Finite Group Databasebeta

← #26233901

Subgroups — #26233901

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D47C211yesmaximal, characteristic
2#20246261C211yesmaximal, characteristic
3#20246256C211yesmaximal, characteristic
4#20244115C211yesmaximal, characteristic, complemented
5#20244113C211yesmaximal, characteristic, complemented
6#20243974C211yesmaximal, characteristic, complemented
7#20243972C211yesmaximal, characteristic, complemented
8#20243968C211yesmaximal, characteristic, complemented
9#20243477C211yesmaximal, characteristic
10#20243473C211yesmaximal, characteristic
11#20243412C211yesmaximal, characteristic
12C4×#13093955C211yesmaximal, characteristic, complemented
13#20243276C211yesmaximal, characteristic, complemented
14#20243158C211yesmaximal, characteristic
15C47×#1919C211yesmaximal, characteristic, complemented
16C22×C8×D47C2211yescharacteristic
17C2×#13093982—21no—
18C2×C4⋊C4×D47C2211yescharacteristic
19C2×C4⋊C4×D47C2211yescharacteristic
20C2×#13093956C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25#106×D47C418yescomplemented
26C2×(C2×D47)⋊C8C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689022C2211yescharacteristic
30#15689021C2211yescharacteristic
31#15689019C2211yescharacteristic
32#15689010—21no—
33#15689007—21no—
34#15689005—21no—
35#15689004—21no—
36#15688985—21no—
37#15688982—21no—
38#15688980C2211yescharacteristic
39#15688980C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44C4×(C2×D47)⋊C4C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#106×C47C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C47×#340C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C2×C8×D47—22no—
66C2×C8×D47—22no—
67C2×C8×D47C2×C414yes—
68#13093982—44no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C4⋊C4×D47—22no—
79C4⋊C4×D47—22no—
80C4⋊C4×D47C2×C414yes—
81C4⋊C4×D47C2×C414yes—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4D411yescharacteristic
85C2×(C2×D47)⋊C4D411yescharacteristic
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2311yescharacteristic
90#13093956C2×C414yes—
91#13093955—22no—
92#13093955—22no—
93#13093955C2×C414yes—
94#13093955C2×C414yes—
95(C2×D47)⋊C8—22no—
96C4×C47⋊C8—22no—
97C8×Dic47—22no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95D411yescharacteristic
107C47⋊#95D411yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47C2311yescharacteristic
117#106×C47C2×C414yes—
118#95×C47—21no—
119#95×C47—21no—
120C23×D47C2×D411yescharacteristic
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2411yescharacteristic
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135C2×C4×D47—21no—
136C2×C4×D47—21no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—24no—
140C2×C4×D47—24no—
141C2×C4×D47—42no—
142C2×C4×D47—42no—
143C2×C4×D47—81no—
144C2×C4×D47—81no—
145C2×C4×D47C22×C412yes—
146C2×C4×D47#4112yes—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47C22×C412yes—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—42no—
160C4⋊C4×C47C22×C412yes—
161C4⋊C4×C47C22×C412yes—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—81no—
165(C2×D47)⋊C4—22no—
166(C2×D47)⋊C4—42no—
167(C2×D47)⋊C4—42no—
168(C2×D47)⋊C4SD1614yes—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—42no—
176C4⋊Dic47C22×C412yes—
177C4⋊Dic47C22×C412yes—
178Dic47⋊C4—24no—
179Dic47⋊C4—24no—
180Dic47⋊C4—81no—
181Dic47⋊C4—81no—
182C4×Dic47—21no—
183C4×Dic47—21no—
184C4×Dic47—42no—
185C4×Dic47—42no—
186C4×Dic47—81no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8C22×C412yes—
191C2×C47⋊C8#4112yes—
192C2×C47⋊C8#4112yes—
193C8×D47—28no—
194C23×C47C22×D411yescharacteristic
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47C4×D412yes—
200C22×D47C2×SD1612yes—
201C22×D47#13512yes—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47C2×SD1612yes—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C472−1+411yescharacteristic
225C2×C4×C472−1+411yescharacteristic
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—41no—
236C2×Dic47—41no—
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×Dic47C4×D412yes—
246C2×Dic47C2×SD1612yes—
247C4×D47—22no—
248C4×D47—22no—
249C4×D47—22no—
250C4×D47—22no—
251C4×D47—48no—
252C4×D47—84no—
253C8×C47—22no—
254C8×C47—41no—
255C47⋊C8—41no—
256C47⋊C8#11714yescomplemented
257#22609—471nomaximal, Sylow
258C22×C47C2×C4×D411yescharacteristic
259C22×C47#54711yescharacteristic
260C22×C47#54711yescharacteristic
261C22×C47C22×SD1611yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47#57711yescharacteristic
264C22×C47#57711yescharacteristic
265C2×D47—22no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—22no—
270C2×D47—22no—
271C2×D47C4×SD1614yes—
272C4×C47—22no—
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—22no—
283Dic47—22no—
284Dic47—41no—
285Dic47—42no—
286Dic47—42no—
287Dic47—81no—
288Dic47—81no—
289C22×#106—471no—
290#2345—471no—
291#2344—471no—
292#1941—471no—
293#1941—471no—
294#1920—471no—
295#1920—471no—
296#1919—471no—
297#1919D4711yescharacteristic, complemented
298#1824—471no—
299#1823—471no—
300#1807—471no—
301C4×#106—471no—
302C4×#106—471no—
303C8×C22⋊C4—471no—
304C2×C47#234411yescharacteristic
305C2×C47C2×C4×SD1611yescharacteristic
306C2×C47#294911yescharacteristic
307C2×C47#333511yescharacteristic
308C2×C47#333511yescharacteristic
309C2×C47#333711yescharacteristic
310C2×C47#333711yescharacteristic
311D47—24no—
312C23×C8—471no—
313C2×#117—941no—
314C22×C4⋊C4—471no—
315C22×C4⋊C4—471no—
316C2×#106—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—478no—
320C2×#106—471no—
321C2×#106—471no—
322C2×#106C2×D4711yescharacteristic
323C2×C22⋊C8—471no—
324C2×C4×C8—471no—
325C2×C4×C8—471no—
326#389—471no—
327#389—471no—
328#389—471no—
329#389C2×D4711yescharacteristic
330#387—941no—
331#387—941no—
332#387—941no—
333#387—941no—
334#381—941no—
335#380—471no—
336#380—471no—
337#380—941no—
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—471no—
341C4×C4⋊C4C2×D4711yescharacteristic
342C4×C22⋊C4—471no—
343C4×C22⋊C4—941no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340C2×D4711yescharacteristic
352#340C2×D4711yescharacteristic
353#340C2×D4711yescharacteristic
354#336—471no—
355#336C2×D4711yescharacteristic
356C47#2260911yescharacteristic, complemented, Sylow
357C23×C4—471no—
358C23×C4—941no—
359C23×C4—941no—
360C22×C8—471no—
361C22×C8—474no—
362C22×C8—471no—
363C22×C8—942no—
364C22×C8—942no—
365C22×C8C22×D4711yescharacteristic
366C22×C8C22×D4711yescharacteristic
367#117—1884no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—474no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C22⋊C4—471no—
387C2×C22⋊C4—471no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C42—21no—
391C2×C42—471no—
392C2×C42—941no—
393C2×C42—941no—
394C2×C42—941no—
395C2×C42C22×D4711yescharacteristic
396#106—474no—
397#106—474no—
398#106—474no—
399#106—942no—
400#106—942no—
401#106C4×D4714yes—
402C22⋊C8—942no—
403C4×C8—942no—
404C4×C8—942no—
405#95—21no—
406#95—21no—
407#95—471no—
408#95—471no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417C24—471no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—471no—
427C22×C4—944no—
428C22×C4—944no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—3761no—
438C22×C4—3761no—
439C22×C4—1882no—
440C22×C4—1882no—
441C22×C4D4×D4711yescharacteristic
442C22×C4D4×D4711yescharacteristic
443C22×C4C23×D4711yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—472no—
450C2×C8—472no—
451C2×C8—472no—
452C2×C8—948no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8C2×C4×D4712yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—472no—
465C4⋊C4—472no—
466C4⋊C4—944no—
467C4⋊C4—944no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—3761no—
475C4⋊C4—3761no—
476C4⋊C4—1882no—
477C4⋊C4C2×C4×D4712yes—
478C4⋊C4C2×C4×D4712yes—
479C22⋊C4—474no—
480C22⋊C4—942no—
481C22⋊C4—1882no—
482C22⋊C4—1882no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—941no—
487C42—941no—
488C42—3761no—
489C42—1882no—
490C42—1882no—
491C23—472no—
492C23—472no—
493C23—472no—
494C23—941no—
495C23—941no—
496C23—941no—
497C23—941no—
498C23C2×D4×D4711yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—941no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—3764no—
532C2×C4—1888no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4SD16×D4712yes—
544C2×C4C22×C4×D4711yescharacteristic
545C2×C4#41×D4711yescharacteristic
546C2×C4#41×D4711yescharacteristic
547C2×C4#41×D4711yescharacteristic
548C2×C4#1309407111yescharacteristic
549C2×C4#1309407211yescharacteristic
550C8—22no—
551C8—41no—
552C8—474no—
553C8—1881no—
554C22—474no—
555C22—942no—
556C22—942no—
557C22—942no—
558C22—942no—
559C22—942no—
560C22—942no—
561C22C4×D4×D4711yescharacteristic
562C22#1568977111yescharacteristic
563C22#1568977511yescharacteristic
564C22C2×SD16×D4711yescharacteristic
565C22#135×D4711yescharacteristic
566C22#1568993211yescharacteristic
567C22#1568993311yescharacteristic
568C4—22no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—942no—
575C4—942no—
576C4—3761no—
577C4—3761no—
578C4—1881no—
579C4—1882no—
580C4—1882no—
581C4#117×D4711yescharacteristic
582C4#1568959911yescharacteristic
583C4#1568980511yescharacteristic
584C4#1568980911yescharacteristic
585C2—944no—
586C2#2024625411yescharacteristic
587C2C4×SD16×D4711yescharacteristic
588C2#2025224411yescharacteristic
589C2#2025574311yescharacteristic
590C2#2025574811yescharacteristic
591C2#2025577011yescharacteristic
592C2#2025577411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1919C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1919D47
hypocenterC47#22609
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #20244115C2semidirect
5. #20244113C2semidirect
6. #20243974C2semidirect
7. #20243972C2semidirect
8. #20243968C2semidirect
12. C4×#13093955C2semidirect
13. #20243276C2semidirect
15. C47×#1919C2semidirect
25. #106×D47C4semidirect
256. C47⋊C8#117semidirect
297. #1919D47semidirect
356. C47#22609semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D472
#202462612
#202462562
#202441152
#202441132
#202439742
#202439722
#202439682
#202434772
#202434732
#202434122
C4×#130939552
#202432762
#202431582
C47×#19192
#2260947