Finite Group Databasebeta

← #27085941

Subgroups — #27085941

Order 12032.

1890 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D47C212yesmaximal
2#20253939C218yesmaximal, complemented
3#20253933C218yesmaximal, complemented
4#20253912C211yesmaximal, characteristic, complemented
5#20253910C212yesmaximal, complemented
6#20253900C214yesmaximal, complemented
7#20251976C214yesmaximal
8C2×#15689075C211yesmaximal, characteristic
9C47×#3162C211yesmaximal, characteristic, complemented
10C22×#10625324C2212yes—
11#15689680C2212yes—
12#15689679C2214yes—
13#15689678C2214yes—
14#15689677—24no—
15#15689676C2214yes—
16#15689675—24no—
17C4⋊Q8×D47C2214yes—
18#125×D47C2214yes—
19C22⋊Q8×D47—24no—
20C4×Q8×D47—24no—
21#117×D47C2214yes—
22C2×C42×D47C2211yescharacteristic
23#15689533C2218yescomplemented
24#15689532C2218yescomplemented
25#15689529C2218yescomplemented
26#15689527C2218yescomplemented
27#15689524C2218yescomplemented
28#15689523C2218yescomplemented
29#15689520C2218yescomplemented
30#15689164C22116yes—
31#15689162C22116yes—
32#15689161C2214yes—
33#15689159C2218yes—
34#15689075C2214yes—
35#15689075C2214yes—
36C2×#13093935C2211yescharacteristic
37C2×C8×Dic47C2212yes—
38C47×#533C2212yes—
39C47×#501C2218yescomplemented
40C47×#427C2214yes—
41C2×#97×C47C2211yescharacteristic
42C2×Q8×D47—24no—
43C22×C4×D47D412yes—
44C22×C4×D47C2311yescharacteristic
45C2×#10625324Q8116yescomplemented
46C2×#10625324C2318yes—
47C22×C47⋊C8C2312yes—
48#13094024C2314yes—
49#13094023—24no—
50#13094023C2318yes—
51#13094022—24no—
52#13094016C2314yes—
53#13094015C2318yes—
54#13094014—28no—
55#13094004—24no—
56#13094003—24no—
57#13094002—28no—
58#13094001—28no—
59#13094000—24no—
60Q8×Dic47—24no—
61C4×Dic94—24no—
62C4×Dic94—28no—
63#13093989—24no—
64#13093989—28no—
65#13093983C2314yes—
66#13093982C2318yes—
67#13093981—24no—
68#13093980—24no—
69C4⋊C4×D47—28no—
70C4⋊C4×D47—24no—
71C4⋊C4×D47C2318yes—
72C22⋊C4×D47—24no—
73C2×C4×Dic47D412yes—
74C2×C4×Dic47C2311yescharacteristic
75C42×D47—24no—
76C42×D47D414yes—
77C42×D47C2312yes—
78C42×D47C2312yes—
79#13093958C2318yes—
80#13093957C23116yes—
81#13093956C2318yes—
82#13093955C23116yes—
83#13093935C2314yes—
84C8×Dic47C2314yes—
85C8×Dic47C2314yes—
86C22×C8×C47C2312yes—
87C4⋊Q8×C47C2314yes—
88#125×C47C2314yes—
89C22⋊Q8×C47—24no—
90C4×Q8×C47—24no—
91#117×C47C2314yes—
92C2×C42×C47C2311yescharacteristic
93#107×C47C2318yes—
94#106×C47C2318yes—
95#97×C47C2314yes—
96C23×D47C2×D411yescharacteristic
97C2×Q8×C47—24no—
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2411yescharacteristic
100C22×Dic47C2×D412yes—
101C22×Dic47C2×D411yescharacteristic
102Q8×D47—416no—
103C2×C4×D47—28no—
104C2×C4×D47—44no—
105C2×C4×D47C2×D414yes—
106C2×C4×D47C2×D414yes—
107C2×C4×D47C2×D412yes—
108C2×C4×D47C2×Q818yes—
109C2×C4×D47C2412yes—
110C2×Dic94—24no—
111C2×Dic94—28no—
112C2×C8×C47C2×Q818yescomplemented
113C2×C8×C47C2414yes—
114C4⋊C4×C47—28no—
115C4⋊C4×C47—24no—
116C4⋊C4×C47C2418yes—
117C22⋊C4×C47—24no—
118C42×C47—24no—
119C42×C47C2×D412yes—
120C42×C47C2412yes—
121C22⋊Dic47—24no—
122(C2×D47)⋊C4—28no—
123C4⋊Dic47—24no—
124C4⋊Dic47—28no—
125C4⋊Dic47C2418yes—
126Dic47⋊C4—28no—
127Dic47⋊C4—28no—
128Dic47⋊C4—28no—
129Dic47⋊C4—28no—
130C4×Dic47—24no—
131C4×Dic47—28no—
132C4×Dic47C2×D412yes—
133C4×Dic47C2×D414yes—
134C4×Dic47C2×D414yes—
135C4×Dic47C2411yescharacteristic
136C4×Dic47C2411yescharacteristic
137C2×C47⋊C8C2×Q818yescomplemented
138C2×C47⋊C8C2414yes—
139#10625324C2×Q8132yescomplemented
140C23×C47C22×D411yescharacteristic
141C22×D47—42no—
142C22×D47C4⋊Q814yes—
143C22×D47C22×D412yes—
144Q8×C47—48no—
145C2×C4×C47—28no—
146C2×C4×C47—42no—
147C2×C4×C47C22×D414yes—
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47C22×Q814yes—
150C2×C4×C47C2511yescharacteristic
151C2×Dic47—24no—
152C2×Dic47—28no—
153C2×Dic47C4⋊Q814yes—
154C2×Dic47C22×D412yes—
155C2×Dic47C22×D412yes—
156C2×Dic47C22×D412yes—
157C4×D47—28no—
158C4×D47—44no—
159C4×D47—416no—
160C4×D47C4⋊Q818yes—
161C4×D47C22×Q818yes—
162Dic94—48no—
163Dic94—416no—
164C8×C47C22×Q818yes—
165C47⋊C8C22×Q818yes—
166#60663—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47C2×C4⋊Q812yes—
169C22×C47C23×D411yescharacteristic
170C2×D47—24no—
171C2×D47—44no—
172C2×D47C2×C4⋊Q814yes—
173C4×C47—24no—
174C4×C47—48no—
175C4×C47C2×C4⋊Q812yes—
176C4×C47C23×Q812yes—
177Dic47—22no—
178Dic47—48no—
179Dic47C2×C4⋊Q814yes—
180Dic47#57514yes—
181C2×#533—472no—
182#3166—478no—
183#3165—478no—
184#3162—471no—
185#3162—472no—
186#3162D4711yescharacteristic, complemented
187#3160—474no—
188#2911—474no—
189C2×C4×M16—471no—
190C2×C47—22no—
191C2×C47C22×C4⋊Q811yescharacteristic
192C2×C47C2×#57512yes—
193D47—24no—
194C22×M16—472no—
195#533—474no—
196#533—474no—
197#533—474no—
198#533—472no—
199#533—944no—
200#533—944no—
201#533C2×D4712yes—
202C2×C4⋊Q8—474no—
203C2×#125—474no—
204C2×C22⋊Q8—944no—
205C2×C4×Q8—944no—
206C2×#117—474no—
207C22×C42—471no—
208#501—478no—
209#501—478no—
210#501—478no—
211#501C2×D4718yescomplemented
212#500—478no—
213#499—478no—
214#499—478no—
215#498—478no—
216#428—4716no—
217#428—4716no—
218#427—474no—
219#427—478no—
220#427C2×D4714yes—
221C4×M16—474no—
222C4×M16—474no—
223C2×#97—471no—
224C2×#97C2×D4711yescharacteristic
225C2×C4×C8—472no—
226C47#6066311yescharacteristic, complemented, Sylow
227C22×Q8—944no—
228C23×C4—471no—
229C23×C4—472no—
230C2×M16—4716no—
231C2×M16—478no—
232C22×C8—472no—
233C22×C8C22×D4712yes—
234C4⋊Q8—478no—
235C4⋊Q8—474no—
236C4⋊Q8—944no—
237C4⋊Q8—944no—
238C4⋊Q8C22×D4714yes—
239#125—478no—
240#125—474no—
241#125—948no—
242#125C22×D4714yes—
243C22⋊Q8—24no—
244C22⋊Q8—948no—
245C22⋊Q8—948no—
246C22⋊Q8—944no—
247C22⋊Q8—944no—
248C22⋊Q8—944no—
249C4×Q8—24no—
250C4×Q8—948no—
251C4×Q8—944no—
252C4×Q8—948no—
253C4×Q8—944no—
254C4×Q8—944no—
255#117—474no—
256#117—478no—
257#117—944no—
258#117—944no—
259#117C22×D4714yes—
260C2×C4⋊C4—478no—
261C2×C4⋊C4—948no—
262C2×C4⋊C4—944no—
263C2×C22⋊C4—944no—
264C2×C42—472no—
265C2×C42—471no—
266C2×C42—474no—
267C2×C42—472no—
268C2×C42—472no—
269C2×C42—944no—
270C2×C42C22×D4711yescharacteristic
271#107—478no—
272#107—4716no—
273#107C22×D4718yes—
274#106—478no—
275#106—4716no—
276#106C22×D4718yes—
277#97—474no—
278#97C22×D4714yes—
279C4×C8—474no—
280C4×C8—474no—
281C24—471no—
282C2×Q8—24no—
283C2×Q8—948no—
284C2×Q8—944no—
285C2×Q8—18816no—
286C22×C4—471no—
287C22×C4—472no—
288C22×C4—478no—
289C22×C4—472no—
290C22×C4—472no—
291C22×C4—474no—
292C22×C4—474no—
293C22×C4—948no—
294C22×C4—1884no—
295C22×C4D4×D4712yes—
296C22×C4C23×D4711yescharacteristic
297M16—4732no—
298C2×C8—478no—
299C2×C8—474no—
300C2×C8Q8×D4718yescomplemented
301C2×C8C23×D4714yes—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—478no—
305C4⋊C4—948no—
306C4⋊C4—948no—
307C4⋊C4—948no—
308C4⋊C4—948no—
309C4⋊C4—948no—
310C4⋊C4—944no—
311C4⋊C4C23×D4718yes—
312C22⋊C4—24no—
313C22⋊C4—948no—
314C22⋊C4—944no—
315C42—24no—
316C42—474no—
317C42—474no—
318C42—472no—
319C42—471no—
320C42—471no—
321C42—948no—
322C42—944no—
323C42D4×D4712yes—
324C42C23×D4712yes—
325C23—472no—
326C23—474no—
327C23—1882no—
328C23C2×D4×D4711yescharacteristic
329Q8—48no—
330Q8—1888no—
331Q8—18816no—
332C2×C4—28no—
333C2×C4—42no—
334C2×C4—474no—
335C2×C4—472no—
336C2×C4—472no—
337C2×C4—478no—
338C2×C4—478no—
339C2×C4—472no—
340C2×C4—948no—
341C2×C4—944no—
342C2×C4—948no—
343C2×C4—1884no—
344C2×C4—18816no—
345C2×C4C2×D4×D4714yes—
346C2×C4C2×D4×D4711yescharacteristic
347C2×C4C2×Q8×D4714yes—
348C2×C4C24×D4711yescharacteristic
349C8—478no—
350C8C2×Q8×D4718yes—
351C22—22no—
352C22—474no—
353C22—944no—
354C22—1884no—
355C22C4⋊Q8×D4712yes—
356C22C22×D4×D4711yescharacteristic
357C4—24no—
358C4—48no—
359C4—474no—
360C4—474no—
361C4—942no—
362C4—1888no—
363C4C4⋊Q8×D4712yes—
364C4C22×Q8×D4712yes—
365C2—22no—
366C2—944no—
367C2C2×C4⋊Q8×D4711yescharacteristic
368C2#575×D4712yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3162C2
socleC22×C47C23×D4
prime radicalC23×D47C2×D4
hypercenter#3162D47
hypocenterC47#60663
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
2. #20253939C2semidirect
3. #20253933C2semidirect
4. #20253912C2semidirect
5. #20253910C2semidirect
6. #20253900C2semidirect
9. C47×#3162C2semidirect
23. #15689533C22semidirect
24. #15689532C22semidirect
25. #15689529C22semidirect
26. #15689527C22semidirect
27. #15689524C22semidirect
28. #15689523C22semidirect
29. #15689520C22semidirect
39. C47×#501C22semidirect
45. C2×#10625324Q8semidirect
112. C2×C8×C47C2×Q8semidirect
137. C2×C47⋊C8C2×Q8semidirect
139. #10625324C2×Q8semidirect
186. #3162D47semidirect
211. #501C2×D47semidirect
226. C47#60663semidirect
300. C2×C8Q8×D47semidirect

Maximal subgroups

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

subgroupindex
#533×D472
#202539392
#202539332
#202539122
#202539102
#202539002
#202519762
C2×#156890752
C47×#31622
#6066347