Finite Group Databasebeta

← #26131250

Subgroups — #26131250

Order 12032.

1026 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
1C22×#13093939C211yesmaximal, characteristic
2#20245970C211yesmaximal, characteristic
3#20241512C214yesmaximal, complemented
4C22×#99×C47C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic, complemented
6C2×#13093983C411yescharacteristic, complemented
7C22×C22⋊Dic47—21no—
8C2×#13093939—21no—
9C2×#13093939—22no—
10C2×#13093939C414yescomplemented
11C2×#13093939C2212yes—
12#15689026—21no—
13#15689026—21no—
14#15689026C412yescomplemented
15#15689026C2212yes—
16#15688979—21no—
17#15688979—21no—
18C23×D4×C47C2211yescharacteristic
19C22×C22⋊C4×C47—21no—
20C2×#99×C47—22no—
21C2×#99×C47—21no—
22C2×#99×C47C414yescomplemented
23C2×#99×C47C2212yes—
24C23×Dic47—41no—
25#13093983—41no—
26#13093983—41no—
27#13093983—42no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47C2×C412yescomplemented
30C2×C22⋊Dic47—24no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—41no—
39C2×C22⋊Dic47—42no—
40C2×C4×Dic47—21no—
41C2×C4×Dic47C2×C412yescomplemented
42#13093939—24no—
43#13093939—22no—
44#13093939C2×C414yescomplemented
45C25×C47D411yescharacteristic
46C25×C47D411yescharacteristic
47C22×D4×C47—41no—
48C22×D4×C47—41no—
49C22×D4×C47—42no—
50C22×D4×C47C2×C412yes—
51C22×D4×C47C2×C412yes—
52C22×D4×C47C2×C411yescharacteristic
53C22×D4×C47D411yescharacteristic
54C22×D4×C47D411yescharacteristic
55C22×D4×C47D412yes—
56C22×D4×C47Q812yes—
57C22×D4×C47C2311yescharacteristic
58C23×C4×C47—41no—
59C23×C4×C47C2×C411yescharacteristic
60C2×C22⋊C4×C47—24no—
61C2×C22⋊C4×C47—22no—
62C2×C22⋊C4×C47—41no—
63C2×C22⋊C4×C47—42no—
64#99×C47—24no—
65#99×C47—22no—
66#99×C47C2×C414yescomplemented
67C24×C47—21no—
68C24×C47—21no—
69C24×C47—22no—
70C24×C47—21no—
71C24×C47—22no—
72C24×C47—21no—
73C24×C47—82no—
74C24×C47—82no—
75C24×C47C22⋊C411yescharacteristic
76C24×C47C22⋊C411yescharacteristic
77C24×C47C22⋊C412yes—
78C24×C47C22⋊C412yes—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C4⋊C412yes—
81C24×C47C4⋊C412yes—
82C24×C47C2×D411yescharacteristic
83C24×C47C2×D411yescharacteristic
84C2×D4×C47—42no—
85C2×D4×C47—42no—
86C2×D4×C47—42no—
87C2×D4×C47—42no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47—42no—
97C2×D4×C47—42no—
98C2×D4×C47—41no—
99C2×D4×C47—41no—
100C2×D4×C47C4212yes—
101C2×D4×C47C22⋊C412yes—
102C2×D4×C47C22⋊C412yes—
103C2×D4×C47C4⋊C412yes—
104C2×D4×C47C4⋊C412yes—
105C2×D4×C47C4⋊C412yes—
106C2×D4×C47C22×C411yescharacteristic
107C2×D4×C47C22×C411yescharacteristic
108C2×D4×C47C2×D411yescharacteristic
109C2×D4×C47C2×Q811yescharacteristic
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—44no—
114C22×C4×C47—42no—
115C22×C4×C47—84no—
116C22×C4×C47C4212yes—
117C22×C4×C47C22⋊C411yescharacteristic
118C22×C4×C47C4⋊C412yes—
119C22×C4×C47C22×C411yescharacteristic
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—44no—
123C22×Dic47—42no—
124C22×Dic47—84no—
125C22⋊C4×C47—24no—
126C22⋊C4×C47—44no—
127C22⋊C4×C47—42no—
128C22⋊Dic47—24no—
129C22⋊Dic47—44no—
130C22⋊Dic47—42no—
131C22⋊Dic47—41no—
132C22⋊Dic47—41no—
133C22⋊Dic47—42no—
134C22⋊Dic47—82no—
135C22⋊Dic47—82no—
136C22⋊Dic47—82no—
137C22⋊Dic47—82no—
138C4⋊Dic47—42no—
139C4⋊Dic47—42no—
140C4×Dic47—42no—
141C4×Dic47—42no—
142C23×C47—22no—
143C23×C47—22no—
144C23×C47—22no—
145C23×C47—22no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—22no—
151C23×C47—22no—
152C23×C47—22no—
153C23×C47—22no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—42no—
159C23×C47—44no—
160C23×C47—44no—
161C23×C47—42no—
162C23×C47—42no—
163C23×C47—42no—
164C23×C47—42no—
165C23×C47—82no—
166C23×C47—82no—
167C23×C47—82no—
168C23×C47—82no—
169C23×C47—82no—
170C23×C47—82no—
171C23×C47—84no—
172C23×C47—84no—
173C23×C47#9512yes—
174C23×C47#9512yes—
175C23×C47#9512yes—
176C23×C47#9911yescharacteristic
177C23×C47#9911yescharacteristic
178C23×C47#9912yes—
179C23×C47C2×C22⋊C411yescharacteristic
180C23×C47C2×C22⋊C411yescharacteristic
181C23×C47C2×C22⋊C411yescharacteristic
182C23×C47C2×C4⋊C411yescharacteristic
183C23×C47C2×C4⋊C411yescharacteristic
184D4×C47—42no—
185D4×C47—42no—
186D4×C47—42no—
187D4×C47—42no—
188D4×C47—42no—
189D4×C47—42no—
190D4×C47—41no—
191D4×C47—41no—
192D4×C47—41no—
193D4×C47—41no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—44no—
203C2×C4×C47—88no—
204C2×C4×C47—84no—
205C2×C4×C47#9512yes—
206C2×C4×C47C2×C4211yescharacteristic
207C2×C4×C47C2×C4⋊C411yescharacteristic
208C2×Dic47—44no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—88no—
216C2×Dic47—84no—
217#12582—471nomaximal, Sylow
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—21no—
222C22×C47—21no—
223C22×C47—22no—
224C22×C47—22no—
225C22×C47—21no—
226C22×C47—21no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—41no—
232C22×C47—41no—
233C22×C47—41no—
234C22×C47—41no—
235C22×C47—44no—
236C22×C47—44no—
237C22×C47—44no—
238C22×C47—44no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—42no—
243C22×C47—81no—
244C22×C47—81no—
245C22×C47—81no—
246C22×C47—81no—
247C22×C47—82no—
248C22×C47—82no—
249C22×C47—82no—
250C22×C47—82no—
251C22×C47#34212yes—
252C22×C47#34212yes—
253C22×C47C2×#9511yescharacteristic
254C22×C47C2×#9911yescharacteristic
255C22×C47C2×#9911yescharacteristic
256C4×C47—22no—
257C4×C47—21no—
258C4×C47—21no—
259C4×C47—84no—
260Dic47—82no—
261Dic47—84no—
262Dic47—82no—
263C22×#99—471no—
264C22×#99D4711yescharacteristic
265#2309—471no—
266#1393—474no—
267C2×C47—21no—
268C2×C47—21no—
269C2×C47—22no—
270C2×C47—22no—
271C2×C47—41no—
272C2×C47—41no—
273C2×C47—42no—
274C2×C47—42no—
275C2×C47—41no—
276C2×C47—41no—
277C2×C47C2×#34211yescharacteristic
278C2×C47#212712yes—
279C23×D4C2×D4711yescharacteristic
280C2×#117—471no—
281C2×#117—471no—
282C22×C22⋊C4—21no—
283C22×C22⋊C4—941no—
284C2×#99—21no—
285C2×#99—22no—
286C2×#99—474no—
287C2×#99—472no—
288C2×#99—941no—
289C2×#99—942no—
290C2×#99Dic4714yescomplemented
291C2×#99C2×D4712yes—
292#390—472no—
293#390—472no—
294#390—941no—
295#390—941no—
296#379—941no—
297#379—941no—
298C47#1258211yescharacteristic, complemented, Sylow
299C25D18811yescharacteristic
300C25C22⋊D4711yescharacteristic
301C22×D4—41no—
302C22×D4—41no—
303C22×D4—42no—
304C22×D4Dic9412yes—
305C22×D4C4×D4712yes—
306C22×D4C4×D4711yescharacteristic
307C22×D4D18811yescharacteristic
308C22×D4C2×Dic4712yes—
309C22×D4C22⋊D4712yes—
310C22×D4C22⋊D4711yescharacteristic
311C22×D4C22×D4711yescharacteristic
312C23×C4—41no—
313C23×C4—1881no—
314C23×C4C4×D4711yescharacteristic
315#117—1882no—
316#117—1881no—
317#117—1881no—
318C2×C4⋊C4—472no—
319C2×C4⋊C4—941no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—41no—
324C2×C22⋊C4—944no—
325C2×C22⋊C4—942no—
326C2×C22⋊C4—942no—
327C2×C22⋊C4—942no—
328C2×C22⋊C4—942no—
329C2×C22⋊C4—942no—
330C2×C22⋊C4—941no—
331C2×C22⋊C4—941no—
332C2×C22⋊C4—1882no—
333C2×C22⋊C4—1881no—
334C2×C42—472no—
335C2×C42—941no—
336#99—24no—
337#99—22no—
338#99—474no—
339#99—944no—
340#99—942no—
341#99C2×Dic4714yescomplemented
342C24—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—82no—
349C24—82no—
350C24Dic47⋊C412yes—
351C24C4⋊Dic4712yes—
352C24(C2×D47)⋊C411yescharacteristic
353C24(C2×D47)⋊C411yescharacteristic
354C24(C2×D47)⋊C412yes—
355C24(C2×D47)⋊C411yescharacteristic
356C24C22⋊Dic4712yes—
357C24C2×D18811yescharacteristic
358C24C2×C22⋊D4711yescharacteristic
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C4×Dic4712yes—
376C2×D4Dic47⋊C412yes—
377C2×D4Dic47⋊C412yes—
378C2×D4C4⋊Dic4712yes—
379C2×D4(C2×D47)⋊C412yes—
380C2×D4C22⋊Dic4712yes—
381C2×D4C2×Dic9411yescharacteristic
382C2×D4C2×C4×D4711yescharacteristic
383C2×D4C22×Dic4711yescharacteristic
384C2×D4C2×C22⋊D4711yescharacteristic
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4—942no—
392C22×C4—942no—
393C22×C4—3764no—
394C22×C4—1882no—
395C22×C4—1884no—
396C22×C4C4×Dic4712yes—
397C22×C4Dic47⋊C412yes—
398C22×C4(C2×D47)⋊C411yescharacteristic
399C22×C4C2×C4×D4711yescharacteristic
400C4⋊C4—1882no—
401C4⋊C4—1882no—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—944no—
406C22⋊C4—3762no—
407C22⋊C4—3762no—
408C22⋊C4—3762no—
409C22⋊C4—3762no—
410C22⋊C4—1882no—
411C22⋊C4—1884no—
412C22⋊C4—1882no—
413C22⋊C4—1881no—
414C22⋊C4—1881no—
415C42—1882no—
416C42—1882no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—84no—
447C23—84no—
448C23C47⋊#9512yes—
449C23C47⋊#9512yes—
450C23C47⋊#9512yes—
451C23#1309393811yescharacteristic
452C23#1309393811yescharacteristic
453C23#1309393812yes—
454C23C2×(C2×D47)⋊C411yescharacteristic
455C23C2×(C2×D47)⋊C411yescharacteristic
456C23C2×C22⋊Dic4711yescharacteristic
457C23C2×Dic47⋊C411yescharacteristic
458C23C2×C4⋊Dic4711yescharacteristic
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—88no—
479C2×C4—84no—
480C2×C4—3768no—
481C2×C4—3764no—
482C2×C4—1884no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4—1882no—
486C2×C4—1882no—
487C2×C4—1882no—
488C2×C4—1882no—
489C2×C4C47⋊#9512yes—
490C2×C4C2×C4×Dic4711yescharacteristic
491C2×C4C2×Dic47⋊C411yescharacteristic
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22#1568888912yes—
526C22#1568888912yes—
527C22C2×C47⋊#9511yescharacteristic
528C22C2×#1309393811yescharacteristic
529C22C2×#1309393811yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—84no—
534C4—3762no—
535C4—3764no—
536C4—3762no—
537C2—21no—
538C2—21no—
539C2—22no—
540C2—22no—
541C2—41no—
542C2—41no—
543C2—42no—
544C2—42no—
545C2—41no—
546C2—41no—
547C2C2×#1568888911yescharacteristic
548C2#2024505512yes—
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC22×#99×C47C2
socleC22×C47C2×#99
prime radicalC23×D4×C47C22
cosocleC22×D4×C47C23
hypercenterC22×#99D47
hypocenterC47#12582
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. #20241512C2semidirect
5. C2×#13093983C4semidirect
6. C2×#13093983C4semidirect
10. C2×#13093939C4semidirect
14. #15689026C4semidirect
22. C2×#99×C47C4semidirect
29. C2×C4⋊Dic47C2×C4semidirect
41. C2×C4×Dic47C2×C4semidirect
44. #13093939C2×C4semidirect
66. #99×C47C2×C4semidirect
290. C2×#99Dic47semidirect
298. C47#12582semidirect
341. #99C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939392
#202459702
#202415122
C22×#99×C472
#1258247