Finite Group Databasebeta

← C31×#23424 #21480183

Subgroups — C31×#23424 #21480183

Order 7936.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C23×SD16×C31C211yesmaximal, characteristic, complemented
3C2×C31×#536C211yesmaximal, characteristic, complemented
4C2×C31×#419C211yesmaximal, characteristic, complemented
5C2×C31×#416C211yesmaximal, characteristic, complemented
6C31×#2309C211yesmaximal, characteristic, complemented
7C31×#2013C214yesmaximal, complemented
8C31×#2012C214yesmaximal, complemented
9C31×#1729C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C23×D4×C31C2211yescharacteristic
13C2×C31×AGL(1,ℤ/8)—24no—
14C2×C31×AGL(1,ℤ/8)—24no—
15C2×C31×AGL(1,ℤ/8)—24no—
16C22×SD16×C31—24no—
17C22×SD16×C31—24no—
18C22×SD16×C31—24no—
19C22×SD16×C31C2214yes—
20C22×SD16×C31C2212yes—
21C22×D8×C31C2212yes—
22C22×M16×C31C2211yescharacteristic
23C23×C8×C31C2211yescharacteristic
24C31×#536—24no—
25C31×#536—24no—
26C2×C31×C4⋊Q8C2212yescomplemented
27C2×C31×#124C2212yescomplemented
28C2×C31×#123—22no—
29C2×C31×C22⋊Q8—22no—
30C2×C31×#117C2211yescharacteristic
31C2×C31×#117C2211yescharacteristic
32C31×#419—24no—
33C31×#416—24no—
34C2×C31×#103C2212yescomplemented
35C2×C31×#103C2214yescomplemented
36C2×C31×#102C2212yescomplemented
37C31×#390—22no—
38C31×#390C2214yes—
39C31×#379—22no—
40C31×#336C2214yescomplemented
41C25×C31—21no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31D414yes—
48C22×Q8×C31—21no—
49C22×Q8×C31—22no—
50C22×Q8×C31—22no—
51C22×Q8×C31—22no—
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31D414yes—
55C22×Q8×C31C2312yes—
56C22×Q8×C31C2311yescharacteristic
57C22×D4×C31—21no—
58C22×D4×C31—22no—
59C22×D4×C31—22no—
60C22×D4×C31—22no—
61C22×D4×C31—22no—
62C22×D4×C31—21no—
63C22×D4×C31—22no—
64C22×D4×C31D414yes—
65C22×D4×C31C2311yescharacteristic
66C22×D4×C31C2312yes—
67C23×C4×C31—21no—
68C23×C4×C31—21no—
69C23×C4×C31C2311yescharacteristic
70C31×AGL(1,ℤ/8)—232no—
71C2×SD16×C31—232no—
72C2×SD16×C31—28no—
73C2×SD16×C31—24no—
74C2×SD16×C31—28no—
75C2×SD16×C31—24no—
76C2×SD16×C31—28no—
77C2×SD16×C31—24no—
78C2×D8×C31—24no—
79C2×D8×C31—24no—
80C2×D8×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×C31—28no—
105C2×C22⋊C4×C31D412yes—
106C2×C42×C31—21no—
107C2×C42×C31C2312yes—
108C31×#103—24no—
109C31×#103—28no—
110C31×#102—24no—
111C24×C31—21no—
112C24×C31—22no—
113C24×C31—24no—
114C24×C31—42no—
115C24×C31—42no—
116C24×C31—84no—
117C24×C31C2×D411yescharacteristic
118#41×C31—28no—
119#41×C31—28no—
120#41×C31—44no—
121#41×C31—44no—
122C2×Q8×C31—24no—
123C2×Q8×C31—22no—
124C2×Q8×C31—21no—
125C2×Q8×C31—28no—
126C2×Q8×C31—28no—
127C2×Q8×C31—22no—
128C2×Q8×C31—21no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31—22no—
134C2×Q8×C31—21no—
135C2×Q8×C31—44no—
136C2×Q8×C31—42no—
137C2×Q8×C31—44no—
138C2×Q8×C31—42no—
139C2×Q8×C31C2×D414yes—
140C2×Q8×C31C2×D412yes—
141C2×D4×C31—24no—
142C2×D4×C31—28no—
143C2×D4×C31—28no—
144C2×D4×C31—24no—
145C2×D4×C31—21no—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—22no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—22no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—22no—
155C2×D4×C31—42no—
156C2×D4×C31—42no—
157C2×D4×C31—42no—
158C2×D4×C31—44no—
159C2×D4×C31—44no—
160C2×D4×C31—42no—
161C2×D4×C31C2×D412yes—
162C2×D4×C31C2×D414yes—
163C22×C4×C31—24no—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—24no—
167C22×C4×C31—22no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—42no—
171C22×C4×C31—42no—
172C22×C4×C31C2×D414yes—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2×D412yes—
179C22×C4×C31#4112yes—
180C22×C4×C31C2411yescharacteristic
181SD16×C31—232no—
182SD16×C31—216no—
183D8×C31—216no—
184M16×C31—28no—
185C2×C8×C31—28no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C2×C8×C31C2×D414yes—
191C2×C8×C31C2×D414yes—
192C4⋊C4×C31—216no—
193C4⋊C4×C31—24no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C22⋊C4×C31—216no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—42no—
199C22⋊C4×C31—88no—
200C42×C31—22no—
201C42×C31—22no—
202C42×C31—42no—
203#23424C3111yesmaximal, characteristic, direct product factor, Sylow
204C23×C31—24no—
205C23×C31—22no—
206C23×C31—24no—
207C23×C31—42no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—42no—
211C23×C31—48no—
212C23×C31—44no—
213C23×C31—48no—
214C23×C31—88no—
215C23×C31—88no—
216C23×C31—88no—
217C23×C31#12012yes—
218C23×C31#12112yes—
219C23×C31#12712yes—
220C23×C31C22×D411yescharacteristic
221Q8×C31—24no—
222Q8×C31—22no—
223Q8×C31—24no—
224Q8×C31—22no—
225Q8×C31—42no—
226Q8×C31—41no—
227Q8×C31—42no—
228Q8×C31—41no—
229Q8×C31—48no—
230D4×C31—22no—
231D4×C31—24no—
232D4×C31—24no—
233D4×C31—22no—
234D4×C31—48no—
235D4×C31—41no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—41no—
239D4×C31—48no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—22no—
243C2×C4×C31—28no—
244C2×C4×C31—41no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—48no—
249C2×C4×C31—48no—
250C2×C4×C31—42no—
251C2×C4×C31—44no—
252C2×C4×C31#12014yes—
253C2×C4×C31#12014yes—
254C2×C4×C31#12012yes—
255C2×C4×C31#12114yes—
256C2×C4×C31#12114yes—
257C2×C4×C31#12112yes—
258C2×C4×C31#12712yes—
259C2×C4×C31C22×D411yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C2×#4111yescharacteristic
265C8×C31—24no—
266C8×C31—24no—
267C22×AGL(1,ℤ/8)C2×C3111yescharacteristic
268C23×SD16C2×C3111yescharacteristic, complemented
269C2×#536C2×C3111yescharacteristic, complemented
270C2×#419C2×C3111yescharacteristic, complemented
271C2×#416C2×C3111yescharacteristic, complemented
272#2309C2×C3111yescharacteristic, complemented
273#2013C2×C3114yescomplemented
274#2012C2×C3114yescomplemented
275#1729C2×C3111yescharacteristic, complemented
276C22×C31—24no—
277C22×C31—24no—
278C22×C31—24no—
279C22×C31—42no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—48no—
283C22×C31—816no—
284C22×C31—82no—
285C22×C31—84no—
286C22×C31—84no—
287C22×C31—82no—
288C22×C31#39214yes—
289C22×C31C2×#12011yescharacteristic
290C22×C31C2×#12111yescharacteristic
291C22×C31C2×#12711yescharacteristic
292C22×C31#57714yes—
293C4×C31—42no—
294C4×C31—44no—
295C4×C31—44no—
296C4×C31—82no—
297C4×C31#39214yes—
298C4×C31C2×#12011yescharacteristic
299C4×C31C2×#12011yescharacteristic
300C4×C31C2×#12111yescharacteristic
301C4×C31C2×#12111yescharacteristic
302C22×#41C22×C3111yescharacteristic
303C23×Q8C22×C3111yescharacteristic
304C23×D4C22×C3111yescharacteristic
305C2×AGL(1,ℤ/8)—24no—
306C2×AGL(1,ℤ/8)—24no—
307C2×AGL(1,ℤ/8)—24no—
308C22×SD16—24no—
309C22×SD16—24no—
310C22×SD16—24no—
311C22×SD16C22×C3114yes—
312C22×SD16C22×C3112yes—
313C22×D8C22×C3112yes—
314C22×M16C22×C3111yescharacteristic
315C23×C8C22×C3111yescharacteristic
316#536—24no—
317#536—24no—
318C2×C4⋊Q8C22×C3112yescomplemented
319C2×#124C22×C3112yescomplemented
320C2×#123—22no—
321C2×C22⋊Q8—22no—
322C2×#117C22×C3111yescharacteristic
323C2×#117C22×C3111yescharacteristic
324#419—24no—
325#416—24no—
326C2×#103C22×C3112yescomplemented
327C2×#103C22×C3114yescomplemented
328C2×#102C22×C3112yescomplemented
329#390—22no—
330#390C22×C3114yes—
331#379—22no—
332#336C22×C3114yescomplemented
333C2×C31—24no—
334C2×C31—82no—
335C2×C31—84no—
336C2×C31C2×#39211yescharacteristic
337C2×C31#301012yes—
338C2×C31#304212yes—
339C2×C31#315412yes—
340C25—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41D4×C3114yes—
347C22×Q8—21no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8D4×C3114yes—
354C22×Q8C23×C3112yes—
355C22×Q8C23×C3111yescharacteristic
356C22×D4—21no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—22no—
363C22×D4D4×C3114yes—
364C22×D4C23×C3111yescharacteristic
365C22×D4C23×C3112yes—
366C23×C4—21no—
367C23×C4—21no—
368C23×C4C23×C3111yescharacteristic
369AGL(1,ℤ/8)—232no—
370C2×SD16—232no—
371C2×SD16—28no—
372C2×SD16—24no—
373C2×SD16—28no—
374C2×SD16—24no—
375C2×SD16—28no—
376C2×SD16—24no—
377C2×D8—24no—
378C2×D8—24no—
379C2×D8—24no—
380C2×M16—22no—
381C2×M16—22no—
382C2×M16D4×C3114yes—
383C22×C8—22no—
384C22×C8—22no—
385C22×C8D4×C3114yes—
386C22×C8C23×C3112yes—
387C22×C8C23×C3112yes—
388C4⋊Q8—24no—
389C4⋊Q8—24no—
390#124—24no—
391#124—24no—
392#123—216no—
393C22⋊Q8—216no—
394#117—22no—
395#117—22no—
396#117—22no—
397#117—22no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4D4×C3112yes—
400C2×C4⋊C4C23×C3112yes—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—28no—
404C2×C22⋊C4D4×C3112yes—
405C2×C42—21no—
406C2×C42C23×C3112yes—
407#103—24no—
408#103—28no—
409#102—24no—
410C31#2342411yescharacteristic, direct product factor, Sylow
411C24—21no—
412C24—22no—
413C24—24no—
414C24—42no—
415C24—42no—
416C24—84no—
417C24C2×D4×C3111yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8C2×D4×C3114yes—
440C2×Q8C2×D4×C3112yes—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—24no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4C2×D4×C3112yes—
462C2×D4C2×D4×C3114yes—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4C2×D4×C3114yes—
473C22×C4C2×D4×C3112yes—
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3112yes—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3112yes—
478C22×C4C2×D4×C3112yes—
479C22×C4#41×C3112yes—
480C22×C4C24×C3111yescharacteristic
481SD16—232no—
482SD16—216no—
483D8—216no—
484M16—28no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8C2×D4×C3114yes—
491C2×C8C2×D4×C3114yes—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C22⋊C4—216no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—88no—
500C42—22no—
501C42—22no—
502C42—42no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—48no—
511C23—44no—
512C23—48no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23C31×#12012yes—
517C23C31×#12112yes—
518C23C31×#12712yes—
519C23C22×D4×C3111yescharacteristic
520Q8—24no—
521Q8—22no—
522Q8—24no—
523Q8—22no—
524Q8—42no—
525Q8—41no—
526Q8—42no—
527Q8—41no—
528Q8—48no—
529D4—22no—
530D4—24no—
531D4—24no—
532D4—22no—
533D4—48no—
534D4—41no—
535D4—42no—
536D4—42no—
537D4—41no—
538D4—48no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4C31×#12014yes—
552C2×C4C31×#12014yes—
553C2×C4C31×#12012yes—
554C2×C4C31×#12114yes—
555C2×C4C31×#12114yes—
556C2×C4C31×#12112yes—
557C2×C4C31×#12712yes—
558C2×C4C22×D4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C2×C4C22×D4×C3111yescharacteristic
561C2×C4C22×D4×C3111yescharacteristic
562C2×C4C22×D4×C3111yescharacteristic
563C2×C4C2×#41×C3111yescharacteristic
564C8—24no—
565C8—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—816no—
574C22—84no—
575C22—82no—
576C22—84no—
577C22—82no—
578C22C31×#39214yes—
579C22C2×C31×#12011yescharacteristic
580C22C2×C31×#12111yescharacteristic
581C22C2×C31×#12711yescharacteristic
582C22C31×#57714yes—
583C4—42no—
584C4—44no—
585C4—44no—
586C4—82no—
587C4C31×#39214yes—
588C4C2×C31×#12011yescharacteristic
589C4C2×C31×#12011yescharacteristic
590C4C2×C31×#12111yescharacteristic
591C4C2×C31×#12111yescharacteristic
592C2—24no—
593C2—84no—
594C2—82no—
595C2C2×C31×#39211yescharacteristic
596C2C31×#301012yes—
597C2C31×#304212yes—
598C2C31×#315412yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×C31×AGL(1,ℤ/8)C2
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×SD16×C31C2semidirect
3. C2×C31×#536C2semidirect
4. C2×C31×#419C2semidirect
5. C2×C31×#416C2semidirect
6. C31×#2309C2semidirect
7. C31×#2013C2semidirect
8. C31×#2012C2semidirect
9. C31×#1729C2semidirect
26. C2×C31×C4⋊Q8C22semidirect
27. C2×C31×#124C22semidirect
34. C2×C31×#103C22semidirect
35. C2×C31×#103C22semidirect
36. C2×C31×#102C22semidirect
40. C31×#336C22semidirect
203. #23424C31direct (N × H)
268. C23×SD16C2×C31semidirect
269. C2×#536C2×C31semidirect
270. C2×#419C2×C31semidirect
271. C2×#416C2×C31semidirect
272. #2309C2×C31semidirect
273. #2013C2×C31semidirect
274. #2012C2×C31semidirect
275. #1729C2×C31semidirect
318. C2×C4⋊Q8C22×C31semidirect
319. C2×#124C22×C31semidirect
326. C2×#103C22×C31semidirect
327. C2×#103C22×C31semidirect
328. C2×#102C22×C31semidirect
332. #336C22×C31semidirect
410. C31#23424direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×AGL(1,ℤ/8)2
C23×SD16×C312
C2×C31×#5362
C2×C31×#4192
C2×C31×#4162
C31×#23092
C31×#20132
C31×#20122
C31×#17292
#2342431