Finite Group Databasebeta

← #26268081

Subgroups — #26268081

Order 12032.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689533C211yesmaximal, characteristic, complemented
2#20248561C211yesmaximal, characteristic
3#20246488C211yesmaximal, characteristic, complemented
4#20244873C211yesmaximal, characteristic, complemented
5#20244872C211yesmaximal, characteristic, complemented
6#20244772C211yesmaximal, characteristic
7#20244758C211yesmaximal, characteristic
8#20244339C211yesmaximal, characteristic, complemented
9#20244157C211yesmaximal, characteristic, complemented
10#20244153C211yesmaximal, characteristic, complemented
11#20244109C211yesmaximal, characteristic, complemented
12#20244018C211yesmaximal, characteristic
13#20244011C211yesmaximal, characteristic
14#20243213C211yesmaximal, characteristic
15C2×C47×#478C211yesmaximal, characteristic
16C2×#13094024C2211yescharacteristic
17#15689533—24no—
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×Dic47⋊C8C2211yescharacteristic
23C2×Dic47⋊C8C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27#15689035—21no—
28#15689035—21no—
29#15689026C2211yescharacteristic
30#15689026C2211yescharacteristic
31#15689022—21no—
32#15689022C2211yescharacteristic
33#15689022C2211yescharacteristic
34#15689009—21no—
35#15689009—21no—
36#15688988C2211yescharacteristic
37#15688988C2211yescharacteristic
38#15688983—21no—
39#15688983—21no—
40C4⋊C4×Dic47—21no—
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#125×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C47×#478—24no—
56C2×#107×C47C2211yescharacteristic
57C2×#106×C47C2211yescharacteristic
58C2×C4⋊C8×C47C2211yescharacteristic
59C2×#102×C47C2211yescharacteristic
60C2×#102×C47C2211yescharacteristic
61C22×Dic94—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094024—22no—
65#13094024—22no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47D411yescharacteristic
70C2×C4⋊Dic47D411yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84#13093957—22no—
85#13093957—22no—
86#13093955—22no—
87#13093955—22no—
88Dic47⋊C8—22no—
89Dic47⋊C8—22no—
90#13093945—22no—
91#13093945Q814yescomplemented
92#13093935—22no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95D411yescharacteristic
103C47⋊#95D411yescharacteristic
104C22×D4×C47C2311yescharacteristic
105C23×C4×C47—21no—
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108#125×C47—22no—
109#125×C47D414yes—
110C4×D4×C47—22no—
111C4×D4×C47—22no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#107×C47—22no—
120#106×C47—22no—
121C4⋊C8×C47—22no—
122#102×C47—22no—
123#102×C47—22no—
124C24×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47C2×Q812yes—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47—42no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C2×Dic94—42no—
149C2×Dic94—81no—
150C2×Dic94—81no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C4⋊C4×C47—24no—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47C2×D412yes—
164C4⋊C4×C47C2×D412yes—
165C22⋊C4×C47—42no—
166C42×C47C2×D412yes—
167C42×C47#4112yes—
168C22⋊Dic47—81no—
169C22⋊Dic47—81no—
170C22⋊Dic47—81no—
171C22⋊Dic47—81no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—41no—
180C4⋊Dic47—41no—
181C4⋊Dic47—42no—
182C4⋊Dic47—42no—
183C4⋊Dic47—81no—
184C4⋊Dic47C2×Q812yes—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—21no—
189C4×Dic47—42no—
190C4×Dic47—81no—
191C4×Dic47—81no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8C2×Q812yes—
197C2×C47⋊C8#4112yes—
198C23×C47—22no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47C22×D411yescharacteristic
206D4×C47—22no—
207D4×C47—22no—
208D4×C47—41no—
209D4×C47—41no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—84no—
224C2×C4×C47#12112yes—
225C2×C4×C47#12112yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×Q811yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C472−1+411yescharacteristic
231C2×C4×C472−1+411yescharacteristic
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—22no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256Dic94—84no—
257C8×C47—41no—
258C8×C47—41no—
259C47⋊C8—22no—
260C47⋊C8—22no—
261#25466—471nomaximal, Sylow
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—81no—
268C22×C47—81no—
269C22×C47C2×#12111yescharacteristic
270C22×C47#54711yescharacteristic
271C22×C47D4×Q811yescharacteristic
272C22×C47#57511yescharacteristic
273C22×C47#57511yescharacteristic
274C22×C47#57611yescharacteristic
275C22×C47#57711yescharacteristic
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—42no—
279C4×C47—42no—
280C4×C47—81no—
281C4×C47#53311yescharacteristic
282C4×C47#54111yescharacteristic
283C4×C47#54811yescharacteristic
284C4×C47D4×Q811yescharacteristic
285Dic47—41no—
286Dic47—41no—
287Dic47—41no—
288Dic47—41no—
289Dic47—42no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293C2×#501—471no—
294C2×#478D4711yescharacteristic
295#2563—471no—
296#2370—471no—
297#2085—471no—
298#2085—471no—
299#2067—471no—
300#2065—471no—
301#1988—471no—
302#1949—471no—
303#1949—471no—
304#1940—471no—
305#1926—471no—
306#1925—471no—
307#1762—471no—
308C2×C47—82no—
309C2×C47#242911yescharacteristic
310C2×C47#304011yescharacteristic
311C2×C47#304511yescharacteristic
312C2×C47#334711yescharacteristic
313C2×C47#334811yescharacteristic
314C2×C47#338811yescharacteristic
315C2×C47#338911yescharacteristic
316C2×C4⋊Q8—471no—
317C2×#125C2×D4711yescharacteristic
318C2×C4×D4C2×D4711yescharacteristic
319#501—944no—
320#478—24no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#107C2×D4711yescharacteristic
324C2×#106—471no—
325C2×#106—471no—
326C2×#106C2×D4711yescharacteristic
327C2×C4⋊C8—471no—
328C2×C4⋊C8—471no—
329C2×C4⋊C8C2×D4711yescharacteristic
330C2×#102—471no—
331C2×#102—471no—
332C2×#102C2×D4711yescharacteristic
333C2×#102C2×D4711yescharacteristic
334C2×#97—471no—
335#393—941no—
336#393—941no—
337#390—471no—
338#390—471no—
339#389—471no—
340#389—471no—
341#389—941no—
342#387—941no—
343#387—941no—
344#382—471no—
345#382—471no—
346#380—941no—
347#380—941no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—941no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#336—471no—
360#336—471no—
361C47#2546611yescharacteristic, complemented, Sylow
362C22×Q8—941no—
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C22×C8—471no—
366C22×C8—471no—
367C22×C8C22×D4711yescharacteristic
368C22×C8C22×D4711yescharacteristic
369C4⋊Q8—942no—
370C4⋊Q8—942no—
371#125—22no—
372#125C22⋊D4714yes—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42—941no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42C22×D4711yescharacteristic
400#107—22no—
401#107—942no—
402#107—942no—
403#106—22no—
404#106—942no—
405#106—942no—
406C4⋊C8—22no—
407C4⋊C8—942no—
408C4⋊C8—942no—
409#102—22no—
410#102—22no—
411#102—474no—
412#102—942no—
413#97—942no—
414#95—471no—
415#95—471no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425C24—21no—
426C2×Q8—3761no—
427C2×Q8—3761no—
428C2×Q8—1882no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D4712yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—471no—
444C22×C4—471no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4C23×D4711yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—472no—
459C2×C8—472no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—472no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—3761no—
479C4⋊C4—1881no—
480C4⋊C4—1881no—
481C4⋊C4—1882no—
482C4⋊C4—1882no—
483C4⋊C4C2×C22⋊D4712yes—
484C4⋊C4C2×C22⋊D4712yes—
485C22⋊C4—42no—
486C22⋊C4—3761no—
487C22⋊C4—3761no—
488C22⋊C4—3761no—
489C22⋊C4—3761no—
490C42—941no—
491C42—941no—
492C42—941no—
493C42—941no—
494C42—3761no—
495C42—3761no—
496C42—1882no—
497C42#1062535212yes—
498C42C2×C22⋊D4712yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D4711yescharacteristic
507Q8—3764no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—942no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4#1309399612yes—
551C2×C4#1309399612yes—
552C2×C4C22×C22⋊D4711yescharacteristic
553C2×C4C2×Q8×D4711yescharacteristic
554C2×C4#41×D4711yescharacteristic
555C2×C4C2×#1062535211yescharacteristic
556C2×C4#1309407111yescharacteristic
557C2×C4#1309407211yescharacteristic
558C8—41no—
559C8—41no—
560C8—942no—
561C8—942no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1309399611yescharacteristic
570C22#1568977511yescharacteristic
571C22#1568978611yescharacteristic
572C22#1568991911yescharacteristic
573C22#1568991911yescharacteristic
574C22#1568992511yescharacteristic
575C22#1568993011yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—3761no—
582C4—3761no—
583C4—3761no—
584C4—1882no—
585C4—1881no—
586C4—1881no—
587C4—1881no—
588C4—1881no—
589C4#1568967511yescharacteristic
590C4#1568973311yescharacteristic
591C4#1568978011yescharacteristic
592C4Q8×C22⋊D4711yescharacteristic
593C2—82no—
594C2#2024698911yescharacteristic
595C2#2025288011yescharacteristic
596C2#2025292511yescharacteristic
597C2#2025588511yescharacteristic
598C2#2025590011yescharacteristic
599C2#2025633011yescharacteristic
600C2#2025634511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#478C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenterC2×#478D47
hypocenterC47#25466
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#15689533C2semidirect
3. #20246488C2semidirect
4. #20244873C2semidirect
5. #20244872C2semidirect
8. #20244339C2semidirect
9. #20244157C2semidirect
10. #20244153C2semidirect
11. #20244109C2semidirect
91. #13093945Q8semidirect
361. C47#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#156895332
#202485612
#202464882
#202448732
#202448722
#202447722
#202447582
#202443392
#202441572
#202441532
#202441092
#202440182
#202440112
#202432132
C2×C47×#4782
#2546647