Finite Group Databasebeta

← #26316446

Subgroups — #26316446

Order 12032.

728 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246070C211yesmaximal, characteristic
2#20244955C211yesmaximal, characteristic, complemented
3#20244939C211yesmaximal, characteristic, complemented
4#20244713C211yesmaximal, characteristic, complemented
5#20244526C211yesmaximal, characteristic, complemented
6#20243963C211yesmaximal, characteristic, complemented
7#20243956C211yesmaximal, characteristic, complemented
8#20243819C211yesmaximal, characteristic
9#20243809C211yesmaximal, characteristic
10#20243641C211yesmaximal, characteristic, complemented
11#20243641C211yesmaximal, characteristic, complemented
12#20243419C211yesmaximal, characteristic
13#20243415C211yesmaximal, characteristic
14#20243085C211yesmaximal, characteristic
15C47×#2628C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28#15689029C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689018—21no—
31#15689017—21no—
32#15689011—21no—
33#15689004—21no—
34#15688983C2211yescharacteristic
35C4×C4⋊Dic47C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4⋊Q8×C47C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C47×#400—21no—
51C47×#399—21no—
52C47×#395—21no—
53C47×#393—21no—
54C47×#389C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#380C2211yescharacteristic
59C22×Dic94C2311yescharacteristic
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—41no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8—41no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094003—22no—
71#13094003Q814yescomplemented
72C4×Dic94—41no—
73C4×Dic94—41no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84#13093957—22no—
85#13093957—41no—
86#13093957—41no—
87#13093955—22no—
88#13093947—22no—
89#13093947—22no—
90#13093947—41no—
91#13093947—41no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×Q8×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C4⋊Q8×C47—41no—
100C4⋊Q8×C47—41no—
101#117×C47—41no—
102#117×C47—41no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47D411yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C22⋊C4×C47D411yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124C24×C47C2×D411yescharacteristic
125C2×Q8×C47—42no—
126C2×Q8×C47—81no—
127C2×Q8×C47—81no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×Q812yes—
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94C2×Q812yes—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—41no—
163C4⋊C4×C47—41no—
164C4⋊C4×C47—42no—
165C4⋊C4×C47—81no—
166C4⋊C4×C47—81no—
167C22⋊C4×C47—41no—
168C22⋊C4×C47—41no—
169C22⋊C4×C47—81no—
170C22⋊C4×C47—81no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C22⋊Dic47—24no—
176C22⋊Dic47—81no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47C2×Q812yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—42no—
185Dic47⋊C4—81no—
186C4×Dic47—42no—
187C4×Dic47—81no—
188#10625329—44no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—44no—
200C2×C47⋊C8#4112yes—
201C2×C47⋊C8#4112yes—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47C22⋊Q812yes—
209C23×C47C22×D411yescharacteristic
210Q8×C47—84no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
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—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47C22⋊Q812yes—
242C2×C4×C47C22×Q811yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47H(5,𝔽2)11yescharacteristic
247C2×C4×C47H(5,𝔽2)11yescharacteristic
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260Dic94—22no—
261Dic94—22no—
262Dic94—41no—
263Dic94—41no—
264C47⋊C8—22no—
265C47⋊C8—22no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268#29171—471nomaximal, Sylow
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47C2×C22⋊Q811yescharacteristic
277C22×C47#53211yescharacteristic
278C22×C47#53411yescharacteristic
279C22×C47#57611yescharacteristic
280C22×C47#57711yescharacteristic
281C22×C47#57811yescharacteristic
282C22×C47#57811yescharacteristic
283C4×C47—22no—
284C4×C47—41no—
285C4×C47—41no—
286C4×C47—41no—
287C4×C47—41no—
288C4×C47—81no—
289C4×C47—81no—
290C4×C47—81no—
291C4×C47—81no—
292C4×C47#55111yescharacteristic
293C4×C47#55111yescharacteristic
294C4×C47#55411yescharacteristic
295C4×C47#55411yescharacteristic
296Dic47—42no—
297Dic47—42no—
298Dic47—81no—
299Dic47—81no—
300#2628D4711yescharacteristic
301#2323—471no—
302#2101—471no—
303#2097—471no—
304#2057—471no—
305#2024—471no—
306#1919—471no—
307#1918—471no—
308#1894—471no—
309#1892—471no—
310#1863—471no—
311#1863—471no—
312#1809—471no—
313#1808—471no—
314#1729—471no—
315C2×C47—42no—
316C2×C47#256411yescharacteristic
317C2×C47#308211yescharacteristic
318C2×C47#308311yescharacteristic
319C2×C47#317511yescharacteristic
320C2×C47#317811yescharacteristic
321C2×C47#319811yescharacteristic
322C2×C47#320211yescharacteristic
323C22×M16—471no—
324C23×C8—471no—
325C2×C4⋊Q8C2×D4711yescharacteristic
326C2×C22⋊Q8—471no—
327C2×C4×Q8—471no—
328C2×#117C2×D4711yescharacteristic
329C2×#107—471no—
330C2×#107—471no—
331C2×#107—471no—
332C2×#106—471no—
333C2×#103—471no—
334C2×#103—471no—
335C2×#103—471no—
336C2×#103—471no—
337#400—21no—
338#399—21no—
339#395—21no—
340#393—21no—
341#391—471no—
342#389—471no—
343#389—941no—
344#389C2×D4711yescharacteristic
345#389C2×D4711yescharacteristic
346#388—941no—
347#388—941no—
348#387—941no—
349#382C2×D4711yescharacteristic
350#380—471no—
351#380C2×D4711yescharacteristic
352#380C2×D4711yescharacteristic
353C4×C4⋊C4—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#336—471no—
363#336—471no—
364#336—471no—
365#336—471no—
366C47#2917111yescharacteristic, complemented, Sylow
367C22×Q8—21no—
368C22×Q8—471no—
369C23×C4C22×D4711yescharacteristic
370C2×M16—942no—
371C2×M16—942no—
372C2×M16—1881no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8—471no—
377C22×C8—942no—
378C22×C8—942no—
379C22×C8—1881no—
380C4⋊Q8—41no—
381C4⋊Q8—41no—
382C22⋊Q8—474no—
383C22⋊Q8—942no—
384C4×Q8—1881no—
385C4×Q8—1881no—
386#117—41no—
387#117—41no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4C22⋊D4711yescharacteristic
398C2×C4⋊C4C22⋊D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4C22⋊D4711yescharacteristic
408C2×C22⋊C4C22⋊D4711yescharacteristic
409C2×C42—941no—
410C2×C42—941no—
411C2×C42C22×D4711yescharacteristic
412C2×C42C22×D4711yescharacteristic
413#107—942no—
414#107—1881no—
415#107—1881no—
416#106—942no—
417#103—942no—
418#103—942no—
419#103—1881no—
420#103—1881no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432C24C2×C22⋊D4711yescharacteristic
433C2×Q8—42no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×Q8—472no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4Q8×D4712yes—
455C22×C4#1062535412yes—
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458C22×C4C2×C22⋊D4711yescharacteristic
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C23×D4711yescharacteristic
462M16—1884no—
463C2×C8—472no—
464C2×C8—472no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—1884no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—472no—
490C4⋊C4—944no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—3761no—
497C4⋊C4—1882no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—944no—
503C22⋊C4—3761no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—3761no—
509C42—1882no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1309400012yes—
517C23C22×C22⋊D4711yescharacteristic
518Q8—84no—
519Q8—942no—
520Q8—942no—
521Q8—1881no—
522Q8—1881no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—942no—
554C2×C4—942no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4#1309400012yes—
566C2×C4C2×Q8×D4711yescharacteristic
567C2×C4#41×D4711yescharacteristic
568C2×C4#41×D4711yescharacteristic
569C2×C4C2×#1062535411yescharacteristic
570C2×C4#1309406911yescharacteristic
571C2×C4#1309406911yescharacteristic
572C8—942no—
573C8—942no—
574C8—1881no—
575C8—1881no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1309400011yescharacteristic
584C22#1568966911yescharacteristic
585C22#1568968211yescharacteristic
586C22#1568992611yescharacteristic
587C22#1568993411yescharacteristic
588C22#1568993811yescharacteristic
589C22#1568993811yescharacteristic
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—3761no—
600C4—3761no—
601C4—1882no—
602C4—1882no—
603C4#1568980011yescharacteristic
604C4#1568980011yescharacteristic
605C4#1568982711yescharacteristic
606C4#1568982711yescharacteristic
607C2—42no—
608C2#2024857011yescharacteristic
609C2#2025322411yescharacteristic
610C2#2025323511yescharacteristic
611C2#2025402111yescharacteristic
612C2#2025405011yescharacteristic
613C2#2025424011yescharacteristic
614C2#2025427911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2628C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2628D47
hypocenterC47#29171
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #20244955C2semidirect
3. #20244939C2semidirect
4. #20244713C2semidirect
5. #20244526C2semidirect
6. #20243963C2semidirect
7. #20243956C2semidirect
10. #20243641C2semidirect
11. #20243641C2semidirect
71. #13094003Q8semidirect
366. C47#29171semidirect

Maximal subgroups

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

subgroupindex
#202460702
#202449552
#202449392
#202447132
#202445262
#202439632
#202439562
#202438192
#202438092
#202436412
#202434192
#202434152
#202430852
C47×#26282
#2917147