Finite Group Databasebeta

← #26127438

Subgroups — #26127438

Order 12032.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#20248564C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20241419C212yesmaximal, complemented
4#20241416C212yesmaximal, complemented
5C47×#1459C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7#15689147—21no—
8C2×#13093941—21no—
9#15689062C2211yescharacteristic
10#15689059C411yescharacteristic
11#15689043C2211yescharacteristic
12#15689035C411yescharacteristic
13#15689007—21no—
14#15689003—21no—
15#15688996—21no—
16#15688981—21no—
17C2×C47⋊#95C411yescharacteristic
18#15688892C2212yes—
19C2×C4×D4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#324C2212yes—
22C47×#323—21no—
23C22×Dic94C2×C411yescharacteristic
24C23×Dic47—21no—
25C2×#10625329—22no—
26#13094003—41no—
27#13094003—41no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C4⋊(C47⋊C8)—41no—
36C4⋊(C47⋊C8)—41no—
37#13093941—42no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95C2×C411yescharacteristic
44C47⋊#95C2×C411yescharacteristic
45C22×D4×C47D411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—41no—
49C4×D4×C47—41no—
50C4×D4×C47—41no—
51C2×C4⋊C4×C47C2311yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C22⋊C8×C47—22no—
56C22⋊C8×C47—41no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C22⋊C411yescharacteristic
59C2×D4×C47—41no—
60C2×D4×C47—41no—
61C2×D4×C47SD1612yes—
62C2×D4×C47Q1612yes—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—81no—
68C22×C4×C47—81no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—42no—
79C22×Dic47—42no—
80C22×Dic47—81no—
81C2×Dic94—41no—
82C2×Dic94—81no—
83C2×C8×C47—42no—
84C2×C8×C47—82no—
85C4⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C42×C47—21no—
89C42×C47—21no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C22⋊Dic47—81no—
93C4⋊Dic47—41no—
94Dic47⋊C4—42no—
95Dic47⋊C4—41no—
96Dic47⋊C4—41no—
97Dic47⋊C4—81no—
98#10625329—42no—
99#10625329—42no—
100C2×C47⋊C8—42no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47#10312yes—
110C23×C47#10312yes—
111C23×C47C2×C22⋊C411yescharacteristic
112D4×C47—42no—
113D4×C47—42no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—81no—
125C2×C4×C47—81no—
126C2×C4×C47—81no—
127C2×C4×C47#9911yescharacteristic
128C2×C4×C47#9911yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47C2×Q1611yescharacteristic
131C2×Dic47—22no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—84no—
144Dic94—82no—
145C8×C47—82no—
146C47⋊C8—82no—
147#11906—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—42no—
153C22×C47—42no—
154C22×C47—42no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—81no—
160C22×C47C2×#9911yescharacteristic
161C22×C47C2×#10311yescharacteristic
162C22×C47#42111yescharacteristic
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—41no—
170Dic47—42no—
171Dic47—81no—
172Dic47—81no—
173Dic47—81no—
174#2564—471no—
175#1800—471no—
176#1459D4711yescharacteristic
177#1354—472no—
178#1353—472no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—42no—
182C2×C47—42no—
183C2×C47#152011yescharacteristic
184C2×C47#212111yescharacteristic
185C2×C47#213211yescharacteristic
186C2×C22⋊Q8—471no—
187C2×C4×D4C2×D4711yescharacteristic
188#423—941no—
189C2×#100—941no—
190C2×C22⋊C8—21no—
191#400—471no—
192#399—471no—
193#395—471no—
194#393—471no—
195#387—941no—
196#386—941no—
197#385—941no—
198#380—941no—
199C2×#95—471no—
200#343—472no—
201#324C2×D4712yes—
202#323—21no—
203C47#1190611yescharacteristic, complemented, Sylow
204C22×Q8—471no—
205C22×D4D18811yescharacteristic
206C23×C4—941no—
207C23×C4C4×D4711yescharacteristic
208C23×C4C22⋊D4711yescharacteristic
209C2×M16—942no—
210C22×C8—41no—
211C22⋊Q8—1881no—
212C22⋊Q8—1881no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—471no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4—941no—
218C2×C4⋊C4—941no—
219C2×C4⋊C4C22×D4711yescharacteristic
220C2×C22⋊C4—941no—
221C2×C22⋊C4—941no—
222C2×C22⋊C4—941no—
223C2×C22⋊C4C4×D4711yescharacteristic
224C2×C22⋊C4C22⋊D4711yescharacteristic
225C2×C42D18811yescharacteristic
226C4⋊C8—1881no—
227C4⋊C8—1881no—
228#100—1882no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231#95—471no—
232#95—471no—
233#95—941no—
234#95—941no—
235#95—941no—
236#95—941no—
237#95—941no—
238C24(C2×D47)⋊C411yescharacteristic
239C24(C2×D47)⋊C411yescharacteristic
240C2×Q8—3761no—
241C2×Q8—1881no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1062532512yes—
245C2×D4Dic18812yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—941no—
253C22×C4—941no—
254C22×C4—941no—
255C22×C4—941no—
256C22×C4—3761no—
257C22×C4—1882no—
258C22×C4—1882no—
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4(C2×D47)⋊C411yescharacteristic
261C22×C4C2×C4×D4711yescharacteristic
262C22×C4C2×D18811yescharacteristic
263C22×C4C2×C22⋊D4711yescharacteristic
264M16—1882no—
265M16—1882no—
266C2×C8—42no—
267C2×C8—82no—
268C2×C8—1882no—
269C4⋊C4—41no—
270C4⋊C4—3761no—
271C4⋊C4—1881no—
272C4⋊C4—1882no—
273C4⋊C4—1881no—
274C4⋊C4—1881no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—3761no—
278C22⋊C4—1882no—
279C22⋊C4—1882no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1309394912yes—
291C23#1309394912yes—
292C23C2×(C2×D47)⋊C411yescharacteristic
293Q8—3762no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—942no—
310C2×C4—3764no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4#1309393811yescharacteristic
323C2×C4#1309393811yescharacteristic
324C2×C4C2×#1062532511yescharacteristic
325C2×C4C2×Dic18811yescharacteristic
326C8—82no—
327C8—3762no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×#1309393811yescharacteristic
341C22C2×#1309394911yescharacteristic
342C22#1568913911yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3761no—
351C4—3761no—
352C4—3761no—
353C4—1882no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2024196111yescharacteristic
359C2#2024503211yescharacteristic
360C2#2024507111yescharacteristic
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#11906
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
4. #20241416C2semidirect
203. C47#11906semidirect

Maximal subgroups

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

subgroupindex
#202485642
#202433802
#202414192
#202414162
C47×#14592
#1190647