Finite Group Databasebeta

← #26185679

Subgroups — #26185679

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#10625354C211yesmaximal, characteristic, complemented
2C8×#10625352C211yesmaximal, characteristic, complemented
3C4×#13094045C211yesmaximal, characteristic
4#20242210C211yesmaximal, characteristic, complemented
5#20242209C211yesmaximal, characteristic, complemented
6#20242205C211yesmaximal, characteristic, complemented
7#20242204C211yesmaximal, characteristic, complemented
8#20242202C211yesmaximal, characteristic, complemented
9SD16⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
10#20242199C211yesmaximal, characteristic, complemented
11#20242135C211yesmaximal, characteristic, complemented
12#20241769C211yesmaximal, characteristic, complemented
13#20241767C211yesmaximal, characteristic, complemented
14#322×D47C211yesmaximal, characteristic, complemented
15C47×SD16⋊C8C211yesmaximal, characteristic, complemented
16C2×#13094045C411yescharacteristic
17C4×#10625354C2211yescharacteristic
18C4×#10625352C2211yescharacteristic
19Q8×C47⋊C8C2211yescharacteristic
20C8×Dic94C2211yescharacteristic
21#15689247C2211yescharacteristic
22C4×#10625337C2211yescharacteristic
23#15689215C411yescharacteristic
24C4×#10625325C2211yescharacteristic
25SD16×Dic47C411yescharacteristic
26#15689210C411yescharacteristic
27C4×#10625334C2211yescharacteristic
28#15689203C411yescharacteristic
29C8×D188—21no—
30C8×D188C2211yescharacteristic
31D4×C47⋊C8C2211yescharacteristic
32#15689188—21no—
33#15689188C2211yescharacteristic
34C8×C22⋊D47—21no—
35(C22⋊D47)⋊C8—21no—
36#15689175—21no—
37#15689175C2211yescharacteristic
38#15689175C2211yescharacteristic
39#15689173—21no—
40#15689158C411yescharacteristic
41#15689118C411yescharacteristic
42#15689116C411yescharacteristic
43C2×C8×Dic47—21no—
44C4×C8×D47—21no—
45C4×C8×D47C2211yescharacteristic
46C4×C8×D47C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688873C2211yescharacteristic
50#15688873C2211yescharacteristic
51#15688855C2211yescharacteristic
52Q8⋊(C47⋊C8)C2211yescharacteristic
53#15688853C2211yescharacteristic
54#15688852C2211yescharacteristic
55D4⋊(C47⋊C8)C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic
58#15688845C2211yescharacteristic
59#15688844C2211yescharacteristic
60C8×Q8×C47C2211yescharacteristic
61C4×SD16×C47C2211yescharacteristic
62C8×D4×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#10625354C2×C411yescharacteristic
68C2×#10625352C2×C411yescharacteristic
69#13094045C818yescomplemented
70C2×#10625337C2×C411yescharacteristic
71C2×#10625325C2×C411yescharacteristic
72C2×#10625334C2×C411yescharacteristic
73C22×C47⋊C8—21no—
74C2×C8×D47—21no—
75C2×C8×D47—21no—
76C2×C8×D47—21no—
77C2×C8×D47C2×C411yescharacteristic
78C2×C8×D47C2×C411yescharacteristic
79C2×C8×D47D411yescharacteristic
80C2×C8×D47D411yescharacteristic
81Q8×Dic47C2×C411yescharacteristic
82C4×Dic94C2311yescharacteristic
83#13093989C2×C411yescharacteristic
84C4×D188—21no—
85C4×D188C2311yescharacteristic
86D4×Dic47C2×C411yescharacteristic
87#13093986—21no—
88#13093986C2×C411yescharacteristic
89C4×C22⋊D47—21no—
90(C22⋊D47)⋊C4—21no—
91#13093982—21no—
92#13093982C2×C411yescharacteristic
93#13093982C2×C411yescharacteristic
94#13093980—21no—
95C2×C4×Dic47—21no—
96C42×D47—21no—
97C42×D47C2311yescharacteristic
98#13093957C2×C411yescharacteristic
99#13093957C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101C4⋊(C47⋊C8)—21no—
102C4⋊(C47⋊C8)C2311yescharacteristic
103C4⋊(C47⋊C8)C2311yescharacteristic
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
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—
117C4×C47⋊C8—21no—
118C4×C47⋊C8C2311yescharacteristic
119C4×C47⋊C8C2311yescharacteristic
120C4×C47⋊C8C2311yescharacteristic
121C8×Dic47—21no—
122C8×Dic47—21no—
123C8×Dic47—21no—
124C8×Dic47—21no—
125C8×Dic47—21no—
126C8×Dic47C2×C411yescharacteristic
127C8×Dic47C2×C411yescharacteristic
128C8×Dic47D411yescharacteristic
129C8×Dic47D411yescharacteristic
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
145C2×Q8×C47C22×C411yescharacteristic
146C2×D4×C47C22×C411yescharacteristic
147C22×C4×C47—21no—
148C2×C22⋊D47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151#10625354—22no—
152#10625354C2×C814yescomplemented
153#10625352—22no—
154#10625352C2×C814yescomplemented
155C2×D188—21no—
156C2×D188C22×C411yescharacteristic
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47C22×C411yescharacteristic
161C2×C4×D47C2×D411yescharacteristic
162C2×C4×D47#4111yescharacteristic
163C2×Dic94C22×C411yescharacteristic
164SD16×C47C2×C814yescomplemented
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—42no—
169C2×C8×C47C22×C411yescharacteristic
170C2×C8×C47C22×C411yescharacteristic
171C2×C8×C47C2×D411yescharacteristic
172C2×C8×C47C2×D411yescharacteristic
173C4⋊C4×C47—21no—
174C4⋊C4×C47C22×C411yescharacteristic
175C4⋊C4×C47C22×C411yescharacteristic
176C22⋊C4×C47—21no—
177C42×C47—21no—
178C42×C47C2411yescharacteristic
179C22⋊Dic47—21no—
180#10625337C2×C814yescomplemented
181#10625334C2×C814yescomplemented
182(C2×D47)⋊C4—21no—
183(C2×D47)⋊C4—21no—
184(C2×D47)⋊C4—21no—
185(C2×D47)⋊C4—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47C22×C411yescharacteristic
188C4⋊Dic47C22×C411yescharacteristic
189Dic47⋊C4—21no—
190Dic47⋊C4—21no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47C22×C411yescharacteristic
197C4×Dic47C2×D411yescharacteristic
198C4×Dic47#4111yescharacteristic
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—42no—
203C2×C47⋊C8C22×C411yescharacteristic
204C2×C47⋊C8C22×C411yescharacteristic
205C2×C47⋊C8C2×D411yescharacteristic
206C2×C47⋊C8C2×D411yescharacteristic
207#10625325C2×C814yescomplemented
208C8×D47—22no—
209C8×D47—41no—
210C8×D47—41no—
211C8×D47—42no—
212C8×D47—42no—
213C8×D47—42no—
214C8×D47C2×C814yescomplemented
215C23×C47—21no—
216C22×D47—21no—
217C22×D47C4×D411yescharacteristic
218Q8×C47—21no—
219Q8×C47C22×C812yes—
220D4×C47—21no—
221D4×C47C22×C812yes—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C22⋊D47—24no—
231C2×Dic47—21no—
232C2×Dic47—24no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—42no—
236C2×Dic47C4×D411yescharacteristic
237C2×Dic47C4×D411yescharacteristic
238C2×Dic47C4×D411yescharacteristic
239D188—24no—
240D188—21no—
241D188C22×C812yes—
242C4×D47—21no—
243C4×D47—21no—
244C4×D47—24no—
245C4×D47—22no—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—42no—
249C4×D47C22×C812yes—
250C4×D47#13112yes—
251Dic94—21no—
252Dic94C22×C812yes—
253C8×C47—21no—
254C8×C47—21no—
255C8×C47—21no—
256C8×C47—41no—
257C8×C47—41no—
258C8×C47—41no—
259C8×C47C22×C812yes—
260C47⋊C8—21no—
261C47⋊C8—21no—
262C47⋊C8—21no—
263C47⋊C8—41no—
264C47⋊C8—41no—
265C47⋊C8—41no—
266C47⋊C8C22×C812yes—
267#17849—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47C2×C4×D411yescharacteristic
272C2×D47—21no—
273C2×D47—21no—
274C2×D47—22no—
275C2×D47—41no—
276C2×D47—41no—
277C2×D47C8×D412yes—
278C4×C47—22no—
279C4×C47—21no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47C23×C811yescharacteristic
283C4×C47C2×#13111yescharacteristic
284C4×C47#57511yescharacteristic
285C4×C47#57511yescharacteristic
286Dic47—22no—
287Dic47—21no—
288Dic47—41no—
289Dic47—41no—
290Dic47C8×D412yes—
291Dic47#44412yes—
292C8×#41—471no—
293C8×#41—471no—
294C4×#135—471no—
295#1580—471no—
296#1580—471no—
297#1579—471no—
298#1579—471no—
299SD16⋊C8—471no—
300SD16⋊C8—471no—
301SD16⋊C8—471no—
302SD16⋊C8D4711yescharacteristic, complemented
303#1564—471no—
304#1476—471no—
305#1476—471no—
306C2×#322—471no—
307C2×C47—42no—
308C2×C47C2×C8×D411yescharacteristic
309C2×C47#294311yescharacteristic
310C2×C47C2×#44411yescharacteristic
311D47—22no—
312D47—42no—
313C2×#135—471no—
314C4×#41—471no—
315C4×#41—471no—
316C8×Q8—471no—
317C8×Q8—471no—
318C8×Q8—471no—
319C8×Q8C2×D4711yescharacteristic
320C4×Q16—471no—
321C4×Q16—471no—
322C4×SD16—471no—
323C4×SD16—471no—
324C4×SD16—471no—
325C4×SD16C2×D4711yescharacteristic
326C4×D8—471no—
327C4×D8—471no—
328C8×D4—471no—
329C8×D4—471no—
330C8×D4—471no—
331C8×D4—941no—
332C8×D4—941no—
333C8×D4—941no—
334C8×D4—941no—
335C8×D4C2×D4711yescharacteristic
336#431—471no—
337#431—471no—
338#431—941no—
339#431—941no—
340#427—471no—
341#416—471no—
342#416—471no—
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345C2×C4×C8—471no—
346C2×C4×C8—941no—
347C2×C4×C8—941no—
348#335—471no—
349#335C2×D4711yescharacteristic
350#334—471no—
351#334—471no—
352Q8⋊C8—471no—
353Q8⋊C8—471no—
354Q8⋊C8—471no—
355Q8⋊C8C2×D4711yescharacteristic
356D4⋊C8—471no—
357D4⋊C8—471no—
358D4⋊C8—471no—
359D4⋊C8C2×D4711yescharacteristic
360#322—471no—
361#322—471no—
362#322—471no—
363#322C2×D4711yescharacteristic
364C47#1784911yescharacteristic, complemented, Sylow
365C2×#41—471no—
366C2×#41—471no—
367#135—478no—
368C2×Q16—471no—
369C2×SD16—471no—
370C2×SD16C4×D4711yescharacteristic
371C2×D8—471no—
372C22×C8—21no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8—471no—
377C22×C8—941no—
378C22×C8—941no—
379C22×C8—941no—
380C22×C8—941no—
381C4×Q8—471no—
382C4×Q8—471no—
383C4×Q8—471no—
384C4×Q8C22×D4711yescharacteristic
385C4×D4—471no—
386C4×D4—471no—
387C4×D4—471no—
388C4×D4—941no—
389C4×D4—941no—
390C4×D4—941no—
391C4×D4—941no—
392C4×D4C22×D4711yescharacteristic
393#117—471no—
394#117—471no—
395#117—941no—
396#117—941no—
397C2×C42—471no—
398C2×C42—941no—
399C2×C42—941no—
400#107—471no—
401#107—471no—
402#106—471no—
403#106C4×D4711yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—471no—
406C4⋊C8—471no—
407C4⋊C8—941no—
408C4⋊C8—941no—
409C4⋊C8—941no—
410C4⋊C8C22×D4711yescharacteristic
411C4⋊C8C22×D4711yescharacteristic
412#103—471no—
413#103—471no—
414#103—471no—
415#103C4×D4711yescharacteristic
416#102—471no—
417#102—471no—
418#102—471no—
419#102C4×D4711yescharacteristic
420C22⋊C8—21no—
421C22⋊C8—941no—
422C22⋊C8—941no—
423C22⋊C8—941no—
424C22⋊C8—941no—
425C22⋊C8—941no—
426C4×C8—21no—
427C4×C8—471no—
428C4×C8—471no—
429C4×C8—471no—
430C4×C8—471no—
431C4×C8—471no—
432C4×C8—471no—
433C4×C8—471no—
434C4×C8—941no—
435C4×C8—941no—
436C4×C8—941no—
437C4×C8—941no—
438C4×C8—941no—
439C4×C8—941no—
440C4×C8C22×D4711yescharacteristic
441C4×C8C22×D4711yescharacteristic
442C4×C8C22×D4711yescharacteristic
443#41—474no—
444#41—474no—
445#41—942no—
446#41—942no—
447C2×Q8—471no—
448C2×Q8C2×C4×D4711yescharacteristic
449C2×D4—471no—
450C2×D4—941no—
451C2×D4—941no—
452C2×D4C2×C4×D4711yescharacteristic
453C22×C4—21no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462Q16—474no—
463SD16—474no—
464SD16C8×D4714yescomplemented
465D8—474no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—42no—
470C2×C8—471no—
471C2×C8—474no—
472C2×C8—471no—
473C2×C8—471no—
474C2×C8—471no—
475C2×C8—942no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—1881no—
480C2×C8—1881no—
481C2×C8—1882no—
482C2×C8—1882no—
483C2×C8—1882no—
484C2×C8—1882no—
485C2×C8C2×C4×D4711yescharacteristic
486C2×C8C2×C4×D4711yescharacteristic
487C2×C8D4×D4711yescharacteristic
488C2×C8D4×D4711yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—471no—
491C4⋊C4—471no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4C2×C4×D4711yescharacteristic
496C4⋊C4C2×C4×D4711yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—941no—
499C22⋊C4—941no—
500C22⋊C4—941no—
501C22⋊C4—941no—
502C22⋊C4—941no—
503C42—21no—
504C42—471no—
505C42—471no—
506C42—471no—
507C42—941no—
508C42—941no—
509C42—941no—
510C42—941no—
511C42—941no—
512C42C23×D4711yescharacteristic
513C23—21no—
514C23—471no—
515C23—941no—
516Q8—21no—
517Q8—472no—
518Q8—941no—
519Q8C2×C8×D4712yes—
520D4—21no—
521D4—472no—
522D4—944no—
523D4—944no—
524D4—941no—
525D4C2×C8×D4712yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—471no—
532C2×C4—471no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—471no—
536C2×C4—942no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4C22×C4×D4711yescharacteristic
549C2×C4C2×D4×D4711yescharacteristic
550C2×C4#41×D4711yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—472no—
558C8—941no—
559C8—941no—
560C8—941no—
561C8—1881no—
562C8—1881no—
563C8—1881no—
564C8C2×C8×D4712yes—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—472no—
569C22—941no—
570C22—941no—
571C22—942no—
572C22—1881no—
573C22—1881no—
574C22C4×D4×D4711yescharacteristic
575C4—21no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—472no—
580C4—472no—
581C4—941no—
582C4—942no—
583C4—1881no—
584C4—1881no—
585C4C22×C8×D4711yescharacteristic
586C4#131×D4711yescharacteristic
587C4#1568991911yescharacteristic
588C4#1568992211yescharacteristic
589C2—42no—
590C2—942no—
591C2—1882no—
592C2C8×D4×D4711yescharacteristic
593C2#2025218311yescharacteristic
594C2#444×D4711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC4×C47C23×C8
FrattiniC42C23×D47
FittingC47×SD16⋊C8C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094045C4
cosocleC42×C47C24
hypercenterSD16⋊C8D47
hypocenterC47#17849
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
1. C8×#10625354C2semidirect
2. C8×#10625352C2semidirect
4. #20242210C2semidirect
5. #20242209C2semidirect
6. #20242205C2semidirect
7. #20242204C2semidirect
8. #20242202C2semidirect
9. SD16⋊(C47⋊C8)C2semidirect
10. #20242199C2semidirect
11. #20242135C2semidirect
12. #20241769C2semidirect
13. #20241767C2semidirect
14. #322×D47C2semidirect
15. C47×SD16⋊C8C2semidirect
69. #13094045C8semidirect
152. #10625354C2×C8semidirect
154. #10625352C2×C8semidirect
164. SD16×C47C2×C8semidirect
180. #10625337C2×C8semidirect
181. #10625334C2×C8semidirect
207. #10625325C2×C8semidirect
214. C8×D47C2×C8semidirect
302. SD16⋊C8D47semidirect
364. C47#17849semidirect
464. SD16C8×D47semidirect

Maximal subgroups

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

subgroupindex
C8×#106253542
C8×#106253522
C4×#130940452
#202422102
#202422092
#202422052
#202422042
#202422022
SD16⋊(C47⋊C8)2
#202421992
#202421352
#202417692
#202417672
#322×D472
C47×SD16⋊C82
#1784947