Finite Group Databasebeta

← #26248218

Subgroups — #26248218

Order 12032.

1286 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 195 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×#15689728C211yesmaximal, characteristic
2C2×#15689262C212yesmaximal, complemented
3#20244539C212yesmaximal, complemented
4#20244475C212yesmaximal, complemented
5#20243725C214yesmaximal, complemented
6#20243710C212yesmaximal, complemented
7#20243459C211yesmaximal, characteristic, complemented
8C47×#2310C211yesmaximal, characteristic, complemented
9C22×#10625336C2214yes—
10#15689728—28no—
11C2×#13094021C2211yescharacteristic
12C2×#13094016C2211yescharacteristic
13C2×#13094009—22no—
14C2×#13093998C2212yes—
15C2×#13093997—22no—
16C2×C4×D188C2212yes—
17#15689262—28no—
18C2×#13093948C2214yes—
19C2×#13093944C2214yes—
20C2×C22⋊(C47⋊C8)C2212yes—
21#15688885C2214yes—
22#15688878C2212yes—
23#15688878C2212yes—
24C23×Q8×C47C2211yescharacteristic
25C2×#117×C47C2211yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C47×#391—22no—
28C47×#391C2212yes—
29C47×#391C2212yes—
30C47×#381—22no—
31C22×D188—22no—
32C22×D188C2312yes—
33C22×C22⋊D47—22no—
34C22×C4×D47—22no—
35C2×#10625336—28no—
36C2×#10625336—28no—
37C2×#10625336—44no—
38C22×C47⋊C8C2314yes—
39#13094021—24no—
40#13094016—24no—
41#13094009—216no—
42#13093998—24no—
43#13093998D418yes—
44#13093997—216no—
45C4×D188—24no—
46C4×D188—24no—
47C2×C4⋊Dic47C2312yes—
48C2×Dic47⋊C4—22no—
49C2×(C2×D47)⋊C4—22no—
50C2×(C2×D47)⋊C4—22no—
51#13093948—28no—
52#13093944—44no—
53C22⋊(C47⋊C8)—24no—
54C22×Q8×C47—24no—
55C22×Q8×C47—21no—
56C22×Q8×C47—24no—
57C22×Q8×C47—42no—
58C22×Q8×C47C2312yes—
59C23×C4×C47—21no—
60C23×C4×C47C2311yescharacteristic
61#117×C47—24no—
62#117×C47—42no—
63C2×C4⋊C4×C47D412yes—
64C2×C4⋊C4×C47C2312yes—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47D412yes—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C2×C42×C47C2311yescharacteristic
70#95×C47—24no—
71#95×C47—24no—
72C24×C47C2×D411yescharacteristic
73C23×D47—22no—
74C2×Q8×C47—24no—
75C2×Q8×C47—22no—
76C2×Q8×C47—24no—
77C2×Q8×C47—24no—
78C2×Q8×C47—416no—
79C2×Q8×C47—42no—
80C2×Q8×C47—42no—
81C2×Q8×C47#4114yes—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—24no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—42no—
88C22×C4×C47—42no—
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D414yes—
93C22×C4×C47C2411yescharacteristic
94C2×C22⋊D47—28no—
95C2×C22⋊D47—84no—
96C22×Dic47—22no—
97C2×D188—24no—
98C2×D188—24no—
99C2×D188—22no—
100C2×D188—28no—
101C2×D188—84no—
102C2×D188C2×D414yes—
103C2×C4×D47—28no—
104C2×C4×D47—84no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C22⋊C4×C47—24no—
108C22⋊C4×C47—42no—
109C42×C47—22no—
110C42×C47—22no—
111C42×C47—42no—
112#10625336—416no—
113(C2×D47)⋊C4—28no—
114(C2×D47)⋊C4—28no—
115C4⋊Dic47—22no—
116C4⋊Dic47C2×D414yes—
117Dic47⋊C4—28no—
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—42no—
128C22×D47—42no—
129C22×D47—44no—
130C22×D47—44no—
131Q8×C47—28no—
132Q8×C47—48no—
133Q8×C47—44no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—28no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—24no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—48no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—44no—
147C2×C4×C47#12012yes—
148C2×C4×C47C22×D412yes—
149C2×C4×C47C2×#4112yes—
150C2×C4×C47H(5,𝔽2)12yes—
151C22⋊D47—88no—
152C2×Dic47—24no—
153C2×Dic47—44no—
154D188—28no—
155D188—44no—
156D188—88no—
157C4×D47—88no—
158C47⋊C8—44no—
159#23940—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—44no—
169C2×D47—44no—
170C2×D47—44no—
171C2×D47—84no—
172C2×D47—84no—
173C4×C47—22no—
174C4×C47—44no—
175C4×C47—42no—
176C4×C47—42no—
177C4×C47—82no—
178C4×C47#54614yes—
179Dic47—44no—
180C2×#540—471no—
181C2×#448—472no—
182#2310D4711yescharacteristic, complemented
183#2026—472no—
184#2014—472no—
185#1877—474no—
186#1875—472no—
187#1818—471no—
188C2×C47—42no—
189C2×C47#240411yescharacteristic
190C2×C47#300912yes—
191C2×C47#312612yes—
192C2×C47#322112yes—
193D47—84no—
194C23×Q8C2×D4711yescharacteristic
195C22×SD16—474no—
196#540—948no—
197C2×#127—471no—
198C2×#125—471no—
199C2×#123—942no—
200C2×#121—472no—
201C2×#121—942no—
202C2×C4×D4—472no—
203C2×#117C2×D4711yescharacteristic
204C2×#117C2×D4711yescharacteristic
205#448—948no—
206C2×#103—474no—
207C2×#102—474no—
208C2×C22⋊C8—472no—
209#391—22no—
210#391C2×D4712yes—
211#391C2×D4712yes—
212#381—22no—
213#340—474no—
214#336—472no—
215#336—472no—
216C47#2394011yescharacteristic, complemented, Sylow
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8C22×D4712yes—
222C22×D4—472no—
223C22×D4—942no—
224C22×D4—942no—
225C23×C4—21no—
226C23×C4—942no—
227C23×C4C22×D4711yescharacteristic
228C2×SD16—948no—
229C2×SD16—948no—
230C2×SD16—1884no—
231C22×C8—474no—
232#127—944no—
233#125—944no—
234#123—9416no—
235#121—478no—
236#121—9416no—
237#121—944no—
238C4×D4—944no—
239C4×D4—944no—
240#117—24no—
241#117—42no—
242C2×C4⋊C4—472no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4C22⋊D4712yes—
245C2×C4⋊C4C22×D4712yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—942no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4C22⋊D4712yes—
250C2×C42—21no—
251C2×C42C22×D4711yescharacteristic
252C2×C42C22×D4711yescharacteristic
253#103—948no—
254#102—1884no—
255C22⋊C8—944no—
256#95—24no—
257#95—24no—
258C24—942no—
259C24C2×C22⋊D4711yescharacteristic
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#1062535014yes—
268C2×D4—474no—
269C2×D4—948no—
270C2×D4—948no—
271C2×D4—942no—
272C2×D4—944no—
273C2×D4—944no—
274C2×D4—3764no—
275C2×D4—3764no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—948no—
284C22×C4—942no—
285C22×C4—3764no—
286C22×C4D4×D4714yes—
287C22×C4C2×C22⋊D4712yes—
288C22×C4C2×C22⋊D4712yes—
289C22×C4C2×C22⋊D4711yescharacteristic
290C22×C4C23×D4711yescharacteristic
291SD16—18816no—
292C2×C8—944no—
293C2×C8—944no—
294C2×C8—944no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—474no—
298C4⋊C4—948no—
299C4⋊C4—942no—
300C22⋊C4—24no—
301C22⋊C4—42no—
302C22⋊C4—948no—
303C22⋊C4—948no—
304C42—22no—
305C42—22no—
306C42—42no—
307C23—21no—
308C23—24no—
309C23—21no—
310C23—944no—
311C23—1882no—
312C23—1884no—
313C23—1884no—
314C23—1882no—
315C23#1309399212yes—
316C23C22×C22⋊D4711yescharacteristic
317Q8—28no—
318Q8—48no—
319Q8—44no—
320D4—948no—
321D4—3768no—
322D4—3768no—
323D4—1884no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—944no—
338C2×C4—3768no—
339C2×C4—1884no—
340C2×C4#1309399212yes—
341C2×C4C2×D4×D4712yes—
342C2×C4C2×#1062535012yes—
343C2×C4#1309406912yes—
344C8—1884no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—3764no—
350C22—3764no—
351C22—1884no—
352C22—1884no—
353C22—1884no—
354C22C2×#1309399211yescharacteristic
355C22#1568963411yescharacteristic
356C22#1568969011yescharacteristic
357C22#1568993114yes—
358C4—22no—
359C4—44no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—1884no—
364C4#1568976314yes—
365C2—42no—
366C2—3764no—
367C2#2024670711yescharacteristic
368C2#2025266312yes—
369C2#2025367912yes—
370C2#2025444912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2310C2
socleC23×C47C22×D4
prime radicalC2×#15689728C2
hypercenter#2310D47
hypocenterC47#23940
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#15689262C2semidirect
3. #20244539C2semidirect
4. #20244475C2semidirect
5. #20243725C2semidirect
6. #20243710C2semidirect
7. #20243459C2semidirect
8. C47×#2310C2semidirect
182. #2310D47semidirect
216. C47#23940semidirect

Maximal subgroups

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

subgroupindex
C2×#156897282
C2×#156892622
#202445392
#202444752
#202437252
#202437102
#202434592
C47×#23102
#2394047