Finite Group Databasebeta

← #26219952

Subgroups — #26219952

Order 12032.

1638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689671C211yesmaximal, characteristic
2C2×#15689121C214yesmaximal, complemented
3C2×#15689075C211yesmaximal, characteristic, complemented
4#382×D47C211yesmaximal, characteristic, complemented
5#20244213C211yesmaximal, characteristic, complemented
6#20244208C211yesmaximal, characteristic, complemented
7#20244207C211yesmaximal, characteristic, complemented
8#20244198C214yesmaximal, complemented
9C47×#1961C211yesmaximal, characteristic, complemented
10C22×#10625352C2212yes—
11C22×#10625324C2212yes—
12#15689671—24no—
13#15689671C418yescomplemented
14C2×#13094024C2211yescharacteristic
15C2×#13094011C2212yes—
16C2×#13093995—22no—
17C2×D4×Dic47—22no—
18C2×C4⋊C4×D47C2212yes—
19C2×C42×D47C2211yescharacteristic
20#15689121—216no—
21C2×#13093949C2214yes—
22C2×#13093947C2214yes—
23C2×#13093945C2214yes—
24#15689075—24no—
25C2×#13093935C2211yescharacteristic
26C2×C8×Dic47C2212yes—
27#15688989C2211yescharacteristic
28#15688988C2212yes—
29#15688987—22no—
30#95×D47—22no—
31C2×#127×C47C2211yescharacteristic
32C2×#102×C47C2214yes—
33C2×#97×C47C2211yescharacteristic
34C47×#382C2211yescharacteristic
35C2×#10625352—28no—
36C2×#10625352—24no—
37C2×#10625352—28no—
38C2×#10625352C2×C418yes—
39C22×Dic94C2312yes—
40C22×C22⋊D47—22no—
41C23×Dic47—22no—
42C22×C4×D47—22no—
43C22×C4×D47D412yes—
44C22×C4×D47C2311yescharacteristic
45C2×#10625324—24no—
46C2×#10625324—24no—
47C2×#10625324D418yescomplemented
48C22×C47⋊C8C2312yes—
49#13094024—22no—
50#13094024C2×C414yescomplemented
51#13094011—24no—
52#13094011C2×C418yescomplemented
53#13093995—216no—
54D4×Dic47—216no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47C2312yes—
57C2×Dic47⋊C4—22no—
58C4⋊C4×D47—28no—
59C2×C22⋊Dic47—24no—
60C2×C4×Dic47D412yes—
61C2×C4×Dic47C2311yescharacteristic
62C42×D47C2×C414yes—
63C42×D47D414yes—
64#13093949—28no—
65#13093947—28no—
66#13093945—28no—
67#13093935—22no—
68C8×Dic47—24no—
69C47⋊#95—22no—
70C47⋊#95—24no—
71C22×D4×C47—22no—
72C22×D4×C47C2312yes—
73C22×C8×C47C2312yes—
74#127×C47—22no—
75#127×C47C2×C414yescomplemented
76C2×C4⋊C4×C47C2312yes—
77C2×C42×C47C2311yescharacteristic
78#102×C47—28no—
79#97×C47—22no—
80#95×C47—22no—
81C24×C47—22no—
82C23×D47C2×D411yescharacteristic
83C2×D4×C47—28no—
84C2×D4×C47—24no—
85C2×D4×C47—22no—
86C2×D4×C47—24no—
87C2×D4×C47—48no—
88C2×D4×C47C22×C414yes—
89C22×C4×C47—22no—
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2411yescharacteristic
92C2×C22⋊D47—28no—
93C2×C22⋊D47—44no—
94C2×C22⋊D47—44no—
95C22×Dic47—22no—
96C22×Dic47—28no—
97C22×Dic47—22no—
98C22×Dic47—44no—
99C22×Dic47—44no—
100C22×Dic47C2×D412yes—
101C22×Dic47C2×D411yescharacteristic
102#10625352—232no—
103#10625352—416no—
104C2×C4×D47—44no—
105C2×C4×D47—48no—
106C2×C4×D47—44no—
107C2×C4×D47C22⋊C418yes—
108C2×C4×D47C22×C412yes—
109C2×C4×D47C2×D414yes—
110C2×C4×D47C2×D412yes—
111C2×C4×D47#4114yes—
112C2×Dic94—24no—
113C2×Dic94—24no—
114C2×Dic94—22no—
115C2×Dic94C22×C414yes—
116C2×C8×C47—22no—
117C2×C8×C47—22no—
118C2×C8×C47C2×D414yes—
119C4⋊C4×C47—24no—
120C42×C47C22×C412yes—
121C42×C47C2×D412yes—
122C22⋊Dic47—216no—
123C4⋊Dic47—28no—
124C4⋊Dic47—24no—
125Dic47⋊C4—44no—
126C4×Dic47C22⋊C418yes—
127C4×Dic47C22×C412yes—
128C4×Dic47C2×D412yes—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8C2×D414yes—
132#10625324—216no—
133C23×C47—24no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—48no—
137C23×C47C22×D411yescharacteristic
138C22×D47—24no—
139C22×D47C2×C22⋊C412yes—
140C22×D47#12412yes—
141C22×D47#12712yes—
142D4×C47—28no—
143D4×C47—416no—
144D4×C47—44no—
145C2×C4×C47—24no—
146C2×C4×C47—42no—
147C2×C4×C47—44no—
148C2×C4×C47C2×C22⋊C414yes—
149C2×C4×C47C23×C411yescharacteristic
150C2×C4×C47C22×D412yes—
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C2×#4112yes—
153C22⋊D47—416no—
154C2×Dic47—24no—
155C2×Dic47—24no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—416no—
159C2×Dic47—42no—
160C2×Dic47—44no—
161C2×Dic47C2×C22⋊C414yes—
162C2×Dic47C2×C22⋊C412yes—
163C2×Dic47#12412yes—
164C2×Dic47#12712yes—
165C4×D47—48no—
166C4×D47—88no—
167C4×D47C4×D418yes—
168C4×D47#12118yes—
169Dic94—28no—
170Dic94—44no—
171C8×C47—24no—
172C47⋊C8—24no—
173#21201—471nomaximal, Sylow
174C22×C47—48no—
175C22×C47—44no—
176C22×C47—84no—
177C22×C47—84no—
178C22×C47C22×C22⋊C411yescharacteristic
179C22×C47C2×#12411yescharacteristic
180C22×C47C2×#12711yescharacteristic
181C22×C47#57514yes—
182C2×D47—24no—
183C2×D47—24no—
184C2×D47—24no—
185C2×D47#39014yes—
186C4×C47—24no—
187C4×C47—82no—
188C4×C47C2×C4×D412yes—
189C4×C47C2×#12112yes—
190Dic47—24no—
191Dic47—44no—
192Dic47—82no—
193Dic47#39014yes—
194C2×#532—471no—
195C2×#417—474no—
196C2×C4×M16—471no—
197C2×#382—471no—
198#1961—471no—
199#1961D4711yescharacteristic, complemented
200#1960—471no—
201#1960—471no—
202#1958—474no—
203C2×C47—84no—
204C2×C47C2×#39011yescharacteristic
205C2×C47#289612yes—
206C2×C47#314212yes—
207C2×C47#315312yes—
208D47—24no—
209C22×#41—472no—
210C22×M16—472no—
211#532—478no—
212#532—944no—
213C2×C4⋊Q8—471no—
214C2×#127C2×D4711yescharacteristic
215C2×#124—472no—
216C2×#121—942no—
217C2×C4×D4—942no—
218C22×C4⋊C4—472no—
219C22×C42—471no—
220#417—9416no—
221C2×#103—474no—
222C2×#103—474no—
223C2×#102—474no—
224C2×#102C2×D4714yes—
225C4×M16—944no—
226C2×#97—471no—
227C2×#97C2×D4711yescharacteristic
228C2×C4×C8—472no—
229#382—472no—
230#382—471no—
231#382—942no—
232#382C2×D4711yescharacteristic
233C2×#95—942no—
234C47#2120111yescharacteristic, complemented, Sylow
235C2×#41—478no—
236C2×#41—944no—
237C2×#41—948no—
238C2×#41—948no—
239C22×Q8—472no—
240C22×D4—22no—
241C22×D4—942no—
242C22×D4C22×D4712yes—
243C23×C4—471no—
244C23×C4—472no—
245C23×C4—942no—
246C23×C4—942no—
247C2×M16—478no—
248C2×M16—944no—
249C2×M16—944no—
250C22×C8—472no—
251C22×C8C22×D4712yes—
252C4⋊Q8—474no—
253C4⋊Q8—942no—
254#127—22no—
255#127C4×D4714yescomplemented
256#124—478no—
257#124—944no—
258#121—9416no—
259C4×D4—9416no—
260C2×C4⋊C4—472no—
261C2×C4⋊C4—948no—
262C2×C4⋊C4—942no—
263C2×C4⋊C4—942no—
264C2×C4⋊C4C22×D4712yes—
265C2×C22⋊C4—944no—
266C2×C42—472no—
267C2×C42—471no—
268C2×C42—474no—
269C2×C42—474no—
270C2×C42C22×D4711yescharacteristic
271#103—948no—
272#103—948no—
273#102—28no—
274#102—948no—
275#97—22no—
276#97—942no—
277C4×C8—944no—
278#95—22no—
279#95—944no—
280#95—942no—
281C24—22no—
282C24—471no—
283#41—9432no—
284#41—18816no—
285C2×Q8—474no—
286C2×Q8—942no—
287C2×Q8—944no—
288C2×Q8—944no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—48no—
294C2×D4—948no—
295C2×D4—1884no—
296C2×D4—1884no—
297C2×D4C2×C4×D4714yes—
298C22×C4—22no—
299C22×C4—471no—
300C22×C4—472no—
301C22×C4—472no—
302C22×C4—474no—
303C22×C4—474no—
304C22×C4—472no—
305C22×C4—478no—
306C22×C4—948no—
307C22×C4—942no—
308C22×C4—942no—
309C22×C4—1884no—
310C22×C4—1884no—
311C22×C4—1888no—
312C22×C4—1884no—
313C22×C4—1884no—
314C22×C4D4×D4712yes—
315C22×C4C23×D4711yescharacteristic
316M16—9416no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—474no—
320C2×C8—942no—
321C2×C8—942no—
322C2×C8D4×D4714yes—
323C4⋊C4—24no—
324C4⋊C4—948no—
325C4⋊C4—944no—
326C4⋊C4—1884no—
327C22⋊C4—9416no—
328C42—478no—
329C42—472no—
330C42—472no—
331C42C2×C4×D4712yes—
332C42D4×D4712yes—
333C23—24no—
334C23—42no—
335C23—42no—
336C23—48no—
337C23—472no—
338C23—472no—
339C23—472no—
340C23—944no—
341C23C2×D4×D4711yescharacteristic
342Q8—948no—
343Q8—1884no—
344D4—28no—
345D4—416no—
346D4—44no—
347D4—18816no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—472no—
352C2×C4—472no—
353C2×C4—472no—
354C2×C4—478no—
355C2×C4—478no—
356C2×C4—474no—
357C2×C4—944no—
358C2×C4—944no—
359C2×C4—3768no—
360C2×C4—1888no—
361C2×C4—18816no—
362C2×C4—1882no—
363C2×C4—1884no—
364C2×C4—1882no—
365C2×C4—1882no—
366C2×C4C22⋊C4×D4714yes—
367C2×C4C22×C4×D4711yescharacteristic
368C2×C4C2×D4×D4712yes—
369C2×C4C2×D4×D4711yescharacteristic
370C2×C4#41×D4712yes—
371C8—24no—
372C8—944no—
373C22—44no—
374C22—48no—
375C22—84no—
376C22—84no—
377C22—474no—
378C22—944no—
379C22—944no—
380C22—944no—
381C22C2×C22⋊C4×D4711yescharacteristic
382C22#124×D4711yescharacteristic
383C22#127×D4711yescharacteristic
384C22#1568991914yes—
385C4—24no—
386C4—82no—
387C4—474no—
388C4—944no—
389C4—3762no—
390C4—1884no—
391C4C4×D4×D4712yes—
392C4#121×D4712yes—
393C2—84no—
394C2—944no—
395C2#390×D4711yescharacteristic
396C2#2025190212yes—
397C2#2025378712yes—
398C2#2025385312yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1961C2
socleC23×C47C22×D4
prime radicalC2×#15689671C2
cosocleC22×C4×C47C24
hypercenter#1961D47
hypocenterC47#21201
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. C2×#15689121C2semidirect
3. C2×#15689075C2semidirect
4. #382×D47C2semidirect
5. #20244213C2semidirect
6. #20244208C2semidirect
7. #20244207C2semidirect
8. #20244198C2semidirect
9. C47×#1961C2semidirect
13. #15689671C4semidirect
47. C2×#10625324D4semidirect
50. #13094024C2×C4semidirect
52. #13094011C2×C4semidirect
75. #127×C47C2×C4semidirect
199. #1961D47semidirect
234. C47#21201semidirect
255. #127C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896712
C2×#156891212
C2×#156890752
#382×D472
#202442132
#202442082
#202442072
#202441982
C47×#19612
#2120147