Finite Group Databasebeta

← #26238157

Subgroups — #26238157

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689411C211yesmaximal, characteristic
2#20246474C211yesmaximal, characteristic, complemented
3D4×Dic47⋊C4C211yesmaximal, characteristic
4#20244347C211yesmaximal, characteristic, complemented
5#20244143C211yesmaximal, characteristic, complemented
6#20244129C211yesmaximal, characteristic, complemented
7#20244128C211yesmaximal, characteristic, complemented
8#20243974C211yesmaximal, characteristic, complemented
9#20243971C211yesmaximal, characteristic, complemented
10#20243633C211yesmaximal, characteristic
11#20243254C211yesmaximal, characteristic, complemented
12#20243215C211yesmaximal, characteristic
13#20243213C211yesmaximal, characteristic
14#102×Dic47C211yesmaximal, characteristic
15C47×#1862C211yesmaximal, characteristic
16C2×#13094023C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C22×Dic47⋊C4—21no—
20#15689411C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×Dic47⋊C8C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093935C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15689028—21no—
31#15689027—21no—
32#15689021C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689019C2211yescharacteristic
35#15689018—21no—
36#15688996—21no—
37#15688995—21no—
38#15688992C2211yescharacteristic
39#15688981—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×Dic47⋊C4C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44#15688974—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×Dic94—21no—
61C23×Dic47—21no—
62C23×Dic47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094023—22no—
66#13094023C2×C414yescomplemented
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47—41no—
70D4×Dic47C2×C414yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—44no—
82C2×Dic47⋊C4D411yescharacteristic
83C2×Dic47⋊C4D411yescharacteristic
84C2×C22⋊Dic47—21no—
85C2×C22⋊Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093957—41no—
92#13093956C2×C414yescomplemented
93#13093955C2×C414yescomplemented
94Dic47⋊C8C2×C414yescomplemented
95#13093945C2×C414yes—
96#13093945Q814yescomplemented
97#13093935—41no—
98#13093934—41no—
99C8×Dic47—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95D411yescharacteristic
107C47⋊#95D411yescharacteristic
108C22×D4×C47C2311yescharacteristic
109C23×C4×C47—21no—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C4×D4×C47—22no—
113C4×D4×C47—41no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#107×C47—41no—
121#102×C47—22no—
122#102×C47C2×C414yes—
123#95×C47—21no—
124C24×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47C22×C412yes—
129C2×D4×C47C2×Q812yes—
130C2×D4×C47#4112yes—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47—81no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—24no—
144C22×Dic47—21no—
145C22×Dic47—42no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47—81no—
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C2×Dic94—24no—
155C2×Dic94—81no—
156C2×Dic94—81no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47C22×C412yes—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—81no—
169C4⋊C4×C47C22×C412yes—
170C22⋊C4×C47—42no—
171C42×C47—21no—
172C42×C47—21no—
173C22⋊Dic47—24no—
174C22⋊Dic47—42no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C22×C412yes—
182C4⋊Dic47C2×Q812yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—22no—
185Dic47⋊C4—22no—
186Dic47⋊C4—42no—
187Dic47⋊C4—42no—
188Dic47⋊C4—42no—
189Dic47⋊C4—81no—
190Dic47⋊C4—81no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—42no—
194C4×Dic47—81no—
195C4×Dic47—81no—
196C4×Dic47C22×C412yes—
197C4×Dic47#4112yes—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8C22×C412yes—
203C2×C47⋊C8C2×Q812yes—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47C22×D411yescharacteristic
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—22no—
215D4×C47C4×Q814yes—
216C2×C4×C47—22no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
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×C47—41no—
231C2×C4×C47—84no—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×Q811yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47H(5,𝔽2)11yescharacteristic
237C2×C4×C472−1+411yescharacteristic
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—22no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—48no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—84no—
260C2×Dic47C4×D412yes—
261C2×Dic47C4×D412yes—
262C2×Dic47AGL(1,ℤ/8)12yes—
263C2×Dic47#13712yes—
264Dic94—84no—
265C8×C47—22no—
266C8×C47—41no—
267C47⋊C8—22no—
268C47⋊C8—22no—
269#22969—471nomaximal, Sylow
270C22×C47—24no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47C2×C4×D411yescharacteristic
278C22×C47#54611yescharacteristic
279C22×C47D4×Q811yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C47C2×#13711yescharacteristic
282C22×C47#57611yescharacteristic
283C22×C47#57711yescharacteristic
284C4×C47—41no—
285C4×C47—42no—
286C4×C47—41no—
287C4×C47—81no—
288C4×C47—81no—
289C4×C47C2×C4×Q811yescharacteristic
290C4×C47#52011yescharacteristic
291C4×C47#55211yescharacteristic
292C4×C47#55411yescharacteristic
293Dic47—22no—
294Dic47—41no—
295Dic47—42no—
296Dic47—42no—
297Dic47—42no—
298Dic47—41no—
299Dic47—81no—
300Dic47#44014yes—
301C2×#476—471no—
302#2369—471no—
303D4×C4⋊C4—471no—
304#1990—471no—
305#1947—471no—
306#1944—471no—
307#1944—471no—
308#1920—471no—
309#1920—471no—
310#1862—471no—
311#1862D4711yescharacteristic
312#1770—471no—
313#1762—471no—
314#1762—471no—
315C4×#102—471no—
316C2×C47—44no—
317C2×C47#236011yescharacteristic
318C2×C47C2×#44011yescharacteristic
319C2×C47#294611yescharacteristic
320C2×C47#332511yescharacteristic
321C2×C47#333111yescharacteristic
322C2×C47#338411yescharacteristic
323C2×C47#338511yescharacteristic
324C2×C4⋊Q8—471no—
325C2×C4×D4—471no—
326C2×C4×D4—471no—
327C2×C4×D4C2×D4711yescharacteristic
328C22×C4⋊C4—941no—
329#476—478no—
330C2×#107—471no—
331C2×#107C2×D4711yescharacteristic
332C2×#106—471no—
333C2×#106—471no—
334C2×C4⋊C8—471no—
335C2×#102—471no—
336C2×#102—471no—
337C2×#102C2×D4711yescharacteristic
338C2×#102C2×D4711yescharacteristic
339C2×#97—471no—
340C2×#97—471no—
341C2×C4×C8—471no—
342#391—941no—
343#391—941no—
344#389—471no—
345#389—471no—
346#389—471no—
347#389—941no—
348#389C2×D4711yescharacteristic
349#385—941no—
350#385—941no—
351#384—471no—
352#380—941no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4—941no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340C2×D4711yescharacteristic
366#336—471no—
367#336C2×D4711yescharacteristic
368C47#2296911yescharacteristic, complemented, Sylow
369C22×Q8—941no—
370C22×D4C22×D4711yescharacteristic
371C23×C4—21no—
372C23×C4—941no—
373C23×C4—941no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8C22×D4711yescharacteristic
377C22×C8C22×D4711yescharacteristic
378C4⋊Q8—474no—
379C4⋊Q8—942no—
380C4×D4—22no—
381C4×D4—41no—
382C4×D4—474no—
383C4×D4—942no—
384C4×D4—942no—
385C4×D4—1881no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—1884no—
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C42—471no—
407C2×C42—471no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42—941no—
411C2×C42C22×D4711yescharacteristic
412#107—41no—
413#107—1881no—
414#106—474no—
415#106—474no—
416C4⋊C8—474no—
417#102—22no—
418#102—474no—
419#102—474no—
420#102C4×D4714yes—
421#97—1881no—
422#97—1881no—
423C4×C8—942no—
424#95—21no—
425#95—471no—
426#95—471no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433C24—21no—
434C2×Q8—944no—
435C2×Q8—3761no—
436C2×Q8—3761no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×C4×D4712yes—
441C2×D4#1062535212yes—
442C2×D4Q8×D4712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—944no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—3761no—
459C22×C4—1882no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4D4×D4711yescharacteristic
464C22×C4D4×D4711yescharacteristic
465C22×C4C23×D4711yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—472no—
472C2×C8—472no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8C2×C4×D4712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—472no—
485C4⋊C4—472no—
486C4⋊C4—472no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—944no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—3761no—
496C4⋊C4—3761no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4—1882no—
500C4⋊C4C2×C4×D4712yes—
501C22⋊C4—42no—
502C22⋊C4—944no—
503C22⋊C4—1882no—
504C42—21no—
505C42—21no—
506C42—472no—
507C42—472no—
508C42—941no—
509C42—941no—
510C42—3761no—
511C42—3761no—
512C42—1882no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D4711yescharacteristic
521Q8—3764no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#1309398914yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—472no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—942no—
551C2×C4—942no—
552C2×C4—942no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—3764no—
558C2×C4—1888no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4C2×Q8×D4711yescharacteristic
570C2×C4#41×D4711yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C2×C4#1309406911yescharacteristic
573C2×C4#1309407111yescharacteristic
574C8—22no—
575C8—41no—
576C8—942no—
577C8—942no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C4×D4×D4711yescharacteristic
586C22#1568976011yescharacteristic
587C22#1568978611yescharacteristic
588C22AGL(1,ℤ/8)×D4711yescharacteristic
589C22#137×D4711yescharacteristic
590C22#1568992411yescharacteristic
591C22#1568993011yescharacteristic
592C4—42no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—474no—
598C4—942no—
599C4—3761no—
600C4—1882no—
601C4—1882no—
602C4—1882no—
603C4—1881no—
604C4—1881no—
605C4C2×#1309398911yescharacteristic
606C4#1568960211yescharacteristic
607C4#1568981011yescharacteristic
608C4#1568982711yescharacteristic
609C2—44no—
610C2#2024638811yescharacteristic
611C2#440×D4711yescharacteristic
612C2#2025221311yescharacteristic
613C2#2025562311yescharacteristic
614C2#2025569711yescharacteristic
615C2#2025629011yescharacteristic
616C2#2025629311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1862C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1862D47
hypocenterC47#22969
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #20246474C2semidirect
4. #20244347C2semidirect
5. #20244143C2semidirect
6. #20244129C2semidirect
7. #20244128C2semidirect
8. #20243974C2semidirect
9. #20243971C2semidirect
11. #20243254C2semidirect
20. #15689411C4semidirect
66. #13094023C2×C4semidirect
92. #13093956C2×C4semidirect
93. #13093955C2×C4semidirect
94. Dic47⋊C8C2×C4semidirect
96. #13093945Q8semidirect
368. C47#22969semidirect

Maximal subgroups

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

subgroupindex
C2×#156894112
#202464742
D4×Dic47⋊C42
#202443472
#202441432
#202441292
#202441282
#202439742
#202439712
#202436332
#202432542
#202432152
#202432132
#102×Dic472
C47×#18622
#2296947