Finite Group Databasebeta

← #26248233

Subgroups — #26248233

Order 12032.

1318 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689273C212yesmaximal
2#20245968C211yesmaximal, characteristic
3#20244531C212yesmaximal, complemented
4#20244471C212yesmaximal, complemented
5#20243743C214yesmaximal, complemented
6#20243701C212yesmaximal, complemented
7#20243459C211yesmaximal, characteristic, complemented
8C2×C47×#541C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C22×#10625336C2214yes—
11C2×#13093983C2211yescharacteristic
12#15689273—28no—
13C2×#13093948C2214yescomplemented
14C2×#13093944C2214yes—
15C2×C22⋊(C47⋊C8)C2212yes—
16#15689025C2212yes—
17#15689025C2212yes—
18#15689024—22no—
19#15688978—22no—
20#15688885C2214yescomplemented
21#15688878C2212yescomplemented
22#15688878C2212yescomplemented
23C47×#541—28no—
24C2×C4⋊Q8×C47C2211yescharacteristic, complemented
25C2×#126×C47—22no—
26C2×#125×C47—22no—
27C2×#125×C47C2211yescharacteristic, complemented
28C2×C22⋊Q8×C47C2212yes—
29C2×C4×Q8×C47C2212yescomplemented
30C2×#117×C47C2211yescharacteristic
31C24×D47—21no—
32C22×D188—21no—
33C22×D188—24no—
34C22×D188—24no—
35C22×D188—42no—
36C22×D188C2312yes—
37C2×#10625336—28no—
38C2×#10625336—28no—
39C2×#10625336—44no—
40C22×C47⋊C8C2314yes—
41#13093983—42no—
42C2×C4⋊Dic47C2312yes—
43C2×C22⋊Dic47—21no—
44C2×(C2×D47)⋊C4—24no—
45C2×(C2×D47)⋊C4—24no—
46C2×C4×Dic47—21no—
47#13093948—44no—
48#13093944—28no—
49C22⋊(C47⋊C8)—24no—
50C22×Q8×C47C2312yes—
51C23×C4×C47C2311yescharacteristic
52C4⋊Q8×C47—24no—
53#126×C47—216no—
54#125×C47—216no—
55#125×C47—24no—
56C22⋊Q8×C47—24no—
57C22⋊Q8×C47D418yes—
58C4×Q8×C47—24no—
59C4×Q8×C47—24no—
60#117×C47—24no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47D412yes—
66C2×C4⋊C4×C47C2312yes—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47D412yes—
69C2×C42×C47—22no—
70C2×C42×C47C2311yescharacteristic
71C2×C42×C47C2311yescharacteristic
72C24×C47C2×D411yescharacteristic
73C23×D47—22no—
74C23×D47—24no—
75C23×D47—42no—
76C23×D47—42no—
77C23×D47—84no—
78C2×Q8×C47—22no—
79C2×Q8×C47—24no—
80C2×Q8×C47—24no—
81C2×Q8×C47C2×D414yes—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D414yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47—22no—
92C2×D188—24no—
93C2×D188—24no—
94C2×D188—24no—
95C2×D188—22no—
96C2×D188—416no—
97C2×D188—42no—
98C2×D188—42no—
99C2×D188#4114yes—
100C4⋊C4×C47—28no—
101C4⋊C4×C47—28no—
102C4⋊C4×C47—28no—
103C4⋊C4×C47—28no—
104C4⋊C4×C47—24no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47C2×D414yes—
107C22⋊C4×C47—28no—
108C22⋊C4×C47—24no—
109C42×C47—28no—
110C42×C47—22no—
111C42×C47—22no—
112C22⋊Dic47—42no—
113#10625336—416no—
114(C2×D47)⋊C4—84no—
115(C2×D47)⋊C4—84no—
116C4⋊Dic47—24no—
117C4×Dic47—42no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—24no—
120C2×C47⋊C8—24no—
121C23×C47—21no—
122C23×C47—21no—
123C23×C47—24no—
124C23×C47#12012yes—
125C23×C47C22×D411yescharacteristic
126C22×D47—24no—
127C22×D47—48no—
128C22×D47—42no—
129C22×D47—42no—
130C22×D47—44no—
131C22×D47—44no—
132C22×D47—88no—
133C22×D47—88no—
134C22×D47—88no—
135Q8×C47—28no—
136Q8×C47—44no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—28no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—24no—
143C2×C4×C47—24no—
144C2×C4×C47—24no—
145C2×C4×C47—24no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—44no—
149C2×C4×C47#12012yes—
150C2×C4×C47C22×D412yes—
151C2×C4×C47C2×#4112yes—
152C2×C4×C47H(5,𝔽2)12yes—
153C2×Dic47—42no—
154C2×Dic47—44no—
155D188—28no—
156D188—48no—
157D188—44no—
158C47⋊C8—44no—
159#23942—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—24no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47C2×#12011yescharacteristic
165C22×C47#52511yescharacteristic
166C22×C47#53511yescharacteristic
167C22×C47#57714yes—
168C2×D47—48no—
169C2×D47—44no—
170C2×D47—816no—
171C2×D47—82no—
172C2×D47—82no—
173C2×D47—84no—
174C4×C47—22no—
175C4×C47—44no—
176C4×C47—42no—
177C4×C47—42no—
178C4×C47—44no—
179C4×C47#54614yes—
180Dic47—82no—
181C2×#541D4711yescharacteristic, complemented
182C2×#450—472no—
183#2309—471no—
184#2025—472no—
185#2013—472no—
186#1880—474no—
187#1873—472no—
188#1818—471no—
189C2×C47—42no—
190C2×C47#240411yescharacteristic
191C2×C47#301012yes—
192C2×C47#312312yes—
193C2×C47#322212yes—
194D47—84no—
195C23×D4—471no—
196C22×SD16—474no—
197#541—28no—
198C2×C4⋊Q8C2×D4711yescharacteristic, complemented
199C2×#126—22no—
200C2×#125—22no—
201C2×#125C2×D4711yescharacteristic, complemented
202C2×C22⋊Q8C2×D4712yes—
203C2×C4×Q8C2×D4712yescomplemented
204C2×#117—471no—
205C2×#117C2×D4711yescharacteristic
206#450—948no—
207C2×#103—474no—
208C2×#102—474no—
209C2×C22⋊C8—472no—
210#390—472no—
211#390—472no—
212#390—942no—
213#379—942no—
214#340—474no—
215#336—472no—
216#336—472no—
217C47#2394211yescharacteristic, complemented, Sylow
218C25—941no—
219C22×Q8C22×D4712yes—
220C22×D4—472no—
221C22×D4—944no—
222C22×D4—941no—
223C22×D4—944no—
224C22×D4—1882no—
225C23×C4C22×D4711yescharacteristic
226C2×SD16—948no—
227C2×SD16—948no—
228C2×SD16—1884no—
229C22×C8—474no—
230C4⋊Q8—24no—
231#126—216no—
232#125—216no—
233#125—24no—
234C22⋊Q8—24no—
235C22⋊Q8C22⋊D4718yes—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—24no—
239#117—1882no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—472no—
245C2×C4⋊C4C22⋊D4712yes—
246C2×C4⋊C4C22×D4712yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—944no—
249C2×C22⋊C4—944no—
250C2×C22⋊C4—941no—
251C2×C22⋊C4C22⋊D4712yes—
252C2×C42—22no—
253C2×C42—941no—
254C2×C42C22×D4711yescharacteristic
255C2×C42C22×D4711yescharacteristic
256#103—1884no—
257#102—948no—
258C22⋊C8—944no—
259C24—944no—
260C24—942no—
261C24—3764no—
262C24—1882no—
263C24—1882no—
264C24C2×C22⋊D4711yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8C2×C22⋊D4714yes—
269C2×D4—474no—
270C2×D4—944no—
271C2×D4—944no—
272C2×D4—942no—
273C2×D4—944no—
274C2×D4—18816no—
275C2×D4—1882no—
276C2×D4—1882no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—942no—
282C22×C4C2×C22⋊D4712yes—
283C22×C4C2×C22⋊D4712yes—
284C22×C4C2×C22⋊D4711yescharacteristic
285C22×C4C2×C22⋊D4714yes—
286C22×C4C23×D4711yescharacteristic
287SD16—18816no—
288C2×C8—944no—
289C2×C8—944no—
290C2×C8—944no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—944no—
298C4⋊C4C2×C22⋊D4714yes—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C22⋊C4—3764no—
302C22⋊C4—3764no—
303C22⋊C4—1882no—
304C42—28no—
305C42—22no—
306C42—22no—
307C42—1882no—
308C23—21no—
309C23—24no—
310C23—21no—
311C23—944no—
312C23—3768no—
313C23—3768no—
314C23—3768no—
315C23—1888no—
316C23—1882no—
317C23—1884no—
318C23—1884no—
319C23—1882no—
320C23#1309399312yes—
321C23C22×C22⋊D4711yescharacteristic
322Q8—28no—
323Q8—44no—
324D4—948no—
325D4—1888no—
326D4—1884no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—1884no—
340C2×C4—1882no—
341C2×C4#1309399312yes—
342C2×C4C22×C22⋊D4712yes—
343C2×C4#41×D4712yes—
344C2×C4#1309407012yes—
345C8—1884no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—37616no—
351C22—3764no—
352C22—3762no—
353C22—3762no—
354C22—1884no—
355C22—1888no—
356C22C2×#1309399311yescharacteristic
357C22#1568963011yescharacteristic
358C22#1568968711yescharacteristic
359C22#1568993114yes—
360C4—22no—
361C4—42no—
362C4—42no—
363C4—44no—
364C4—44no—
365C4—3762no—
366C4#1568976114yes—
367C2—42no—
368C2—3764no—
369C2#2024670311yescharacteristic
370C2#2025267412yes—
371C2#2025363512yes—
372C2#2025445412yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#541C2
socleC23×C47C22×D4
prime radicalC23×D188C22
hypercenterC2×#541D47
hypocenterC47#23942
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
3. #20244531C2semidirect
4. #20244471C2semidirect
5. #20243743C2semidirect
6. #20243701C2semidirect
7. #20243459C2semidirect
8. C2×C47×#541C2semidirect
13. C2×#13093948C22semidirect
20. #15688885C22semidirect
21. #15688878C22semidirect
22. #15688878C22semidirect
24. C2×C4⋊Q8×C47C22semidirect
27. C2×#125×C47C22semidirect
29. C2×C4×Q8×C47C22semidirect
181. C2×#541D47semidirect
198. C2×C4⋊Q8C2×D47semidirect
201. C2×#125C2×D47semidirect
203. C2×C4×Q8C2×D47semidirect
217. C47#23942semidirect

Maximal subgroups

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

subgroupindex
C2×#156892732
#202459682
#202445312
#202444712
#202437432
#202437012
#202434592
C2×C47×#5412
#2394247