
266C23×D4C22×C3111yescharacteristic
267C22×SD16—22no—
268#554—24no—
269#554—22no—
270D4×Q8—22no—
271#544C22×C3112yescomplemented
272#543—21no—
273#543C22×C3112yescomplemented
274#533—21no—
275C2×C22⋊Q8—21no—
276C2×C22⋊Q8C22×C3112yes—
277C2×#121—21no—
278C2×#120—21no—
279#518—21no—
280C2×C4×D4C22×C3111yescharacteristic
281#450—24no—
282#450C22×C3118yes—
283C2×#102—22no—
284C2×C22⋊C8C22×C3112yes—
285#394C22×C3112yes—
286#392C22×C3112yes—
287#390—21no—
288#380C22×C3111yescharacteristic
289#379—21no—
290#328C22×C3118yescomplemented
291#324C22×C3112yescomplemented
292#323C22×C3112yescomplemented
293C2×C31—21no—
294C2×C31—21no—
295C2×C31—24no—
296C2×C31—42no—
297C2×C31—84no—
298C2×C31C2×#44811yescharacteristic
299C2×C31#302111yescharacteristic
300C2×C31C2×B(4,𝔽2)11yescharacteristic
301C25—21no—
302C22×Q8—21no—
303C22×Q8D4×C3112yescomplemented
304C22×D4—22no—
305C22×D4—22no—
306C22×D4—21no—
307C22×D4—22no—
308C22×D4—44no—
309C22×D4D4×C3111yescharacteristic
310C22×D4C23×C3112yes—
311C23×C4D4×C3111yescharacteristic
312C23×C4C23×C3111yescharacteristic
313C2×SD16—28no—
314C2×SD16—48no—
315C22×C8—22no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—24no—
319#125—22no—
320#125—24no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8D4×C3114yescomplemented
332C22⋊Q8C23×C3114yes—
333#121—44no—
334#120—44no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—22no—
340C4×D4—22no—
341#117—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4D4×C3112yescomplemented
347C2×C4⋊C4C23×C3111yescharacteristic
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4D4×C3111yescharacteristic
355C2×C22⋊C4C23×C3111yescharacteristic
356C2×C42D4×C3111yescharacteristic
357#102—28no—
358C22⋊C8—22no—
359C22⋊C8C23×C3114yes—
360#95C23×C3112yes—
361C31#1819611yescharacteristic, direct product factor, Sylow
362C24—22no—
363C24—44no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—82no—
368C24C2×D4×C3111yescharacteristic
369C24C2×D4×C3111yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8C2×D4×C3114yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—82no—
390C2×D4SD16×C3114yescomplemented
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×D4×C3111yescharacteristic
401C22×C4C2×D4×C3112yes—
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3112yes—
406C22×C4C24×C3111yescharacteristic
407SD16—416no—
408C2×C8—24no—
409C2×C8—44no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4C2×D4×C3114yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C42—24no—
432C42—21no—
433C42—21no—
434C23—21no—
435C23—21no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—48no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—88no—
450C23—88no—
451C23—84no—
452C23C31×#12012yes—
453C23C2×SD16×C3112yes—
454C23C2×SD16×C3112yescomplemented
455C23C31×#13712yes—
456C23C22×D4×C3111yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—82no—
460Q8—82no—
461D4—28no—
462D4—28no—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—88no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4C31×#12011yescharacteristic
489C2×C4C31×#12011yescharacteristic
490C2×C4C2×SD16×C3112yescomplemented
491C2×C4C22×D4×C3112yes—
492C8—44no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—44no—
503C22—41no—
504C22—41no—
505C22—44no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22—816no—
510C22—82no—
511C22—82no—
512C22C31×#44814yes—
513C22C2×C31×#12011yescharacteristic
514C22C22×SD16×C3111yescharacteristic
515C22C2×C31×#13711yescharacteristic
516C4—22no—
517C4—22no—
518C4—21no—
519C4—41no—
520C4—44no—
521C4—41no—
522C4—44no—
523C4—82no—
524C4C31×B(4,𝔽2)12yes—
525C2—21no—
526C2—21no—
527C2—24no—
528C2—42no—
529C2—84no—
530C2C2×C31×#44811yescharacteristic
531C2C31×#302111yescharacteristic
532C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2408C2
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
1. C31×#3501C2semidirect
2. C2×C31×#450C2semidirect
4. C31×#1599C2semidirect
5. C31×#1596C2semidirect
6. C31×#1520C2semidirect
7. C31×#1459C2semidirect
13. C31×#544C22semidirect
15. C31×#543C22semidirect
32. C31×#328C22semidirect
33. C31×#324C22semidirect
34. C31×#323C22semidirect
37. C22×Q8×C31D4semidirect
65. C31×C22⋊Q8D4semidirect
80. C2×C4⋊C4×C31D4semidirect
123. C2×D4×C31SD16semidirect
167. #18196C31direct (N × H)
188. C23×C31C2×SD16semidirect
224. C2×C4×C31C2×SD16semidirect
227. #3501C2×C31semidirect
228. C2×#450C2×C31semidirect
230. #1599C2×C31semidirect
231. #1596C2×C31semidirect
232. #1520C2×C31semidirect
233. #1459C2×C31semidirect
271. #544C22×C31semidirect
273. #543C22×C31semidirect
290. #328C22×C31semidirect
291. #324C22×C31semidirect
292. #323C22×C31semidirect
303. C22×Q8D4×C31semidirect
331. C22⋊Q8D4×C31semidirect
346. C2×C4⋊C4D4×C31semidirect
361. C31#18196direct (N × H)
390. C2×D4SD16×C31semidirect
454. C23C2×SD16×C31semidirect
490. C2×C4C2×SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C2×C31×#4502
C31×#24082
C31×#15992
C31×#15962
C31×#15202
C31×#14592
#1819631