Finite Group Databasebeta

← #26221639

Subgroups — #26221639

Order 12032.

1162 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689718C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic, complemented
3#20245743C211yesmaximal, characteristic, complemented
4#20244373C211yesmaximal, characteristic, complemented
5#20244372C211yesmaximal, characteristic, complemented
6#20244225C211yesmaximal, characteristic
7#20243843C214yesmaximal, complemented
8#99×Dic47C214yesmaximal, complemented
9C47×#1964C211yesmaximal, characteristic
10#15689718—22no—
11#15689718—22no—
12#15689718—22no—
13#15689718—22no—
14C2×#13094017C412yescomplemented
15C2×#13094014C412yescomplemented
16C2×#13094005C412yescomplemented
17C2×#13094005C2212yes—
18C2×#13094003C2211yescharacteristic
19C2×#13094001C412yescomplemented
20C2×D4×Dic47C2212yes—
21C2×#13093980C2212yes—
22C22×Dic47⋊C4C2211yescharacteristic
23C2×#13093939C2214yes—
24#15689010—22no—
25#15689004—22no—
26#15688984C2212yes—
27#15688983C2211yescharacteristic
28C4×Dic47⋊C4—22no—
29#15688974—22no—
30C22⋊C4×Dic47C2212yes—
31C22⋊C4×Dic47C2212yes—
32C4×C22⋊Dic47C2211yescharacteristic
33#15688889C2214yes—
34#15688889C2214yes—
35C2×#121×C47C2211yescharacteristic
36C2×#99×C47C2214yes—
37C47×#380C2211yescharacteristic
38C4×C22⋊C4×C47C2211yescharacteristic
39C22×Dic94C2×C411yescharacteristic
40C23×Dic47C2×C412yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43#13094017—24no—
44#13094017—24no—
45#13094014—24no—
46#13094014—24no—
47#13094005—24no—
48#13094005—24no—
49#13094005—24no—
50#13094005—24no—
51#13094003—21no—
52#13094003—21no—
53#13094003—21no—
54#13094003—21no—
55#13094001—24no—
56#13094001—24no—
57D4×Dic47—24no—
58D4×Dic47—24no—
59#13093980—24no—
60#13093980—24no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47C2×C412yes—
63C2×C4⋊Dic47C2×C411yescharacteristic
64C2×Dic47⋊C4—22no—
65C2×Dic47⋊C4C2×C412yes—
66C2×Dic47⋊C4C2×C412yes—
67C2×Dic47⋊C4C2×C412yes—
68C2×Dic47⋊C4C2×C412yes—
69C2×Dic47⋊C4C2×C412yes—
70C2×Dic47⋊C4D412yes—
71C2×Dic47⋊C4D412yes—
72C2×Dic47⋊C4D412yes—
73C2×Dic47⋊C4D412yes—
74C2×C22⋊Dic47C2×C412yes—
75C2×C22⋊Dic47C2×C412yes—
76C2×C22⋊Dic47C2×C412yes—
77C2×C22⋊Dic47C2312yes—
78C2×C22⋊Dic47C2312yes—
79C2×C22⋊Dic47C2312yes—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47C2×C412yes—
85#13093939—28no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—22no—
89C47⋊#95—22no—
90C47⋊#95—24no—
91C22×D4×C47C2×C411yescharacteristic
92C22×D4×C47C2312yes—
93C23×C4×C47C2311yescharacteristic
94#121×C47—21no—
95#121×C47—21no—
96#121×C47—21no—
97#121×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C22⋊C4×C47C2×C412yes—
101C2×C22⋊C4×C47C2312yes—
102C2×C22⋊C4×C47C2312yes—
103C2×C42×C47—21no—
104#99×C47—28no—
105#95×C47—21no—
106#95×C47—21no—
107C24×C47C22×C412yes—
108C24×C47C2411yescharacteristic
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47—24no—
112C2×D4×C47—24no—
113C2×D4×C47—21no—
114C2×D4×C47—21no—
115C2×D4×C47—81no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—81no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—24no—
130C22×Dic47—22no—
131C22×Dic47—81no—
132C22×Dic47—82no—
133C22×Dic47C22×C412yes—
134C22×Dic47C22×C412yes—
135C22×Dic47C22×C412yes—
136C22×Dic47C22×C412yes—
137C22×Dic47C2×D412yes—
138C22×Dic47C2×D412yes—
139C22×Dic47C2×D412yes—
140C22×Dic47C2×D412yes—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—81no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—81no—
147C22⋊C4×C47—22no—
148C22⋊C4×C47—22no—
149C22⋊C4×C47—22no—
150C22⋊C4×C47—24no—
151C22⋊C4×C47#4114yes—
152C42×C47—81no—
153C22⋊Dic47—22no—
154C22⋊Dic47—22no—
155C22⋊Dic47—22no—
156C22⋊Dic47—24no—
157C22⋊Dic47—24no—
158C22⋊Dic47—24no—
159C22⋊Dic47—24no—
160C22⋊Dic47#4114yes—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47—24no—
164C4⋊Dic47—81no—
165Dic47⋊C4—28no—
166Dic47⋊C4—22no—
167Dic47⋊C4—22no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4C22⋊C414yes—
171Dic47⋊C4C22⋊C414yes—
172Dic47⋊C4C22⋊C414yes—
173Dic47⋊C4C22⋊C414yes—
174C4×Dic47—28no—
175C4×Dic47—24no—
176C4×Dic47—82no—
177C4×Dic47—82no—
178C4×Dic47—81no—
179C23×C47—22no—
180C23×C47—24no—
181C23×C47—41no—
182C23×C47—41no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47C23×C411yescharacteristic
186C23×C47C22×D411yescharacteristic
187C23×C47C22×D411yescharacteristic
188C23×C47C2×#4112yes—
189C23×C47C2×#4112yes—
190D4×C47—44no—
191D4×C47—42no—
192D4×C47—42no—
193D4×C47—81no—
194D4×C47—81no—
195C2×C4×C47—22no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47C2×C22⋊C411yescharacteristic
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47C2×C22⋊C411yescharacteristic
212C2×C4×C47C2×C22⋊C411yescharacteristic
213C2×Dic47—24no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—24no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—84no—
232C2×Dic47C2×C22⋊C414yes—
233C2×Dic47C2×C22⋊C414yes—
234Dic94—81no—
235Dic94—81no—
236#21423—471nomaximal, Sylow
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—42no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—84no—
249C22×C47C22×C22⋊C411yescharacteristic
250C22×C47C4×#4112yes—
251C22×C47#52512yes—
252C22×C47#53211yescharacteristic
253C22×C47#53211yescharacteristic
254C4×C47—21no—
255C4×C47—21no—
256C4×C47—21no—
257C4×C47—21no—
258C4×C47—44no—
259C4×C47—42no—
260C4×C47—82no—
261Dic47—24no—
262Dic47—24no—
263Dic47—44no—
264Dic47—42no—
265Dic47—42no—
266Dic47—82no—
267C2×#539—471no—
268#2313—471no—
269#2274—471no—
270#1996—471no—
271#1996—471no—
272#1964—471no—
273#1964D4711yescharacteristic
274#1899—474no—
275C4×#99—474no—
276C2×C47—22no—
277C2×C47—21no—
278C2×C47—21no—
279C2×C47—22no—
280C2×C47—82no—
281C2×C47#229711yescharacteristic
282C2×C47#288311yescharacteristic
283C2×C47#288411yescharacteristic
284#539—942no—
285#539—942no—
286#539—942no—
287#539—942no—
288C2×#126—472no—
289C2×#125—472no—
290C2×#123—472no—
291C2×#123—472no—
292C2×C22⋊Q8—471no—
293C2×C22⋊Q8—472no—
294C2×#121C2×D4711yescharacteristic
295C2×C4×D4—472no—
296C2×#117—472no—
297C22×C4⋊C4—471no—
298C2×#99—474no—
299C2×#99C2×D4714yes—
300#387—942no—
301#387—942no—
302#381—472no—
303#380—471no—
304#380C2×D4711yescharacteristic
305C4×C4⋊C4—942no—
306C4×C4⋊C4—942no—
307C4×C22⋊C4—471no—
308C4×C22⋊C4—472no—
309C4×C22⋊C4—472no—
310C4×C22⋊C4C2×D4711yescharacteristic
311#342—474no—
312#342—474no—
313C47#2142311yescharacteristic, complemented, Sylow
314C22×Q8—471no—
315C22×D4C4×D4711yescharacteristic
316C22×D4C22×D4712yes—
317C23×C4—471no—
318C23×C4—471no—
319C23×C4—472no—
320C23×C4C22×D4711yescharacteristic
321#126—944no—
322#126—944no—
323#125—944no—
324#125—944no—
325#123—944no—
326#123—944no—
327#123—944no—
328#123—944no—
329C22⋊Q8—944no—
330C22⋊Q8—944no—
331C22⋊Q8—941no—
332C22⋊Q8—941no—
333C22⋊Q8—941no—
334C22⋊Q8—941no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339C4×D4—944no—
340C4×D4—944no—
341#117—944no—
342#117—944no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—472no—
345C2×C4⋊C4—472no—
346C2×C4⋊C4—472no—
347C2×C4⋊C4—472no—
348C2×C4⋊C4—472no—
349C2×C4⋊C4—472no—
350C2×C4⋊C4—472no—
351C2×C4⋊C4—472no—
352C2×C4⋊C4—472no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4C4×D4711yescharacteristic
358C2×C22⋊C4—472no—
359C2×C22⋊C4—472no—
360C2×C22⋊C4—472no—
361C2×C22⋊C4—472no—
362C2×C22⋊C4—472no—
363C2×C22⋊C4—472no—
364C2×C22⋊C4C4×D4712yes—
365C2×C22⋊C4C22×D4712yes—
366C2×C22⋊C4C22×D4712yes—
367C2×C42—21no—
368C2×C42—472no—
369C2×C42—942no—
370C2×C42—942no—
371C2×C42—942no—
372C2×C42—941no—
373#99—28no—
374#99—948no—
375#95—21no—
376#95—21no—
377#95—942no—
378#95—942no—
379#95—944no—
380#95—941no—
381#95—941no—
382C24C2×C4×D4712yes—
383C24C23×D4711yescharacteristic
384C2×Q8—941no—
385C2×Q8—941no—
386C2×Q8—3761no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—81no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—81no—
397C22×C4—472no—
398C22×C4—472no—
399C22×C4—472no—
400C22×C4—472no—
401C22×C4—472no—
402C22×C4—472no—
403C22×C4—472no—
404C22×C4—472no—
405C22×C4—944no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—3762no—
411C22×C4—3761no—
412C22×C4C2×C4×D4712yes—
413C22×C4C2×C4×D4711yescharacteristic
414C22×C4C2×C4×D4711yescharacteristic
415C22×C4D4×D4711yescharacteristic
416C22×C4D4×D4711yescharacteristic
417C22×C4D4×D4711yescharacteristic
418C22×C4D4×D4711yescharacteristic
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—81no—
422C4⋊C4—474no—
423C4⋊C4—474no—
424C4⋊C4—474no—
425C4⋊C4—474no—
426C4⋊C4—944no—
427C4⋊C4—944no—
428C4⋊C4—944no—
429C4⋊C4—948no—
430C4⋊C4—942no—
431C4⋊C4—942no—
432C4⋊C4—941no—
433C4⋊C4—941no—
434C4⋊C4—3761no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—474no—
440C22⋊C4—944no—
441C22⋊C4—944no—
442C22⋊C4—944no—
443C22⋊C4—942no—
444C22⋊C4—942no—
445C22⋊C4—942no—
446C22⋊C4—944no—
447C22⋊C4#1062535214yes—
448C42—81no—
449C42—944no—
450C42—948no—
451C42—3761no—
452C42—3762no—
453C42—3762no—
454C23—22no—
455C23—24no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23C22×C4×D4711yescharacteristic
461C23C2×D4×D4711yescharacteristic
462C23C2×D4×D4711yescharacteristic
463C23#41×D4712yes—
464C23C2×#1062535212yes—
465Q8—3761no—
466Q8—3761no—
467D4—44no—
468D4—42no—
469D4—42no—
470D4—81no—
471D4—81no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—474no—
487C2×C4—474no—
488C2×C4—944no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—942no—
494C2×C4—942no—
495C2×C4—944no—
496C2×C4—3764no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4C22⋊C4×D4711yescharacteristic
508C2×C4C22⋊C4×D4711yescharacteristic
509C2×C4C22⋊C4×D4711yescharacteristic
510C2×C4C22⋊C4×D4711yescharacteristic
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—84no—
523C22C2×C22⋊C4×D4711yescharacteristic
524C22C4×#1062535212yes—
525C22#1568963312yes—
526C22#1568967111yescharacteristic
527C22#1568967211yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—44no—
533C4—42no—
534C4—82no—
535C4—944no—
536C4—944no—
537C4—3762no—
538C4—1884no—
539C4—1882no—
540C4—1882no—
541C2—22no—
542C2—21no—
543C2—21no—
544C2—22no—
545C2—82no—
546C2#2024587511yescharacteristic
547C2#2025184611yescharacteristic
548C2#2025185011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1964C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC24×C47C24
hypercenter#1964D47
hypocenterC47#21423
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #20245990C2semidirect
3. #20245743C2semidirect
4. #20244373C2semidirect
5. #20244372C2semidirect
7. #20243843C2semidirect
8. #99×Dic47C2semidirect
14. C2×#13094017C4semidirect
15. C2×#13094014C4semidirect
16. C2×#13094005C4semidirect
19. C2×#13094001C4semidirect
313. C47#21423semidirect

Maximal subgroups

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

subgroupindex
C2×#156897182
#202459902
#202457432
#202443732
#202443722
#202442252
#202438432
#99×Dic472
C47×#19642
#2142347