Finite Group Databasebeta

← #26234056

Subgroups — #26234056

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689163C211yesmaximal, characteristic
2#20246271C211yesmaximal, characteristic
3#20244128C211yesmaximal, characteristic, complemented
4#20244113C211yesmaximal, characteristic, complemented
5#20243977C211yesmaximal, characteristic, complemented
6#20243973C211yesmaximal, characteristic, complemented
7#20243966C211yesmaximal, characteristic, complemented
8#20243956C211yesmaximal, characteristic, complemented
9#20243480C211yesmaximal, characteristic
10#20243478C211yesmaximal, characteristic
11#20243419C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13C4×#13093957C211yesmaximal, characteristic, complemented
14#20243172C211yesmaximal, characteristic
15C47×#2344C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18#15689163C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689011—21no—
31#15689011—21no—
32#15688987—21no—
33#15688983C2211yescharacteristic
34C4×C4⋊Dic47C2211yescharacteristic
35C4×C4⋊Dic47C2211yescharacteristic
36#15688968—21no—
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×#117×C47—21no—
50C22×C4⋊C4×C47C2211yescharacteristic
51C47×#389C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#387—21no—
54C47×#387—21no—
55C47×#381—21no—
56C47×#380C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C22⋊C4×C47C2211yescharacteristic
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329C2×C414yes—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093983—22no—
68#13093983C2×C414yes—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80#13093957—22no—
81#13093957C2×C414yes—
82#13093957C2×C414yes—
83#13093955—22no—
84#13093955C2×C414yes—
85#13093955C2×C414yes—
86C22⋊(C47⋊C8)—22no—
87#13093935—22no—
88C4×C47⋊C8—22no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C23×C4×C47—21no—
95C23×C4×C47C2311yescharacteristic
96#117×C47—44no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2×C414yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47D411yescharacteristic
108C2×C22⋊C4×C47D411yescharacteristic
109C2×C42×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112C2×C42×C47C2311yescharacteristic
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47C2×D411yescharacteristic
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—24no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47#4112yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C4⋊C4×C47—24no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—42no—
152C4⋊C4×C47—42no—
153C4⋊C4×C47C22×C412yes—
154C4⋊C4×C47C22×C412yes—
155C22⋊C4×C47—22no—
156C22⋊C4×C47—41no—
157C22⋊C4×C47—42no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—42no—
163C42×C47—42no—
164C22⋊Dic47—24no—
165C22⋊Dic47—81no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47C22×C412yes—
173C4⋊Dic47C22×C412yes—
174Dic47⋊C4—81no—
175Dic47⋊C4—81no—
176C4×Dic47—24no—
177C4×Dic47—42no—
178C4×Dic47—42no—
179#10625329—28no—
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⋊C8C22×C412yes—
189C2×C47⋊C8C22×C412yes—
190C2×C47⋊C8#4112yes—
191C2×C47⋊C8#4112yes—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47C4×D412yes—
199C23×C47C22×D411yescharacteristic
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
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×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47C4×D412yes—
231C2×C4×C47AGL(1,ℤ/8)12yes—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C472−1+411yescharacteristic
237C2×C4×C472−1+411yescharacteristic
238C2×Dic47—22no—
239C2×Dic47—22no—
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—
249C2×Dic47—41no—
250C47⋊C8—22no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8#11714yescomplemented
254#22624—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47C2×C4×D411yescharacteristic
263C22×C47#54711yescharacteristic
264C22×C47#54711yescharacteristic
265C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C47#57511yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57811yescharacteristic
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—41no—
272C4×C47—41no—
273C4×C47—41no—
274C4×C47—42no—
275C4×C47—42no—
276C4×C47—81no—
277C4×C47—81no—
278C4×C47C2×#11711yescharacteristic
279C4×C47#51911yescharacteristic
280C4×C47#55211yescharacteristic
281C4×C47#55211yescharacteristic
282Dic47—42no—
283Dic47—42no—
284Dic47—81no—
285Dic47—81no—
286C2×#428—471no—
287#2346—471no—
288#2344D4711yescharacteristic
289#1944—471no—
290#1941—471no—
291#1921—471no—
292#1920—471no—
293#1919—471no—
294#1918—471no—
295#1825—471no—
296#1824—471no—
297#1809—471no—
298#1776—471no—
299C4×#107—471no—
300#1753—471no—
301C2×C47—42no—
302C2×C47#234411yescharacteristic
303C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C47#294711yescharacteristic
305C2×C47#334311yescharacteristic
306C2×C47#334811yescharacteristic
307C2×C47#335211yescharacteristic
308C2×C47#335411yescharacteristic
309C22×M16—471no—
310C2×#117—21no—
311C2×#117—471no—
312C22×C4⋊C4C2×D4711yescharacteristic
313#428—478no—
314C2×#107—471no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—471no—
320C2×C22⋊C8—471no—
321C2×#97—471no—
322C2×C4×C8—471no—
323#389—471no—
324#389—471no—
325#389C2×D4711yescharacteristic
326#389C2×D4711yescharacteristic
327#388—941no—
328#388—941no—
329#387—21no—
330#387—21no—
331#382—941no—
332#381—21no—
333#380—471no—
334#380C2×D4711yescharacteristic
335C4×C4⋊C4—471no—
336C4×C4⋊C4—471no—
337C4×C4⋊C4C2×D4711yescharacteristic
338C4×C4⋊C4C2×D4711yescharacteristic
339C4×C22⋊C4C2×D4711yescharacteristic
340#376—941no—
341#340—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#336—471no—
352#336—471no—
353C47#2262411yescharacteristic, complemented, Sylow
354C23×C4—21no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—474no—
357C2×M16—942no—
358C2×M16—942no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—471no—
362C22×C8—471no—
363#117—44no—
364#117—474no—
365#117—942no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C4×D4714yes—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4C22⋊D4711yescharacteristic
385C2×C22⋊C4C22⋊D4711yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42C22×D4711yescharacteristic
392C2×C42C22×D4711yescharacteristic
393#107—474no—
394#107—474no—
395#107—942no—
396#106—474no—
397#106—474no—
398#106—942no—
399C22⋊C8—942no—
400#97—942no—
401C4×C8—942no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95C22⋊D4711yescharacteristic
413#95C22⋊D4711yescharacteristic
414C24C2×C22⋊D4711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4C2×C4×D4712yes—
431C22×C4#1062535012yes—
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4711yescharacteristic
437C22×C4C23×D4711yescharacteristic
438M16—948no—
439C2×C8—472no—
440C2×C8—472no—
441C2×C8—472no—
442C2×C8—472no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—472no—
461C4⋊C4—472no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—3761no—
469C4⋊C4—3761no—
470C4⋊C4C2×C4×D4712yes—
471C4⋊C4C2×C4×D4712yes—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—944no—
476C22⋊C4—3761no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—42no—
483C42—944no—
484C42—1882no—
485C42—1882no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D4712yes—
493C23C22×C22⋊D4711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4C4×C22⋊D4712yes—
537C2×C4#1309405012yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4#41×D4711yescharacteristic
540C2×C4#41×D4711yescharacteristic
541C2×C4C2×#1062535011yescharacteristic
542C2×C4#1309407111yescharacteristic
543C2×C4#1309407111yescharacteristic
544C8—474no—
545C8—942no—
546C8—1881no—
547C8—1881no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D4711yescharacteristic
556C22#1568977211yescharacteristic
557C22#1568977211yescharacteristic
558C22C2×#1309405011yescharacteristic
559C22#1568992111yescharacteristic
560C22#1568993111yescharacteristic
561C22#1568993811yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—3761no—
572C4—3761no—
573C4—1882no—
574C4—1882no—
575C4#117×D4711yescharacteristic
576C4#1568960011yescharacteristic
577C4#1568980611yescharacteristic
578C4#1568980611yescharacteristic
579C2—42no—
580C2#2024625811yescharacteristic
581C2C4×#1309405011yescharacteristic
582C2#2025223111yescharacteristic
583C2#2025582211yescharacteristic
584C2#2025589711yescharacteristic
585C2#2025595711yescharacteristic
586C2#2025598311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2344C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2344D47
hypocenterC47#22624
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #20244128C2semidirect
4. #20244113C2semidirect
5. #20243977C2semidirect
6. #20243973C2semidirect
7. #20243966C2semidirect
8. #20243956C2semidirect
12. #20243286C2semidirect
13. C4×#13093957C2semidirect
18. #15689163C4semidirect
253. C47⋊C8#117semidirect
353. C47#22624semidirect

Maximal subgroups

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

subgroupindex
C2×#156891632
#202462712
#202441282
#202441132
#202439772
#202439732
#202439662
#202439562
#202434802
#202434782
#202434192
#202432862
C4×#130939572
#202431722
C47×#23442
#2262447