Finite Group Databasebeta

← #26223404

Subgroups — #26223404

Order 12032.

1060 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689442C211yesmaximal, characteristic
2#20245894C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
4#20244014C212yesmaximal, complemented
5#20243965C212yesmaximal, complemented
6#20243705C211yesmaximal, characteristic
7#20243476C211yesmaximal, characteristic, complemented
8#106×Dic47C212yesmaximal, complemented
9#20243202C212yesmaximal, complemented
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C47×#1853C211yesmaximal, characteristic, complemented
12C2×#13093998C2211yescharacteristic
13C2×#13093986C2211yescharacteristic
14C2×(C22⋊D47)⋊C4—21no—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4×Dic47—21no—
17C22×C4×Dic47C2211yescharacteristic
18#15689442C418yescomplemented
19C2×#13093956C2212yes—
20C2×#13093946C2212yes—
21C2×#13093944C2212yes—
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C8×Dic47C2212yes—
24#15689024C2211yescharacteristic
25#15689019C2211yescharacteristic
26#15689012—21no—
27#15689000—21no—
28C4⋊C4×Dic47—21no—
29C4⋊C4×Dic47—21no—
30C4⋊C4×Dic47C2212yes—
31C4⋊C4×Dic47C2212yes—
32C4⋊C4×Dic47C2211yescharacteristic
33#15688970—21no—
34C2×C47⋊#95—21no—
35#15688885C2212yes—
36#15688885C2212yes—
37#15688885C2212yes—
38#15688877C2212yes—
39C22×C4⋊C4×C47C2211yescharacteristic
40C2×#106×C47C2212yes—
41C2×C22⋊C8×C47C2211yescharacteristic
42C47×#380C2211yescharacteristic
43C47×#340C2212yes—
44C22×D188C2311yescharacteristic
45C22×C22⋊D47—21no—
46C23×Dic47—21no—
47C23×Dic47D411yescharacteristic
48C23×Dic47D411yescharacteristic
49C22×C4×D47—21no—
50C22×C47⋊C8C2312yes—
51#13093998—22no—
52#13093998C2×C414yes—
53#13093986—24no—
54(C22⋊D47)⋊C4—44no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47C2×C414yescomplemented
58C2×C4⋊Dic47C2312yes—
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—21no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—21no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—44no—
70C2×C4×Dic47D411yescharacteristic
71C2×C4×Dic47D411yescharacteristic
72C2×C4×Dic47C2311yescharacteristic
73C2×C4×Dic47C2311yescharacteristic
74#13093956C2×C418yescomplemented
75#13093946C2×C418yescomplemented
76#13093944—24no—
77C22⋊(C47⋊C8)—22no—
78C8×Dic47—24no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C47⋊#95—21no—
82C47⋊#95—21no—
83C23×C4×C47—21no—
84C23×C4×C47C2311yescharacteristic
85C22×C8×C47C2312yes—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—24no—
88C2×C4⋊C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47C2311yescharacteristic
90C2×C4⋊C4×C47C2312yes—
91C2×C22⋊C4×C47—21no—
92#106×C47—24no—
93C22⋊C8×C47C2×C414yescomplemented
94#95×C47—21no—
95C24×C47C2×D411yescharacteristic
96C23×D47—21no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—24no—
100C22×C4×C47—22no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—42no—
104C22×C4×C47—42no—
105C22×C4×C47C22×C412yes—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47#4112yes—
108C22×C4×C47C2411yescharacteristic
109C2×C22⋊D47—24no—
110C2×C22⋊D47—42no—
111C2×C22⋊D47—42no—
112C22×Dic47—22no—
113C22×Dic47—22no—
114C22×Dic47—24no—
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—21no—
118C22×Dic47—22no—
119C22×Dic47—41no—
120C22×Dic47—42no—
121C22×Dic47—42no—
122C22×Dic47SD1614yes—
123C22×Dic47C2×D411yescharacteristic
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C2×D188—22no—
128C2×D188—22no—
129C2×D188—21no—
130C2×D188C22×C412yes—
131C2×C4×D47—42no—
132C2×C4×D47—42no—
133C2×C4×D47—42no—
134C2×C8×C47—22no—
135C2×C8×C47—22no—
136C2×C8×C47C22×C414yes—
137C4⋊C4×C47—24no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47#4114yes—
142C22⋊C4×C47—42no—
143(C2×D47)⋊C4—24no—
144(C2×D47)⋊C4—42no—
145(C2×D47)⋊C4—42no—
146(C2×D47)⋊C4—42no—
147C4⋊Dic47—24no—
148C4⋊Dic47—22no—
149C4⋊Dic47—21no—
150C4⋊Dic47C22×C414yes—
151C4⋊Dic47C22×C412yes—
152Dic47⋊C4—42no—
153C4×Dic47—22no—
154C4×Dic47—22no—
155C4×Dic47—22no—
156C4×Dic47—44no—
157C4×Dic47—42no—
158C4×Dic47—44no—
159C4×Dic47#4114yes—
160C2×C47⋊C8—22no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8—22no—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47C4×D412yes—
168C23×C47C2×SD1612yes—
169C23×C47#13512yes—
170C23×C47C22×D411yescharacteristic
171C22×D47—22no—
172C22×D47—41no—
173C22×D47—41no—
174C22×D47—41no—
175C22×D47—41no—
176C22×D47—41no—
177C22×D47—41no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—24no—
181C2×C4×C47—24no—
182C2×C4×C47—24no—
183C2×C4×C47—48no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47C4×D412yes—
193C2×C4×C47C23×C411yescharacteristic
194C2×C4×C47C2×#4112yes—
195C2×C4×C47C2×#4112yes—
196C2×C4×C47C2×#4111yescharacteristic
197C22⋊D47—48no—
198C2×Dic47—21no—
199C2×Dic47—21no—
200C2×Dic47—21no—
201C2×Dic47—21no—
202C2×Dic47—28no—
203C2×Dic47—24no—
204C2×Dic47—22no—
205C2×Dic47—21no—
206C2×Dic47—21no—
207C2×Dic47—21no—
208C2×Dic47—44no—
209C2×Dic47—48no—
210C2×Dic47—41no—
211C2×Dic47—41no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47#12312yes—
215C2×Dic47#12312yes—
216C2×Dic47C2×SD1612yes—
217C2×Dic47C2×SD1612yes—
218C2×Dic47AGL(1,ℤ/8)12yes—
219D188—24no—
220D188—42no—
221C4×D47—84no—
222C8×C47—24no—
223C47⋊C8—42no—
224C4×#481—471nomaximal, Sylow
225C22×C47—22no—
226C22×C47—22no—
227C22×C47—22no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—22no—
231C22×C47C4×SD1614yes—
232C22×C47C2×C4×D411yescharacteristic
233C22×C47C2×#12311yescharacteristic
234C22×C47#52511yescharacteristic
235C22×C47C22×SD1611yescharacteristic
236C22×C47C2×#13511yescharacteristic
237C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C47#57511yescharacteristic
239C2×D47—42no—
240C2×D47—42no—
241C2×D47—42no—
242C2×D47—81no—
243C2×D47—81no—
244C2×D47—81no—
245C2×D47—81no—
246C4×C47—22no—
247C4×C47—44no—
248C4×C47—81no—
249C4×C47—81no—
250C4×C47C4×#4112yes—
251C4×C47#52912yes—
252Dic47—22no—
253Dic47—22no—
254Dic47—41no—
255Dic47—42no—
256Dic47—44no—
257Dic47#48114yescomplemented
258C2×#481—471no—
259C4×#121—471no—
260C2×C4×C4⋊C4—471no—
261#1926—472no—
262#1919—472no—
263#1874—471no—
264#1853D4711yescharacteristic, complemented
265#1824—471no—
266C4×#106—472no—
267C4×#102—472no—
268C4×C22⋊C8—471no—
269C2×C47—24no—
270C2×C47C4×#12311yescharacteristic
271C2×C47C2×C4×SD1611yescharacteristic
272C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
273C2×C47C2×#48111yescharacteristic
274C2×C47#309111yescharacteristic
275C2×C47#310311yescharacteristic
276C2×C47#310511yescharacteristic
277D47—82no—
278C2×#121—471no—
279C2×C4×D4—471no—
280C2×C4×D4—941no—
281C22×C4⋊C4—471no—
282C22×C4⋊C4C2×D4711yescharacteristic
283C22×C42—471no—
284C22×C42—941no—
285#481—478no—
286C2×#106—472no—
287C2×#106C2×D4712yes—
288C2×#102—472no—
289C2×#102—472no—
290C2×C22⋊C8—471no—
291C2×C22⋊C8C2×D4711yescharacteristic
292C2×C4×C8—472no—
293#390—471no—
294#389—471no—
295#388—941no—
296#386—941no—
297#380C2×D4711yescharacteristic
298C4×C4⋊C4—472no—
299C4×C4⋊C4—472no—
300C4×C4⋊C4—471no—
301C4×C4⋊C4—941no—
302C4×C4⋊C4—941no—
303C4×C22⋊C4—941no—
304C2×#95—941no—
305#340—472no—
306#340—472no—
307#340—472no—
308#340C2×D4712yes—
309#336—472no—
310C47C4×#48111yescharacteristic, complemented, Sylow
311C22×D4—471no—
312C22×D4—941no—
313C23×C4—21no—
314C23×C4—471no—
315C23×C4—471no—
316C23×C4—941no—
317C23×C4—941no—
318C23×C4C22×D4711yescharacteristic
319C22×C8—472no—
320C22×C8C22×D4712yes—
321#121—474no—
322#121—942no—
323C4×D4—944no—
324C4×D4—1884no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—474no—
328C2×C4⋊C4—472no—
329C2×C4⋊C4—471no—
330C2×C4⋊C4—941no—
331C2×C4⋊C4—942no—
332C2×C4⋊C4—941no—
333C2×C4⋊C4C22×D4711yescharacteristic
334C2×C4⋊C4C22×D4711yescharacteristic
335C2×C4⋊C4C22×D4712yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—941no—
338C2×C22⋊C4—941no—
339C2×C22⋊C4—941no—
340C2×C22⋊C4—941no—
341C2×C42—471no—
342C2×C42—471no—
343C2×C42—471no—
344C2×C42—471no—
345C2×C42—944no—
346C2×C42—941no—
347C2×C42—942no—
348C2×C42—942no—
349C2×C42—1884no—
350#106—24no—
351#106—478no—
352#102—478no—
353#102—944no—
354C22⋊C8—942no—
355C22⋊C8C4×D4714yescomplemented
356C4×C8—944no—
357#95—21no—
358#95—942no—
359#95—942no—
360#95—941no—
361#95—941no—
362C24—941no—
363C24D4×D4711yescharacteristic
364C2×D4—472no—
365C2×D4—944no—
366C2×D4—942no—
367C2×D4—942no—
368C2×D4—941no—
369C2×D4—1882no—
370C2×D4—1882no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—471no—
380C22×C4—474no—
381C22×C4—471no—
382C22×C4—471no—
383C22×C4—471no—
384C22×C4—942no—
385C22×C4—942no—
386C22×C4—942no—
387C22×C4—944no—
388C22×C4—942no—
389C22×C4—942no—
390C22×C4—941no—
391C22×C4—1882no—
392C22×C4—1882no—
393C22×C4—1882no—
394C22×C4—1881no—
395C22×C4—1882no—
396C22×C4—1882no—
397C22×C4C2×C4×D4712yes—
398C22×C4D4×D4711yescharacteristic
399C22×C4#1062535412yes—
400C22×C4C23×D4711yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—942no—
404C2×C8—942no—
405C2×C8—942no—
406C2×C8C2×C4×D4714yes—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—474no—
412C4⋊C4—472no—
413C4⋊C4—944no—
414C4⋊C4—942no—
415C4⋊C4—941no—
416C4⋊C4—1882no—
417C4⋊C4#1062535214yes—
418C22⋊C4—42no—
419C22⋊C4—944no—
420C22⋊C4—1882no—
421C22⋊C4—1882no—
422C22⋊C4—1882no—
423C42—474no—
424C42—942no—
425C42—942no—
426C42—942no—
427C42—1884no—
428C42—1882no—
429C42—1884no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—942no—
435C23—1881no—
436C23—1881no—
437C23—1881no—
438C23—1881no—
439C23—1881no—
440C23—1881no—
441C23#1309398612yes—
442C23SD16×D4712yes—
443C23#1309404512yes—
444C23C2×D4×D4711yescharacteristic
445D4—944no—
446D4—1888no—
447D4—1882no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—472no—
463C2×C4—472no—
464C2×C4—472no—
465C2×C4—472no—
466C2×C4—472no—
467C2×C4—948no—
468C2×C4—941no—
469C2×C4—941no—
470C2×C4—941no—
471C2×C4—941no—
472C2×C4—942no—
473C2×C4—944no—
474C2×C4—941no—
475C2×C4—941no—
476C2×C4—941no—
477C2×C4—3764no—
478C2×C4—1884no—
479C2×C4—1888no—
480C2×C4—1881no—
481C2×C4—1881no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4#1309398612yes—
485C2×C4C22×C4×D4711yescharacteristic
486C2×C4#41×D4712yes—
487C2×C4C2×#1062535212yes—
488C2×C4C2×#1062535411yescharacteristic
489C8—24no—
490C8—1882no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—3761no—
498C22—3761no—
499C22—3761no—
500C22—3761no—
501C22—1882no—
502C22—1882no—
503C22—1882no—
504C22#1568921014yes—
505C22C2×#1309398611yescharacteristic
506C22#123×D4711yescharacteristic
507C22#1568962911yescharacteristic
508C22C2×SD16×D4711yescharacteristic
509C22C2×#1309404511yescharacteristic
510C22AGL(1,ℤ/8)×D4711yescharacteristic
511C22#1568992211yescharacteristic
512C4—22no—
513C4—44no—
514C4—81no—
515C4—81no—
516C4—474no—
517C4—942no—
518C4—942no—
519C4—1884no—
520C4—1881no—
521C4—1882no—
522C4C4×#1062535212yes—
523C4#1568965912yes—
524C2—24no—
525C2—3762no—
526C2#2024590711yescharacteristic
527C2C2×#1568921011yescharacteristic
528C2#2025219511yescharacteristic
529C2#481×D4711yescharacteristic
530C2#2025330211yescharacteristic
531C2#2025341411yescharacteristic
532C2#2025343611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1853C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1853D47
hypocenterC47C4×#481
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×Dic47C2semidirect
4. #20244014C2semidirect
5. #20243965C2semidirect
7. #20243476C2semidirect
8. #106×Dic47C2semidirect
9. #20243202C2semidirect
10. C22⋊C8×Dic47C2semidirect
11. C47×#1853C2semidirect
18. #15689442C4semidirect
57. C2×C4⋊Dic47C2×C4semidirect
74. #13093956C2×C4semidirect
75. #13093946C2×C4semidirect
93. C22⋊C8×C47C2×C4semidirect
257. Dic47#481semidirect
264. #1853D47semidirect
310. C47C4×#481semidirect
355. C22⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894422
#202458942
C2×C4⋊C4×Dic472
#202440142
#202439652
#202437052
#202434762
#106×Dic472
#202432022
C22⋊C8×Dic472
C47×#18532
C4×#48147