Finite Group Databasebeta

← #26226873

Subgroups — #26226873

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689181C211yesmaximal, characteristic
2#20246008C211yesmaximal, characteristic
3#20245995C211yesmaximal, characteristic
4#20244159C211yesmaximal, characteristic, complemented
5#20244157C211yesmaximal, characteristic, complemented
6#20244145C211yesmaximal, characteristic, complemented
7#20244143C211yesmaximal, characteristic, complemented
8#20243587C211yesmaximal, characteristic
9#20243578C211yesmaximal, characteristic
10#20243575C211yesmaximal, characteristic
11#20243553C211yesmaximal, characteristic, complemented
12#20243546C211yesmaximal, characteristic, complemented
13#20243520C211yesmaximal, characteristic, complemented
14#20243517C211yesmaximal, characteristic, complemented
15C47×#1854C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18#15689181C418yescomplemented
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15689021—21no—
31#15689021—21no—
32#15689017—21no—
33#15689011—21no—
34#15689008—21no—
35#15689006—21no—
36#15688995—21no—
37#15688995—21no—
38#15688992C2211yescharacteristic
39#15688992C2211yescharacteristic
40#15688992C2211yescharacteristic
41#15688988C2211yescharacteristic
42#15688988C2211yescharacteristic
43#15688988C2211yescharacteristic
44#15688983C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46C4⋊C4×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63#13093983C2×C414yes—
64#13093983Q814yes—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—41no—
67C2×C4⋊Dic47—41no—
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47D411yescharacteristic
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—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47D411yescharacteristic
82C2×C22⋊Dic47D411yescharacteristic
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093958—41no—
88#13093958—41no—
89#13093957—41no—
90#13093957—41no—
91#13093957—41no—
92#13093955—41no—
93Dic47⋊C8C2×C414yescomplemented
94Dic47⋊C8C2×C414yescomplemented
95C22⋊(C47⋊C8)C2×C414yes—
96#13093934C2×C414yescomplemented
97C8×Dic47C2×C414yescomplemented
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⋊#95—21no—
107C23×C4×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110#117×C47—41no—
111#117×C47—41no—
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—21no—
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—21no—
119#107×C47—41no—
120#106×C47—41no—
121C22⋊C8×C47C2×C414yes—
122#95×C47—21no—
123C24×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×Q812yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—42no—
141C22×Dic47—81no—
142C22×Dic47—81no—
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47C22×C412yes—
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—21no—
160C4⋊C4×C47—21no—
161C22⋊C4×C47—42no—
162C22⋊C4×C47—81no—
163C42×C47—42no—
164C22⋊Dic47—81no—
165C22⋊Dic47C4⋊C414yes—
166C22⋊Dic47C4⋊C414yes—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—42no—
176C4⋊Dic47C4⋊C414yes—
177C4⋊Dic47C4⋊C414yes—
178Dic47⋊C4—81no—
179Dic47⋊C4—81no—
180Dic47⋊C4—81no—
181Dic47⋊C4—81no—
182Dic47⋊C4—81no—
183Dic47⋊C4—81no—
184C4×Dic47—81no—
185C4×Dic47—81no—
186C4×Dic47C22×C412yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C2×Q812yes—
189C4×Dic47C2×Q812yes—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8C22×C412yes—
195C2×C47⋊C8C22×C412yes—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C2×C4⋊C412yes—
203C23×C47C22×D411yescharacteristic
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
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×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47C2×C4⋊C412yes—
223C2×C4×C47C23×C411yescharacteristic
224C2×C4×C47C22×Q811yescharacteristic
225C2×C4×C47H(5,𝔽2)11yescharacteristic
226C2×C4×C47H(5,𝔽2)11yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×C4×C472−1+411yescharacteristic
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
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×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—84no—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C2×Dic47C2×C4⋊C412yes—
253C2×Dic47C2×C4⋊C412yes—
254C8×C47—22no—
255C8×C47—22no—
256C47⋊C8—22no—
257C47⋊C8—22no—
258#21991—471nomaximal, Sylow
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47C22×C4⋊C411yescharacteristic
267C22×C47#53511yescharacteristic
268C22×C47#53611yescharacteristic
269C22×C47#57611yescharacteristic
270C22×C47#57611yescharacteristic
271C22×C47#57811yescharacteristic
272C22×C47#57811yescharacteristic
273C4×C47—22no—
274C4×C47—81no—
275C4×C47—81no—
276C4×C47—81no—
277C4×C47—81no—
278C4×C47#52011yescharacteristic
279C4×C47#52011yescharacteristic
280C4×C47#54411yescharacteristic
281C4×C47#54411yescharacteristic
282Dic47—22no—
283Dic47—22no—
284Dic47—22no—
285Dic47—22no—
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289Dic47—81no—
290C2×#433—471no—
291#2317—471no—
292#2314—471no—
293#1949—471no—
294#1949—471no—
295#1947—471no—
296#1947—471no—
297#1854—471no—
298#1854D4711yescharacteristic
299#1852—471no—
300#1852—471no—
301#1847—471no—
302#1845—471no—
303#1836—471no—
304#1835—471no—
305C2×C47—42no—
306C2×C47#231711yescharacteristic
307C2×C47#291311yescharacteristic
308C2×C47#291311yescharacteristic
309C2×C47#322011yescharacteristic
310C2×C47#322311yescharacteristic
311C2×C47#324311yescharacteristic
312C2×C47#324311yescharacteristic
313C2×#117—471no—
314C2×#117C2×D4711yescharacteristic
315C22×C4⋊C4—471no—
316#433—478no—
317C2×#107—471no—
318C2×#107—471no—
319C2×#107—471no—
320C2×#107—471no—
321C2×#107—471no—
322C2×#107C2×D4711yescharacteristic
323C2×#106—471no—
324C2×#106C2×D4711yescharacteristic
325C2×C4⋊C8—471no—
326C2×C4⋊C8—471no—
327C2×C22⋊C8—471no—
328C2×C22⋊C8C2×D4711yescharacteristic
329C2×#97—471no—
330C2×C4×C8—471no—
331#389—941no—
332#389—941no—
333#388—941no—
334#388—941no—
335#387—941no—
336#387—941no—
337#385—941no—
338#385—941no—
339#384—471no—
340#384—471no—
341#384—471no—
342#382—471no—
343#382—471no—
344#382—471no—
345#380—471no—
346#380C2×D4711yescharacteristic
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340C2×D4711yescharacteristic
356#340C2×D4711yescharacteristic
357C47#2199111yescharacteristic, complemented, Sylow
358C23×C4—941no—
359C23×C4C22×D4711yescharacteristic
360C22×C8—471no—
361C22×C8—471no—
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4711yescharacteristic
364#117—41no—
365#117—41no—
366#117—474no—
367#117—474no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
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⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—1881no—
382C2×C4⋊C4—1881no—
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—471no—
390C2×C22⋊C4—471no—
391C2×C22⋊C4—941no—
392C2×C42—21no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—941no—
396C2×C42—941no—
397#107—41no—
398#107—1881no—
399#107—1881no—
400#107—1881no—
401#107—1881no—
402#107—1881no—
403#106—41no—
404#106—1881no—
405C4⋊C8—474no—
406C4⋊C8—474no—
407C22⋊C8—474no—
408C22⋊C8C4×D4714yes—
409#97—474no—
410C4×C8—474no—
411#95—21no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421C24D4×D4711yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—471no—
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—471no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—3761no—
439C22×C4—3761no—
440C22×C4—1882no—
441C22×C4C2×C4×D4712yes—
442C22×C4D4×D4711yescharacteristic
443C22×C4Q8×D4712yes—
444C22×C4C23×D4711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—472no—
450C2×C8—472no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8C2×C4×D4712yes—
456C2×C8C2×C4×D4712yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—474no—
466C4⋊C4—474no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—3761no—
476C4⋊C4—3761no—
477C4⋊C4—3761no—
478C4⋊C4—3761no—
479C4⋊C4—3761no—
480C4⋊C4—3761no—
481C4⋊C4—1882no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—474no—
485C22⋊C4—474no—
486C22⋊C4—3761no—
487C42—42no—
488C42—472no—
489C42—472no—
490C42—472no—
491C42—472no—
492C42—3761no—
493C42—3761no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D4712yes—
501C23C2×D4×D4711yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
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—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—472no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—472no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—3764no—
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×C4—1881no—
544C2×C4—1881no—
545C2×C4C4⋊C4×D4712yes—
546C2×C4C22×C4×D4711yescharacteristic
547C2×C4C2×Q8×D4711yescharacteristic
548C2×C4#1309406911yescharacteristic
549C2×C4#1309406911yescharacteristic
550C2×C4#1309407111yescharacteristic
551C2×C4#1309407111yescharacteristic
552C8—22no—
553C8—22no—
554C8—942no—
555C8—942no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D4711yescharacteristic
564C22#1568968811yescharacteristic
565C22#1568969511yescharacteristic
566C22#1568992511yescharacteristic
567C22#1568992711yescharacteristic
568C22#1568993711yescharacteristic
569C22#1568993911yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—942no—
576C4—942no—
577C4—942no—
578C4—942no—
579C4—3761no—
580C4—3761no—
581C4—3761no—
582C4—3761no—
583C4#1568960211yescharacteristic
584C4#1568960211yescharacteristic
585C4#1568975211yescharacteristic
586C4#1568975211yescharacteristic
587C2—42no—
588C2#2024600711yescharacteristic
589C2#2025198511yescharacteristic
590C2#2025198711yescharacteristic
591C2#2025443611yescharacteristic
592C2#2025446911yescharacteristic
593C2#2025466911yescharacteristic
594C2#2025467711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1854C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1854D47
hypocenterC47#21991
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #20244159C2semidirect
5. #20244157C2semidirect
6. #20244145C2semidirect
7. #20244143C2semidirect
11. #20243553C2semidirect
12. #20243546C2semidirect
13. #20243520C2semidirect
14. #20243517C2semidirect
18. #15689181C4semidirect
93. Dic47⋊C8C2×C4semidirect
94. Dic47⋊C8C2×C4semidirect
96. #13093934C2×C4semidirect
97. C8×Dic47C2×C4semidirect
357. C47#21991semidirect

Maximal subgroups

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

subgroupindex
C2×#156891812
#202460082
#202459952
#202441592
#202441572
#202441452
#202441432
#202435872
#202435782
#202435752
#202435532
#202435462
#202435202
#202435172
C47×#18542
#2199147