Finite Group Databasebeta

← #26250239

Subgroups — #26250239

Order 12032.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689834C211yesmaximal, characteristic
2C2×C4×#10625337C211yesmaximal, characteristic, complemented
3#20248306C211yesmaximal, characteristic
4#20244593C212yesmaximal, complemented
5#20244397C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244181C211yesmaximal, characteristic
8#20243692C212yesmaximal, complemented
9#20243574C212yesmaximal, complemented
10#20243394C211yesmaximal, characteristic, complemented
11C2×C47×#477C211yesmaximal, characteristic, complemented
12C22×#10625337C2211yescharacteristic
13#15689834—24no—
14#15689834—24no—
15C2×#13094015—21no—
16C2×#13094004C2212yes—
17C2×#13094000—22no—
18C2×#13094000C2212yes—
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21C2×#13093986—21no—
22C2×C4⋊C4×D47C2212yes—
23C4×#10625337—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093949C2212yes—
26C2×#13093948C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×(C2×D47)⋊C8C2212yes—
29C2×(C2×D47)⋊C8C2212yes—
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689045—22no—
32#15689025—21no—
33#15689021—21no—
34#15688993C2211yescharacteristic
35#15688980C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C4×Q8×C47C2211yescharacteristic
41C47×#477—24no—
42C2×#107×C47C2212yes—
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#103×C47C2212yes—
45C22×Dic94C2311yescharacteristic
46C22×D188D411yescharacteristic
47C22×C4×D47—22no—
48C22×C4×D47C2312yes—
49C2×#10625337—22no—
50C2×#10625337—22no—
51C2×#10625337—22no—
52C22×C47⋊C8C2311yescharacteristic
53C22×C47⋊C8C2311yescharacteristic
54#13094015—28no—
55#13094004—24no—
56#13094004—24no—
57#13094000—216no—
58#13094000—24no—
59#13094000—42no—
60C4×Dic94—22no—
61C4×Dic94D414yes—
62C4×D188—22no—
63C4×D188—22no—
64#13093986—28no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—24no—
75C4⋊C4×D47—24no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4—22no—
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4D412yes—
80C2×C4×Dic47—21no—
81#13093957—22no—
82#13093949—42no—
83#13093948—22no—
84#13093947—22no—
85(C2×D47)⋊C8—24no—
86(C2×D47)⋊C8—42no—
87C4×C47⋊C8—22no—
88C47⋊#95—22no—
89C22×Q8×C47—21no—
90C22×Q8×C47C2311yescharacteristic
91C22×C8×C47C2312yes—
92C4⋊Q8×C47—22no—
93C4⋊Q8×C47D414yes—
94C4×Q8×C47—22no—
95C4×Q8×C47—22no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47C2312yes—
100C2×C42×C47—21no—
101C2×C42×C47C2311yescharacteristic
102#107×C47—24no—
103C4⋊C8×C47—22no—
104#103×C47—24no—
105C23×D47C2×D412yes—
106C2×Q8×C47—24no—
107C2×Q8×C47—21no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—81no—
113C2×Q8×C47—81no—
114C2×Q8×C47#4112yes—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47—22no—
124C22×Dic47C2×D412yes—
125C2×D188—22no—
126C2×D188—22no—
127C2×D188Q1614yescomplemented
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—28no—
135C2×C4×D47—82no—
136C2×C4×D47—82no—
137C2×Dic94—21no—
138C2×Dic94—21no—
139C2×Dic94—21no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94C2×D412yes—
143C2×C8×C47—22no—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—42no—
151C4⋊C4×C47C2×D414yes—
152C42×C47—42no—
153C42×C47C2×D412yes—
154C42×C47C2×D412yes—
155#10625337—28no—
156(C2×D47)⋊C4—28no—
157(C2×D47)⋊C4—24no—
158(C2×D47)⋊C4—44no—
159(C2×D47)⋊C4—82no—
160C4⋊Dic47—28no—
161C4⋊Dic47—22no—
162C4⋊Dic47—21no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—42no—
166C4⋊Dic47—81no—
167C4⋊Dic47C2×D412yes—
168Dic47⋊C4—28no—
169Dic47⋊C4—24no—
170Dic47⋊C4—44no—
171C4×Dic47—24no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8#4112yes—
178C23×C47C22×D411yescharacteristic
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47C2×Q1614yes—
186Q8×C47—22no—
187Q8×C47—22no—
188Q8×C47—41no—
189Q8×C47—41no—
190Q8×C47—84no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—24no—
194C2×C4×C47—22no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47#12012yes—
203C2×C4×C47#12012yes—
204C2×C4×C47C2×Q1612yes—
205C2×C4×C47#13712yes—
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C2×#4111yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C2×C4×C47H(5,𝔽2)11yescharacteristic
211C2×C4×C472−1+411yescharacteristic
212C2×Dic47—22no—
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224D188—28no—
225C4×D47—42no—
226C4×D47—42no—
227C4×D47—42no—
228C4×D47—42no—
229C4×D47—88no—
230Dic94—22no—
231Dic94—22no—
232Dic94—41no—
233Dic94—41no—
234C8×C47—42no—
235C47⋊C8—22no—
236C47⋊C8—41no—
237#24122—471nomaximal, Sylow
238C22×C47C2×#12011yescharacteristic
239C22×C47#54611yescharacteristic
240C22×C47#54711yescharacteristic
241C22×C47C22×Q1611yescharacteristic
242C22×C47C2×#13711yescharacteristic
243C22×C47#57611yescharacteristic
244C22×C47#57711yescharacteristic
245C2×D47—24no—
246C2×D47—24no—
247C2×D47—24no—
248C2×D47—42no—
249C2×D47—42no—
250C2×D47—42no—
251C2×D47—42no—
252C4×C47—22no—
253C4×C47—42no—
254C4×C47—44no—
255C4×C47—81no—
256C4×C47#45114yes—
257C4×C47#52511yescharacteristic
258C4×C47#53711yescharacteristic
259C4×C47#54811yescharacteristic
260C4×C47#55011yescharacteristic
261Dic47—42no—
262Dic47—42no—
263Dic47—42no—
264Dic47—82no—
265C2×#554—471no—
266C2×#477D4711yescharacteristic, complemented
267C2×C4×Q16—471no—
268#2538—471no—
269#2036—472no—
270#2001—472no—
271#1990—471no—
272#1954—471no—
273#1872—472no—
274#1852—472no—
275#1803—471no—
276C2×C47#240711yescharacteristic
277C2×C47C2×#45111yescharacteristic
278C2×C47#301211yescharacteristic
279C2×C47#329911yescharacteristic
280C2×C47#333311yescharacteristic
281C2×C47#333811yescharacteristic
282C2×C47#335611yescharacteristic
283D47—44no—
284C22×Q16—471no—
285#554—944no—
286#554—944no—
287C2×C4⋊Q8C2×D4711yescharacteristic
288C2×#125—941no—
289C2×C22⋊Q8—472no—
290C2×C22⋊Q8—472no—
291C2×C22⋊Q8—942no—
292C2×C4×Q8—471no—
293C2×C4×Q8C2×D4711yescharacteristic
294C2×C4×D4—471no—
295C2×C4×D4—941no—
296C22×C4⋊C4—472no—
297#477—24no—
298C4×Q16—944no—
299C2×#107—471no—
300C2×#107C2×D4712yes—
301C2×C4⋊C8C2×D4711yescharacteristic
302C2×#103—471no—
303C2×#103—471no—
304C2×#103—472no—
305C2×#103C2×D4712yes—
306C2×C22⋊C8—472no—
307C2×C22⋊C8—472no—
308C2×C4×C8—471no—
309#396—942no—
310#390—941no—
311#389—941no—
312#384—471no—
313#380—472no—
314#340—472no—
315#340—472no—
316#340—472no—
317C47#2412211yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8—471no—
320C22×Q8C22×D4711yescharacteristic
321C22×D4—471no—
322C23×C4—472no—
323C23×C4—942no—
324C2×Q16—942no—
325C2×Q16—942no—
326C2×Q16—942no—
327C22×C8—471no—
328C22×C8—471no—
329C22×C8C22×D4712yes—
330C4⋊Q8—22no—
331C4⋊Q8C22⋊D4714yes—
332#125—948no—
333C22⋊Q8—944no—
334C22⋊Q8—9416no—
335C22⋊Q8—944no—
336C22⋊Q8—944no—
337C22⋊Q8—1882no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—474no—
341C4×Q8—942no—
342C4×D4—948no—
343C4×D4—942no—
344C4×D4—942no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—471no—
348C2×C4⋊C4—471no—
349C2×C4⋊C4—472no—
350C2×C4⋊C4—944no—
351C2×C4⋊C4—942no—
352C2×C4⋊C4—942no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—944no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4C22×D4711yescharacteristic
359C2×C4⋊C4C22×D4712yes—
360C2×C22⋊C4—472no—
361C2×C22⋊C4—942no—
362C2×C22⋊C4—942no—
363C2×C22⋊C4—942no—
364C2×C42—21no—
365C2×C42—941no—
366C2×C42C22×D4711yescharacteristic
367#107—24no—
368#107—942no—
369C4⋊C8—22no—
370#103—24no—
371#103—942no—
372#103—942no—
373#103—1882no—
374C22⋊C8—944no—
375C22⋊C8—1882no—
376C4×C8—942no—
377#95—942no—
378C24—472no—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—472no—
388C2×Q8—941no—
389C2×Q8—941no—
390C2×Q8—941no—
391C2×Q8—941no—
392C2×Q8—941no—
393C2×Q8#1062535412yes—
394C2×D4—474no—
395C2×D4—942no—
396C2×D4—942no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—472no—
401C22×C4—948no—
402C22×C4—941no—
403C22×C4—941no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—3762no—
412C22×C4—3762no—
413C22×C4D4×D4711yescharacteristic
414C22×C4D4×D4711yescharacteristic
415C22×C4C23×D4711yescharacteristic
416Q16—948no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—472no—
421C2×C8—941no—
422C2×C8—941no—
423C2×C8—941no—
424C2×C8—941no—
425C2×C8—941no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—472no—
432C4⋊C4—948no—
433C4⋊C4—948no—
434C4⋊C4—944no—
435C4⋊C4—942no—
436C4⋊C4—941no—
437C4⋊C4—942no—
438C4⋊C4—942no—
439C4⋊C4—3761no—
440C4⋊C4—1884no—
441C4⋊C4—1882no—
442C4⋊C4C2×C22⋊D4714yes—
443C22⋊C4—948no—
444C22⋊C4—944no—
445C22⋊C4—3762no—
446C22⋊C4—1884no—
447C42—42no—
448C42—944no—
449C42D4×D4712yes—
450C42C2×C22⋊D4712yes—
451C23—474no—
452C23—942no—
453C23—942no—
454C23—942no—
455C23—942no—
456C23—942no—
457C23—942no—
458C23C2×D4×D4711yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—84no—
464Q8—942no—
465Q8—942no—
466Q8—1881no—
467Q8—1881no—
468D4—948no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—942no—
481C2×C4—942no—
482C2×C4—942no—
483C2×C4—942no—
484C2×C4—942no—
485C2×C4—3768no—
486C2×C4—1882no—
487C2×C4—1882no—
488C2×C4—1882no—
489C2×C4—1881no—
490C2×C4—1881no—
491C2×C4—1881no—
492C2×C4—1881no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4#1309399212yes—
498C2×C4#1309399212yes—
499C2×C4Q16×D4712yes—
500C2×C4#1309405612yes—
501C2×C4C2×D4×D4711yescharacteristic
502C2×C4C22×C22⋊D4711yescharacteristic
503C2×C4#41×D4711yescharacteristic
504C2×C4C2×#1062535411yescharacteristic
505C2×C4#1309407011yescharacteristic
506C2×C4#1309407211yescharacteristic
507C8—42no—
508C8—942no—
509C8—1881no—
510C22—944no—
511C22—944no—
512C22—944no—
513C22—1882no—
514C22—1882no—
515C22—1882no—
516C22—1882no—
517C22C2×#1309399211yescharacteristic
518C22#1568976411yescharacteristic
519C22#1568977511yescharacteristic
520C22C2×Q16×D4711yescharacteristic
521C22C2×#1309405611yescharacteristic
522C22#1568992711yescharacteristic
523C22#1568993311yescharacteristic
524C4—22no—
525C4—44no—
526C4—42no—
527C4—81no—
528C4—3762no—
529C4—1882no—
530C4—1882no—
531C4—1882no—
532C4#1568927714yes—
533C4#1568963611yescharacteristic
534C4#1568970311yescharacteristic
535C4#1568978411yescharacteristic
536C4#1568979211yescharacteristic
537C2—1884no—
538C2#2024673511yescharacteristic
539C2C2×#1568927711yescharacteristic
540C2#2025269111yescharacteristic
541C2#2025526111yescharacteristic
542C2#2025572311yescharacteristic
543C2#2025577911yescharacteristic
544C2#2025600611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#477C2
socleC23×C47C22×D4
prime radicalC22×D188D4
hypercenterC2×#477D47
hypocenterC47#24122
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625337C2semidirect
4. #20244593C2semidirect
5. #20244397C2semidirect
6. #20244350C2semidirect
8. #20243692C2semidirect
9. #20243574C2semidirect
10. #20243394C2semidirect
11. C2×C47×#477C2semidirect
127. C2×D188Q16semidirect
266. C2×#477D47semidirect
317. C47#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#156898342
C2×C4×#106253372
#202483062
#202445932
#202443972
#202443502
#202441812
#202436922
#202435742
#202433942
C2×C47×#4772
#2412247