Finite Group Databasebeta

← #26275381

Subgroups — #26275381

Order 12032.

708 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20248622C211yesmaximal, characteristic, complemented
2#20244916C211yesmaximal, characteristic
3#20244909C211yesmaximal, characteristic
4#20244776C211yesmaximal, characteristic
5#20244762C211yesmaximal, characteristic
6#20244156C211yesmaximal, characteristic, complemented
7#20244146C211yesmaximal, characteristic, complemented
8#20244144C211yesmaximal, characteristic, complemented
9#20244112C211yesmaximal, characteristic, complemented
10#20243980C211yesmaximal, characteristic, complemented
11#20243973C211yesmaximal, characteristic, complemented
12#20243554C211yesmaximal, characteristic, complemented
13#20243398C211yesmaximal, characteristic
14#20243213C211yesmaximal, characteristic
15C47×#2354C211yesmaximal, characteristic
16C2×#13093957C2211yescharacteristic
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30#15689062—21no—
31#15689056—21no—
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689010—21no—
35#15689006—21no—
36#15688993C2211yescharacteristic
37#15688989C2211yescharacteristic
38#15688989C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4×D4×C47C2211yescharacteristic
49C2×#117×C47—21no—
50C47×#390C2211yescharacteristic
51C47×#390C2211yescharacteristic
52C47×#388—21no—
53C47×#388—21no—
54C47×#384C2211yescharacteristic
55C47×#380—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×C4×Dic47—21no—
72#13093957—22no—
73#13093957—41no—
74#13093957—41no—
75#13093957—41no—
76#13093955—22no—
77#13093955—41no—
78C4⋊(C47⋊C8)—22no—
79C4⋊(C47⋊C8)—22no—
80C4⋊(C47⋊C8)—22no—
81#13093945—22no—
82#13093945—22no—
83#13093945—41no—
84#13093945—41no—
85#13093935—22no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C22×D4×C47C2311yescharacteristic
94C23×C4×C47—21no—
95C4×D4×C47—22no—
96C4×D4×C47Q814yes—
97#117×C47—44no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115#95×C47—21no—
116#95×C47—21no—
117#95×C47D411yescharacteristic
118#95×C47D411yescharacteristic
119C24×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47C2×Q812yes—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—41no—
150C4⋊C4×C47—42no—
151C4⋊C4×C47—81no—
152C4⋊C4×C47C2×Q812yes—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—42no—
155C22⋊C4×C47—42no—
156C22⋊C4×C47—81no—
157C22⋊C4×C47—81no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—42no—
163C42×C47—42no—
164C42×C47C2×Q812yes—
165C42×C47#4112yes—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—42no—
175C4⋊Dic47—81no—
176Dic47⋊C4—42no—
177Dic47⋊C4—81no—
178C4×Dic47—81no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8#4112yes—
190C2×C47⋊C8#4112yes—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47C22×D411yescharacteristic
199D4×C47—22no—
200D4×C47—22no—
201D4×C47—41no—
202D4×C47—41no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—48no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47C22⋊Q812yes—
229C2×C4×C47C22⋊Q812yes—
230C2×C4×C47AGL(1,ℤ/8)12yes—
231C2×C4×C47C22×Q811yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47H(5,𝔽2)11yescharacteristic
236C2×C4×C472−1+411yescharacteristic
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C47⋊C8—22no—
250C47⋊C8—22no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253#26098—471nomaximal, Sylow
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47C2×C22⋊Q811yescharacteristic
262C22×C47#54711yescharacteristic
263C22×C47#55011yescharacteristic
264C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C47#57511yescharacteristic
266C22×C47#57611yescharacteristic
267C22×C47#57811yescharacteristic
268C4×C47—22no—
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—42no—
272C4×C47—41no—
273C4×C47—41no—
274C4×C47—41no—
275C4×C47—81no—
276C4×C47—81no—
277C4×C47#53311yescharacteristic
278C4×C47#54411yescharacteristic
279C4×C47#55211yescharacteristic
280C4×C47#55311yescharacteristic
281Dic47—81no—
282Dic47—81no—
283Dic47—81no—
284Dic47—81no—
285#2570—471no—
286#2354D4711yescharacteristic
287#2093—471no—
288#2092—471no—
289#2067—471no—
290#2065—471no—
291#1949—471no—
292#1947—471no—
293#1947—471no—
294#1940—471no—
295#1921—471no—
296#1920—471no—
297#1847—471no—
298#1804—471no—
299#1762—471no—
300C2×C47—82no—
301C2×C47#244111yescharacteristic
302C2×C47#307711yescharacteristic
303C2×C47#308311yescharacteristic
304C2×C47#334311yescharacteristic
305C2×C47#334711yescharacteristic
306C2×C47#339911yescharacteristic
307C2×C47#340211yescharacteristic
308C2×C4×D4C2×D4711yescharacteristic
309C2×#117—21no—
310C2×#107—471no—
311C2×#107—471no—
312C2×#107—471no—
313C2×#107—471no—
314C2×#106—471no—
315C2×#106—471no—
316C2×C4⋊C8—471no—
317C2×C4⋊C8—471no—
318C2×C4⋊C8—471no—
319C2×#102—471no—
320C2×#102—471no—
321C2×#102—471no—
322C2×#102—471no—
323C2×#97—471no—
324#400—941no—
325#398—941no—
326#390C2×D4711yescharacteristic
327#390C2×D4711yescharacteristic
328#389—471no—
329#389—471no—
330#388—21no—
331#388—21no—
332#387—941no—
333#387—941no—
334#384—471no—
335#384C2×D4711yescharacteristic
336#382—471no—
337#382—471no—
338#380—21no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4C2×D4711yescharacteristic
341C4×C4⋊C4C2×D4711yescharacteristic
342C4×C4⋊C4C2×D4711yescharacteristic
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#336—471no—
350#336—471no—
351C47#2609811yescharacteristic, complemented, Sylow
352C22×D4C22×D4711yescharacteristic
353C23×C4—21no—
354C22×C8—471no—
355C22×C8—471no—
356C22×C8—471no—
357C22×C8—471no—
358C4×D4—22no—
359C4×D4Dic9414yes—
360#117—44no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22⋊D4711yescharacteristic
372C2×C4⋊C4C22⋊D4711yescharacteristic
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—941no—
384C2×C42C22×D4711yescharacteristic
385C2×C42C22×D4711yescharacteristic
386C2×C42C22×D4711yescharacteristic
387#107—942no—
388#107—1881no—
389#107—1881no—
390#107—1881no—
391#106—942no—
392#106—1881no—
393C4⋊C8—942no—
394C4⋊C8—942no—
395C4⋊C8—942no—
396#102—942no—
397#102—942no—
398#102—1881no—
399#102—1881no—
400#97—942no—
401#95—21no—
402#95—21no—
403#95—941no—
404#95—941no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95C22⋊D4711yescharacteristic
411#95C22⋊D4711yescharacteristic
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×Dic9412yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C23×D4711yescharacteristic
436C2×C8—472no—
437C2×C8—472no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—3761no—
466C4⋊C4—3761no—
467C4⋊C4—1882no—
468C4⋊C4—1882no—
469C4⋊C4C2×Dic9412yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—3761no—
482C42C2×Dic9412yes—
483C42#1062535012yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C22×C22⋊D4711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4#1309400312yes—
534C2×C4#1309400312yes—
535C2×C4#1309405012yes—
536C2×C4C22×Dic9411yescharacteristic
537C2×C4#41×D4711yescharacteristic
538C2×C4#41×D4711yescharacteristic
539C2×C4C2×#1062535011yescharacteristic
540C2×C4#1309407011yescharacteristic
541C2×C4#1309407111yescharacteristic
542C8—942no—
543C8—942no—
544C8—1881no—
545C8—1881no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1309400311yescharacteristic
554C22#1568977211yescharacteristic
555C22#1568979211yescharacteristic
556C22C2×#1309405011yescharacteristic
557C22#1568992111yescharacteristic
558C22#1568992611yescharacteristic
559C22#1568993811yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—3761no—
570C4—3761no—
571C4—3761no—
572C4—3761no—
573C4#1568967611yescharacteristic
574C4#1568975111yescharacteristic
575C4#1568981211yescharacteristic
576C4#1568981711yescharacteristic
577C2—82no—
578C2#2024709511yescharacteristic
579C2#2025318511yescharacteristic
580C2#2025323911yescharacteristic
581C2#2025582211yescharacteristic
582C2#2025588811yescharacteristic
583C2#2025643011yescharacteristic
584C2#2025646311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2354C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2354D47
hypocenterC47#26098
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. #20248622C2semidirect
6. #20244156C2semidirect
7. #20244146C2semidirect
8. #20244144C2semidirect
9. #20244112C2semidirect
10. #20243980C2semidirect
11. #20243973C2semidirect
12. #20243554C2semidirect
351. C47#26098semidirect

Maximal subgroups

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

subgroupindex
#202486222
#202449162
#202449092
#202447762
#202447622
#202441562
#202441462
#202441442
#202441122
#202439802
#202439732
#202435542
#202433982
#202432132
C47×#23542
#2609847