Finite Group Databasebeta

← #26122470

Subgroups — #26122470

Order 12032.

1266 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688996C211yesmaximal, characteristic
2C2×#15688889C212yesmaximal
3C2×#15688889C212yesmaximal
4#20241611C211yesmaximal, characteristic
5C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
6C22×Dic47⋊C4C412yescomplemented
7C22×C22⋊Dic47C2211yescharacteristic
8C22×C22⋊Dic47C2212yes—
9C22×C22⋊Dic47C2211yescharacteristic
10#15688996—21no—
11#15688996—22no—
12#15688996—21no—
13C2×C47⋊#95—22no—
14C2×C47⋊#95C412yescomplemented
15#15688889—24no—
16#15688889C418yescomplemented
17C24×C4×C47C2211yescharacteristic
18C22×C22⋊C4×C47C2212yes—
19C22×C42×C47—21no—
20C4×C22⋊C4×C47—24no—
21C2×#95×C47—21no—
22C23×Dic47—22no—
23C23×Dic47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C412yes—
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—44no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47—22no—
40C2×C22⋊Dic47—22no—
41C2×C22⋊Dic47C2×C414yes—
42C2×C22⋊Dic47C2×C414yes—
43C47⋊#95—22no—
44C47⋊#95—22no—
45C47⋊#95—44no—
46C25×C47C2311yescharacteristic
47C23×C4×C47—22no—
48C23×C4×C47—22no—
49C23×C4×C47—22no—
50C23×C4×C47—44no—
51C23×C4×C47C2×C412yes—
52C23×C4×C47D411yescharacteristic
53C23×C4×C47D412yes—
54C23×C4×C47D411yescharacteristic
55C2×C22⋊C4×C47—22no—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47C2×C414yes—
59C2×C22⋊C4×C47D414yes—
60C2×C22⋊C4×C47Q814yes—
61C2×C42×C47—24no—
62C2×C42×C47—44no—
63#95×C47—24no—
64C24×C47—42no—
65C24×C47—42no—
66C24×C47C22×C411yescharacteristic
67C24×C47C22×C412yes—
68C24×C47C2×D411yescharacteristic
69C24×C47C2×D411yescharacteristic
70C24×C47C2×D412yes—
71C24×C47C2×Q812yes—
72C24×C47#4112yes—
73C24×C47#4112yes—
74C24×C47#4112yes—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—24no—
78C22×C4×C47—24no—
79C22×C4×C47—24no—
80C22×C4×C47—24no—
81C22×C4×C47—21no—
82C22×C4×C47—24no—
83C22×C4×C47—24no—
84C22×C4×C47—21no—
85C22×C4×C47—24no—
86C22×C4×C47—24no—
87C22×C4×C47—42no—
88C22×C4×C47—42no—
89C22×C4×C47—42no—
90C22×C4×C47—48no—
91C22×C4×C47—48no—
92C22×C4×C47—44no—
93C22×C4×C47—44no—
94C22×C4×C47C22⋊C412yes—
95C22×C4×C47C22⋊C412yes—
96C22×Dic47—22no—
97C22×Dic47—22no—
98C22×Dic47—21no—
99C22×Dic47—21no—
100C22×Dic47—24no—
101C22×Dic47—22no—
102C22×Dic47—22no—
103C22×Dic47—42no—
104C22×Dic47—42no—
105C22×Dic47—44no—
106C22×Dic47—44no—
107C22×Dic47—42no—
108C22×Dic47—42no—
109C22×Dic47—42no—
110C22×Dic47—42no—
111C22×Dic47C4214yescomplemented
112C22⋊C4×C47—24no—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—24no—
115C22⋊C4×C47C4⋊C418yescomplemented
116C42×C47—48no—
117C22⋊Dic47—24no—
118C22⋊Dic47—24no—
119C22⋊Dic47—24no—
120C22⋊Dic47—24no—
121C22⋊Dic47—41no—
122C22⋊Dic47—42no—
123C22⋊Dic47—41no—
124C22⋊Dic47—42no—
125C22⋊Dic47—42no—
126Dic47⋊C4—44no—
127Dic47⋊C4—44no—
128C23×C47—22no—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47—22no—
132C23×C47—22no—
133C23×C47—22no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—44no—
137C23×C47—42no—
138C23×C47—44no—
139C23×C47—42no—
140C23×C47—44no—
141C23×C47—44no—
142C23×C47C2×C4211yescharacteristic
143C23×C47C2×C22⋊C411yescharacteristic
144C23×C47C2×C4⋊C412yes—
145C23×C47#11711yescharacteristic
146C23×C47#11712yes—
147C23×C47C4×D412yes—
148C23×C47C4×D412yes—
149C23×C47C4×D412yes—
150C23×C47C4×D412yes—
151C23×C47C4×Q812yes—
152C23×C47C4×Q812yes—
153C23×C47#12011yescharacteristic
154C23×C47#12112yes—
155C23×C47C22⋊Q812yes—
156C23×C47C22⋊Q812yes—
157C23×C47#12311yescharacteristic
158C23×C47#12312yes—
159C23×C47#12411yescharacteristic
160C23×C47#12512yes—
161C23×C47#12612yes—
162C23×C47C4⋊Q811yescharacteristic
163C2×C4×C47—24no—
164C2×C4×C47—24no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—28no—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—22no—
171C2×C4×C47—48no—
172C2×C4×C47—48no—
173C2×C4×C47—44no—
174C2×C4×C47—44no—
175C2×C4×C47—48no—
176C2×C4×C47—48no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47#9914yes—
180C2×Dic47—24no—
181C2×Dic47—44no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—44no—
185C2×Dic47—48no—
186C2×Dic47—41no—
187C2×Dic47—41no—
188C2×Dic47—44no—
189C2×Dic47—42no—
190C2×Dic47—42no—
191C2×Dic47—82no—
192C2×Dic47—82no—
193C2×Dic47—84no—
194C2×Dic47—82no—
195C2×Dic47—82no—
196#10912—471nomaximal, Sylow
197C22×C47—22no—
198C22×C47—22no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—22no—
202C22×C47—22no—
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—22no—
206C22×C47—24no—
207C22×C47—24no—
208C22×C47—22no—
209C22×C47—22no—
210C22×C47—22no—
211C22×C47—24no—
212C22×C47—24no—
213C22×C47—42no—
214C22×C47—42no—
215C22×C47—42no—
216C22×C47—42no—
217C22×C47—42no—
218C22×C47—42no—
219C22×C47—42no—
220C22×C47—42no—
221C22×C47C4×C22⋊C411yescharacteristic
222C22×C47C4×C4⋊C412yes—
223C22×C47#38512yes—
224C22×C47#38611yescharacteristic
225C22×C47#38712yes—
226C22×C47#38712yes—
227C22×C47#38812yes—
228C22×C47#38912yes—
229C22×C47#39111yescharacteristic
230C22×C47C2×#9912yes—
231C22×C47#41012yes—
232C4×C47—24no—
233C4×C47—48no—
234C4×C47—42no—
235Dic47—44no—
236Dic47—84no—
237Dic47—81no—
238Dic47—81no—
239C2×#385—471no—
240C2×C4×C22⋊C4D4711yescharacteristic
241C2×#342—472no—
242C2×#342—472no—
243C22⋊#95—471no—
244C2×C47—22no—
245C2×C47—22no—
246C2×C47—22no—
247C2×C47—22no—
248C2×C47—22no—
249C2×C47—22no—
250C2×C47#143611yescharacteristic
251C2×C47C4×#9912yes—
252C2×C47#189811yescharacteristic
253C2×C47#189911yescharacteristic
254C2×C47#199612yes—
255C24×C4C2×D4711yescharacteristic
256C22×C4⋊C4—472no—
257C22×C22⋊C4—471no—
258C22×C22⋊C4—472no—
259C22×C22⋊C4—471no—
260C22×C22⋊C4C2×D4712yes—
261C22×C42—21no—
262#385—942no—
263#385—941no—
264#385—941no—
265C4×C22⋊C4—24no—
266C2×#95—21no—
267C2×#95—472no—
268C2×#95—942no—
269#342—478no—
270#342—944no—
271C47#1091211yescharacteristic, complemented, Sylow
272C25C22×D4711yescharacteristic
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—44no—
277C23×C4—472no—
278C23×C4—471no—
279C23×C4—471no—
280C23×C4—942no—
281C23×C4C4×D4712yes—
282C23×C4D18812yes—
283C23×C4C22⋊D4711yescharacteristic
284C23×C4C22⋊D4711yescharacteristic
285C2×C4⋊C4—942no—
286C2×C4⋊C4—942no—
287C2×C4⋊C4—1884no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—474no—
292C2×C22⋊C4—474no—
293C2×C22⋊C4—941no—
294C2×C22⋊C4—942no—
295C2×C22⋊C4—941no—
296C2×C22⋊C4—942no—
297C2×C22⋊C4—942no—
298C2×C22⋊C4—942no—
299C2×C22⋊C4—942no—
300C2×C22⋊C4—941no—
301C2×C22⋊C4—942no—
302C2×C22⋊C4—942no—
303C2×C22⋊C4—942no—
304C2×C22⋊C4—941no—
305C2×C22⋊C4Dic9414yes—
306C2×C22⋊C4C4×D4714yes—
307C2×C22⋊C4C22⋊D4714yes—
308C2×C42—24no—
309C2×C42—44no—
310#95—24no—
311#95—942no—
312#95—942no—
313#95—1884no—
314C24—42no—
315C24—42no—
316C24C2×Dic9412yes—
317C24C2×C4×D4711yescharacteristic
318C24C2×C4×D4712yes—
319C24C2×D18811yescharacteristic
320C24#1062535012yes—
321C24#1062535012yes—
322C24#1062535012yes—
323C24C2×C22⋊D4711yescharacteristic
324C24C2×C22⋊D4712yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—474no—
345C22×C4—942no—
346C22×C4—942no—
347C22×C4—942no—
348C22×C4—942no—
349C22×C4—941no—
350C22×C4—941no—
351C22×C4—944no—
352C22×C4—1884no—
353C22×C4—1884no—
354C22×C4—1882no—
355C22×C4—1882no—
356C22×C4—1882no—
357C22×C4—1882no—
358C22×C4—1882no—
359C22×C4—1882no—
360C22×C4(C2×D47)⋊C412yes—
361C22×C4(C2×D47)⋊C412yes—
362C4⋊C4—1884no—
363C4⋊C4—1884no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—944no—
368C22⋊C4—944no—
369C22⋊C4—944no—
370C22⋊C4—944no—
371C22⋊C4—1881no—
372C22⋊C4—1882no—
373C22⋊C4—1881no—
374C22⋊C4—1882no—
375C22⋊C4—1882no—
376C22⋊C4Dic47⋊C418yescomplemented
377C42—48no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23C42×D4711yescharacteristic
393C23C2×(C2×D47)⋊C411yescharacteristic
394C23C2×Dic47⋊C412yes—
395C23#1309398111yescharacteristic
396C23#1309398112yes—
397C23C4×C22⋊D4712yes—
398C23C4×C22⋊D4712yes—
399C23C4×C22⋊D4712yes—
400C23C4×D18812yes—
401C23C4×Dic9412yes—
402C23C4×Dic9412yes—
403C23#1309399311yescharacteristic
404C23#1309399812yes—
405C23#1309400312yes—
406C23#1309400312yes—
407C23#1309400711yescharacteristic
408C23#1309400712yes—
409C23#1309401211yescharacteristic
410C23#1309401612yes—
411C23#1309401812yes—
412C23#1309402411yescharacteristic
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—944no—
430C2×C4—3762no—
431C2×C4—3762no—
432C2×C4—3764no—
433C2×C4—3762no—
434C2×C4—3762no—
435C2×C4—1888no—
436C2×C4—1881no—
437C2×C4—1881no—
438C2×C4—1884no—
439C2×C4—1882no—
440C2×C4—1882no—
441C2×C4—1884no—
442C2×C4—1882no—
443C2×C4—1882no—
444C2×C4—1884no—
445C2×C4#1309393814yes—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—21no—
454C22—24no—
455C22—22no—
456C22—24no—
457C22—22no—
458C22—24no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22C4×(C2×D47)⋊C411yescharacteristic
471C22C4×Dic47⋊C412yes—
472C22#1568899612yes—
473C22#1568900211yescharacteristic
474C22#1568900712yes—
475C22#1568900712yes—
476C22#1568901412yes—
477C22#1568902112yes—
478C22#1568902911yescharacteristic
479C22C2×#1309393812yes—
480C22#1568909612yes—
481C4—24no—
482C4—48no—
483C4—42no—
484C4—3764no—
485C4—3761no—
486C4—3761no—
487C4—1884no—
488C2—22no—
489C2—22no—
490C2—22no—
491C2—22no—
492C2—22no—
493C2—22no—
494C2#2024162111yescharacteristic
495C2C4×#1309393812yes—
496C2#2024384111yescharacteristic
497C2#2024384611yescharacteristic
498C2#2024437312yes—
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C2×C42
FrattiniC25C22×D47
FittingC2×C4×C22⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radical, cosocleC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#10912
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×Dic47⋊C4C4semidirect
14. C2×C47⋊#95C4semidirect
16. #15688889C4semidirect
111. C22×Dic47C42semidirect
115. C22⋊C4×C47C4⋊C4semidirect
271. C47#10912semidirect
376. C22⋊C4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156889962
C2×#156888892
#202416112
C2×C4×C22⋊C4×C472
#1091247