Finite Group Databasebeta

← #26232489

Subgroups — #26232489

Order 12032.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689778C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic, complemented
3#20246288C211yesmaximal, characteristic, complemented
4#20244348C211yesmaximal, characteristic, complemented
5#20244339C211yesmaximal, characteristic, complemented
6#20243813C212yesmaximal, complemented
7#20243702C212yesmaximal, complemented
8#20243473C212yesmaximal, complemented
9C4×#13093945C211yesmaximal, characteristic, complemented
10#340×D47C212yesmaximal, complemented
11C47×#1802C211yesmaximal, characteristic, complemented
12C22×#10625350—21no—
13C22×C8×D47C2212yes—
14#15689778—24no—
15#15689778C418yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094006—22no—
18C2×#13094003—21no—
19C2×#13093995C2212yes—
20C2×C4×D188C2211yescharacteristic
21C2×#13093986—21no—
22C2×C4⋊C4×D47C2212yes—
23C2×C4⋊C4×D47C2212yes—
24#15689188—24no—
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×(C2×D47)⋊C8C2212yes—
28C2×C4×C47⋊C8C2211yescharacteristic
29#15689019—21no—
30#15688994—22no—
31#15688980C2212yes—
32C4×C4⋊Dic47C2211yescharacteristic
33#15688886C2212yes—
34#15688886C2212yes—
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2212yes—
38#15688885C2212yes—
39C2×C4×D4×C47C2211yescharacteristic
40C2×C4⋊C8×C47C2211yescharacteristic
41C2×#102×C47C2212yes—
42C47×#384C2211yescharacteristic
43C47×#340C2212yes—
44C2×#10625350—28no—
45C2×#10625350—44no—
46C2×#10625350—44no—
47C22×Dic94—21no—
48C22×D188D411yescharacteristic
49C22×C22⋊D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47C2312yes—
53C22×C47⋊C8C2311yescharacteristic
54C22×C47⋊C8C2311yescharacteristic
55C2×C8×D47—24no—
56C2×C8×D47—24no—
57C2×C8×D47—42no—
58#13094024—22no—
59#13094024C2×C414yescomplemented
60#13094006—216no—
61#13094003—28no—
62#13093995—24no—
63#13093995C2×C418yescomplemented
64C4×D188C2×C414yes—
65C4×D188D414yes—
66#13093986—44no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—24no—
75C4⋊C4×D47—24no—
76C4⋊C4×D47—42no—
77C4⋊C4×D47C2×C418yescomplemented
78C2×C22⋊Dic47—22no—
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4D412yes—
81C2×C4×Dic47—21no—
82#13093945—22no—
83#13093945—22no—
84(C2×D47)⋊C8—24no—
85C4×C47⋊C8—22no—
86C47⋊#95—21no—
87C47⋊#95—22no—
88C22×D4×C47C2311yescharacteristic
89C23×C4×C47—21no—
90C22×C8×C47C2312yes—
91C4×D4×C47—22no—
92C4×D4×C47C2×C414yescomplemented
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47C2312yes—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C22⋊C4×C47—21no—
98C2×C42×C47C2311yescharacteristic
99C4⋊C8×C47—22no—
100#102×C47—42no—
101C24×C47—21no—
102C23×D47C2×D412yes—
103C2×D4×C47—21no—
104C2×D4×C47—21no—
105C2×D4×C47—21no—
106C2×D4×C47—21no—
107C2×D4×C47—21no—
108C2×D4×C47C22×C412yes—
109C22×C4×C47—21no—
110C22×C4×C47—24no—
111C22×C4×C47—22no—
112C22×C4×C47—42no—
113C22×C4×C47—42no—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2411yescharacteristic
117C2×C22⋊D47—28no—
118C2×C22⋊D47—44no—
119C2×C22⋊D47—44no—
120C22×Dic47—21no—
121C22×Dic47—22no—
122C22×Dic47—21no—
123C22×Dic47C2×D412yes—
124#10625350—432no—
125C2×D188C22⋊C414yes—
126C2×D188D814yes—
127C2×D188SD1614yes—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—28no—
133C2×C4×D47—44no—
134C2×C4×D47—44no—
135C2×C4×D47—44no—
136C2×C4×D47—44no—
137C2×C4×D47—82no—
138C2×C4×D47C22×C414yes—
139C2×C4×D47C2×D414yes—
140C2×Dic94—24no—
141C2×Dic94—42no—
142C2×Dic94—42no—
143C2×C8×C47—22no—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—42no—
150C4⋊C4×C47—81no—
151C4⋊C4×C47C22×C412yes—
152C22⋊C4×C47—24no—
153C42×C47C22×C412yes—
154C42×C47C2×D412yes—
155C22⋊Dic47—28no—
156(C2×D47)⋊C4—44no—
157(C2×D47)⋊C4C22⋊C418yes—
158C4⋊Dic47—24no—
159C4⋊Dic47—22no—
160C4⋊Dic47—21no—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47C22⋊C414yes—
164C4⋊Dic47C22×C414yes—
165C4⋊Dic47#4112yescomplemented
166Dic47⋊C4—28no—
167Dic47⋊C4—44no—
168C4×Dic47—42no—
169C2×C47⋊C8—21no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8#4112yescomplemented
175C8×D47—48no—
176C23×C47—22no—
177C23×C47—41no—
178C23×C47—41no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—41no—
183C23×C47C22×D411yescharacteristic
184C22×D47—22no—
185C22×D47—22no—
186C22×D47—22no—
187C22×D47—22no—
188C22×D47C2×C22⋊C414yes—
189C22×D47C2×D814yes—
190C22×D47C2×SD1614yes—
191D4×C47—22no—
192D4×C47—22no—
193D4×C47—41no—
194D4×C47—41no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—22no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—48no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47C2×C22⋊C412yes—
205C2×C4×C47C2×C22⋊C412yes—
206C2×C4×C47C2×D812yes—
207C2×C4×C47C2×SD1612yes—
208C2×C4×C47C23×C411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C2×C4×C47H(5,𝔽2)11yescharacteristic
213C2×C4×C472−1+411yescharacteristic
214C22⋊D47—416no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—24no—
218C2×Dic47—22no—
219C2×Dic47—41no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47C2×C22⋊C414yes—
227D188#102116yescomplemented
228C4×D47—24no—
229C4×D47—24no—
230C4×D47—24no—
231C4×D47—24no—
232C4×D47—416no—
233C4×D47—88no—
234Dic94—48no—
235C8×C47—42no—
236C47⋊C8—22no—
237C47⋊C8—41no—
238#22485—471nomaximal, Sylow
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—81no—
243C22×C47—81no—
244C22×C47—81no—
245C22×C47—81no—
246C22×C47C22×C22⋊C411yescharacteristic
247C22×C47#54611yescharacteristic
248C22×C47#54711yescharacteristic
249C22×C47C22×D811yescharacteristic
250C22×C47C22×SD1611yescharacteristic
251C22×C47#57711yescharacteristic
252C22×C47#57811yescharacteristic
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—24no—
258C2×D47—24no—
259C2×D47C2×#10218yes—
260C4×C47—22no—
261C4×C47—42no—
262C4×C47—82no—
263C4×C47C2×#10214yes—
264C4×C47C4×#4111yescharacteristic
265C4×C47#52011yescharacteristic
266C4×C47#54611yescharacteristic
267C4×C47#54711yescharacteristic
268Dic47—24no—
269Dic47—44no—
270Dic47—42no—
271Dic47—81no—
272C2×#547—471no—
273C2×C8×D4—471no—
274D4×C4⋊C4—471no—
275#1990—471no—
276#1988—471no—
277#1893—472no—
278#1874—472no—
279#1823—472no—
280#1802D4711yescharacteristic, complemented
281C4×#102—471no—
282C2×#340—472no—
283C2×C47—82no—
284C2×C47#233911yescharacteristic
285C2×C47C22×#10211yescharacteristic
286C2×C47#289811yescharacteristic
287C2×C47#329411yescharacteristic
288C2×C47#329811yescharacteristic
289C2×C47#333411yescharacteristic
290C2×C47#333511yescharacteristic
291D47—28no—
292C22×#41—941no—
293C23×C8—472no—
294#547—478no—
295#547—944no—
296C2×C4⋊Q8—471no—
297C2×#123—942no—
298C2×C22⋊Q8—941no—
299C2×#121—472no—
300C2×C4×D4—471no—
301C2×C4×D4—941no—
302C2×C4×D4C2×D4711yescharacteristic
303C22×C4⋊C4—472no—
304C22×C4⋊C4—472no—
305C8×D4—944no—
306C2×C4⋊C8C2×D4711yescharacteristic
307C2×#102—471no—
308C2×#102—471no—
309C2×#102C2×D4712yes—
310C2×C22⋊C8—472no—
311C2×C4×C8—471no—
312#389—941no—
313#385—942no—
314#384C2×D4711yescharacteristic
315#380—472no—
316C4×C4⋊C4—471no—
317#340—472no—
318#340—472no—
319#340—471no—
320#340—472no—
321#340—472no—
322#340—471no—
323#340C2×D4712yes—
324C47#2248511yescharacteristic, complemented, Sylow
325C2×#41—948no—
326C2×#41—1884no—
327C2×#41—1884no—
328C22×Q8—941no—
329C22×D4—471no—
330C22×D4—942no—
331C22×D4C22×D4711yescharacteristic
332C23×C4—21no—
333C23×C4—472no—
334C23×C4—942no—
335C23×C4—942no—
336C22×C8—471no—
337C22×C8—471no—
338C22×C8—944no—
339C22×C8—944no—
340C22×C8—1882no—
341C22×C8C22×D4712yes—
342C4⋊Q8—474no—
343C4⋊Q8—942no—
344#123—9416no—
345C22⋊Q8—948no—
346#121—478no—
347#121—944no—
348C4×D4—22no—
349C4×D4—474no—
350C4×D4—474no—
351C4×D4—1884no—
352C4×D4C4×D4714yescomplemented
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—478no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—1882no—
366C2×C4⋊C4C22×D4712yes—
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—472no—
370C2×C22⋊C4—942no—
371C2×C22⋊C4—942no—
372C2×C42—941no—
373C2×C42C22×D4711yescharacteristic
374C4⋊C8—22no—
375#102—42no—
376#102—942no—
377#102—942no—
378C22⋊C8—944no—
379C4×C8—942no—
380#95—942no—
381#95—941no—
382C24—21no—
383C24—472no—
384#41—18832no—
385C2×Q8—944no—
386C2×Q8—1882no—
387C2×Q8—1882no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—474no—
394C2×D4—474no—
395C2×D4—474no—
396C2×D4—948no—
397C2×D4—1884no—
398C2×D4—1884no—
399C2×D4C2×C4×D4712yes—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—472no—
406C22×C4—474no—
407C22×C4—474no—
408C22×C4—948no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—941no—
414C22×C4—942no—
415C22×C4—941no—
416C22×C4—3762no—
417C22×C4—1884no—
418C22×C4—1884no—
419C22×C4—1884no—
420C22×C4—1884no—
421C22×C4D4×D4711yescharacteristic
422C22×C4D4×D4711yescharacteristic
423C22×C4C23×D4711yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—472no—
428C2×C8—941no—
429C2×C8—941no—
430C2×C8—941no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—1888no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—81no—
439C4⋊C4—474no—
440C4⋊C4—474no—
441C4⋊C4—472no—
442C4⋊C4—948no—
443C4⋊C4—944no—
444C4⋊C4—942no—
445C4⋊C4—941no—
446C4⋊C4—941no—
447C4⋊C4—941no—
448C4⋊C4—1884no—
449C4⋊C4C2×C4×D4712yes—
450C22⋊C4—24no—
451C22⋊C4—478no—
452C22⋊C4—948no—
453C22⋊C4—1884no—
454C42—1882no—
455C42C2×C4×D4712yes—
456C42D4×D4712yes—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—474no—
465C23—474no—
466C23—474no—
467C23—942no—
468C23—942no—
469C23—942no—
470C23—942no—
471C23C2×D4×D4711yescharacteristic
472Q8—1888no—
473D4—22no—
474D4—22no—
475D4—41no—
476D4—41no—
477D4—4716no—
478D4—18816no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—474no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—944no—
492C2×C4—944no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—942no—
496C2×C4—944no—
497C2×C4—3768no—
498C2×C4—18816no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1881no—
506C2×C4C22⋊C4×D4712yes—
507C2×C4C22⋊C4×D4712yes—
508C2×C4D8×D4712yes—
509C2×C4SD16×D4712yes—
510C2×C4C22×C4×D4711yescharacteristic
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4#41×D4711yescharacteristic
513C2×C4#41×D4711yescharacteristic
514C2×C4#1309406911yescharacteristic
515C2×C4#1309407211yescharacteristic
516C8—42no—
517C8—942no—
518C8—1881no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—478no—
527C22—944no—
528C22—944no—
529C22—944no—
530C22—944no—
531C22—944no—
532C22—944no—
533C22C2×C22⋊C4×D4711yescharacteristic
534C22#1568977011yescharacteristic
535C22#1568977511yescharacteristic
536C22C2×D8×D4711yescharacteristic
537C22C2×SD16×D4711yescharacteristic
538C22#1568993311yescharacteristic
539C22#1568993911yescharacteristic
540C4—22no—
541C4—42no—
542C4—82no—
543C4—944no—
544C4—3761no—
545C4—1884no—
546C4—1882no—
547C4#102×D4714yes—
548C4#1568959111yescharacteristic
549C4#1568960411yescharacteristic
550C4#1568977011yescharacteristic
551C4#1568977511yescharacteristic
552C2—82no—
553C2—948no—
554C2#2024621611yescharacteristic
555C2C2×#102×D4711yescharacteristic
556C2#2025191911yescharacteristic
557C2#2025520411yescharacteristic
558C2#2025525611yescharacteristic
559C2#2025573411yescharacteristic
560C2#2025574211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC2×#15689778C2
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#15689188C2semidirect
3. #20246288C2semidirect
4. #20244348C2semidirect
5. #20244339C2semidirect
6. #20243813C2semidirect
7. #20243702C2semidirect
8. #20243473C2semidirect
9. C4×#13093945C2semidirect
10. #340×D47C2semidirect
11. C47×#1802C2semidirect
15. #15689778C4semidirect
59. #13094024C2×C4semidirect
63. #13093995C2×C4semidirect
77. C4⋊C4×D47C2×C4semidirect
92. C4×D4×C47C2×C4semidirect
165. C4⋊Dic47#41semidirect
174. C2×C47⋊C8#41semidirect
227. D188#102semidirect
280. #1802D47semidirect
324. C47#22485semidirect
352. C4×D4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897782
C2×#156891882
#202462882
#202443482
#202443392
#202438132
#202437022
#202434732
C4×#130939452
#340×D472
C47×#18022
#2248547