Finite Group Databasebeta

← #26127027

Subgroups — #26127027

Order 12032.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#20249201C211yesmaximal, characteristic
2#20243468C211yesmaximal, characteristic
3#20241419C212yesmaximal, complemented
4#20241416C212yesmaximal, complemented
5C47×#1463C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7#15689087—21no—
8#15689062C411yescharacteristic
9#15689062C2211yescharacteristic
10#15689056—21no—
11#15689056—21no—
12#15689043C411yescharacteristic
13#15689043C2211yescharacteristic
14#15688996—21no—
15#15688986—21no—
16#15688983C411yescharacteristic
17#15688892—21no—
18#15688892C2212yes—
19C22×C4⋊C4×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C2×#10625329—22no—
25#13094003—41no—
26#13094003—41no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C22⋊(C47⋊C8)—41no—
35C22⋊(C47⋊C8)—41no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—41no—
48C2×C4⋊C4×C47—41no—
49C2×C4⋊C4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C4⋊C4×C47C2311yescharacteristic
54C22⋊C8×C47—22no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—41no—
65C22×C4×C47—81no—
66C22×C4×C47—81no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C2×Dic94—41no—
78C2×Dic94—81no—
79M16×C47—82no—
80C2×C8×C47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C22⋊Dic47—42no—
86C22⋊Dic47—41no—
87C22⋊Dic47—41no—
88C4⋊Dic47—41no—
89C4⋊Dic47—81no—
90Dic47⋊C4—42no—
91Dic47⋊C4—81no—
92Dic47⋊C4—81no—
93#10625329—42no—
94#10625329—42no—
95C2×C47⋊C8—42no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47#9911yescharacteristic
101C23×C47#9911yescharacteristic
102C23×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47—81no—
122C2×C4×C47#13712yes—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—42no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134Dic94—82no—
135C8×C47—82no—
136C47⋊C8—82no—
137#11817—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47C2×#9911yescharacteristic
149C22×C47#41911yescharacteristic
150C22×C47#42111yescharacteristic
151C4×C47—22no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157Dic47—81no—
158Dic47—81no—
159Dic47—81no—
160Dic47—81no—
161#2627—471no—
162#1821—471no—
163#1463D4711yescharacteristic
164#1354—472no—
165#1353—472no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—41no—
169C2×C47—41no—
170C2×C47#151111yescharacteristic
171C2×C47#212111yescharacteristic
172C2×C47#213311yescharacteristic
173C2×C22⋊Q8—471no—
174C22×C4⋊C4C2×D4711yescharacteristic
175#407—21no—
176#407—941no—
177#400—471no—
178#400—471no—
179#398—941no—
180#398—941no—
181#395—471no—
182#395—471no—
183#385—941no—
184#381—941no—
185#380—471no—
186#343—472no—
187#343—941no—
188#324—21no—
189#324C2×D4712yes—
190C47#1181711yescharacteristic, complemented, Sylow
191C22×Q8—471no—
192C23×C4C4×D4711yescharacteristic
193C23×C4D18811yescharacteristic
194C23×C4C22⋊D4711yescharacteristic
195C2×M16—41no—
196C2×M16—942no—
197C22⋊Q8—1881no—
198C22⋊Q8—1881no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—471no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4C4×D4711yescharacteristic
207C2×C4⋊C4D18811yescharacteristic
208C2×C4⋊C4C22⋊D4711yescharacteristic
209C2×C4⋊C4C22×D4711yescharacteristic
210C2×C22⋊C4—941no—
211C2×C22⋊C4—941no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—1881no—
215C22⋊C8—1881no—
216#95—471no—
217#95—471no—
218#95—941no—
219#95—941no—
220#95—941no—
221#95—941no—
222#95—941no—
223#95—941no—
224C24(C2×D47)⋊C411yescharacteristic
225C2×Q8—3761no—
226C2×Q8—1881no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—941no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4C2×C4×D4711yescharacteristic
245C22×C4C2×D18811yescharacteristic
246C22×C4C2×C22⋊D4711yescharacteristic
247M16—82no—
248M16—1882no—
249M16—1882no—
250C2×C8—42no—
251C2×C8—1882no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3761no—
257C4⋊C4—3761no—
258C4⋊C4—3761no—
259C4⋊C4—1881no—
260C4⋊C4—1882no—
261C22⋊C4—1882no—
262C22⋊C4—1881no—
263C22⋊C4—1881no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1309393811yescharacteristic
269C23#1309393811yescharacteristic
270C23C2×(C2×D47)⋊C411yescharacteristic
271Q8—3762no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1882no—
298C2×C4—1881no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4#1309405712yes—
303C8—82no—
304C8—3762no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1309393811yescharacteristic
316C22#1568913311yescharacteristic
317C22#1568913911yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3761no—
325C4—3761no—
326C4—3761no—
327C4—3761no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2024192211yescharacteristic
333C2#2024503211yescharacteristic
334C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1463C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1463D47
hypocenterC47#11817
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20241419C2semidirect
4. #20241416C2semidirect
190. C47#11817semidirect

Maximal subgroups

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

subgroupindex
#202492012
#202434682
#202414192
#202414162
C47×#14632
#1181747