Finite Group Databasebeta

← #26232547

Subgroups — #26232547

Order 12032.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689839C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic, complemented
3#20246335C211yesmaximal, characteristic, complemented
4#20244350C211yesmaximal, characteristic, complemented
5#20243815C212yesmaximal, complemented
6#20243709C212yesmaximal, complemented
7#20243482C212yesmaximal, complemented
8#20243389C211yesmaximal, characteristic, complemented
9C4×#13093944C211yesmaximal, characteristic, complemented
10#20243103C212yesmaximal, complemented
11C47×#1987C211yesmaximal, characteristic, complemented
12C22×C8×D47C2212yes—
13#15689839—24no—
14#15689839C418yescomplemented
15C2×#13094013—22no—
16C2×#13093999—21no—
17C2×#13093999C2212yes—
18C2×C4×D188C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20C2×#13093982—21no—
21C2×#13093982C2212yes—
22C2×#13093982C2212yes—
23#15689188—24no—
24C2×#13093946C2212yes—
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×(C2×D47)⋊C8C2212yes—
28C2×C4×C47⋊C8C2211yescharacteristic
29#15689025—21no—
30#15689015—22no—
31#15688993C2211yescharacteristic
32#15688980C2212yes—
33C4×C4⋊Dic47—21no—
34#15688886C2212yes—
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2212yes—
38#15688885C2212yes—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C4⋊C8×C47C2211yescharacteristic
41C4×C4⋊C4×C47C2211yescharacteristic
42C47×#340C2212yes—
43C47×#340C2212yes—
44C22×D188—22no—
45C22×D188D411yescharacteristic
46C22×D188C2311yescharacteristic
47C22×C4×D47—21no—
48C22×C4×D47C2312yes—
49C22×C47⋊C8C2311yescharacteristic
50C22×C47⋊C8C2311yescharacteristic
51C2×C8×D47—24no—
52C2×C8×D47—24no—
53C2×C8×D47—42no—
54#13094013—216no—
55#13093999—28no—
56#13093999—24no—
57#13093999C2×C418yes—
58C4×D188—22no—
59C4×D188C2×C414yes—
60C4×D188C2×C414yes—
61C4×D188D414yes—
62#13093982—24no—
63#13093982—28no—
64#13093982—24no—
65#13093982—42no—
66#13093982C2×C418yes—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47D411yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—22no—
75C2×(C2×D47)⋊C4—22no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4D412yes—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—22no—
80#13093946—42no—
81#13093944—22no—
82#13093944—22no—
83(C2×D47)⋊C8—24no—
84C4×C47⋊C8—22no—
85C47⋊#95—22no—
86C22×Q8×C47—21no—
87C22×C8×C47C2312yes—
88C4⋊Q8×C47—22no—
89C4⋊Q8×C47C2×C414yes—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47C2311yescharacteristic
92C2×C4⋊C4×C47C2312yes—
93C2×C4⋊C4×C47C2311yescharacteristic
94C2×C42×C47—21no—
95C2×C42×C47C2311yescharacteristic
96C4⋊C8×C47—22no—
97#95×C47—21no—
98C23×D47—21no—
99C23×D47C2×D412yes—
100C2×Q8×C47—24no—
101C2×Q8×C47—81no—
102C2×Q8×C47—81no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2411yescharacteristic
109C22×Dic47—21no—
110C22×Dic47—22no—
111C22×Dic47C2×D412yes—
112C2×D188—22no—
113C2×D188—22no—
114C2×D188—21no—
115C2×D188—21no—
116C2×D188—21no—
117C2×D188—21no—
118C2×D188—21no—
119C2×D188—28no—
120C2×D188—82no—
121C2×D188—82no—
122C2×D188C22⋊C414yes—
123C2×D188C22×C412yes—
124C2×C4×D47—22no—
125C2×C4×D47—22no—
126C2×C4×D47—22no—
127C2×C4×D47—22no—
128C2×C4×D47—24no—
129C2×C4×D47—42no—
130C2×C4×D47—42no—
131C2×C4×D47C22×C414yes—
132C2×C4×D47C2×D414yes—
133C2×C8×C47—22no—
134C2×C8×C47—22no—
135C2×C8×C47—22no—
136C4⋊C4×C47—24no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47C22×C414yes—
142C4⋊C4×C47#4112yes—
143C42×C47—42no—
144C42×C47C22×C412yes—
145C42×C47C2×D412yes—
146(C2×D47)⋊C4—28no—
147(C2×D47)⋊C4—24no—
148(C2×D47)⋊C4—28no—
149(C2×D47)⋊C4—44no—
150(C2×D47)⋊C4—82no—
151(C2×D47)⋊C4C22⋊C418yes—
152C4⋊Dic47—21no—
153C4⋊Dic47—22no—
154C4⋊Dic47—22no—
155C4⋊Dic47—42no—
156C4⋊Dic47—81no—
157C4⋊Dic47C22⋊C414yes—
158C4⋊Dic47C22×C412yes—
159C4×Dic47—28no—
160C4×Dic47—44no—
161C2×C47⋊C8—21no—
162C2×C47⋊C8—21no—
163C2×C47⋊C8—21no—
164C2×C47⋊C8—21no—
165C2×C47⋊C8—21no—
166C2×C47⋊C8#4112yes—
167C8×D47—48no—
168C23×C47C22×D411yescharacteristic
169C22×D47—22no—
170C22×D47—22no—
171C22×D47—22no—
172C22×D47—22no—
173C22×D47—22no—
174C22×D47—22no—
175C22×D47—22no—
176C22×D47—41no—
177C22×D47—41no—
178C22×D47—41no—
179C22×D47—41no—
180C22×D47—41no—
181C22×D47—41no—
182C22×D47C2×C22⋊C414yes—
183Q8×C47—84no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—41no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47C2×C22⋊C412yes—
196C2×C4×C47C2×C22⋊C412yes—
197C2×C4×C47#13512yes—
198C2×C4×C47#13512yes—
199C2×C4×C47C23×C411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C2×#4111yescharacteristic
202C2×C4×C47C2×#4111yescharacteristic
203C2×C4×C47H(5,𝔽2)11yescharacteristic
204C2×C4×C472−1+411yescharacteristic
205C2×Dic47—22no—
206C2×Dic47—22no—
207C2×Dic47—22no—
208C2×Dic47—41no—
209C2×Dic47—41no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47C2×C22⋊C414yes—
214D188—22no—
215D188—22no—
216D188—28no—
217D188—41no—
218D188—41no—
219D188—88no—
220C4×D47—24no—
221C4×D47—24no—
222C4×D47—24no—
223C4×D47—24no—
224C4×D47—48no—
225C8×C47—42no—
226C47⋊C8—22no—
227C47⋊C8—41no—
228#22489—471nomaximal, Sylow
229C22×C47C22×C22⋊C411yescharacteristic
230C22×C47#54611yescharacteristic
231C22×C47#54711yescharacteristic
232C22×C47C2×#13511yescharacteristic
233C22×C47C2×#13511yescharacteristic
234C22×C47#57611yescharacteristic
235C22×C47#57711yescharacteristic
236C2×D47—24no—
237C2×D47—24no—
238C2×D47—24no—
239C2×D47—42no—
240C2×D47—42no—
241C2×D47—42no—
242C2×D47—42no—
243C2×D47—42no—
244C2×D47—42no—
245C2×D47—42no—
246C2×D47—81no—
247C2×D47—81no—
248C2×D47—81no—
249C2×D47—81no—
250C4×C47—22no—
251C4×C47—44no—
252C4×C47—42no—
253C4×C47—81no—
254C4×C47#41614yes—
255C4×C47C4×#4111yescharacteristic
256C4×C47#52011yescharacteristic
257C4×C47#54611yescharacteristic
258C4×C47#54711yescharacteristic
259Dic47—24no—
260Dic47—42no—
261Dic47—82no—
262C2×#555—471no—
263C2×C8×D4—471no—
264#2354—471no—
265#1990—471no—
266#1987D4711yescharacteristic, complemented
267#1893—472no—
268#1875—472no—
269#1825—472no—
270#1802—471no—
271C4×#102—471no—
272#1735—472no—
273C2×C47#233911yescharacteristic
274C2×C47C2×#41611yescharacteristic
275C2×C47#289711yescharacteristic
276C2×C47#330311yescharacteristic
277C2×C47#330511yescharacteristic
278C2×C47#334011yescharacteristic
279C2×C47#334111yescharacteristic
280D47—44no—
281D47—82no—
282C23×C8—472no—
283#555—478no—
284#555—944no—
285C2×C4⋊Q8C2×D4711yescharacteristic
286C2×#124—942no—
287C2×#121—472no—
288C2×#121—941no—
289C2×C4×D4—471no—
290C2×C4×D4—471no—
291C2×#117—472no—
292C2×#117—472no—
293C2×#117—941no—
294C8×D4—944no—
295C2×C4⋊C8C2×D4711yescharacteristic
296C2×#102—471no—
297C2×#102—471no—
298C2×#102—472no—
299C2×C22⋊C8—472no—
300C2×C4×C8—471no—
301#390—941no—
302#388—942no—
303#384—471no—
304#380—472no—
305C4×C4⋊C4—941no—
306C4×C4⋊C4C2×D4711yescharacteristic
307#340—472no—
308#340—471no—
309#340—472no—
310#340—472no—
311#340—471no—
312#340C2×D4712yes—
313#340C2×D4712yes—
314C47#2248911yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×D4—471no—
317C22×D4—471no—
318C22×D4—942no—
319C23×C4—472no—
320C23×C4—941no—
321C22×C8—471no—
322C22×C8—471no—
323C22×C8—944no—
324C22×C8—944no—
325C22×C8—1882no—
326C22×C8C22×D4712yes—
327C4⋊Q8—22no—
328C4⋊Q8C4×D4714yes—
329#124—9416no—
330#121—478no—
331#121—944no—
332#121—948no—
333C4×D4—474no—
334C4×D4—474no—
335C4×D4—474no—
336C4×D4—942no—
337#117—478no—
338#117—948no—
339#117—944no—
340#117—944no—
341#117—1882no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—471no—
344C2×C4⋊C4—471no—
345C2×C4⋊C4—472no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4C22×D4711yescharacteristic
349C2×C4⋊C4C22×D4712yes—
350C2×C4⋊C4C22×D4711yescharacteristic
351C2×C22⋊C4—472no—
352C2×C22⋊C4—942no—
353C2×C22⋊C4—942no—
354C2×C22⋊C4—941no—
355C2×C22⋊C4—942no—
356C2×C22⋊C4—942no—
357C2×C42—21no—
358C2×C42—942no—
359C2×C42—942no—
360C2×C42C22×D4711yescharacteristic
361C4⋊C8—22no—
362#102—942no—
363#102—942no—
364#102—1882no—
365C22⋊C8—944no—
366C4×C8—942no—
367#95—21no—
368#95—942no—
369C24—472no—
370C24—941no—
371C2×Q8—24no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×D4—474no—
375C2×D4—472no—
376C2×D4—948no—
377C2×D4—942no—
378C2×D4—942no—
379C2×D4—941no—
380C2×D4—941no—
381C2×D4—941no—
382C2×D4—941no—
383C2×D4—941no—
384C2×D4—3762no—
385C2×D4—3762no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—472no—
390C22×C4—474no—
391C22×C4—474no—
392C22×C4—942no—
393C22×C4—942no—
394C22×C4—942no—
395C22×C4—942no—
396C22×C4—941no—
397C22×C4—944no—
398C22×C4—942no—
399C22×C4—1882no—
400C22×C4—1882no—
401C22×C4D4×D4711yescharacteristic
402C22×C4D4×D4711yescharacteristic
403C22×C4C23×D4711yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—472no—
408C2×C8—941no—
409C2×C8—941no—
410C2×C8—941no—
411C2×C8—941no—
412C2×C8—941no—
413C2×C8—1888no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—474no—
420C4⋊C4—472no—
421C4⋊C4—941no—
422C4⋊C4—942no—
423C4⋊C4—942no—
424C4⋊C4—3761no—
425C4⋊C4—1882no—
426C4⋊C4C2×C4×D4714yes—
427C4⋊C4#1062535412yes—
428C22⋊C4—478no—
429C22⋊C4—948no—
430C22⋊C4—948no—
431C22⋊C4—944no—
432C22⋊C4—3762no—
433C22⋊C4—1884no—
434C42—42no—
435C42—948no—
436C42—1884no—
437C42C2×C4×D4712yes—
438C42D4×D4712yes—
439C23—474no—
440C23—942no—
441C23—942no—
442C23—942no—
443C23—942no—
444C23—942no—
445C23—942no—
446C23—942no—
447C23—1881no—
448C23—1881no—
449C23—1881no—
450C23—1881no—
451C23—1881no—
452C23—1881no—
453C23C2×D4×D4711yescharacteristic
454Q8—84no—
455D4—942no—
456D4—942no—
457D4—948no—
458D4—3768no—
459D4—1881no—
460D4—1881no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—474no—
473C2×C4—942no—
474C2×C4—942no—
475C2×C4—944no—
476C2×C4—944no—
477C2×C4—944no—
478C2×C4—944no—
479C2×C4—942no—
480C2×C4—1882no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4—1881no—
484C2×C4—1881no—
485C2×C4—1888no—
486C2×C4C22⋊C4×D4712yes—
487C2×C4C22⋊C4×D4712yes—
488C2×C4#1309404512yes—
489C2×C4#1309404712yes—
490C2×C4C22×C4×D4711yescharacteristic
491C2×C4C2×D4×D4711yescharacteristic
492C2×C4#41×D4711yescharacteristic
493C2×C4C2×#1062535411yescharacteristic
494C2×C4#1309407011yescharacteristic
495C2×C4#1309407211yescharacteristic
496C8—42no—
497C8—942no—
498C8—1881no—
499C22—944no—
500C22—944no—
501C22—944no—
502C22—3761no—
503C22—3761no—
504C22—3761no—
505C22—3761no—
506C22—1882no—
507C22—1882no—
508C22—1882no—
509C22—1882no—
510C22—1882no—
511C22—1882no—
512C22—1882no—
513C22C2×C22⋊C4×D4711yescharacteristic
514C22#1568976411yescharacteristic
515C22#1568977511yescharacteristic
516C22C2×#1309404511yescharacteristic
517C22C2×#1309404711yescharacteristic
518C22#1568992711yescharacteristic
519C22#1568993311yescharacteristic
520C4—22no—
521C4—44no—
522C4—42no—
523C4—81no—
524C4—944no—
525C4—3762no—
526C4—1882no—
527C4#1568911814yes—
528C4C4×#1062535411yescharacteristic
529C4#1568960511yescharacteristic
530C4#1568976611yescharacteristic
531C4#1568977611yescharacteristic
532C2—3762no—
533C2—1884no—
534C2#2024621711yescharacteristic
535C2C2×#1568911811yescharacteristic
536C2#2025191311yescharacteristic
537C2#2025531311yescharacteristic
538C2#2025533511yescharacteristic
539C2#2025580411yescharacteristic
540C2#2025580911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1987C2
socleC23×C47C22×D4
prime radicalC2×#15689839C2
cosocleC22×C4×C47C24
hypercenter#1987D47
hypocenterC47#22489
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
2. C2×#15689188C2semidirect
3. #20246335C2semidirect
4. #20244350C2semidirect
5. #20243815C2semidirect
6. #20243709C2semidirect
7. #20243482C2semidirect
8. #20243389C2semidirect
9. C4×#13093944C2semidirect
10. #20243103C2semidirect
11. C47×#1987C2semidirect
14. #15689839C4semidirect
266. #1987D47semidirect
314. C47#22489semidirect

Maximal subgroups

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

subgroupindex
C2×#156898392
C2×#156891882
#202463352
#202443502
#202438152
#202437092
#202434822
#202433892
C4×#130939442
#202431032
C47×#19872
#2248947