Finite Group Databasebeta

← #26244417

Subgroups — #26244417

Order 12032.

1334 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689749C211yesmaximal, characteristic
2#20246002C212yesmaximal
3#20246000C211yesmaximal, characteristic
4#20244742C218yesmaximal, complemented
5C2×#15688889C212yesmaximal
6C2×C47×#342C211yesmaximal, characteristic
7#15689749—24no—
8#15689749—22no—
9#15689749C2214yes—
10C2×#13094003—22no—
11C2×#13094001—24no—
12C22×C22⋊Dic47C2211yescharacteristic
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2211yescharacteristic
15#15688998—24no—
16#15688998C2218yes—
17#15688996—22no—
18#15688995C2214yes—
19#15688983—24no—
20#15688981—22no—
21#15688889C2218yes—
22C22×C22⋊C4×C47C2211yescharacteristic
23C22×C22⋊C4×C47C2212yes—
24C47×#342C2214yes—
25C22×Dic94—22no—
26C23×Dic47—21no—
27C23×Dic47D414yes—
28C23×Dic47C2312yes—
29#13094003—28no—
30#13094003—28no—
31#13094001—216no—
32#13094001—216no—
33C2×C4⋊Dic47—22no—
34C2×C4⋊Dic47—28no—
35C2×Dic47⋊C4—24no—
36C2×Dic47⋊C4—24no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—22no—
41C2×C22⋊Dic47—22no—
42C2×C22⋊Dic47—22no—
43C2×C22⋊Dic47—21no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—21no—
46C2×C22⋊Dic47—22no—
47C2×C22⋊Dic47D414yes—
48C2×C22⋊Dic47Q814yescomplemented
49C2×C22⋊Dic47C2314yes—
50C2×C22⋊Dic47C2312yes—
51C47⋊#95—24no—
52C47⋊#95—28no—
53C25×C47C2311yescharacteristic
54C23×C4×C47—22no—
55C23×C4×C47D412yes—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47—22no—
59C2×C22⋊C4×C47—24no—
60C2×C22⋊C4×C47D412yes—
61C2×C22⋊C4×C47Q818yescomplemented
62C2×C22⋊C4×C47C2312yes—
63C2×C22⋊C4×C47C2314yes—
64C24×C47—82no—
65C24×C47C2×D411yescharacteristic
66C24×C47C2×D411yescharacteristic
67C24×C47C2×D412yes—
68C24×C47C2×D412yes—
69C24×C47C2×Q814yes—
70C24×C47C2×Q812yes—
71C24×C47#4112yes—
72C24×C47C2411yescharacteristic
73C22×C4×C47—22no—
74C22×C4×C47—24no—
75C22×C4×C47—22no—
76C22×C4×C47—28no—
77C22×C4×C47—82no—
78C22×C4×C47—84no—
79C22×C4×C47C2×D414yes—
80C22×Dic47—24no—
81C22×Dic47—22no—
82C22×Dic47—24no—
83C22×Dic47—24no—
84C22×Dic47—22no—
85C22×Dic47—24no—
86C22×Dic47—82no—
87C22×Dic47—84no—
88C22×Dic47—82no—
89C22×Dic47C2×D418yes—
90C2×Dic94—28no—
91C2×Dic94—84no—
92C22⋊C4×C47—24no—
93C22⋊C4×C47—28no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47C2×Q818yescomplemented
97C22⋊Dic47—24no—
98C22⋊Dic47—24no—
99C22⋊Dic47—24no—
100C22⋊Dic47—24no—
101C22⋊Dic47—24no—
102C22⋊Dic47—22no—
103C22⋊Dic47—41no—
104C22⋊Dic47—42no—
105C22⋊Dic47—41no—
106C22⋊Dic47C2×Q814yescomplemented
107C4⋊Dic47—28no—
108C4⋊Dic47—88no—
109Dic47⋊C4—216no—
110Dic47⋊C4—84no—
111C23×C47—42no—
112C23×C47—44no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—44no—
116C23×C47—84no—
117C23×C47—82no—
118C23×C47—82no—
119C23×C47#12011yescharacteristic
120C23×C47#12012yes—
121C23×C47#12011yescharacteristic
122C23×C47C22⋊Q814yes—
123C23×C47C22⋊Q814yes—
124C23×C47C22⋊Q812yes—
125C23×C47C22⋊Q812yes—
126C23×C47C4⋊Q814yes—
127C23×C47C4⋊Q814yes—
128C23×C47C4⋊Q812yes—
129C23×C47C4⋊Q812yes—
130C23×C47C22×D412yes—
131C23×C47C22×D411yescharacteristic
132C23×C47C22×Q811yescharacteristic
133C23×C47C22×Q812yes—
134C23×C47C2×#4111yescharacteristic
135C2×C4×C47—24no—
136C2×C4×C47—28no—
137C2×C4×C47—48no—
138C2×C4×C47—44no—
139C2×C4×C47—44no—
140C2×C4×C47—82no—
141C2×C4×C47—88no—
142C2×Dic47—28no—
143C2×Dic47—24no—
144C2×Dic47—44no—
145C2×Dic47—48no—
146C2×Dic47—48no—
147C2×Dic47—44no—
148C2×Dic47—48no—
149C2×Dic47—42no—
150C2×Dic47—84no—
151C2×Dic47—84no—
152Dic94—88no—
153#23565—471nomaximal, Sylow
154C22×C47—22no—
155C22×C47—24no—
156C22×C47—22no—
157C22×C47—24no—
158C22×C47—22no—
159C22×C47—42no—
160C22×C47—44no—
161C22×C47—44no—
162C22×C47—42no—
163C22×C47—44no—
164C22×C47—44no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47—41no—
168C22×C47—42no—
169C22×C47—41no—
170C22×C47#39512yes—
171C22×C47#39514yes—
172C22×C47#39512yes—
173C22×C47C2×#12011yescharacteristic
174C22×C47C2×C22⋊Q812yes—
175C22×C47C2×C22⋊Q811yescharacteristic
176C22×C47C2×C4⋊Q812yes—
177C22×C47C2×C4⋊Q811yescharacteristic
178C4×C47—48no—
179C4×C47—44no—
180Dic47—44no—
181Dic47—48no—
182Dic47—82no—
183C2×#543—471no—
184#2315—471no—
185#2315—472no—
186#2062—478no—
187C2×#342—472no—
188C2×#342D4711yescharacteristic
189C2×C47—21no—
190C2×C47—22no—
191C2×C47—22no—
192C2×C47—21no—
193C2×C47—22no—
194C2×C47—24no—
195C2×C47—22no—
196C2×C47C2×#39511yescharacteristic
197C2×C47#302612yes—
198#543—474no—
199#543—944no—
200#543—942no—
201C2×C22⋊Q8—944no—
202C2×C22⋊Q8—942no—
203C22×C22⋊C4—471no—
204C22×C22⋊C4—472no—
205C22×C22⋊C4—471no—
206C22×C22⋊C4C2×D4711yescharacteristic
207C22×C22⋊C4C2×D4712yes—
208#385—474no—
209#385—478no—
210#385—944no—
211#385—942no—
212#380—944no—
213#380—942no—
214#342—478no—
215#342C2×D4714yes—
216C47#2356511yescharacteristic, complemented, Sylow
217C25C22×D4711yescharacteristic
218C22×Q8—942no—
219C23×C4—22no—
220C23×C4—474no—
221C23×C4—472no—
222C23×C4—941no—
223C23×C4D18812yes—
224C22⋊Q8—9416no—
225C22⋊Q8—9416no—
226C22⋊Q8—948no—
227C22⋊Q8—948no—
228C2×C4⋊C4—944no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4—948no—
231C2×C4⋊C4—944no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—474no—
237C2×C22⋊C4—472no—
238C2×C22⋊C4—474no—
239C2×C22⋊C4—474no—
240C2×C22⋊C4—942no—
241C2×C22⋊C4—942no—
242C2×C22⋊C4—941no—
243C2×C22⋊C4—942no—
244C2×C22⋊C4—942no—
245C2×C22⋊C4—942no—
246C2×C22⋊C4—941no—
247C2×C22⋊C4—941no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4—942no—
250C2×C22⋊C4Dic9418yescomplemented
251C2×C22⋊C4D18812yes—
252C2×C22⋊C4C22×D4712yes—
253C2×C22⋊C4C22×D4714yes—
254#95—948no—
255#95—944no—
256C24—82no—
257C24C2×Dic9414yes—
258C24C2×D18811yescharacteristic
259C24C2×D18811yescharacteristic
260C24D4×D4712yes—
261C24D4×D4712yes—
262C24#1062535212yes—
263C24Q8×D4712yes—
264C24C23×D4711yescharacteristic
265C2×Q8—948no—
266C2×Q8—3764no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—478no—
274C22×C4—944no—
275C22×C4—944no—
276C22×C4—944no—
277C22×C4—942no—
278C22×C4—944no—
279C22×C4—942no—
280C22×C4—3762no—
281C22×C4—3764no—
282C22×C4—3762no—
283C22×C4C2×D18814yes—
284C4⋊C4—9416no—
285C4⋊C4—948no—
286C4⋊C4—3764no—
287C4⋊C4—3768no—
288C22⋊C4—24no—
289C22⋊C4—28no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—474no—
293C22⋊C4—944no—
294C22⋊C4—944no—
295C22⋊C4—944no—
296C22⋊C4—944no—
297C22⋊C4—944no—
298C22⋊C4—942no—
299C22⋊C4—1882no—
300C22⋊C4—1881no—
301C22⋊C4—1881no—
302C22⋊C4C2×Dic9418yescomplemented
303C23—42no—
304C23—44no—
305C23—42no—
306C23—44no—
307C23—42no—
308C23—84no—
309C23—82no—
310C23—82no—
311C23#1309399412yes—
312C23#1309399411yescharacteristic
313C23#1309399411yescharacteristic
314C23#1309400114yes—
315C23#1309400114yes—
316C23#1309400412yes—
317C23#1309400412yes—
318C23#1309402314yes—
319C23#1309402314yes—
320C23#1309402412yes—
321C23#1309402412yes—
322C23C2×D4×D4712yes—
323C23C22×D18811yescharacteristic
324C23C2×Q8×D4711yescharacteristic
325C23C22×Dic9412yes—
326C23C2×#1062535211yescharacteristic
327Q8—3768no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—948no—
336C2×C4—944no—
337C2×C4—3764no—
338C2×C4—3764no—
339C2×C4—1888no—
340C2×C4—1884no—
341C2×C4—1882no—
342C2×C4—1884no—
343C2×C4—1888no—
344C2×C4—1888no—
345C22—22no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—44no—
353C22—42no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22#1568904414yes—
362C22#1568904412yes—
363C22#1568904412yes—
364C22C2×#1309399411yescharacteristic
365C22C2×#1309400112yes—
366C22C2×#1309400411yescharacteristic
367C22C2×#1309402312yes—
368C22C2×#1309402411yescharacteristic
369C4—48no—
370C4—44no—
371C4—3762no—
372C4—1884no—
373C4—1888no—
374C2—22no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—24no—
379C2—22no—
380C2—21no—
381C2C2×#1568904411yescharacteristic
382C2#2025278712yes—
subgroupG/H
centerC22C2×#13093994
commutator, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC2×C47×#342C2
socleC22×C47C2×#120
prime radicalC25×C47C23
hypercenterC2×#342D47
hypocenterC47#23565
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #20244742C2semidirect
48. C2×C22⋊Dic47Q8semidirect
61. C2×C22⋊C4×C47Q8semidirect
96. C22⋊C4×C47C2×Q8semidirect
106. C22⋊Dic47C2×Q8semidirect
216. C47#23565semidirect
250. C2×C22⋊C4Dic94semidirect
302. C22⋊C4C2×Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202460022
#202460002
#202447422
C2×#156888892
C2×C47×#3422
#2356547