Finite Group Databasebeta

← #26268480

Subgroups — #26268480

Order 12032.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689354C212yesmaximal
2#20251719C211yesmaximal, characteristic
3#20244718C211yesmaximal, characteristic, complemented
4#20244631C211yesmaximal, characteristic, complemented
5#20244521C211yesmaximal, characteristic, complemented
6#20244471C211yesmaximal, characteristic, complemented
7#20243812C212yesmaximal, complemented
8#20243743C212yesmaximal, complemented
9#20243656C212yesmaximal, complemented
10C2×#15688878C211yesmaximal, characteristic, complemented
11C2×C47×#533C211yesmaximal, characteristic, complemented
12C22×#10625336C2212yes—
13C23×C47⋊C8C2212yes—
14C2×#13094007—21no—
15C2×#13093998C2211yescharacteristic
16#15689354—28no—
17C2×#13093955C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093944C2212yes—
22#15689025C2211yescharacteristic
23#15689025C2211yescharacteristic
24#15689021C2211yescharacteristic
25#15689021C2211yescharacteristic
26#15689014—21no—
27#15689014—21no—
28#15689002—21no—
29#15688983C2211yescharacteristic
30#15688885C2212yes—
31#15688885C2212yes—
32#15688878C2212yes—
33#15688878C2211yescharacteristic
34#15688878C2211yescharacteristic
35C47×#533—28no—
36C2×C4⋊Q8×C47C2211yescharacteristic
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#125×C47—21no—
39C2×C22⋊Q8×C47—21no—
40C2×C22⋊Q8×C47C2212yes—
41C2×C4×Q8×C47—21no—
42C2×C4×Q8×C47C2212yes—
43C2×#117×C47—21no—
44C22×C42×C47C2211yescharacteristic
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—21no—
47C2×#10625336—24no—
48C2×#10625336—24no—
49C2×#10625336—42no—
50C22×C47⋊C8—24no—
51C22×C47⋊C8—24no—
52C22×C47⋊C8—42no—
53C22×C47⋊C8C2312yes—
54C22×C47⋊C8C2312yes—
55#13094007—44no—
56#13093998—24no—
57C2×C4⋊Dic47C2311yescharacteristic
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—21no—
61C2×Dic47⋊C4—21no—
62C2×C22⋊Dic47—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—21no—
65C2×(C2×D47)⋊C4—21no—
66C2×(C2×D47)⋊C4—21no—
67C2×(C2×D47)⋊C4—21no—
68#13093955—24no—
69#13093948—24no—
70#13093948—24no—
71#13093948—42no—
72#13093944—24no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C22×Q8×C47—21no—
77C22×Q8×C47C2312yes—
78C23×C4×C47D411yescharacteristic
79C23×C4×C47D411yescharacteristic
80C23×C4×C47C2311yescharacteristic
81C4⋊Q8×C47—24no—
82C4⋊Q8×C47—24no—
83#125×C47—28no—
84C22⋊Q8×C47—28no—
85C22⋊Q8×C47—24no—
86C22⋊Q8×C47D418yes—
87C4×Q8×C47—28no—
88C4×Q8×C47—24no—
89C4×Q8×C47—24no—
90#117×C47—28no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47C2312yes—
97C2×C22⋊C4×C47—22no—
98C2×C42×C47—24no—
99C2×C42×C47—22no—
100C2×C42×C47D411yescharacteristic
101C2×C42×C47D411yescharacteristic
102C2×C42×C47C2311yescharacteristic
103C2×C42×C47C2311yescharacteristic
104C24×C47C2×D411yescharacteristic
105C23×D47—21no—
106C2×Q8×C47—24no—
107C2×Q8×C47—22no—
108C2×Q8×C47—22no—
109C2×Q8×C47—22no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×Q8×C47—42no—
113C2×Q8×C47—42no—
114C2×Q8×C47C2×D414yes—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—41no—
125C22×C4×C47SD1614yescomplemented
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D414yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—42no—
134C2×C22⋊D47—42no—
135C2×C22⋊D47—42no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—21no—
142C2×D188—21no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47—24no—
145C4⋊C4×C47—28no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—28no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47C2×D414yes—
150C22⋊C4×C47—28no—
151C42×C47—28no—
152C42×C47—22no—
153C42×C47—22no—
154C42×C47—22no—
155C42×C47—22no—
156C22⋊Dic47—42no—
157#10625336—48no—
158(C2×D47)⋊C4—42no—
159(C2×D47)⋊C4—42no—
160(C2×D47)⋊C4—42no—
161(C2×D47)⋊C4—42no—
162(C2×D47)⋊C4—42no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167Dic47⋊C4—42no—
168Dic47⋊C4—42no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—48no—
175C2×C47⋊C8#4114yes—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47#12012yes—
181C23×C47C2×SD1612yes—
182C23×C47#13712yes—
183C23×C47C22×D411yescharacteristic
184C22×D47—41no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190C22×D47—41no—
191Q8×C47—24no—
192Q8×C47—24no—
193Q8×C47—48no—
194Q8×C47—42no—
195Q8×C47—42no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—28no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—21no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—44no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47#12012yes—
218C2×C4×C47C2×SD1612yes—
219C2×C4×C47C2×SD1612yes—
220C2×C4×C47#13512yes—
221C2×C4×C47C22×D412yes—
222C2×C4×C47C2×#4112yes—
223C2×C4×C47H(5,𝔽2)11yescharacteristic
224C2×C4×C472−1+411yescharacteristic
225C22⋊D47—84no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235D188—42no—
236D188—42no—
237C47⋊C8—24no—
238C47⋊C8—42no—
239#25497—471nomaximal, Sylow
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47#44814yes—
247C22×C47C2×#12011yescharacteristic
248C22×C47#53211yescharacteristic
249C22×C47#53711yescharacteristic
250C22×C47C22×SD1611yescharacteristic
251C22×C47C2×#13511yescharacteristic
252C22×C47C2×#13711yescharacteristic
253C22×C47#57511yescharacteristic
254C2×D47—81no—
255C2×D47—81no—
256C2×D47—81no—
257C2×D47—81no—
258C2×D47—81no—
259C2×D47—81no—
260C2×D47—81no—
261C4×C47—22no—
262C4×C47—22no—
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—44no—
266C4×C47—41no—
267C4×C47—44no—
268C4×C47#54612yes—
269C4×C47#54712yes—
270Dic47—81no—
271Dic47—81no—
272Dic47—81no—
273C2×#533D4711yescharacteristic, complemented
274C2×#465—472no—
275#2854—471no—
276#2058—471no—
277#2043—471no—
278#2023—471no—
279#2013—471no—
280#1892—472no—
281#1880—472no—
282#1865—472no—
283C2×#336—471no—
284C2×C47—24no—
285C2×C47#243011yescharacteristic
286C2×C47C2×#44811yescharacteristic
287C2×C47#300211yescharacteristic
288C2×C47#316711yescharacteristic
289C2×C47#318111yescharacteristic
290C2×C47#324611yescharacteristic
291C2×C47#324811yescharacteristic
292D47—161no—
293C22×SD16—472no—
294C23×C8—472no—
295#533—28no—
296C2×C4⋊Q8C2×D4711yescharacteristic
297C2×C4⋊Q8C2×D4711yescharacteristic
298C2×#125—21no—
299C2×#123—941no—
300C2×C22⋊Q8—21no—
301C2×C22⋊Q8C2×D4712yes—
302C2×#121—471no—
303C2×C4×Q8—21no—
304C2×C4×Q8C2×D4712yes—
305C2×#117—21no—
306C22×C42C2×D4711yescharacteristic
307#465—948no—
308C2×#106—472no—
309C2×#103—472no—
310C2×#103—472no—
311C2×#103—472no—
312C2×#102—472no—
313#390—471no—
314#390—471no—
315#389—471no—
316#389—471no—
317#388—941no—
318#388—941no—
319#386—941no—
320#380—471no—
321#340—472no—
322#340—472no—
323#336—471no—
324#336—472no—
325#336—471no—
326C47#2549711yescharacteristic, complemented, Sylow
327C22×Q8—21no—
328C22×Q8C22×D4712yes—
329C22×D4—471no—
330C22×D4—941no—
331C23×C4C22⋊D4711yescharacteristic
332C23×C4C22⋊D4711yescharacteristic
333C23×C4C22×D4711yescharacteristic
334C2×SD16—944no—
335C2×SD16—944no—
336C2×SD16—1882no—
337C22×C8—472no—
338C22×C8—472no—
339C22×C8—944no—
340C22×C8—944no—
341C22×C8—1882no—
342C4⋊Q8—24no—
343C4⋊Q8—24no—
344#125—28no—
345#123—1884no—
346C22⋊Q8—28no—
347C22⋊Q8—24no—
348C22⋊Q8C22⋊D4718yes—
349#121—944no—
350C4×Q8—28no—
351C4×Q8—24no—
352C4×Q8—24no—
353#117—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—471no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4C22×D4712yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4—941no—
368C2×C22⋊C4—941no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—941no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42C22⋊D4711yescharacteristic
375C2×C42C22⋊D4711yescharacteristic
376C2×C42C22×D4711yescharacteristic
377C2×C42C22×D4711yescharacteristic
378#106—944no—
379#103—944no—
380#103—944no—
381#103—1882no—
382#102—944no—
383#95—941no—
384#95—941no—
385#95—941no—
386C24—941no—
387C24C2×C22⋊D4711yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8C2×C22⋊D4714yes—
397C2×D4—942no—
398C2×D4—942no—
399C2×D4—941no—
400C2×D4—941no—
401C2×D4—1882no—
402C2×D4—1882no—
403C2×D4—1882no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—941no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4#1062533614yescomplemented
418C22×C4C2×C22⋊D4711yescharacteristic
419C22×C4C2×C22⋊D4711yescharacteristic
420C22×C4C2×C22⋊D4714yes—
421C22×C4C2×C22⋊D4711yescharacteristic
422C22×C4C2×C22⋊D4711yescharacteristic
423C22×C4C2×C22⋊D4711yescharacteristic
424C22×C4C23×D4711yescharacteristic
425SD16—1888no—
426C2×C8—474no—
427C2×C8—942no—
428C2×C8—942no—
429C2×C8—942no—
430C2×C8—942no—
431C2×C8—942no—
432C2×C8—1888no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—942no—
440C4⋊C4—942no—
441C4⋊C4—941no—
442C4⋊C4—941no—
443C4⋊C4—1882no—
444C4⋊C4—1882no—
445C4⋊C4C2×C22⋊D4714yes—
446C22⋊C4—28no—
447C22⋊C4—1882no—
448C22⋊C4—1882no—
449C22⋊C4—1882no—
450C22⋊C4—1882no—
451C22⋊C4—1882no—
452C22⋊C4—1882no—
453C42—28no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—1881no—
463C23—1881no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23#1309399312yes—
470C23C2×#1062533612yes—
471C23#1309405512yes—
472C23C22×C22⋊D4711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—3764no—
479D4—1882no—
480D4—1882no—
481C2×C4—28no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—1881no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4—1881no—
511C2×C4#1309399312yes—
512C2×C4C2×#1062533612yes—
513C2×C4C2×#1062533612yes—
514C2×C4#1309404412yes—
515C2×C4C22×C22⋊D4712yes—
516C2×C4#41×D4712yes—
517C2×C4#1309407011yescharacteristic
518C2×C4#1309407111yescharacteristic
519C8—944no—
520C8—1882no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—3761no—
528C22—3761no—
529C22—3761no—
530C22—3761no—
531C22—3761no—
532C22—3761no—
533C22—3761no—
534C22#1568926214yes—
535C22C2×#1309399311yescharacteristic
536C22#1568966911yescharacteristic
537C22#1568970111yescharacteristic
538C22C22×#1062533611yescharacteristic
539C22C2×#1309404411yescharacteristic
540C22C2×#1309405511yescharacteristic
541C22#1568992111yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—44no—
548C4—44no—
549C4—3761no—
550C4—3761no—
551C4—3761no—
552C4#1568976112yes—
553C4#1568977212yes—
554C2—24no—
555C2—7521no—
556C2#2024699211yescharacteristic
557C2C2×#1568926211yescharacteristic
558C2#2025260611yescharacteristic
559C2#2025394811yescharacteristic
560C2#2025409111yescharacteristic
561C2#2025470011yescharacteristic
562C2#2025471911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#533C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC2×#533D47
hypocenterC47#25497
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
3. #20244718C2semidirect
4. #20244631C2semidirect
5. #20244521C2semidirect
6. #20244471C2semidirect
7. #20243812C2semidirect
8. #20243743C2semidirect
9. #20243656C2semidirect
10. C2×#15688878C2semidirect
11. C2×C47×#533C2semidirect
125. C22×C4×C47SD16semidirect
273. C2×#533D47semidirect
326. C47#25497semidirect
417. C22×C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893542
#202517192
#202447182
#202446312
#202445212
#202444712
#202438122
#202437432
#202436562
C2×#156888782
C2×C47×#5332
#2549747