Finite Group Databasebeta

← C31×#23425 #21480184

Subgroups — C31×#23425 #21480184

Order 7936.

1938 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
1C22×C31×#137C211yesmaximal, characteristic
2C23×Q16×C31C211yesmaximal, characteristic, complemented
3C2×C31×#536C211yesmaximal, characteristic
4C2×C31×#419C211yesmaximal, characteristic, complemented
5C2×C31×#416C211yesmaximal, characteristic
6C31×#2310C211yesmaximal, characteristic, complemented
7C31×#2016C214yesmaximal, complemented
8C31×#2015C214yesmaximal, complemented
9C31×#1729C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C23×Q8×C31C2211yescharacteristic
13C2×C31×#137—24no—
14C2×C31×#137—24no—
15C2×C31×#137—24no—
16C22×Q16×C31—24no—
17C22×Q16×C31—24no—
18C22×Q16×C31—24no—
19C22×Q16×C31C2214yes—
20C22×Q16×C31C2212yes—
21C22×SD16×C31C2212yes—
22C22×M16×C31C2211yescharacteristic
23C23×C8×C31C2211yescharacteristic
24C31×#536—24no—
25C31×#536—24no—
26C2×C31×C4⋊Q8C2212yes—
27C2×C31×#124C2212yes—
28C2×C31×#123—22no—
29C2×C31×C22⋊Q8—22no—
30C2×C31×#117C2211yescharacteristic
31C2×C31×#117C2211yescharacteristic
32C31×#419—24no—
33C31×#416—24no—
34C2×C31×#103C2212yes—
35C2×C31×#103C2214yes—
36C2×C31×#102C2212yes—
37C31×#391—22no—
38C31×#391C2214yes—
39C31×#381—22no—
40C31×#336C2214yes—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31D414yes—
47C22×Q8×C31—21no—
48C22×Q8×C31—22no—
49C22×Q8×C31—21no—
50C22×Q8×C31—22no—
51C22×Q8×C31—22no—
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31—22no—
55C22×Q8×C31—22no—
56C22×Q8×C31—22no—
57C22×Q8×C31—22no—
58C22×Q8×C31—22no—
59C22×Q8×C31D414yes—
60C22×Q8×C31D414yes—
61C22×Q8×C31C2312yes—
62C22×Q8×C31C2311yescharacteristic
63C22×Q8×C31C2312yes—
64C22×D4×C31—21no—
65C22×D4×C31C2311yescharacteristic
66C23×C4×C31—21no—
67C23×C4×C31—21no—
68C23×C4×C31—21no—
69C23×C4×C31C2311yescharacteristic
70C31×#137—232no—
71C2×Q16×C31—232no—
72C2×Q16×C31—28no—
73C2×Q16×C31—28no—
74C2×Q16×C31—24no—
75C2×Q16×C31—24no—
76C2×Q16×C31—28no—
77C2×Q16×C31—24no—
78C2×SD16×C31—24no—
79C2×SD16×C31—24no—
80C2×SD16×C31—24no—
81C2×M16×C31—22no—
82C2×M16×C31—22no—
83C2×M16×C31D414yes—
84C22×C8×C31—22no—
85C22×C8×C31—22no—
86C22×C8×C31D414yes—
87C22×C8×C31C2312yes—
88C22×C8×C31C2312yes—
89C31×C4⋊Q8—24no—
90C31×C4⋊Q8—24no—
91C31×#124—24no—
92C31×#124—24no—
93C31×#123—216no—
94C31×C22⋊Q8—216no—
95C31×#117—22no—
96C31×#117—22no—
97C31×#117—22no—
98C31×#117—22no—
99C2×C4⋊C4×C31—24no—
100C2×C4⋊C4×C31D412yes—
101C2×C4⋊C4×C31C2312yes—
102C2×C22⋊C4×C31—24no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31D412yes—
105C2×C42×C31—21no—
106C2×C42×C31C2312yes—
107C31×#103—24no—
108C31×#103—28no—
109C31×#102—24no—
110C31×#95—28no—
111C24×C31—21no—
112C24×C31C2×D411yescharacteristic
113#41×C31—28no—
114#41×C31—28no—
115#41×C31—44no—
116#41×C31—44no—
117C2×Q8×C31—24no—
118C2×Q8×C31—24no—
119C2×Q8×C31—22no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—28no—
123C2×Q8×C31—28no—
124C2×Q8×C31—28no—
125C2×Q8×C31—28no—
126C2×Q8×C31—22no—
127C2×Q8×C31—21no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31—22no—
134C2×Q8×C31—21no—
135C2×Q8×C31—21no—
136C2×Q8×C31—22no—
137C2×Q8×C31—21no—
138C2×Q8×C31—42no—
139C2×Q8×C31—44no—
140C2×Q8×C31—42no—
141C2×Q8×C31—44no—
142C2×Q8×C31—44no—
143C2×Q8×C31—44no—
144C2×Q8×C31—42no—
145C2×Q8×C31—42no—
146C2×Q8×C31C2×D414yes—
147C2×Q8×C31C2×D412yes—
148C2×Q8×C31C2×D414yes—
149C2×D4×C31—24no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—42no—
156C2×D4×C31—42no—
157C2×D4×C31C2×D412yes—
158C22×C4×C31—24no—
159C22×C4×C31—24no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—24no—
164C22×C4×C31—22no—
165C22×C4×C31—42no—
166C22×C4×C31—42no—
167C22×C4×C31—42no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—42no—
171C22×C4×C31C2×D414yes—
172C22×C4×C31C2×D412yes—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D412yes—
178C22×C4×C31#4112yes—
179C22×C4×C31C2411yescharacteristic
180Q16×C31—232no—
181Q16×C31—216no—
182SD16×C31—216no—
183M16×C31—28no—
184C2×C8×C31—28no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31C2×D414yes—
190C2×C8×C31C2×D414yes—
191C4⋊C4×C31—216no—
192C4⋊C4×C31—24no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C22⋊C4×C31—216no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—42no—
198C42×C31—22no—
199C42×C31—22no—
200C42×C31—42no—
201#23425C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—24no—
203C23×C31—22no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—44no—
207C23×C31#12012yes—
208C23×C31#12112yes—
209C23×C31#12712yes—
210C23×C31C22×D411yescharacteristic
211Q8×C31—24no—
212Q8×C31—24no—
213Q8×C31—24no—
214Q8×C31—22no—
215Q8×C31—24no—
216Q8×C31—22no—
217Q8×C31—42no—
218Q8×C31—41no—
219Q8×C31—42no—
220Q8×C31—42no—
221Q8×C31—42no—
222Q8×C31—41no—
223Q8×C31—48no—
224Q8×C31—48no—
225D4×C31—22no—
226D4×C31—22no—
227D4×C31—48no—
228D4×C31—41no—
229D4×C31—41no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—24no—
234C2×C4×C31—28no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—48no—
242C2×C4×C31—48no—
243C2×C4×C31—48no—
244C2×C4×C31—42no—
245C2×C4×C31—44no—
246C2×C4×C31#12014yes—
247C2×C4×C31#12014yes—
248C2×C4×C31#12012yes—
249C2×C4×C31#12114yes—
250C2×C4×C31#12114yes—
251C2×C4×C31#12112yes—
252C2×C4×C31#12712yes—
253C2×C4×C31C22×D411yescharacteristic
254C2×C4×C31C22×D411yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C2×C4×C31C22×D411yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C2×#4111yescharacteristic
259C8×C31—24no—
260C8×C31—24no—
261C22×#137C2×C3111yescharacteristic
262C23×Q16C2×C3111yescharacteristic, complemented
263C2×#536C2×C3111yescharacteristic
264C2×#419C2×C3111yescharacteristic, complemented
265C2×#416C2×C3111yescharacteristic
266#2310C2×C3111yescharacteristic, complemented
267#2016C2×C3114yescomplemented
268#2015C2×C3114yescomplemented
269#1729C2×C3111yescharacteristic, complemented
270C22×C31—24no—
271C22×C31—24no—
272C22×C31—24no—
273C22×C31—42no—
274C22×C31—44no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31#39214yes—
278C22×C31C2×#12011yescharacteristic
279C22×C31C2×#12111yescharacteristic
280C22×C31C2×#12711yescharacteristic
281C22×C31#57814yes—
282C4×C31—42no—
283C4×C31—44no—
284C4×C31—44no—
285C4×C31—44no—
286C4×C31—82no—
287C4×C31#39214yes—
288C4×C31C2×#12011yescharacteristic
289C4×C31C2×#12011yescharacteristic
290C4×C31C2×#12111yescharacteristic
291C4×C31C2×#12111yescharacteristic
292C22×#41C22×C3111yescharacteristic
293C23×Q8C22×C3111yescharacteristic
294C23×Q8C22×C3111yescharacteristic
295C2×#137—24no—
296C2×#137—24no—
297C2×#137—24no—
298C22×Q16—24no—
299C22×Q16—24no—
300C22×Q16—24no—
301C22×Q16C22×C3114yes—
302C22×Q16C22×C3112yes—
303C22×SD16C22×C3112yes—
304C22×M16C22×C3111yescharacteristic
305C23×C8C22×C3111yescharacteristic
306#536—24no—
307#536—24no—
308C2×C4⋊Q8C22×C3112yes—
309C2×#124C22×C3112yes—
310C2×#123—22no—
311C2×C22⋊Q8—22no—
312C2×#117C22×C3111yescharacteristic
313C2×#117C22×C3111yescharacteristic
314#419—24no—
315#416—24no—
316C2×#103C22×C3112yes—
317C2×#103C22×C3114yes—
318C2×#102C22×C3112yes—
319#391—22no—
320#391C22×C3114yes—
321#381—22no—
322#336C22×C3114yes—
323C2×C31—24no—
324C2×C31—82no—
325C2×C31C2×#39211yescharacteristic
326C2×C31#301112yes—
327C2×C31#304412yes—
328C2×C31#315612yes—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41D4×C3114yes—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—21no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8D4×C3114yes—
348C22×Q8D4×C3114yes—
349C22×Q8C23×C3112yes—
350C22×Q8C23×C3111yescharacteristic
351C22×Q8C23×C3112yes—
352C22×D4—21no—
353C22×D4C23×C3111yescharacteristic
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4C23×C3111yescharacteristic
358#137—232no—
359C2×Q16—232no—
360C2×Q16—28no—
361C2×Q16—28no—
362C2×Q16—24no—
363C2×Q16—24no—
364C2×Q16—28no—
365C2×Q16—24no—
366C2×SD16—24no—
367C2×SD16—24no—
368C2×SD16—24no—
369C2×M16—22no—
370C2×M16—22no—
371C2×M16D4×C3114yes—
372C22×C8—22no—
373C22×C8—22no—
374C22×C8D4×C3114yes—
375C22×C8C23×C3112yes—
376C22×C8C23×C3112yes—
377C4⋊Q8—24no—
378C4⋊Q8—24no—
379#124—24no—
380#124—24no—
381#123—216no—
382C22⋊Q8—216no—
383#117—22no—
384#117—22no—
385#117—22no—
386#117—22no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4D4×C3112yes—
389C2×C4⋊C4C23×C3112yes—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4D4×C3112yes—
393C2×C42—21no—
394C2×C42C23×C3112yes—
395#103—24no—
396#103—28no—
397#102—24no—
398#95—28no—
399C31#2342511yescharacteristic, direct product factor, Sylow
400C24—21no—
401C24C2×D4×C3111yescharacteristic
402#41—28no—
403#41—28no—
404#41—44no—
405#41—44no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8C2×D4×C3114yes—
436C2×Q8C2×D4×C3112yes—
437C2×Q8C2×D4×C3114yes—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4C2×D4×C3112yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4C2×D4×C3114yes—
461C22×C4C2×D4×C3112yes—
462C22×C4C2×D4×C3112yes—
463C22×C4C2×D4×C3112yes—
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3112yes—
466C22×C4C2×D4×C3112yes—
467C22×C4#41×C3112yes—
468C22×C4C24×C3111yescharacteristic
469Q16—232no—
470Q16—216no—
471SD16—216no—
472M16—28no—
473C2×C8—28no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8C2×D4×C3114yes—
479C2×C8C2×D4×C3114yes—
480C4⋊C4—216no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C22⋊C4—216no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C42—22no—
488C42—22no—
489C42—42no—
490C23—24no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—44no—
495C23C31×#12012yes—
496C23C31×#12112yes—
497C23C31×#12712yes—
498C23C22×D4×C3111yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—24no—
502Q8—22no—
503Q8—24no—
504Q8—22no—
505Q8—42no—
506Q8—41no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—41no—
511Q8—48no—
512Q8—48no—
513D4—22no—
514D4—22no—
515D4—48no—
516D4—41no—
517D4—41no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4C31×#12014yes—
535C2×C4C31×#12014yes—
536C2×C4C31×#12012yes—
537C2×C4C31×#12114yes—
538C2×C4C31×#12114yes—
539C2×C4C31×#12112yes—
540C2×C4C31×#12712yes—
541C2×C4C22×D4×C3111yescharacteristic
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C22×D4×C3111yescharacteristic
545C2×C4C22×D4×C3111yescharacteristic
546C2×C4C2×#41×C3111yescharacteristic
547C8—24no—
548C8—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—44no—
554C22—82no—
555C22—82no—
556C22C31×#39214yes—
557C22C2×C31×#12011yescharacteristic
558C22C2×C31×#12111yescharacteristic
559C22C2×C31×#12711yescharacteristic
560C22C31×#57814yes—
561C4—42no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4C31×#39214yes—
567C4C2×C31×#12011yescharacteristic
568C4C2×C31×#12011yescharacteristic
569C4C2×C31×#12111yescharacteristic
570C4C2×C31×#12111yescharacteristic
571C2—24no—
572C2—82no—
573C2C2×C31×#39211yescharacteristic
574C2C31×#301112yes—
575C2C31×#304412yes—
576C2C31×#315612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×Q16×C31C2semidirect
4. C2×C31×#419C2semidirect
6. C31×#2310C2semidirect
7. C31×#2016C2semidirect
8. C31×#2015C2semidirect
9. C31×#1729C2semidirect
201. #23425C31direct (N × H)
262. C23×Q16C2×C31semidirect
264. C2×#419C2×C31semidirect
266. #2310C2×C31semidirect
267. #2016C2×C31semidirect
268. #2015C2×C31semidirect
269. #1729C2×C31semidirect
399. C31#23425direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1372
C23×Q16×C312
C2×C31×#5362
C2×C31×#4192
C2×C31×#4162
C31×#23102
C31×#20162
C31×#20152
C31×#17292
#2342531