Finite Group Databasebeta

← #26235805

Subgroups — #26235805

Order 12032.

782 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245851C211yesmaximal, characteristic
2#20243996C211yesmaximal, characteristic, complemented
3#20243955C211yesmaximal, characteristic, complemented
4#20243942C211yesmaximal, characteristic, complemented
5#20243682C211yesmaximal, characteristic
6#20243677C211yesmaximal, characteristic
7#20243530C211yesmaximal, characteristic, complemented
8#20243457C211yesmaximal, characteristic, complemented
9#20243369C211yesmaximal, characteristic, complemented
10#20243171C211yesmaximal, characteristic, complemented
11#20243158C211yesmaximal, characteristic, complemented
12#20243156C211yesmaximal, characteristic, complemented
13#20243133C211yesmaximal, characteristic, complemented
14#20243083C211yesmaximal, characteristic, complemented
15C47×#1923C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18C2×C4×C22⋊D47C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×(C2×D47)⋊C8C2211yescharacteristic
25C2×(C2×D47)⋊C8C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689023C411yescharacteristic
30#15689023C411yescharacteristic
31#15689020C411yescharacteristic
32#15689020C411yescharacteristic
33#15689015C411yescharacteristic
34#15689015C411yescharacteristic
35#15689011C411yescharacteristic
36#15689011C411yescharacteristic
37C4×Dic47⋊C4C2211yescharacteristic
38C4×(C2×D47)⋊C4C2211yescharacteristic
39C42×Dic47C2211yescharacteristic
40#15688878C2211yescharacteristic
41#15688878C2211yescharacteristic
42#15688877C2211yescharacteristic
43#15688877C2211yescharacteristic
44#15688877C2211yescharacteristic
45#15688877C2211yescharacteristic
46#15688877C2211yescharacteristic
47#15688877C2211yescharacteristic
48#15688877C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×C4⋊C8×C47C2211yescharacteristic
53C2×C22⋊C8×C47C2211yescharacteristic
54C2×C22⋊C8×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C4×C22⋊C4×C47C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C22×C22⋊D47C2×C411yescharacteristic
60C22×C4×D47C2311yescharacteristic
61C2×#10625324—22no—
62C2×#10625324—22no—
63C2×#10625324—41no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C2×C8×D47—22no—
69C2×C8×D47—22no—
70C2×C8×D47—41no—
71C4×C22⋊D47—22no—
72C4×C22⋊D47—22no—
73#13093983—41no—
74#13093983—41no—
75#13093981—41no—
76#13093981—41no—
77C2×C4⋊Dic47C2×C411yescharacteristic
78C2×C4⋊Dic47C2×C411yescharacteristic
79C2×Dic47⋊C4C2×C411yescharacteristic
80C2×Dic47⋊C4C2×C411yescharacteristic
81C2×Dic47⋊C4C2×C411yescharacteristic
82C2×Dic47⋊C4C2×C411yescharacteristic
83C2×Dic47⋊C4C2×C411yescharacteristic
84C2×C22⋊Dic47C2×C411yescharacteristic
85C2×C22⋊Dic47C2×C411yescharacteristic
86C2×C22⋊Dic47C2×C411yescharacteristic
87C2×(C2×D47)⋊C4C2×C411yescharacteristic
88C2×(C2×D47)⋊C4C2×C411yescharacteristic
89C2×(C2×D47)⋊C4C2×C411yescharacteristic
90C2×(C2×D47)⋊C4C2×C411yescharacteristic
91C2×(C2×D47)⋊C4C2×C411yescharacteristic
92C2×C4×Dic47C2×C411yescharacteristic
93C2×C4×Dic47C2×C411yescharacteristic
94C2×C4×Dic47C2×C411yescharacteristic
95C2×C4×Dic47C2×C411yescharacteristic
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98C2×C4×Dic47C2311yescharacteristic
99C4⋊(C47⋊C8)—41no—
100C22⋊(C47⋊C8)—22no—
101C22⋊(C47⋊C8)—41no—
102(C2×D47)⋊C8—22no—
103(C2×D47)⋊C8—22no—
104#13093934—41no—
105C4×C47⋊C8—22no—
106C8×Dic47—41no—
107C47⋊#95C2×C411yescharacteristic
108C47⋊#95C2×C411yescharacteristic
109C47⋊#95C2×C411yescharacteristic
110C47⋊#95C2×C411yescharacteristic
111C23×C4×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115C22×C8×C47C2311yescharacteristic
116C2×C22⋊C4×C47C2×C411yescharacteristic
117C2×C22⋊C4×C47C2×C411yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C4⋊C8×C47—41no—
121C22⋊C8×C47—22no—
122C22⋊C8×C47—41no—
123C4×C8×C47—22no—
124#95×C47C2×C411yescharacteristic
125#95×C47C2×C411yescharacteristic
126C24×C47C22×C411yescharacteristic
127C23×D47C22×C411yescharacteristic
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C22×C411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C2×C22⋊D47—22no—
141C2×C22⋊D47—22no—
142C2×C22⋊D47—41no—
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C22×Dic47C22×C411yescharacteristic
146C22×Dic47C22×C411yescharacteristic
147C22×Dic47C22×C411yescharacteristic
148C22×Dic47C22×C411yescharacteristic
149C22×Dic47C22×C411yescharacteristic
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47—21no—
154C2×C4×D47#4112yes—
155C2×C4×D47#4112yes—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47#4112yes—
168C22⋊C4×C47—22no—
169C22⋊C4×C47—22no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C22⋊Dic47—22no—
175C22⋊Dic47—41no—
176C22⋊Dic47—41no—
177(C2×D47)⋊C4—22no—
178(C2×D47)⋊C4—22no—
179(C2×D47)⋊C4—22no—
180(C2×D47)⋊C4—41no—
181(C2×D47)⋊C4—41no—
182C4⋊Dic47—41no—
183C4⋊Dic47—41no—
184Dic47⋊C4—22no—
185Dic47⋊C4—22no—
186Dic47⋊C4—41no—
187Dic47⋊C4—41no—
188Dic47⋊C4—41no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—22no—
198C4×Dic47—41no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8#4112yes—
211#10625324—44no—
212C8×D47—44no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—21no—
219C23×C47—21no—
220C23×C47C23×C411yescharacteristic
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—21no—
227C22×D47#11712yes—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47#13112yes—
246C2×C4×C47#13112yes—
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47H(5,𝔽2)11yescharacteristic
252C2×C4×C47H(5,𝔽2)11yescharacteristic
253C22⋊D47—44no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47#11712yes—
274C2×Dic47#13112yes—
275C4×D47—22no—
276C4×D47—22no—
277C4×D47—22no—
278C4×D47—22no—
279C8×C47—22no—
280C8×C47—41no—
281C8×C47—41no—
282C8×C47—41no—
283C47⋊C8—22no—
284C47⋊C8—41no—
285C47⋊C8—41no—
286C47⋊C8—41no—
287#22789—471nomaximal, Sylow
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47—41no—
294C22×C47—41no—
295C22×C47C2×#11711yescharacteristic
296C22×C47C4×#4111yescharacteristic
297C22×C47#51811yescharacteristic
298C22×C47C2×#13111yescharacteristic
299C22×C47C2×#13111yescharacteristic
300C22×C47C2×#13111yescharacteristic
301C22×C47#56811yescharacteristic
302C2×D47—22no—
303C2×D47—22no—
304C2×D47—22no—
305C2×D47—41no—
306C2×D47—41no—
307C2×D47—41no—
308C2×D47—41no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—41no—
312C4×C47—41no—
313C4×C47—41no—
314C4×C47#55111yescharacteristic
315C4×C47#55111yescharacteristic
316C4×C47#55311yescharacteristic
317C4×C47#55311yescharacteristic
318Dic47—22no—
319Dic47—22no—
320Dic47—41no—
321Dic47—41no—
322Dic47—41no—
323Dic47—41no—
324Dic47—41no—
325#2293—471no—
326#1923—471no—
327#1923D4711yescharacteristic, complemented
328#1917—471no—
329#1916—471no—
330#1870—471no—
331#1869—471no—
332#1839—471no—
333#1818—471no—
334#1798—471no—
335#1753—471no—
336C8×C22⋊C4—471no—
337#1750—471no—
338#1744—471no—
339#1729—471no—
340C2×C47—81no—
341C2×C47#235311yescharacteristic
342C2×C47#292211yescharacteristic
343C2×C47#292411yescharacteristic
344C2×C47#296611yescharacteristic
345C2×C47#296811yescharacteristic
346C2×C47#297911yescharacteristic
347C2×C47#298111yescharacteristic
348D47—42no—
349C22×M16—471no—
350C23×C8—471no—
351C2×C4×D4—471no—
352C2×#117—471no—
353C2×#117—471no—
354C2×C4⋊C8—471no—
355C2×C4⋊C8C2×D4711yescharacteristic
356C2×C22⋊C8—471no—
357C2×C22⋊C8—471no—
358C2×C22⋊C8—471no—
359C2×C22⋊C8—471no—
360C2×C22⋊C8C2×D4711yescharacteristic
361C2×C22⋊C8C2×D4711yescharacteristic
362C2×#97—471no—
363C2×C4×C8—471no—
364C2×C4×C8—471no—
365C2×C4×C8C2×D4711yescharacteristic
366#390—471no—
367#390—471no—
368#389—471no—
369#389—471no—
370#388—471no—
371#388—471no—
372#388—471no—
373#388—471no—
374C4×C4⋊C4—471no—
375C4×C22⋊C4—471no—
376C4×C22⋊C4C2×D4711yescharacteristic
377C43—471no—
378#336—471no—
379#336—471no—
380#336—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384#336—471no—
385#336—471no—
386#336—471no—
387#336—471no—
388#336—471no—
389#336—471no—
390#336C2×D4711yescharacteristic
391#336C2×D4711yescharacteristic
392C47#2278911yescharacteristic, complemented, Sylow
393C22×D4—471no—
394C23×C4—471no—
395C23×C4C22×D4711yescharacteristic
396C2×M16—942no—
397C2×M16—942no—
398C2×M16—1881no—
399C22×C8—471no—
400C22×C8—471no—
401C22×C8—471no—
402C22×C8—471no—
403C22×C8—942no—
404C22×C8—942no—
405C22×C8—1881no—
406C22×C8C22×D4711yescharacteristic
407C22×C8C22×D4711yescharacteristic
408C22×C8C22×D4711yescharacteristic
409C22×C8C22×D4711yescharacteristic
410C4×D4—942no—
411C4×D4—942no—
412#117—1881no—
413#117—1881no—
414#117—1881no—
415#117—1881no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C22⋊C4—471no—
424C2×C22⋊C4—471no—
425C2×C22⋊C4—471no—
426C2×C22⋊C4—471no—
427C2×C22⋊C4—471no—
428C2×C22⋊C4—471no—
429C2×C22⋊C4—471no—
430C2×C22⋊C4—471no—
431C2×C22⋊C4C4×D4711yescharacteristic
432C2×C22⋊C4C4×D4711yescharacteristic
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42C22×D4711yescharacteristic
441C2×C42C22×D4711yescharacteristic
442C4⋊C8—41no—
443C4⋊C8—1881no—
444C22⋊C8—22no—
445C22⋊C8—41no—
446C22⋊C8—942no—
447C22⋊C8—942no—
448C22⋊C8—942no—
449C22⋊C8—1881no—
450#97—1881no—
451C4×C8—22no—
452C4×C8—942no—
453C4×C8—1881no—
454#95—471no—
455#95—471no—
456#95—471no—
457#95—471no—
458#95C4×D4711yescharacteristic
459#95C4×D4711yescharacteristic
460C24—471no—
461C24C2×C4×D4711yescharacteristic
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—1881no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—941no—
484C22×C4C2×C4×D4711yescharacteristic
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4C2×C4×D4711yescharacteristic
487C22×C4C2×C4×D4711yescharacteristic
488C22×C4C2×C4×D4711yescharacteristic
489C22×C4C23×D4711yescharacteristic
490M16—1884no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—472no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—1884no—
515C2×C8#1062535212yes—
516C4⋊C4—942no—
517C4⋊C4—942no—
518C4⋊C4—1881no—
519C4⋊C4—1881no—
520C4⋊C4—1881no—
521C4⋊C4—1881no—
522C4⋊C4—1881no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—942no—
526C22⋊C4—942no—
527C22⋊C4—942no—
528C22⋊C4—942no—
529C22⋊C4—1881no—
530C22⋊C4—1881no—
531C22⋊C4—1881no—
532C22⋊C4—1881no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—942no—
538C42—942no—
539C42—942no—
540C42—941no—
541C42—941no—
542C42—941no—
543C42—941no—
544C42—941no—
545C42—941no—
546C42—1881no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—472no—
555C23—941no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23—941no—
561C23C22×C4×D4711yescharacteristic
562D4—1884no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—942no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—941no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—941no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4#1309403012yes—
606C2×C4#1309403212yes—
607C2×C4#41×D4711yescharacteristic
608C2×C4#41×D4711yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4C2×#1062535211yescharacteristic
611C2×C4#1309406911yescharacteristic
612C2×C4#1309407011yescharacteristic
613C8—22no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—942no—
618C8—1881no—
619C8—1881no—
620C8—1881no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—942no—
629C22—942no—
630C22—942no—
631C22—1881no—
632C22—1881no—
633C22—1881no—
634C22—1881no—
635C22#117×D4711yescharacteristic
636C22C4×#1062535211yescharacteristic
637C22#1568959511yescharacteristic
638C22#131×D4711yescharacteristic
639C22C2×#1309403011yescharacteristic
640C22C2×#1309403211yescharacteristic
641C22#1568988811yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—942no—
648C4—942no—
649C4—1881no—
650C4—1881no—
651C4—1881no—
652C4—1881no—
653C4—1881no—
654C4#1568979611yescharacteristic
655C4#1568980311yescharacteristic
656C4#1568982011yescharacteristic
657C4#1568982411yescharacteristic
658C2—81no—
659C2—1882no—
660C2#2024632311yescharacteristic
661C2#2025202711yescharacteristic
662C2#2025204411yescharacteristic
663C2#2025235811yescharacteristic
664C2#2025237611yescharacteristic
665C2#2025245111yescharacteristic
666C2#2025246511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1923C2
prime radicalC22×C22⋊D47C2×C4
cosocleC22×C4×C47C24
hypercenter#1923D47
hypocenterC47#22789
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
2. #20243996C2semidirect
3. #20243955C2semidirect
4. #20243942C2semidirect
7. #20243530C2semidirect
8. #20243457C2semidirect
9. #20243369C2semidirect
10. #20243171C2semidirect
11. #20243158C2semidirect
12. #20243156C2semidirect
13. #20243133C2semidirect
14. #20243083C2semidirect
15. C47×#1923C2semidirect
327. #1923D47semidirect
392. C47#22789semidirect

Maximal subgroups

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

subgroupindex
#202458512
#202439962
#202439552
#202439422
#202436822
#202436772
#202435302
#202434572
#202433692
#202431712
#202431582
#202431562
#202431332
#202430832
C47×#19232
#2278947