Finite Group Databasebeta

← #26127428

Subgroups — #26127428

Order 12032.

568 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#20246753C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20241419C214yesmaximal, complemented
4C47×#1459C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C2×#13094003—21no—
7#15689147—21no—
8C2×#13093941—21no—
9#15689043C2212yes—
10#15689035C412yes—
11#15689029—21no—
12#15689003—21no—
13#15688995—21no—
14#15688983C411yescharacteristic
15#15688892C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C22×Dic94—44no—
22C22×Dic94—41no—
23C22×Dic94—41no—
24C22×Dic94C2×C412yes—
25C23×Dic47—21no—
26C2×#10625329—22no—
27#13094003—44no—
28C2×C4⋊Dic47—21no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—22no—
32C4⋊(C47⋊C8)—41no—
33C4⋊(C47⋊C8)—41no—
34#13093941—42no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95C2×C412yes—
38C22×D4×C47D411yescharacteristic
39C23×C4×C47C2×C411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C22×C8×C47—41no—
42C4×D4×C47—41no—
43C4×D4×C47—41no—
44C2×C4⋊C4×C47C2311yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C22⋊C8×C47—22no—
49C22⋊C8×C47—41no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C22⋊C411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47SD1612yes—
55C2×D4×C47Q1612yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—22no—
68C22×Dic47—22no—
69C22×Dic47—44no—
70C22×Dic47—42no—
71C2×Dic94—24no—
72C2×Dic94—42no—
73C2×Dic94—48no—
74C2×Dic94—44no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—42no—
78C2×Dic94—44no—
79C2×Dic94—44no—
80C2×C8×C47—42no—
81C2×C8×C47—82no—
82C4⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C42×C47—21no—
86C42×C47—21no—
87C22⋊Dic47—44no—
88C22⋊Dic47—42no—
89C4⋊Dic47—42no—
90Dic47⋊C4—44no—
91#10625329—42no—
92#10625329—42no—
93C2×C47⋊C8—42no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47#10312yes—
103C23×C47#10312yes—
104C23×C47C2×C22⋊C411yescharacteristic
105D4×C47—42no—
106D4×C47—42no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—81no—
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47C2×SD1611yescharacteristic
123C2×C4×C47C2×Q1611yescharacteristic
124C2×Dic47—24no—
125C2×Dic47—42no—
126C2×Dic47—48no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131Dic94—44no—
132Dic94—44no—
133Dic94—44no—
134Dic94—44no—
135C8×C47—82no—
136C47⋊C8—82no—
137#11904—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—81no—
150C22×C47C2×#9911yescharacteristic
151C22×C47C2×#10311yescharacteristic
152C22×C47#42111yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—41no—
159C4×C47AΣL(1,𝔽16)12yes—
160Dic47—44no—
161Dic47—82no—
162#2409—471no—
163#1800—471no—
164#1459D4711yescharacteristic
165#1354—474no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—42no—
169C2×C47—42no—
170C2×C47#152011yescharacteristic
171C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
172C2×C47#213511yescharacteristic
173C23×Q8—471no—
174C2×C22⋊Q8—941no—
175C2×C4×D4C2×D4711yescharacteristic
176#423—941no—
177C2×#100—941no—
178C2×C22⋊C8—21no—
179#395—472no—
180#393—472no—
181#391—941no—
182#386—941no—
183#385—941no—
184#380—471no—
185#343—472no—
186#324C2×D4712yes—
187#323—21no—
188C47#1190411yescharacteristic, complemented, Sylow
189C22×Q8—472no—
190C22×Q8—941no—
191C22×Q8—1881no—
192C22×Q8—1881no—
193C22×Q8—1884no—
194C22×D4D18811yescharacteristic
195C23×C4—941no—
196C23×C4C4×D4711yescharacteristic
197C23×C4C22⋊D4711yescharacteristic
198C2×M16—942no—
199C22×C8—41no—
200C22⋊Q8—1884no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4C22×D4711yescharacteristic
206C2×C22⋊C4—942no—
207C2×C22⋊C4—941no—
208C2×C22⋊C4C4×D4711yescharacteristic
209C2×C22⋊C4C22⋊D4711yescharacteristic
210C2×C42D18811yescharacteristic
211C4⋊C8—1881no—
212C4⋊C8—1881no—
213#100—1882no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#95—472no—
217#95—942no—
218#95—942no—
219C24(C2×D47)⋊C411yescharacteristic
220C24(C2×D47)⋊C411yescharacteristic
221C2×Q8—944no—
222C2×Q8—1882no—
223C2×Q8—1882no—
224C2×Q8—1884no—
225C2×Q8—1884no—
226C2×Q8—1888no—
227C2×Q8—1884no—
228C2×Q8—1881no—
229C2×Q8—1881no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1062532512yes—
233C2×D4Dic18812yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—942no—
241C22×C4—942no—
242C22×C4—1884no—
243C22×C4—1882no—
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4C2×C4×D4711yescharacteristic
247C22×C4C2×D18811yescharacteristic
248C22×C4C2×C22⋊D4711yescharacteristic
249M16—1882no—
250M16—1882no—
251C2×C8—42no—
252C2×C8—82no—
253C2×C8—1882no—
254C4⋊C4—41no—
255C4⋊C4—1882no—
256C4⋊C4—1884no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—1884no—
260C22⋊C4—1882no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1309394912yes—
272C23#1309394912yes—
273C23C2×(C2×D47)⋊C411yescharacteristic
274Q8—1884no—
275Q8—1884no—
276Q8—1884no—
277Q8—1884no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—944no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1888no—
299C2×C4—1882no—
300C2×C4#1309393811yescharacteristic
301C2×C4#1309393811yescharacteristic
302C2×C4C2×#1062532511yescharacteristic
303C2×C4C2×Dic18811yescharacteristic
304C8—82no—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1309393811yescharacteristic
319C22C2×#1309394911yescharacteristic
320C22#1568913911yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3762no—
328C4—1884no—
329C4#1568890912yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2024196111yescharacteristic
335C2C2×#1568890911yescharacteristic
336C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11904
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241419C2semidirect
188. C47#11904semidirect

Maximal subgroups

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

subgroupindex
#202467532
#202433802
#202414192
C47×#14592
#1190447