Finite Group Databasebeta

← #26234444

Subgroups — #26234444

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689196C211yesmaximal, characteristic
2C2×#15689164C211yesmaximal, characteristic, complemented
3C2×#15689158C211yesmaximal, characteristic, complemented
4#20246339C211yesmaximal, characteristic
5#20246314C211yesmaximal, characteristic
6#20244154C211yesmaximal, characteristic, complemented
7#20244144C211yesmaximal, characteristic, complemented
8#20243586C211yesmaximal, characteristic, complemented
9#20243586C211yesmaximal, characteristic, complemented
10#20243584C211yesmaximal, characteristic, complemented
11#20243574C211yesmaximal, characteristic, complemented
12#20243418C211yesmaximal, characteristic, complemented
13#20243414C211yesmaximal, characteristic, complemented
14#20243285C211yesmaximal, characteristic, complemented
15C47×#1847C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×#13093982C2211yescharacteristic
21C2×#13093982C2211yescharacteristic
22C2×C4⋊C4×D47C2211yescharacteristic
23#15689196C418yescomplemented
24#15689164—24no—
25#15689158—24no—
26C2×#13093958C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093957C2211yescharacteristic
29C2×#13093957C2211yescharacteristic
30C2×#13093956C2211yescharacteristic
31C2×#13093955C2211yescharacteristic
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33C2×(C2×D47)⋊C8C2211yescharacteristic
34C2×(C2×D47)⋊C8C2211yescharacteristic
35#15689024—21no—
36#15689013—21no—
37#15689013—21no—
38#15689012—21no—
39#15689006—21no—
40#15689001—21no—
41#15688980C2211yescharacteristic
42#15688980C2211yescharacteristic
43#15688980C2211yescharacteristic
44#15688980C2211yescharacteristic
45C4⋊C4×Dic47—21no—
46C4×C4⋊Dic47C2211yescharacteristic
47C4×C4⋊Dic47C2211yescharacteristic
48#15688970—21no—
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×#107×C47C2211yescharacteristic
56C2×#107×C47C2211yescharacteristic
57C2×#106×C47C2211yescharacteristic
58C2×#106×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C47×#384C2211yescharacteristic
61C47×#382C2211yescharacteristic
62C22×D188D411yescharacteristic
63C22×C4×D47—21no—
64C22×C4×D47C2311yescharacteristic
65C22×C4×D47C2311yescharacteristic
66C2×#10625324—22no—
67C2×#10625324—22no—
68C2×#10625324C2×C414yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C2×C8×D47—22no—
72C2×C8×D47—22no—
73C2×C8×D47C2×C414yescomplemented
74C4×D188C2×C414yes—
75C4×D188Q814yes—
76#13093982—22no—
77#13093982—22no—
78#13093982—22no—
79#13093982—41no—
80#13093982—41no—
81#13093982—41no—
82C2×C4⋊Dic47D411yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×Dic47⋊C4—21no—
88C4⋊C4×D47—22no—
89C4⋊C4×D47—41no—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—21no—
93C2×(C2×D47)⋊C4—21no—
94C2×(C2×D47)⋊C4—21no—
95C2×(C2×D47)⋊C4—21no—
96C2×(C2×D47)⋊C4—21no—
97C2×(C2×D47)⋊C4D411yescharacteristic
98C2×(C2×D47)⋊C4D411yescharacteristic
99C2×C4×Dic47—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—21no—
102C2×C4×Dic47—21no—
103C2×C4×Dic47—21no—
104#13093958—22no—
105#13093957—22no—
106#13093957—22no—
107#13093957—22no—
108#13093956D414yescomplemented
109#13093955—22no—
110C4⋊(C47⋊C8)C2×C414yescomplemented
111(C2×D47)⋊C8C2×C414yes—
112(C2×D47)⋊C8C2×C414yescomplemented
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—21no—
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C42×C47C2311yescharacteristic
128#107×C47—22no—
129#107×C47—41no—
130#106×C47—22no—
131#106×C47—41no—
132#97×C47C2×C414yescomplemented
133#95×C47—21no—
134#95×C47—21no—
135C23×D47C2×D411yescharacteristic
136C23×D47C2×D411yescharacteristic
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×D188—41no—
151C2×D188—41no—
152C2×D188C4⋊C414yes—
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—42no—
162C2×C4×D47—42no—
163C2×C4×D47—81no—
164C2×C4×D47C22×C412yes—
165C2×C4×D47C22×C412yes—
166C2×C4×D47C2×Q812yes—
167C2×C4×D47C2×Q812yes—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47C22×C412yes—
171C2×C8×C47C22×C412yes—
172C2×C8×C47C2×D412yes—
173C2×C8×C47#4112yes—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—42no—
183C4⋊C4×C47—81no—
184C42×C47C22×C412yes—
185C42×C47C2×Q812yes—
186(C2×D47)⋊C4—42no—
187(C2×D47)⋊C4—42no—
188(C2×D47)⋊C4—42no—
189(C2×D47)⋊C4—42no—
190(C2×D47)⋊C4—81no—
191(C2×D47)⋊C4—81no—
192(C2×D47)⋊C4—81no—
193(C2×D47)⋊C4C4⋊C414yes—
194(C2×D47)⋊C4C4⋊C414yes—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47C4⋊C414yes—
202C4⋊Dic47C2×D412yes—
203C4⋊Dic47C2×D412yes—
204Dic47⋊C4—42no—
205C4×Dic47—42no—
206C4×Dic47—42no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209C4×Dic47—81no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8C22×C412yes—
215C2×C47⋊C8C22×C412yes—
216#10625324—28no—
217C8×D47—28no—
218C23×C47C22×D411yescharacteristic
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—21no—
227C22×D47—21no—
228C22×D47—21no—
229C22×D47—21no—
230C22×D47—21no—
231C22×D47C2×C4⋊C412yes—
232C22×D47C2×C4⋊C412yes—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47C2×C4⋊C412yes—
250C2×C4×C47C2×C4⋊C412yes—
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×Q811yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47H(5,𝔽2)11yescharacteristic
256C2×C4×C472−1+411yescharacteristic
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47C2×C4⋊C412yes—
274C2×Dic47C2×C4⋊C412yes—
275D188—44no—
276C4×D47—22no—
277C4×D47—22no—
278C4×D47—22no—
279C4×D47—22no—
280C4×D47—22no—
281C4×D47—22no—
282C4×D47—22no—
283C4×D47—22no—
284C4×D47—84no—
285C8×C47—22no—
286C8×C47C4×D414yes—
287C47⋊C8—22no—
288C47⋊C8—22no—
289#22660—471nomaximal, Sylow
290C22×C47C22×C4⋊C411yescharacteristic
291C22×C47D4211yescharacteristic
292C22×C47#54711yescharacteristic
293C22×C47#57511yescharacteristic
294C22×C47#57511yescharacteristic
295C22×C47#57611yescharacteristic
296C22×C47#57811yescharacteristic
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47—41no—
304C2×D47—41no—
305C2×D47—41no—
306C2×D47—41no—
307C2×D47—41no—
308C2×D47—41no—
309C2×D47—41no—
310C2×D47—41no—
311C4×C47—22no—
312C4×C47—22no—
313C4×C47—81no—
314C4×C47—81no—
315C4×C47—81no—
316C4×C47—81no—
317C4×C47C2×C4×D411yescharacteristic
318C4×C47#52011yescharacteristic
319C4×C47D4×Q811yescharacteristic
320C4×C47#55411yescharacteristic
321Dic47—22no—
322Dic47—22no—
323Dic47—42no—
324Dic47—42no—
325Dic47—81no—
326Dic47—81no—
327C2×#435—471no—
328C2×#428—471no—
329C2×#427—471no—
330#2354—471no—
331#2352—471no—
332#1949—471no—
333#1947—471no—
334#1854—471no—
335#1854—471no—
336#1854—471no—
337#1852—471no—
338#1847D4711yescharacteristic, complemented
339#1809—471no—
340#1808—471no—
341#1776—471no—
342C2×C47D4×C4⋊C411yescharacteristic
343C2×C47#291111yescharacteristic
344C2×C47#291311yescharacteristic
345C2×C47#326811yescharacteristic
346C2×C47#326911yescharacteristic
347C2×C47#334711yescharacteristic
348C2×C47#334911yescharacteristic
349D47—42no—
350D47—42no—
351C22×M16—471no—
352C23×C8—471no—
353C2×C4×D4—471no—
354C2×#117—471no—
355C2×#117—471no—
356C2×#117—471no—
357C22×C4⋊C4—471no—
358#435—478no—
359#428—944no—
360#427—944no—
361C2×#107—471no—
362C2×#107—471no—
363C2×#107—471no—
364C2×#107—471no—
365C2×#107C2×D4711yescharacteristic
366C2×#107C2×D4711yescharacteristic
367C2×#106—471no—
368C2×#106—471no—
369C2×#106C2×D4711yescharacteristic
370C2×#106C2×D4711yescharacteristic
371C2×C4⋊C8—471no—
372C2×C22⋊C8—471no—
373C2×C22⋊C8—471no—
374C2×#97C2×D4711yescharacteristic
375#390—941no—
376#388—941no—
377#388—941no—
378#388—941no—
379#387—941no—
380#386—941no—
381#384C2×D4711yescharacteristic
382#382C2×D4711yescharacteristic
383#380—471no—
384#380—471no—
385#380—471no—
386#380—471no—
387C4×C4⋊C4—471no—
388C4×C4⋊C4—471no—
389C4×C4⋊C4—941no—
390C4×C22⋊C4—941no—
391#340—471no—
392#340—471no—
393#340—471no—
394#340—471no—
395#340—471no—
396#340—471no—
397C47#2266011yescharacteristic, complemented, Sylow
398C22×D4—471no—
399C23×C4—471no—
400C23×C4—471no—
401C23×C4—941no—
402C2×M16—474no—
403C2×M16—942no—
404C2×M16—942no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8—474no—
408C22×C8—942no—
409C22×C8—942no—
410C22×C8C22×D4711yescharacteristic
411C22×C8C22×D4711yescharacteristic
412C4×D4—474no—
413C4×D4—474no—
414#117—942no—
415#117—942no—
416#117—942no—
417#117—1881no—
418#117—1881no—
419#117—1881no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—942no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—1881no—
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C4⋊C4C22×D4711yescharacteristic
433C2×C4⋊C4C22×D4711yescharacteristic
434C2×C22⋊C4—471no—
435C2×C22⋊C4—471no—
436C2×C22⋊C4—941no—
437C2×C22⋊C4—941no—
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C22⋊C4—941no—
442C2×C22⋊C4—941no—
443C2×C42—941no—
444C2×C42—941no—
445C2×C42—941no—
446C2×C42—941no—
447C2×C42—941no—
448C2×C42C22×D4711yescharacteristic
449#107—22no—
450#107—41no—
451#107—942no—
452#107—942no—
453#107—942no—
454#107—942no—
455#106—22no—
456#106—41no—
457#106—474no—
458#106—942no—
459C4⋊C8—474no—
460C22⋊C8—474no—
461C22⋊C8—474no—
462#97C4×D4714yescomplemented
463#95—21no—
464#95—21no—
465#95—941no—
466#95—941no—
467#95—941no—
468#95—941no—
469#95—941no—
470#95—941no—
471C24—471no—
472C24—471no—
473C2×D4—474no—
474C2×D4—1881no—
475C2×D4—1881no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—471no—
481C22×C4—471no—
482C22×C4—472no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—3761no—
499C22×C4—1882no—
500C22×C4—1882no—
501C22×C4D4×D4711yescharacteristic
502C22×C4D4×D4711yescharacteristic
503C22×C4C23×D4711yescharacteristic
504M16—948no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—472no—
508C2×C8—472no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—948no—
514C2×C8C2×C4×D4712yes—
515C2×C8C2×C4×D4712yes—
516C2×C8D4×D4712yes—
517C2×C8#1062535412yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—474no—
529C4⋊C4—472no—
530C4⋊C4—472no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—941no—
535C4⋊C4—941no—
536C4⋊C4—941no—
537C4⋊C4—1882no—
538C22⋊C4—474no—
539C22⋊C4—474no—
540C22⋊C4—3761no—
541C22⋊C4—3761no—
542C22⋊C4—3761no—
543C22⋊C4—1882no—
544C22⋊C4—1882no—
545C22⋊C4—1882no—
546C22⋊C4—1882no—
547C42—3761no—
548C42—1882no—
549C42—1882no—
550C42—1882no—
551C42—1882no—
552C42C2×C4×D4712yes—
553C42Q8×D4712yes—
554C23—472no—
555C23—472no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23—941no—
561C23—941no—
562C23—941no—
563C23—941no—
564C23—941no—
565C23—941no—
566C23—941no—
567C23—941no—
568C23C2×D4×D4711yescharacteristic
569D4—1884no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—472no—
587C2×C4—472no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—3764no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4C4⋊C4×D4712yes—
614C2×C4C4⋊C4×D4712yes—
615C2×C4C22×C4×D4711yescharacteristic
616C2×C4C2×D4×D4711yescharacteristic
617C2×C4C2×Q8×D4711yescharacteristic
618C2×C4C2×#1062535411yescharacteristic
619C2×C4#1309406911yescharacteristic
620C2×C4#1309407211yescharacteristic
621C8—22no—
622C8—942no—
623C8—942no—
624C8#1309398614yes—
625C22—942no—
626C22—942no—
627C22—942no—
628C22—942no—
629C22—942no—
630C22—942no—
631C22—1881no—
632C22—1881no—
633C22—1881no—
634C22—1881no—
635C22—1881no—
636C22—1881no—
637C22—1881no—
638C22—1881no—
639C22C2×C4⋊C4×D4711yescharacteristic
640C22#1568975811yescharacteristic
641C22#1568977611yescharacteristic
642C22#1568991911yescharacteristic
643C22#1568992211yescharacteristic
644C22#1568992711yescharacteristic
645C22#1568993911yescharacteristic
646C4—22no—
647C4—22no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—942no—
653C4—942no—
654C4—3761no—
655C4—3761no—
656C4—1882no—
657C4—1882no—
658C4C2×#1309398611yescharacteristic
659C4#1568960411yescharacteristic
660C4#1568978811yescharacteristic
661C4#1568983411yescharacteristic
662C2—1882no—
663C2—1882no—
664C2#2024628811yescharacteristic
665C2#2025197511yescharacteristic
666C2#2025198711yescharacteristic
667C2#2025492711yescharacteristic
668C2#2025494211yescharacteristic
669C2#2025588711yescharacteristic
670C2#2025591711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1847C2
socleC23×C47C22×D4
prime radicalC22×D188D4
cosocleC22×C4×C47C24
hypercenter#1847D47
hypocenterC47#22660
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
2. C2×#15689164C2semidirect
3. C2×#15689158C2semidirect
6. #20244154C2semidirect
7. #20244144C2semidirect
8. #20243586C2semidirect
9. #20243586C2semidirect
10. #20243584C2semidirect
11. #20243574C2semidirect
12. #20243418C2semidirect
13. #20243414C2semidirect
14. #20243285C2semidirect
15. C47×#1847C2semidirect
23. #15689196C4semidirect
73. C2×C8×D47C2×C4semidirect
108. #13093956D4semidirect
110. C4⋊(C47⋊C8)C2×C4semidirect
112. (C2×D47)⋊C8C2×C4semidirect
132. #97×C47C2×C4semidirect
338. #1847D47semidirect
397. C47#22660semidirect
462. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891962
C2×#156891642
C2×#156891582
#202463392
#202463142
#202441542
#202441442
#202435862
#202435842
#202435742
#202434182
#202434142
#202432852
C47×#18472
#2266047