Finite Group Databasebeta

← #26217371

Subgroups — #26217371

Order 12032.

1714 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C211yesmaximal, characteristic
2#20246000C212yesmaximal, complemented
3#20245966C211yesmaximal, characteristic
4#20243669C218yesmaximal, complemented
5C2×#15688890C212yesmaximal, complemented
6C2×C47×#518C211yesmaximal, characteristic
7C22×C22⋊Dic47C2211yescharacteristic
8C22×C22⋊Dic47C2211yescharacteristic
9C22×C22⋊Dic47C2212yes—
10C2×#13093939C418yescomplemented
11C2×#13093939C2214yes—
12#15689003—22no—
13#15689003C2214yes—
14#15688996—24no—
15#15688995C2218yescomplemented
16#15688986—21no—
17#15688981—24no—
18#15688979—21no—
19#15688890C2218yescomplemented
20C23×D4×C47C2211yescharacteristic
21C47×#518—22no—
22C47×#518—24no—
23C47×#518C2214yes—
24C2×C4×D4×C47—24no—
25C2×#117×C47—21no—
26C22×C22⋊C4×C47—21no—
27C22×C22⋊C4×C47C2212yes—
28C23×Dic47—22no—
29C23×Dic47—21no—
30C23×Dic47C2312yes—
31C2×Dic47⋊C4—28no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—24no—
34C2×C22⋊Dic47—24no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47C2×C418yescomplemented
42C2×C22⋊Dic47C2312yes—
43C2×C22⋊Dic47C2314yes—
44#13093939C2×C4116yescomplemented
45C47⋊#95—28no—
46C47⋊#95—24no—
47C25×C47D411yescharacteristic
48C25×C47C2311yescharacteristic
49C22×D4×C47—22no—
50C22×D4×C47—22no—
51C22×D4×C47—24no—
52C22×D4×C47C2×C414yes—
53C22×D4×C47D412yes—
54C22×D4×C47Q814yes—
55C22×D4×C47C2312yes—
56C23×C4×C47—22no—
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D414yes—
59C4×D4×C47—216no—
60C4×D4×C47—216no—
61#117×C47—24no—
62#117×C47—24no—
63C2×C4⋊C4×C47—22no—
64C2×C22⋊C4×C47—22no—
65C2×C22⋊C4×C47—24no—
66C2×C22⋊C4×C47—22no—
67C2×C22⋊C4×C47—24no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47D414yes—
72C2×C22⋊C4×C47C2314yes—
73C2×C42×C47—22no—
74C24×C47—21no—
75C24×C47—21no—
76C24×C47—22no—
77C24×C47—82no—
78C24×C47—82no—
79C24×C47C4⋊C414yescomplemented
80C24×C47C22×C412yes—
81C24×C47C2×D412yes—
82C24×C47C2×D411yescharacteristic
83C24×C47C2×D411yescharacteristic
84C24×C47C2×D412yes—
85C24×C47C2×D412yes—
86C24×C47C2×Q812yes—
87C24×C47#4114yes—
88C24×C47C2411yescharacteristic
89C2×D4×C47—28no—
90C2×D4×C47—28no—
91C2×D4×C47—24no—
92C2×D4×C47—28no—
93C2×D4×C47—28no—
94C2×D4×C47—24no—
95C2×D4×C47—24no—
96C2×D4×C47—24no—
97C2×D4×C47C4⋊C418yescomplemented
98C2×D4×C47C22×C414yes—
99C2×D4×C47C2×Q814yes—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—24no—
104C22×C4×C47—28no—
105C22×C4×C47—24no—
106C22×C4×C47—84no—
107C22×C4×C47—84no—
108C22×C4×C47C4⋊C414yescomplemented
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2×D418yes—
111C22×Dic47—24no—
112C22×Dic47—28no—
113C22×Dic47—22no—
114C22×Dic47—22no—
115C22×Dic47—24no—
116C22×Dic47—42no—
117C22×Dic47—82no—
118C22×Dic47—84no—
119C22×Dic47—82no—
120C4⋊C4×C47—28no—
121C22⋊C4×C47—28no—
122C22⋊C4×C47—24no—
123C22⋊C4×C47—24no—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—24no—
126C42×C47—28no—
127C22⋊Dic47—28no—
128C22⋊Dic47—24no—
129C22⋊Dic47—24no—
130C22⋊Dic47—41no—
131C22⋊Dic47—42no—
132C22⋊Dic47—41no—
133C22⋊Dic47—42no—
134C22⋊Dic47—84no—
135C22⋊Dic47C22×C418yescomplemented
136Dic47⋊C4—88no—
137C23×C47—22no—
138C23×C47—24no—
139C23×C47—22no—
140C23×C47—24no—
141C23×C47—44no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—44no—
145C23×C47—42no—
146C23×C47—44no—
147C23×C47—44no—
148C23×C47—82no—
149C23×C47—82no—
150C23×C47—82no—
151C23×C47—84no—
152C23×C47—84no—
153C23×C47—84no—
154C23×C47C2×C4⋊C414yes—
155C23×C47C2×C4⋊C412yes—
156C23×C47C2×C4⋊C412yes—
157C23×C47C4×D414yes—
158C23×C47C4×D414yes—
159C23×C47#12011yescharacteristic
160C23×C47#12012yes—
161C23×C47#12011yescharacteristic
162C23×C47C22⋊Q814yes—
163C23×C47C22⋊Q814yes—
164C23×C47#12312yes—
165C23×C47#12312yes—
166C23×C47C23×C411yescharacteristic
167C23×C47C22×D411yescharacteristic
168C23×C47C22×D412yes—
169C23×C47C22×Q811yescharacteristic
170C23×C47C2×#4112yes—
171D4×C47—28no—
172D4×C47—28no—
173D4×C47—216no—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—24no—
177C2×C4×C47—24no—
178C2×C4×C47—28no—
179C2×C4×C47—48no—
180C2×C4×C47—48no—
181C2×C4×C47—84no—
182C2×C4×C47—88no—
183C2×C4×C47C2×C4⋊C414yes—
184C2×Dic47—28no—
185C2×Dic47—48no—
186C2×Dic47—48no—
187C2×Dic47—44no—
188C2×Dic47—44no—
189C2×Dic47—42no—
190C2×Dic47—42no—
191C2×Dic47—88no—
192C2×Dic47—84no—
193#20889—471nomaximal, Sylow
194C22×C47—24no—
195C22×C47—22no—
196C22×C47—22no—
197C22×C47—24no—
198C22×C47—22no—
199C22×C47—24no—
200C22×C47—44no—
201C22×C47—42no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—42no—
205C22×C47—44no—
206C22×C47—44no—
207C22×C47—44no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47—42no—
211C22×C47—44no—
212C22×C47—48no—
213C22×C47—44no—
214C22×C47—82no—
215C22×C47—84no—
216C22×C47#38512yes—
217C22×C47#38512yes—
218C22×C47#38514yes—
219C22×C47C22×C4⋊C411yescharacteristic
220C22×C47C2×C4×D412yes—
221C22×C47C2×#12011yescharacteristic
222C22×C47C2×C22⋊Q812yes—
223C22×C47C2×#12311yescharacteristic
224C4×C47—24no—
225C4×C47—48no—
226Dic47—48no—
227Dic47—82no—
228Dic47—82no—
229C2×#518D4711yescharacteristic
230C22×#99—471no—
231#2315—472no—
232#2308—471no—
233#1867—478no—
234C2×#342—472no—
235C2×C47—22no—
236C2×C47—21no—
237C2×C47—21no—
238C2×C47—22no—
239C2×C47—22no—
240C2×C47—22no—
241C2×C47—24no—
242C2×C47—44no—
243C2×C47C2×#38511yescharacteristic
244C2×C47#302612yes—
245C23×D4C2×D4711yescharacteristic
246#518—22no—
247#518—24no—
248#518C2×D4714yes—
249C2×C4×D4—24no—
250C2×#117—21no—
251C22×C22⋊C4—21no—
252C22×C22⋊C4—472no—
253C22×C22⋊C4—471no—
254C22×C22⋊C4—471no—
255C22×C22⋊C4C2×D4712yes—
256C2×#99—478no—
257C2×#99—474no—
258#386—474no—
259#386—942no—
260#385—478no—
261#385—944no—
262#381—941no—
263#380—944no—
264#379—941no—
265#342—478no—
266C47#2088911yescharacteristic, complemented, Sylow
267C25C22⋊D4711yescharacteristic
268C25C22×D4711yescharacteristic
269C22×D4—24no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4Dic9414yes—
273C22×D4C4×D4714yes—
274C22×D4C22⋊D4712yes—
275C22×D4C22×D4712yes—
276C23×C4—22no—
277C23×C4—472no—
278C23×C4—942no—
279C23×C4—941no—
280C23×C4C22⋊D4711yescharacteristic
281C23×C4C22⋊D4714yes—
282C4×D4—216no—
283C4×D4—216no—
284#117—24no—
285#117—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—948no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—478no—
296C2×C22⋊C4—474no—
297C2×C22⋊C4—472no—
298C2×C22⋊C4—944no—
299C2×C22⋊C4—942no—
300C2×C22⋊C4—944no—
301C2×C22⋊C4—944no—
302C2×C22⋊C4—941no—
303C2×C22⋊C4—941no—
304C2×C22⋊C4—942no—
305C2×C22⋊C4—942no—
306C2×C22⋊C4—941no—
307C2×C22⋊C4C22⋊D4714yes—
308C2×C22⋊C4C22×D4714yes—
309C2×C42—22no—
310#99—4716no—
311#95—948no—
312#95—944no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—82no—
317C24—82no—
318C24Dic47⋊C414yescomplemented
319C24C2×Dic9412yes—
320C24C2×C4×D4712yes—
321C24#1062535014yes—
322C24C2×C22⋊D4712yes—
323C24C2×C22⋊D4711yescharacteristic
324C24C2×C22⋊D4711yescharacteristic
325C24C2×C22⋊D4712yes—
326C24C2×C22⋊D4712yes—
327C24C23×D4711yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4Dic47⋊C418yescomplemented
337C2×D4C2×Dic9414yes—
338C2×D4C2×C4×D4714yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—948no—
348C22×C4—944no—
349C22×C4—942no—
350C22×C4—942no—
351C22×C4—944no—
352C22×C4—3762no—
353C22×C4—3764no—
354C22×C4—3762no—
355C22×C4—1882no—
356C22×C4Dic47⋊C414yescomplemented
357C22×C4C2×C22⋊D4712yes—
358C22×C4C2×C22⋊D4718yes—
359C4⋊C4—28no—
360C4⋊C4—3768no—
361C22⋊C4—28no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—478no—
367C22⋊C4—948no—
368C22⋊C4—944no—
369C22⋊C4—944no—
370C22⋊C4—3764no—
371C22⋊C4—1881no—
372C22⋊C4—1882no—
373C22⋊C4—1881no—
374C22⋊C4—1882no—
375C42—28no—
376C23—24no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—44no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23C2×Dic47⋊C414yes—
394C23C2×Dic47⋊C412yes—
395C23C2×Dic47⋊C412yes—
396C23C4×C22⋊D4714yes—
397C23C4×C22⋊D4714yes—
398C23#1309399311yescharacteristic
399C23#1309399312yes—
400C23#1309399311yescharacteristic
401C23#1309400314yes—
402C23#1309400314yes—
403C23#1309400712yes—
404C23#1309400712yes—
405C23C22×C4×D4711yescharacteristic
406C23C22×C22⋊D4711yescharacteristic
407C23C22×C22⋊D4712yes—
408C23C22×Dic9411yescharacteristic
409C23C2×#1062535012yes—
410D4—216no—
411D4—28no—
412D4—28no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—948no—
423C2×C4—3764no—
424C2×C4—3768no—
425C2×C4—1884no—
426C2×C4—1884no—
427C2×C4—1888no—
428C2×C4—1882no—
429C2×C4—1882no—
430C2×C4—1888no—
431C2×C4C2×Dic47⋊C414yes—
432C22—22no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—24no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—44no—
450C22—48no—
451C22—44no—
452C22—82no—
453C22—84no—
454C22#1568899612yes—
455C22#1568899612yes—
456C22#1568899614yes—
457C22C22×Dic47⋊C411yescharacteristic
458C22C2×C4×C22⋊D4712yes—
459C22C2×#1309399311yescharacteristic
460C22C2×#1309400312yes—
461C22C2×#1309400711yescharacteristic
462C4—24no—
463C4—48no—
464C4—3762no—
465C4—3762no—
466C4—1888no—
467C2—21no—
468C2—22no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—24no—
473C2—22no—
474C2—44no—
475C2C2×#1568899611yescharacteristic
476C2#2025278512yes—
subgroupG/H
centerC22C2×#13093993
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#518C2
socleC22×C47C2×#120
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC2×#518D47
hypocenterC47#20889
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
2. #20246000C2semidirect
4. #20243669C2semidirect
5. C2×#15688890C2semidirect
10. C2×#13093939C4semidirect
15. #15688995C22semidirect
19. #15688890C22semidirect
41. C2×C22⋊Dic47C2×C4semidirect
44. #13093939C2×C4semidirect
79. C24×C47C4⋊C4semidirect
97. C2×D4×C47C4⋊C4semidirect
108. C22×C4×C47C4⋊C4semidirect
135. C22⋊Dic47C22×C4semidirect
266. C47#20889semidirect
318. C24Dic47⋊C4semidirect
336. C2×D4Dic47⋊C4semidirect
356. C22×C4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130939392
#202460002
#202459662
#202436692
C2×#156888902
C2×C47×#5182
#2088947