Finite Group Databasebeta

← #26221252

Subgroups — #26221252

Order 12032.

1158 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689651C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic, complemented
3C2×C4×#10625324C211yesmaximal, characteristic, complemented
4#20244326C212yesmaximal, complemented
5#20244184C212yesmaximal, complemented
6#20243684C212yesmaximal, complemented
7#20243682C212yesmaximal, complemented
8#20243157C211yesmaximal, characteristic, complemented
9#20243155C211yesmaximal, characteristic, complemented
10#20243154C211yesmaximal, characteristic, complemented
11C47×#1750C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13#15689651—24no—
14#15689651—24no—
15C2×#13094019C411yescharacteristic
16C2×#13094017C411yescharacteristic
17C2×#13094014C411yescharacteristic
18C2×#13094010C411yescharacteristic
19C2×#13094009C411yescharacteristic
20C2×#13094006C411yescharacteristic
21C2×#13094002C411yescharacteristic
22C2×#13093997C411yescharacteristic
23C2×C4×Dic94C2211yescharacteristic
24C2×C4×D188C2211yescharacteristic
25C2×C4×C22⋊D47C2211yescharacteristic
26C2×C4×C22⋊D47C2211yescharacteristic
27C2×#13093981C2211yescharacteristic
28C2×C42×D47C2211yescharacteristic
29#15689178—24no—
30C2×Dic47⋊C8C2212yes—
31C2×C22⋊(C47⋊C8)C2212yes—
32C2×(C2×D47)⋊C8C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×(C2×D47)⋊C8C2212yes—
35C4×#10625324—24no—
36C2×#13093935C2211yescharacteristic
37C2×#13093934C2212yes—
38C2×C4×C47⋊C8C2211yescharacteristic
39#15688878C2212yes—
40#15688877C2212yes—
41#15688877C2212yes—
42C2×#117×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2212yes—
44C2×#97×C47C2211yescharacteristic
45C2×C4×C8×C47C2211yescharacteristic
46C47×#336C2212yes—
47C22×Dic94C2×C411yescharacteristic
48C22×D188C2×C411yescharacteristic
49C22×C22⋊D47C2×C411yescharacteristic
50C22×C22⋊D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2311yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C2×#10625324—24no—
56C2×#10625324—24no—
57C2×#10625324—24no—
58C22×C47⋊C8C2312yes—
59C22×C47⋊C8C2312yes—
60#13094019—24no—
61#13094017—24no—
62#13094014—24no—
63#13094010—24no—
64#13094009—24no—
65#13094006—24no—
66#13094002—24no—
67#13093997—24no—
68C4×Dic94—22no—
69C4×Dic94—22no—
70C4×D188—22no—
71C4×D188—22no—
72C4×C22⋊D47—22no—
73C4×C22⋊D47—22no—
74C4×C22⋊D47—22no—
75C4×C22⋊D47—22no—
76#13093981—22no—
77#13093981—22no—
78C2×C4⋊Dic47C2×C411yescharacteristic
79C2×C4⋊Dic47C2×C411yescharacteristic
80C2×Dic47⋊C4C2×C411yescharacteristic
81C2×Dic47⋊C4C2×C411yescharacteristic
82C2×Dic47⋊C4C2×C411yescharacteristic
83C2×Dic47⋊C4C2×C411yescharacteristic
84C2×Dic47⋊C4C2×C411yescharacteristic
85C2×Dic47⋊C4C2×C411yescharacteristic
86C2×C22⋊Dic47C2×C411yescharacteristic
87C2×C22⋊Dic47C2×C411yescharacteristic
88C2×(C2×D47)⋊C4C2×C411yescharacteristic
89C2×(C2×D47)⋊C4C2×C411yescharacteristic
90C2×(C2×D47)⋊C4C2×C411yescharacteristic
91C2×(C2×D47)⋊C4C2×C411yescharacteristic
92C2×(C2×D47)⋊C4C2×C411yescharacteristic
93C2×(C2×D47)⋊C4C2×C411yescharacteristic
94C2×C4×Dic47C2×C411yescharacteristic
95C2×C4×Dic47C2×C411yescharacteristic
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98C42×D47D414yes—
99C42×D47D414yes—
100Dic47⋊C8—24no—
101C22⋊(C47⋊C8)—24no—
102(C2×D47)⋊C8—24no—
103(C2×D47)⋊C8—24no—
104(C2×D47)⋊C8—24no—
105#13093935—22no—
106#13093934—24no—
107C4×C47⋊C8—22no—
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2312yes—
110C22×C8×C47C2312yes—
111#117×C47—22no—
112#117×C47—22no—
113C2×C4⋊C4×C47C2×C411yescharacteristic
114C2×C4⋊C4×C47C2×C411yescharacteristic
115C2×C22⋊C4×C47C2×C411yescharacteristic
116C2×C22⋊C4×C47C2×C411yescharacteristic
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C22⋊C8×C47—24no—
120#97×C47—22no—
121C4×C8×C47—22no—
122C24×C47C22×C411yescharacteristic
123C23×D47C22×C411yescharacteristic
124C23×D47C22×C411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C22×C411yescharacteristic
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C2×C22⋊D47—22no—
136C2×C22⋊D47—22no—
137C2×C22⋊D47—42no—
138C2×C22⋊D47—42no—
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C2×D188—22no—
146C2×D188—42no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—21no—
150C2×C4×D47—21no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47C22⋊C414yes—
156C2×C4×D47C22⋊C414yes—
157C2×C4×D47C2×D412yes—
158C2×C4×D47C2×D412yes—
159C2×Dic94—22no—
160C2×Dic94—42no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C2×C8×C47#4114yes—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—22no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47C2×D412yes—
174C42×C47C2×D412yes—
175C22⋊Dic47—22no—
176C22⋊Dic47—22no—
177(C2×D47)⋊C4—22no—
178(C2×D47)⋊C4—22no—
179(C2×D47)⋊C4—22no—
180(C2×D47)⋊C4—22no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185Dic47⋊C4—22no—
186Dic47⋊C4—22no—
187Dic47⋊C4—22no—
188Dic47⋊C4—22no—
189Dic47⋊C4—22no—
190Dic47⋊C4—22no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47C22⋊C414yes—
194C4×Dic47C22⋊C414yes—
195C4×Dic47C2×D412yes—
196C4×Dic47C2×D412yes—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8#4114yes—
203#10625324—216no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47C23×C411yescharacteristic
210C22×D47—21no—
211C22×D47—21no—
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47C2×C22⋊C412yes—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47C2×C22⋊C412yes—
232C2×C4×C47C2×C22⋊C412yes—
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C2×#4112yes—
236C2×C4×C47C2×#4112yes—
237C22⋊D47—44no—
238C22⋊D47—44no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47C2×C22⋊C412yes—
251C2×Dic47C2×C22⋊C412yes—
252C2×Dic47C2×C22⋊C412yes—
253D188—44no—
254C4×D47—24no—
255C4×D47—24no—
256C4×D47—44no—
257C4×D47—44no—
258C4×D47—42no—
259C4×D47—42no—
260Dic94—44no—
261C8×C47—24no—
262C8×C47—42no—
263C47⋊C8—24no—
264C47⋊C8—42no—
265#21376—471nomaximal, Sylow
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—41no—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47C4×#4111yescharacteristic
273C22×C47C4×#4111yescharacteristic
274C22×C47#56812yes—
275C22×C47#56812yes—
276C2×D47—22no—
277C2×D47—22no—
278C2×D47—22no—
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—41no—
282C2×D47—41no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—41no—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47#52512yes—
292C4×C47#53212yes—
293Dic47—22no—
294Dic47—22no—
295Dic47—22no—
296Dic47—41no—
297Dic47—41no—
298Dic47—41no—
299C2×#529—471no—
300C2×#432—471no—
301C2×C4×M16—471no—
302#1985—472no—
303#1955—472no—
304#1870—472no—
305#1870—472no—
306#1750—471no—
307#1750—471no—
308#1750—471no—
309#1750D4711yescharacteristic, complemented
310C2×C47—81no—
311C2×C47#229711yescharacteristic
312C2×C47#287911yescharacteristic
313C2×C47#287911yescharacteristic
314C2×C47#296712yes—
315C2×C47#296712yes—
316D47—42no—
317D47—81no—
318C22×M16—472no—
319#529—944no—
320#529—944no—
321C2×#126—471no—
322C2×#126—471no—
323C2×#125—471no—
324C2×#124—471no—
325C2×#123—471no—
326C2×#123—471no—
327C2×C22⋊Q8—471no—
328C2×#121—471no—
329C2×C4×Q8—471no—
330C2×C4×D4—471no—
331C2×C4×D4—471no—
332C2×C4×D4—471no—
333C2×#117—471no—
334C2×#117C2×D4711yescharacteristic
335C22×C42—471no—
336#432—944no—
337C2×C4⋊C8—472no—
338C2×C22⋊C8—472no—
339C2×C22⋊C8—472no—
340C2×C22⋊C8—472no—
341C2×C22⋊C8—472no—
342C2×C22⋊C8C2×D4712yes—
343C4×M16—944no—
344C2×#97—472no—
345C2×#97—471no—
346C2×#97C2×D4711yescharacteristic
347C2×C4×C8—471no—
348C2×C4×C8C2×D4711yescharacteristic
349#336—472no—
350#336—472no—
351#336—472no—
352#336C2×D4712yes—
353C47#2137611yescharacteristic, complemented, Sylow
354C22×Q8—471no—
355C22×D4—471no—
356C22×D4—471no—
357C22×D4—471no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—471no—
362C23×C4C22×D4711yescharacteristic
363C2×M16—944no—
364C2×M16—944no—
365C2×M16—944no—
366C22×C8—472no—
367C22×C8—472no—
368C22×C8C22×D4712yes—
369C22×C8C22×D4712yes—
370#126—944no—
371#126—944no—
372#125—944no—
373#124—944no—
374#123—944no—
375#123—944no—
376C22⋊Q8—944no—
377#121—944no—
378C4×Q8—942no—
379C4×Q8—942no—
380C4×D4—942no—
381C4×D4—942no—
382C4×D4—942no—
383C4×D4—942no—
384C4×D4—942no—
385C4×D4—942no—
386#117—22no—
387#117—22no—
388#117—942no—
389#117—942no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4C4×D4711yescharacteristic
399C2×C4⋊C4C4×D4711yescharacteristic
400C2×C22⋊C4—471no—
401C2×C22⋊C4—471no—
402C2×C22⋊C4—471no—
403C2×C22⋊C4—471no—
404C2×C22⋊C4—471no—
405C2×C22⋊C4—471no—
406C2×C22⋊C4—471no—
407C2×C22⋊C4—471no—
408C2×C22⋊C4C4×D4711yescharacteristic
409C2×C22⋊C4C4×D4711yescharacteristic
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—474no—
414C2×C42—474no—
415C2×C42—471no—
416C2×C42C22×D4711yescharacteristic
417C2×C42C22×D4711yescharacteristic
418C4⋊C8—944no—
419C22⋊C8—24no—
420C22⋊C8—944no—
421C22⋊C8—944no—
422C22⋊C8—944no—
423C22⋊C8—944no—
424#97—22no—
425#97—944no—
426#97—942no—
427C4×C8—22no—
428C4×C8—942no—
429C24—471no—
430C24—471no—
431C24C2×C4×D4711yescharacteristic
432C2×Q8—942no—
433C2×Q8—1882no—
434C2×D4—942no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—1882no—
438C2×D4—1882no—
439C2×D4—1882no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—472no—
448C22×C4—472no—
449C22×C4—474no—
450C22×C4—474no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4C2×C4×D4711yescharacteristic
463C22×C4C2×C4×D4711yescharacteristic
464C22×C4C2×C4×D4711yescharacteristic
465C22×C4C2×C4×D4711yescharacteristic
466C22×C4C2×C4×D4711yescharacteristic
467C22×C4C23×D4711yescharacteristic
468M16—9416no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—474no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8—942no—
478C2×C8—942no—
479C2×C8—942no—
480C2×C8#1062535014yes—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—942no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—942no—
490C4⋊C4—942no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—942no—
494C22⋊C4—942no—
495C22⋊C4—942no—
496C22⋊C4—942no—
497C22⋊C4—942no—
498C22⋊C4—942no—
499C22⋊C4—942no—
500C22⋊C4—942no—
501C42—21no—
502C42—21no—
503C42—474no—
504C42—474no—
505C42—472no—
506C42—472no—
507C42—941no—
508C42—941no—
509C42D4×D4712yes—
510C42D4×D4712yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—472no—
517C23—941no—
518C23—941no—
519C23—942no—
520C23—942no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23—942no—
525C23—942no—
526C23C22×C4×D4711yescharacteristic
527Q8—1884no—
528D4—1884no—
529D4—1884no—
530D4—1884no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—1884no—
559C2×C4—1884no—
560C2×C4—1882no—
561C2×C4—1882no—
562C2×C4C22⋊C4×D4712yes—
563C2×C4C22⋊C4×D4712yes—
564C2×C4C2×D4×D4711yescharacteristic
565C2×C4C2×D4×D4711yescharacteristic
566C2×C4#41×D4712yes—
567C2×C4C2×#1062535012yes—
568C8—24no—
569C8—42no—
570C8—944no—
571C8—1882no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—942no—
578C22—942no—
579C22—942no—
580C22—1881no—
581C22—1881no—
582C22—1881no—
583C22—1882no—
584C22—1882no—
585C22—1882no—
586C22—1882no—
587C22C2×C22⋊C4×D4711yescharacteristic
588C22#1568959111yescharacteristic
589C22C4×#1062535011yescharacteristic
590C22#1568988812yes—
591C22#1568988812yes—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—942no—
598C4—942no—
599C4—942no—
600C4—1881no—
601C4—1881no—
602C4—1881no—
603C4#1568963212yes—
604C4#1568967012yes—
605C2—81no—
606C2—3761no—
607C2—1882no—
608C2#2024587411yescharacteristic
609C2#2025183011yescharacteristic
610C2#2025183011yescharacteristic
611C2#2025236912yes—
612C2#2025237112yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1750C2
prime radicalC2×#13093997C4
cosocleC22×C4×C47C24
hypercenter#1750D47
hypocenterC47#21376
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#15689178C2semidirect
3. C2×C4×#10625324C2semidirect
4. #20244326C2semidirect
5. #20244184C2semidirect
6. #20243684C2semidirect
7. #20243682C2semidirect
8. #20243157C2semidirect
9. #20243155C2semidirect
10. #20243154C2semidirect
11. C47×#1750C2semidirect
309. #1750D47semidirect
353. C47#21376semidirect

Maximal subgroups

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

subgroupindex
C2×#156896512
C2×#156891782
C2×C4×#106253242
#202443262
#202441842
#202436842
#202436822
#202431572
#202431552
#202431542
C47×#17502
#2137647