Finite Group Databasebeta

← #26142935

Subgroups — #26142935

Order 12032.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094014C211yesmaximal, characteristic
2C2×#15688990C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20241654C212yesmaximal
5#20241654C212yesmaximal
6#20241642C211yesmaximal, characteristic
7#20241641C211yesmaximal, characteristic
8#20241638C211yesmaximal, characteristic
9#20241622C212yesmaximal
10#20241619C212yesmaximal
11C47×#1441C211yesmaximal, characteristic
12C2×#13094014—24no—
13C2×#13094014—24no—
14C2×#13094014C418yescomplemented
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×Dic47⋊C4C2211yescharacteristic
20C22×Dic47⋊C4C2211yescharacteristic
21C22×C4×Dic47C2212yes—
22C22×C4×Dic47C2211yescharacteristic
23#15688990—24no—
24C4⋊C4×Dic47—24no—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46#13094014—232no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—24no—
51C2×C4⋊Dic47C2×C414yes—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4—22no—
59C2×Dic47⋊C4—22no—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4C2×C414yes—
62C2×Dic47⋊C4C2×C414yes—
63C2×Dic47⋊C4C2×C414yes—
64C2×C4×Dic47—24no—
65C2×C4×Dic47—42no—
66C2×C4×Dic47—42no—
67C2×C4×Dic47C2×C414yes—
68C2×C4×Dic47D414yes—
69C2×C4×Dic47D414yes—
70C47⋊#95—22no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—42no—
78C47⋊#95—42no—
79C47⋊#95—42no—
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2312yes—
84C23×C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47—24no—
89C2×C4⋊C4×C47C2×C414yes—
90#95×C47—22no—
91#95×C47—24no—
92#95×C47—42no—
93C24×C47C2411yescharacteristic
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47#4114yes—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—22no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47C22×C412yes—
155C22×Dic47C22×C412yes—
156C22×Dic47C22×C412yes—
157C22×Dic47C22×C412yes—
158C22×Dic47C2×D412yes—
159C22×Dic47C2×D412yes—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×D412yes—
162C22×Dic47#4114yes—
163C4⋊C4×C47—28no—
164C4⋊C4×C47—28no—
165C4⋊Dic47—28no—
166C4⋊Dic47—28no—
167Dic47⋊C4—28no—
168Dic47⋊C4—28no—
169Dic47⋊C4—28no—
170Dic47⋊C4—28no—
171C4×Dic47—48no—
172C4×Dic47C22⋊C4116yes—
173C23×C47C23×C411yescharacteristic
174C23×C47C22×D411yescharacteristic
175C23×C47C22×D411yescharacteristic
176C23×C47C2×#4112yes—
177C23×C47C2×#4112yes—
178C23×C47H(5,𝔽2)11yescharacteristic
179C23×C47H(5,𝔽2)11yescharacteristic
180C23×C472−1+412yes—
181C23×C472−1+411yescharacteristic
182C23×C472−1+412yes—
183C23×C472−1+411yescharacteristic
184C2×C4×C47—24no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—24no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—24no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—24no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47C2×C22⋊C414yes—
215C2×Dic47—24no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—24no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—24no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—42no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47C2×C22⋊C414yes—
252C2×Dic47C2×C22⋊C414yes—
253#14289—471nomaximal, Sylow
254C22×C47C22×C22⋊C411yescharacteristic
255C22×C47C4×#4112yes—
256C22×C47#51912yes—
257C22×C47#52011yescharacteristic
258C22×C47#52011yescharacteristic
259C22×C47#52511yescharacteristic
260C22×C47#52511yescharacteristic
261C22×C47#53212yes—
262C22×C47#53511yescharacteristic
263C22×C47#53611yescharacteristic
264C22×C47#53612yes—
265C22×C47#53611yescharacteristic
266C22×C47#53611yescharacteristic
267C22×C47#53711yescharacteristic
268C22×C47#53711yescharacteristic
269C22×C47#53912yes—
270C22×C47#53912yes—
271C22×C47#53912yes—
272C22×C47#53912yes—
273C22×C47#54112yes—
274C22×C47#54112yes—
275C22×C47#54112yes—
276C22×C47#54112yes—
277C4×C47—24no—
278C4×C47—42no—
279C4×C47—44no—
280C4×C47—42no—
281C4×C47—82no—
282Dic47—24no—
283Dic47—24no—
284Dic47—42no—
285Dic47—44no—
286Dic47—42no—
287Dic47—82no—
288C22×#125—471no—
289C2×#383—471no—
290C2×C4×C4⋊C4—471no—
291#1445—472no—
292#1445—472no—
293#1441—471no—
294#1441—471no—
295#1441—471no—
296#1441D4711yescharacteristic
297#1436—472no—
298#1436—472no—
299C2×C47#229711yescharacteristic
300C2×C47#231011yescharacteristic
301C2×C47#231111yescharacteristic
302C2×C47#233612yes—
303C2×C47#233612yes—
304C2×C47#263311yescharacteristic
305C2×C47#263311yescharacteristic
306C2×C47#263411yescharacteristic
307C2×C47#263411yescharacteristic
308C2×C47#266412yes—
309C2×C47#266412yes—
310C2×#125—478no—
311C2×#125—944no—
312C2×#125—944no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4—471no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4C2×D4711yescharacteristic
320C22×C4⋊C4C2×D4711yescharacteristic
321C22×C42—471no—
322C22×C42—472no—
323#383—944no—
324C4×C4⋊C4—944no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—471no—
330C2×#95—472no—
331C2×#95—472no—
332C2×#95—472no—
333C2×#95—472no—
334C2×#95—472no—
335C2×#95C2×D4712yes—
336C2×#95C2×D4711yescharacteristic
337C2×#95C2×D4712yes—
338C47#1428911yescharacteristic, complemented, Sylow
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—472no—
344C23×C4—472no—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347C23×C4C22×D4711yescharacteristic
348C23×C4C22×D4712yes—
349C23×C4C22×D4711yescharacteristic
350#125—9432no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—474no—
356C2×C4⋊C4—474no—
357C2×C4⋊C4—474no—
358C2×C4⋊C4—474no—
359C2×C4⋊C4—474no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4C4×D4714yes—
373C2×C42—474no—
374C2×C42—474no—
375C2×C42—474no—
376C2×C42—944no—
377C2×C42—1882no—
378C2×C42—1882no—
379#95—22no—
380#95—24no—
381#95—42no—
382#95—944no—
383#95—944no—
384#95—944no—
385#95—944no—
386#95—944no—
387#95—942no—
388#95—944no—
389#95—1882no—
390#95—1882no—
391#95—1882no—
392C24C23×D4711yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—472no—
419C22×C4—472no—
420C22×C4—472no—
421C22×C4—472no—
422C22×C4—472no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—474no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—942no—
430C22×C4—941no—
431C22×C4—942no—
432C22×C4—941no—
433C22×C4—942no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—941no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—942no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4C2×C4×D4712yes—
457C22×C4C2×C4×D4712yes—
458C22×C4C2×C4×D4712yes—
459C22×C4D4×D4712yes—
460C22×C4D4×D4712yes—
461C22×C4#1062535214yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—948no—
465C4⋊C4—948no—
466C4⋊C4—948no—
467C4⋊C4—948no—
468C4⋊C4—948no—
469C4⋊C4—948no—
470C42—4716no—
471C42—1888no—
472C23C22×C4×D4711yescharacteristic
473C23C2×D4×D4711yescharacteristic
474C23C2×D4×D4711yescharacteristic
475C23#41×D4712yes—
476C23C2×#1062535212yes—
477C23#1309406911yescharacteristic
478C23#1309407011yescharacteristic
479C23#1309407111yescharacteristic
480C23#1309407112yes—
481C23#1309407211yescharacteristic
482C23#1309407212yes—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—474no—
514C2×C4—474no—
515C2×C4—944no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—944no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—944no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1882no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4C22⋊C4×D4714yes—
552C22C2×C22⋊C4×D4711yescharacteristic
553C22C4×#1062535212yes—
554C22#1568959912yes—
555C22#1568960311yescharacteristic
556C22#1568960411yescharacteristic
557C22#1568962911yescharacteristic
558C22#1568963111yescharacteristic
559C22#1568966812yes—
560C22#1568968911yescharacteristic
561C22#1568969311yescharacteristic
562C22#1568969611yescharacteristic
563C22#1568969612yes—
564C22#1568969811yescharacteristic
565C22#1568970211yescharacteristic
566C22#1568970311yescharacteristic
567C22#1568971712yes—
568C22#1568971812yes—
569C22#1568971912yes—
570C22#1568972112yes—
571C22#1568973012yes—
572C22#1568973212yes—
573C22#1568973312yes—
574C22#1568973412yes—
575C4—24no—
576C4—42no—
577C4—44no—
578C4—42no—
579C4—82no—
580C4—944no—
581C4—944no—
582C4—3762no—
583C4—1882no—
584C4—1884no—
585C4—1882no—
586C2#2024587311yescharacteristic
587C2#2024597111yescharacteristic
588C2#2024597711yescharacteristic
589C2#2024618612yes—
590C2#2024619312yes—
591C2#2024925511yescharacteristic
592C2#2024925811yescharacteristic
593C2#2024926311yescharacteristic
594C2#2024926611yescharacteristic
595C2#2024954612yes—
596C2#2024954812yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1441C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1441D47
hypocenterC47#14289
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. C2×#13094014C4semidirect
338. C47#14289semidirect

Maximal subgroups

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

subgroupindex
C22×#130940142
C2×#156889902
C2×C4⋊C4×Dic472
#202416542
#202416422
#202416412
#202416382
#202416222
#202416192
C47×#14412
#1428947