Finite Group Databasebeta

← C31×#32938 #21489697

Subgroups — C31×#32938 #21489697

Order 7936.

1436 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#131C211yesmaximal, characteristic, complemented
2C2×C31×#468C212yesmaximal, complemented
3C2×C31×#465C211yesmaximal, characteristic, complemented
4C31×#2616C211yesmaximal, characteristic, complemented
5C31×#2615C211yesmaximal, characteristic
6C31×#1895C211yesmaximal, characteristic, complemented
7C31×#1894C211yesmaximal, characteristic, complemented
8C31×#1893C212yesmaximal, complemented
9C31×#1892C212yesmaximal, complemented
10C31×#1809C212yesmaximal, complemented
11C31×#1808C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×SD16×C31C2211yescharacteristic
14C2×C31×#131—28no—
15C2×C31×#131—24no—
16C22×M16×C31C2211yescharacteristic
17C22×M16×C31C2212yes—
18C23×C8×C31C2212yes—
19C23×C8×C31C2211yescharacteristic
20C2×C31×C22⋊Q8C2212yes—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121C2212yes—
23C2×C31×#121C2211yescharacteristic
24C31×#468—28no—
25C31×#465—24no—
26C2×C31×#107C2212yes—
27C2×C31×#106C2211yescharacteristic
28C2×C31×#103C2211yescharacteristic
29C2×C31×#103C2212yes—
30C2×C31×#102C2211yescharacteristic
31C2×C31×#102C2212yes—
32C31×#395—21no—
33C31×#394—21no—
34C31×#394—22no—
35C31×#393—21no—
36C31×#392—21no—
37C31×#392—22no—
38C31×#380C2211yescharacteristic
39C31×#380C2211yescharacteristic
40C31×#380C2212yes—
41C31×#380C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C2×#41×C31—24no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31—24no—
49C22×Q8×C31D411yescharacteristic
50C22×Q8×C31C2311yescharacteristic
51C22×D4×C31—22no—
52C22×D4×C31—21no—
53C22×D4×C31D412yes—
54C22×D4×C31D411yescharacteristic
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31C2312yes—
57C23×C4×C31C2311yescharacteristic
58C2×SD16×C31—22no—
59C2×SD16×C31—24no—
60C31×#131—232no—
61C2×M16×C31—22no—
62C2×M16×C31—24no—
63C2×M16×C31—24no—
64C2×M16×C31—24no—
65C2×M16×C31D418yescomplemented
66C22×C8×C31—24no—
67C22×C8×C31—24no—
68C22×C8×C31—24no—
69C22×C8×C31—24no—
70C22×C8×C31D414yescomplemented
71C22×C8×C31C2311yescharacteristic
72C22×C8×C31C2312yes—
73C22×C8×C31C2311yescharacteristic
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—22no—
78C31×#121—22no—
79C31×#121—24no—
80C31×#121—24no—
81C31×#121—22no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2312yes—
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C31×#107—24no—
97C31×#106—22no—
98C31×#103—22no—
99C31×#103—24no—
100C31×#102—22no—
101C31×#102—24no—
102C31×#95—21no—
103C31×#95—22no—
104C31×#95—21no—
105C31×#95—22no—
106C24×C31—21no—
107C24×C31C2×D411yescharacteristic
108C24×C31C2×D412yes—
109#41×C31—28no—
110#41×C31—216no—
111#41×C31—28no—
112C2×Q8×C31—22no—
113C2×Q8×C31—24no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—21no—
117C2×Q8×C31—22no—
118C2×D4×C31—24no—
119C2×D4×C31—24no—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—24no—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—21no—
126C2×D4×C31—22no—
127C2×D4×C31—42no—
128C2×D4×C31—44no—
129C2×D4×C31—82no—
130C2×D4×C31—82no—
131C2×D4×C31—82no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D414yes—
146C22×C4×C31C2×D412yes—
147C22×C4×C31#4112yes—
148C22×C4×C31#4114yes—
149C22×C4×C31C2411yescharacteristic
150SD16×C31—44no—
151M16×C31—28no—
152M16×C31—216no—
153C2×C8×C31—216no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—28no—
158C2×C8×C31C2×D412yes—
159C2×C8×C31C2×D414yes—
160C2×C8×C31C2×D412yes—
161C2×C8×C31C2×D414yes—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—44no—
172C22⋊C4×C31—42no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—42no—
175C22⋊C4×C31—44no—
176C22⋊C4×C31—81no—
177C22⋊C4×C31—82no—
178C22⋊C4×C31—81no—
179C22⋊C4×C31—82no—
180#32938C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—21no—
190C23×C31—22no—
191C23×C31—41no—
192C23×C31—42no—
193C23×C31—41no—
194C23×C31—42no—
195C23×C31—41no—
196C23×C31#12112yes—
197C23×C31#12114yes—
198C23×C31C22×D411yescharacteristic
199Q8×C31—28no—
200Q8×C31—41no—
201Q8×C31—41no—
202Q8×C31—42no—
203D4×C31—216no—
204D4×C31—28no—
205D4×C31—41no—
206D4×C31—41no—
207D4×C31—42no—
208D4×C31—84no—
209D4×C31—88no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—41no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—41no—
233C2×C4×C31#12112yes—
234C2×C4×C31#12114yes—
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×D412yes—
237C2×C4×C31C2×#4111yescharacteristic
238C2×C4×C31C2×#4112yes—
239C8×C31—22no—
240C8×C31—24no—
241C8×C31#12014yes—
242C22×#131C2×C3111yescharacteristic, complemented
243C2×#468C2×C3112yescomplemented
244C2×#465C2×C3111yescharacteristic, complemented
245#2616C2×C3111yescharacteristic, complemented
246#2615C2×C3111yescharacteristic
247#1895C2×C3111yescharacteristic, complemented
248#1894C2×C3111yescharacteristic, complemented
249#1893C2×C3112yescomplemented
250#1892C2×C3112yescomplemented
251#1809C2×C3112yescomplemented
252#1808C2×C3111yescharacteristic, complemented
253C22×C31—22no—
254C22×C31—24no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—24no—
258C22×C31—24no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31C2×#12111yescharacteristic
271C22×C31C2×#12112yes—
272C22×C31#57611yescharacteristic
273C22×C31#57712yes—
274C22×C31#57811yescharacteristic
275C4×C31—24no—
276C4×C31—22no—
277C4×C31—81no—
278C4×C31—81no—
279C4×C31—81no—
280C4×C31—82no—
281C4×C31—82no—
282C4×C31C2×#12011yescharacteristic
283C4×C31#52512yes—
284C4×C31#52511yescharacteristic
285C22×#41C22×C3111yescharacteristic
286C22×SD16C22×C3111yescharacteristic
287C2×#131—28no—
288C2×#131—24no—
289C22×M16C22×C3111yescharacteristic
290C22×M16C22×C3112yes—
291C23×C8C22×C3112yes—
292C23×C8C22×C3111yescharacteristic
293C2×C22⋊Q8C22×C3112yes—
294C2×C22⋊Q8C22×C3111yescharacteristic
295C2×#121C22×C3112yes—
296C2×#121C22×C3111yescharacteristic
297#468—28no—
298#465—24no—
299C2×#107C22×C3112yes—
300C2×#106C22×C3111yescharacteristic
301C2×#103C22×C3111yescharacteristic
302C2×#103C22×C3112yes—
303C2×#102C22×C3111yescharacteristic
304C2×#102C22×C3112yes—
305#395—21no—
306#394—21no—
307#394—22no—
308#393—21no—
309#392—21no—
310#392—22no—
311#380C22×C3111yescharacteristic
312#380C22×C3111yescharacteristic
313#380C22×C3112yes—
314#380C22×C3112yes—
315#340C22×C3112yes—
316#340C22×C3112yes—
317#340C22×C3112yes—
318C2×C31—42no—
319C2×C31—44no—
320C2×C31—161no—
321C2×C31#284711yescharacteristic
322C2×C31#305711yescharacteristic
323C2×C31#306111yescharacteristic
324C2×C31#306212yes—
325C2×C31#306312yes—
326C2×#41—24no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—24no—
330C22×Q8D4×C3111yescharacteristic
331C22×Q8C23×C3111yescharacteristic
332C22×D4—22no—
333C22×D4—21no—
334C22×D4D4×C3112yes—
335C22×D4D4×C3111yescharacteristic
336C22×D4C23×C3111yescharacteristic
337C23×C4C23×C3112yes—
338C23×C4C23×C3111yescharacteristic
339C2×SD16—22no—
340C2×SD16—24no—
341#131—232no—
342C2×M16—22no—
343C2×M16—24no—
344C2×M16—24no—
345C2×M16—24no—
346C2×M16D4×C3118yescomplemented
347C22×C8—24no—
348C22×C8—24no—
349C22×C8—24no—
350C22×C8—24no—
351C22×C8D4×C3114yescomplemented
352C22×C8C23×C3111yescharacteristic
353C22×C8C23×C3112yes—
354C22×C8C23×C3111yescharacteristic
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—22no—
359#121—22no—
360#121—24no—
361#121—24no—
362#121—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C23×C3111yescharacteristic
366C2×C4⋊C4C23×C3112yes—
367C2×C4⋊C4C23×C3111yescharacteristic
368C2×C4⋊C4C23×C3112yes—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377#107—24no—
378#106—22no—
379#103—22no—
380#103—24no—
381#102—22no—
382#102—24no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387C31#3293811yescharacteristic, direct product factor, Sylow
388C24—21no—
389C24C2×D4×C3111yescharacteristic
390C24C2×D4×C3112yes—
391#41—28no—
392#41—216no—
393#41—28no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—82no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4C2×D4×C3112yes—
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3114yes—
428C22×C4C2×D4×C3112yes—
429C22×C4#41×C3112yes—
430C22×C4#41×C3114yes—
431C22×C4C24×C3111yescharacteristic
432SD16—44no—
433M16—28no—
434M16—216no—
435C2×C8—216no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—28no—
440C2×C8C2×D4×C3112yes—
441C2×C8C2×D4×C3114yes—
442C2×C8C2×D4×C3112yes—
443C2×C8C2×D4×C3114yes—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—81no—
459C22⋊C4—82no—
460C22⋊C4—81no—
461C22⋊C4—82no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23C31×#12112yes—
478C23C31×#12114yes—
479C23C22×D4×C3111yescharacteristic
480Q8—28no—
481Q8—41no—
482Q8—41no—
483Q8—42no—
484D4—216no—
485D4—28no—
486D4—41no—
487D4—41no—
488D4—42no—
489D4—84no—
490D4—88no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4C31×#12112yes—
515C2×C4C31×#12114yes—
516C2×C4C22×D4×C3111yescharacteristic
517C2×C4C22×D4×C3112yes—
518C2×C4C2×#41×C3111yescharacteristic
519C2×C4C2×#41×C3112yes—
520C8—22no—
521C8—24no—
522C8C31×#12014yes—
523C22—22no—
524C22—24no—
525C22—22no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—82no—
539C22—82no—
540C22C2×C31×#12111yescharacteristic
541C22C2×C31×#12112yes—
542C22C31×#57611yescharacteristic
543C22C31×#57712yes—
544C22C31×#57811yescharacteristic
545C4—24no—
546C4—22no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—82no—
551C4—82no—
552C4C2×C31×#12011yescharacteristic
553C4C31×#52512yes—
554C4C31×#52511yescharacteristic
555C2—42no—
556C2—44no—
557C2—161no—
558C2C31×#284711yescharacteristic
559C2C31×#305711yescharacteristic
560C2C31×#306111yescharacteristic
561C2C31×#306212yes—
562C2C31×#306312yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#131C2semidirect
2. C2×C31×#468C2semidirect
3. C2×C31×#465C2semidirect
4. C31×#2616C2semidirect
6. C31×#1895C2semidirect
7. C31×#1894C2semidirect
8. C31×#1893C2semidirect
9. C31×#1892C2semidirect
10. C31×#1809C2semidirect
11. C31×#1808C2semidirect
65. C2×M16×C31D4semidirect
70. C22×C8×C31D4semidirect
180. #32938C31direct (N × H)
242. C22×#131C2×C31semidirect
243. C2×#468C2×C31semidirect
244. C2×#465C2×C31semidirect
245. #2616C2×C31semidirect
247. #1895C2×C31semidirect
248. #1894C2×C31semidirect
249. #1893C2×C31semidirect
250. #1892C2×C31semidirect
251. #1809C2×C31semidirect
252. #1808C2×C31semidirect
346. C2×M16D4×C31semidirect
351. C22×C8D4×C31semidirect
387. C31#32938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1312
C2×C31×#4682
C2×C31×#4652
C31×#26162
C31×#26152
C31×#18952
C31×#18942
C31×#18932
C31×#18922
C31×#18092
C31×#18082
#3293831