Finite Group Databasebeta

← #26185738

Subgroups — #26185738

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252467C211yesmaximal, characteristic, complemented
2#20252409C211yesmaximal, characteristic
3C4×SD16×D47C211yesmaximal, characteristic
4#20242203C211yesmaximal, characteristic, complemented
5#20242200C211yesmaximal, characteristic, complemented
6#20242198C211yesmaximal, characteristic, complemented
7#20242197C211yesmaximal, characteristic, complemented
8C8×#10625325C211yesmaximal, characteristic, complemented
9SD16×C47⋊C8C211yesmaximal, characteristic, complemented
10#20242187C211yesmaximal, characteristic, complemented
11#20242133C211yesmaximal, characteristic, complemented
12#20241781C211yesmaximal, characteristic, complemented
13#20241777C211yesmaximal, characteristic
14#20241670C211yesmaximal, characteristic, complemented
15C8×SD16×C47C211yesmaximal, characteristic, complemented
16C2×SD16×D47C411yescharacteristic
17C4×Q8×D47C2211yescharacteristic
18C4×D4×D47C2211yescharacteristic
19Q8×C47⋊C8C2211yescharacteristic, complemented
20C8×Dic94—21no—
21C8×Dic94C2211yescharacteristic, complemented
22#15689247—21no—
23#15689247C2211yescharacteristic, complemented
24#15689214C411yescharacteristic
25C4×#10625325C2211yescharacteristic
26SD16×Dic47C411yescharacteristic
27C4×#10625336C2211yescharacteristic
28#15689210C411yescharacteristic
29C4×#10625335C2211yescharacteristic
30#15689208C411yescharacteristic
31C8×D188C2211yescharacteristic
32D4×C47⋊C8C2211yescharacteristic
33#15689188C2211yescharacteristic
34C8×C22⋊D47—21no—
35(C22⋊D47)⋊C8—21no—
36#15689175—21no—
37#15689175C2211yescharacteristic
38#15689175C2211yescharacteristic
39#15689174—21no—
40#15689174C2211yescharacteristic
41#15689174C2211yescharacteristic
42#106×D47C411yescharacteristic
43#103×D47C411yescharacteristic
44#102×D47C411yescharacteristic
45C2×(C2×D47)⋊C8—21no—
46C22⋊C8×D47—21no—
47C4×C8×D47C2211yescharacteristic
48#15688875C2211yescharacteristic, complemented
49#15688875C2211yescharacteristic, complemented
50#15688874C2211yescharacteristic, complemented
51#15688855C2211yescharacteristic, complemented
52Q8⋊(C47⋊C8)C2211yescharacteristic, complemented
53#15688853C2211yescharacteristic, complemented
54#15688852C2211yescharacteristic
55D4⋊(C47⋊C8)C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic, complemented
58#15688845C2211yescharacteristic, complemented
59C8×C47⋊C8C2211yescharacteristic, complemented
60C8×Q8×C47C2211yescharacteristic, complemented
61C4×SD16×C47C2211yescharacteristic
62C8×D4×C47C2211yescharacteristic
63C47×#334C2211yescharacteristic, complemented
64C47×Q8⋊C8C2211yescharacteristic, complemented
65C47×D4⋊C8C2211yescharacteristic
66C82×C47C2211yescharacteristic, complemented
67C2×Q8×D47C2×C411yescharacteristic
68C2×D4×D47C2×C411yescharacteristic
69C22×C4×D47—21no—
70C2×#10625325C2×C411yescharacteristic
71C2×#10625335C2×C411yescharacteristic
72C2×#10625336C2×C411yescharacteristic
73SD16×D47C818yescomplemented
74C22×C47⋊C8—21no—
75C2×C8×D47—21no—
76C2×C8×D47—21no—
77C2×C8×D47C2×C411yescharacteristic
78C2×C8×D47C2×C411yescharacteristic
79Q8×Dic47C2×C411yescharacteristic
80C4×Dic94—21no—
81C4×Dic94C2311yescharacteristic
82#13093989—21no—
83#13093989C2×C411yescharacteristic
84C4×D188C2311yescharacteristic
85D4×Dic47C2×C411yescharacteristic
86#13093986C2×C411yescharacteristic
87C4×C22⋊D47—21no—
88(C22⋊D47)⋊C4—21no—
89C4⋊C4×D47—21no—
90C4⋊C4×D47C2×C411yescharacteristic
91C4⋊C4×D47C2×C411yescharacteristic
92C22⋊C4×D47—21no—
93C42×D47—21no—
94C42×D47C2311yescharacteristic
95#13093956C2×C411yescharacteristic
96#13093955C2×C411yescharacteristic
97#13093955C2×C411yescharacteristic
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)C2311yescharacteristic
100C4⋊(C47⋊C8)C2311yescharacteristic
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8D411yescharacteristic
105Dic47⋊C8D411yescharacteristic
106#13093949C2×C411yescharacteristic
107#13093948C2×C411yescharacteristic
108#13093947C2×C411yescharacteristic
109#13093946C2×C411yescharacteristic
110#13093945C2×C411yescharacteristic
111#13093944C2×C411yescharacteristic
112C22⋊(C47⋊C8)—21no—
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8D411yescharacteristic
119(C2×D47)⋊C8D411yescharacteristic
120C4×C47⋊C8—21no—
121C4×C47⋊C8C2311yescharacteristic
122C4×C47⋊C8C2311yescharacteristic
123C4×C47⋊C8C2311yescharacteristic
124C8×Dic47—21no—
125C8×Dic47—21no—
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47C2×C411yescharacteristic
129C8×Dic47C2×C411yescharacteristic
130C2×SD16×C47C2×C411yescharacteristic
131C22×C8×C47—21no—
132C4×Q8×C47C2311yescharacteristic
133C4×D4×C47C2311yescharacteristic
134#106×C47C2×C411yescharacteristic
135C4⋊C8×C47—21no—
136C4⋊C8×C47C2311yescharacteristic
137C4⋊C8×C47C2311yescharacteristic
138#103×C47C2×C411yescharacteristic
139#102×C47C2×C411yescharacteristic
140C22⋊C8×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C23×D47—21no—
146C2×Q8×C47C22×C411yescharacteristic
147C2×D4×C47C22×C411yescharacteristic
148C22×C4×C47—21no—
149C2×C22⋊D47—21no—
150C22×Dic47—21no—
151Q8×D47—22no—
152Q8×D47C2×C814yescomplemented
153D4×D47—22no—
154D4×D47C2×C814yes—
155C2×D188C22×C411yescharacteristic
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—81no—
162C2×C4×D47—81no—
163C2×C4×D47C22×C411yescharacteristic
164C2×C4×D47C2×D411yescharacteristic
165C2×C4×D47#4111yescharacteristic
166C2×Dic94—21no—
167C2×Dic94C22×C411yescharacteristic
168SD16×C47C2×C814yescomplemented
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—81no—
173C2×C8×C47C22×C411yescharacteristic
174C2×C8×C47C22×C411yescharacteristic
175C2×C8×C47C2×D411yescharacteristic
176C2×C8×C47C2×D411yescharacteristic
177C4⋊C4×C47—21no—
178C4⋊C4×C47C22×C411yescharacteristic
179C4⋊C4×C47C22×C411yescharacteristic
180C22⋊C4×C47—21no—
181C42×C47—21no—
182C42×C47C2411yescharacteristic
183C22⋊Dic47—21no—
184#10625336C2×C814yescomplemented
185#10625335C2×C814yescomplemented
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C411yescharacteristic
190C4⋊Dic47C22×C411yescharacteristic
191Dic47⋊C4—21no—
192Dic47⋊C4—21no—
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47C22×C411yescharacteristic
199C4×Dic47C2×D411yescharacteristic
200C4×Dic47#4111yescharacteristic
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—81no—
205C2×C47⋊C8C22×C411yescharacteristic
206C2×C47⋊C8C22×C411yescharacteristic
207C2×C47⋊C8C2×D411yescharacteristic
208C2×C47⋊C8C2×D411yescharacteristic
209#10625325C2×C814yescomplemented
210C8×D47—22no—
211C8×D47—81no—
212C8×D47—81no—
213C8×D47C2×C814yescomplemented
214C23×C47—21no—
215C22×D47—21no—
216C22×D47—24no—
217C22×D47—81no—
218C22×D47—81no—
219C22×D47C4×D411yescharacteristic
220Q8×C47—21no—
221Q8×C47C22×C812yes—
222D4×C47—21no—
223D4×C47C22×C812yes—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C22⋊D47—24no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—81no—
237C2×Dic47C4×D411yescharacteristic
238C2×Dic47C4×D411yescharacteristic
239C2×Dic47C4×D411yescharacteristic
240D188—21no—
241D188C22×C812yes—
242C4×D47—24no—
243C4×D47—21no—
244C4×D47—21no—
245C4×D47—41no—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—81no—
249C4×D47—81no—
250C4×D47—81no—
251C4×D47C22×C812yes—
252C4×D47#13112yes—
253Dic94—21no—
254Dic94—24no—
255Dic94C22×C812yes—
256C8×C47—21no—
257C8×C47—21no—
258C8×C47—21no—
259C8×C47—41no—
260C8×C47—41no—
261C8×C47—41no—
262C8×C47C22×C812yes—
263C47⋊C8—21no—
264C47⋊C8—21no—
265C47⋊C8—21no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269C47⋊C8C22×C812yes—
270#17855—471nomaximal, Sylow
271C22×C47—22no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47C2×C4×D411yescharacteristic
275C2×D47—21no—
276C2×D47—21no—
277C2×D47—22no—
278C2×D47—41no—
279C2×D47—41no—
280C2×D47—84no—
281C2×D47C8×D412yes—
282C4×C47—22no—
283C4×C47—21no—
284C4×C47—41no—
285C4×C47—41no—
286C4×C47C23×C811yescharacteristic
287C4×C47C2×#13111yescharacteristic
288C4×C47#57711yescharacteristic
289C4×C47#57711yescharacteristic
290Dic47—22no—
291Dic47—21no—
292Dic47—21no—
293Dic47—41no—
294Dic47—41no—
295Dic47C8×D412yes—
296#2982—471no—
297#2972—471no—
298C2×C4×SD16—471no—
299SD16⋊C8—471no—
300SD16⋊C8—471no—
301SD16⋊C8—471no—
302SD16⋊C8—471no—
303C8×SD16—471no—
304C8×SD16—471no—
305C8×SD16—471no—
306C8×SD16D4711yescharacteristic, complemented
307#1564—471no—
308#1479—471no—
309#1478—471no—
310#1450—471no—
311C2×C47—42no—
312C2×C47C2×C8×D411yescharacteristic
313C2×C47#294911yescharacteristic
314C2×C47#295711yescharacteristic
315D47—22no—
316D47—42no—
317C22×SD16—471no—
318C2×C4×Q8—471no—
319C2×C4×D4—471no—
320C8×Q8—471no—
321C8×Q8—471no—
322C8×Q8—471no—
323C8×Q8—941no—
324C8×Q8—941no—
325C8×Q8C2×D4711yescharacteristic, complemented
326C4×SD16—471no—
327C4×SD16—471no—
328C4×SD16—471no—
329C4×SD16—471no—
330C4×SD16—471no—
331C4×SD16—471no—
332C4×SD16—471no—
333C4×SD16C2×D4711yescharacteristic
334C8×D4—471no—
335C8×D4—471no—
336C8×D4—471no—
337C8×D4—941no—
338C8×D4—941no—
339C8×D4C2×D4711yescharacteristic
340#431—471no—
341#431—471no—
342#431—471no—
343#431—471no—
344#431—941no—
345#431—941no—
346C2×#106—471no—
347C2×#103—471no—
348C2×#102—471no—
349C2×C22⋊C8—941no—
350C2×C22⋊C8—941no—
351C2×C4×C8—471no—
352#335—471no—
353#335—471no—
354#334—471no—
355#334C2×D4711yescharacteristic, complemented
356Q8⋊C8—471no—
357Q8⋊C8—471no—
358Q8⋊C8—471no—
359Q8⋊C8C2×D4711yescharacteristic, complemented
360D4⋊C8—471no—
361D4⋊C8—471no—
362D4⋊C8—471no—
363D4⋊C8C2×D4711yescharacteristic
364#322—471no—
365#322—471no—
366C82—471no—
367C82C2×D4711yescharacteristic, complemented
368C47#1785511yescharacteristic, complemented, Sylow
369C22×Q8—471no—
370C22×D4—471no—
371C23×C4—941no—
372C2×SD16—471no—
373C2×SD16—471no—
374C2×SD16—478no—
375C2×SD16—471no—
376C2×SD16C4×D4711yescharacteristic
377C22×C8—21no—
378C22×C8—471no—
379C22×C8—471no—
380C22×C8—941no—
381C22×C8—941no—
382C22×C8—941no—
383C4×Q8—471no—
384C4×Q8—471no—
385C4×Q8—471no—
386C4×Q8—941no—
387C4×Q8—941no—
388C4×Q8C22×D4711yescharacteristic
389C4×D4—471no—
390C4×D4—471no—
391C4×D4—471no—
392C4×D4—941no—
393C4×D4—941no—
394C4×D4C22×D4711yescharacteristic
395C2×C4⋊C4—471no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C42—471no—
400C2×C42—941no—
401#106—471no—
402#106—471no—
403#106—471no—
404#106C4×D4711yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—471no—
407C4⋊C8—471no—
408C4⋊C8—471no—
409C4⋊C8—471no—
410C4⋊C8—941no—
411C4⋊C8—941no—
412C4⋊C8—941no—
413C4⋊C8—941no—
414C4⋊C8C22×D4711yescharacteristic
415C4⋊C8C22×D4711yescharacteristic
416#103—471no—
417#103—471no—
418#103—471no—
419#103C4×D4711yescharacteristic
420#102—471no—
421#102—471no—
422#102—471no—
423#102C4×D4711yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—471no—
426C22⋊C8—471no—
427C22⋊C8—941no—
428C22⋊C8—941no—
429C22⋊C8—941no—
430C22⋊C8—941no—
431C22⋊C8—941no—
432C22⋊C8—941no—
433C4×C8—21no—
434C4×C8—471no—
435C4×C8—471no—
436C4×C8—471no—
437C4×C8—471no—
438C4×C8—471no—
439C4×C8—941no—
440C4×C8—941no—
441C4×C8—941no—
442C4×C8—941no—
443C4×C8—941no—
444C4×C8C22×D4711yescharacteristic
445C4×C8C22×D4711yescharacteristic
446C4×C8C22×D4711yescharacteristic
447C24—941no—
448C2×Q8—474no—
449C2×Q8—471no—
450C2×Q8—941no—
451C2×Q8—942no—
452C2×Q8C2×C4×D4711yescharacteristic
453C2×D4—474no—
454C2×D4—471no—
455C2×D4—942no—
456C2×D4—941no—
457C2×D4C2×C4×D4711yescharacteristic
458C22×C4—21no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—3761no—
469C22×C4—3761no—
470SD16—474no—
471SD16—474no—
472SD16—474no—
473SD16C8×D4714yescomplemented
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—471no—
479C2×C8—474no—
480C2×C8—471no—
481C2×C8—471no—
482C2×C8—471no—
483C2×C8—942no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—3761no—
488C2×C8—3761no—
489C2×C8—3761no—
490C2×C8C2×C4×D4711yescharacteristic
491C2×C8C2×C4×D4711yescharacteristic
492C2×C8D4×D4711yescharacteristic
493C2×C8D4×D4711yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—471no—
496C4⋊C4—471no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4C2×C4×D4711yescharacteristic
503C4⋊C4C2×C4×D4711yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—941no—
506C22⋊C4—941no—
507C22⋊C4—941no—
508C42—21no—
509C42—471no—
510C42—471no—
511C42—471no—
512C42—941no—
513C42—941no—
514C42—941no—
515C42C23×D4711yescharacteristic
516C23—21no—
517C23—471no—
518C23—944no—
519C23—941no—
520C23—3761no—
521C23—3761no—
522Q8—21no—
523Q8—472no—
524Q8—944no—
525Q8—941no—
526Q8C2×C8×D4712yes—
527D4—21no—
528D4—472no—
529D4—944no—
530D4—941no—
531D4C2×C8×D4712yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—471no—
538C2×C4—472no—
539C2×C4—472no—
540C2×C4—471no—
541C2×C4—471no—
542C2×C4—944no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—3761no—
549C2×C4—3761no—
550C2×C4—3761no—
551C2×C4—3761no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4C22×C4×D4711yescharacteristic
556C2×C4C2×D4×D4711yescharacteristic
557C2×C4#41×D4711yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—472no—
565C8—941no—
566C8—941no—
567C8—941no—
568C8—1881no—
569C8—1881no—
570C8—1881no—
571C8C2×C8×D4712yes—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—472no—
576C22—941no—
577C22—941no—
578C22—942no—
579C22—3764no—
580C22—1881no—
581C22—1881no—
582C22C4×D4×D4711yescharacteristic
583C4—21no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—472no—
588C4—941no—
589C4—941no—
590C4—942no—
591C4—1881no—
592C4—1881no—
593C4C22×C8×D4711yescharacteristic
594C4#131×D4711yescharacteristic
595C4#1568992911yescharacteristic
596C4#1568993011yescharacteristic
597C2—42no—
598C2—942no—
599C2—1882no—
600C2C8×D4×D4711yescharacteristic
601C2#2025224211yescharacteristic
602C2#2025230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC4×C47C23×C8
FrattiniC42C23×D47
FittingC8×SD16×C47C2
socleC22×C47C2×C4×D4
prime radicalC2×D4×D47C2×C4
cosocleC42×C47C24
hypercenterC8×SD16D47
hypocenterC47#17855
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
1. #20252467C2semidirect
4. #20242203C2semidirect
5. #20242200C2semidirect
6. #20242198C2semidirect
7. #20242197C2semidirect
8. C8×#10625325C2semidirect
9. SD16×C47⋊C8C2semidirect
10. #20242187C2semidirect
11. #20242133C2semidirect
12. #20241781C2semidirect
14. #20241670C2semidirect
15. C8×SD16×C47C2semidirect
19. Q8×C47⋊C8C22semidirect
21. C8×Dic94C22semidirect
23. #15689247C22semidirect
48. #15688875C22semidirect
49. #15688875C22semidirect
50. #15688874C22semidirect
51. #15688855C22semidirect
52. Q8⋊(C47⋊C8)C22semidirect
53. #15688853C22semidirect
57. #15688845C22semidirect
58. #15688845C22semidirect
59. C8×C47⋊C8C22semidirect
60. C8×Q8×C47C22semidirect
63. C47×#334C22semidirect
64. C47×Q8⋊C8C22semidirect
66. C82×C47C22semidirect
73. SD16×D47C8semidirect
152. Q8×D47C2×C8semidirect
168. SD16×C47C2×C8semidirect
184. #10625336C2×C8semidirect
185. #10625335C2×C8semidirect
209. #10625325C2×C8semidirect
213. C8×D47C2×C8semidirect
306. C8×SD16D47semidirect
325. C8×Q8C2×D47semidirect
355. #334C2×D47semidirect
359. Q8⋊C8C2×D47semidirect
367. C82C2×D47semidirect
368. C47#17855semidirect
473. SD16C8×D47semidirect

Maximal subgroups

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

subgroupindex
#202524672
#202524092
C4×SD16×D472
#202422032
#202422002
#202421982
#202421972
C8×#106253252
SD16×C47⋊C82
#202421872
#202421332
#202417812
#202417772
#202416702
C8×SD16×C472
#1785547