Finite Group Databasebeta

← #26127437

Subgroups — #26127437

Order 12032.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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#20243380C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241418C212yesmaximal, complemented
4#20241415C212yesmaximal, complemented
5C47×#2564C211yesmaximal, characteristic
6#15689147—21no—
7C2×#13093941—21no—
8C2×C22⋊(C47⋊C8)—21no—
9#15688892C2212yes—
10#15688849C2212yes—
11#15688847—21no—
12C2×C22⋊Q8×C47C411yescharacteristic
13C2×C4×D4×C47C2211yescharacteristic
14C47×#400C2211yescharacteristic
15C47×#399C411yescharacteristic
16C47×#395C2211yescharacteristic
17C47×#393C411yescharacteristic
18C47×#387—21no—
19C47×#386—21no—
20C47×#385—21no—
21C47×#380—21no—
22C2×#95×C47C411yescharacteristic
23C2×#10625329—22no—
24C22×C47⋊C8—41no—
25C4⋊(C47⋊C8)—41no—
26C4⋊(C47⋊C8)—41no—
27#13093941—42no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×D4×C47D411yescharacteristic
32C23×C4×C47—21no—
33C23×C4×C47C2×C411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C22⋊Q8×C47—41no—
36C22⋊Q8×C47—41no—
37C4×D4×C47—41no—
38C4×D4×C47—41no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47C22⋊C411yescharacteristic
59C2×Q8×C47—41no—
60C2×Q8×C47—81no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47SD1612yes—
64C2×D4×C47Q1612yes—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—81no—
76C22×C4×C47—81no—
77C22×C4×C47—81no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C4⋊C4×C47—41no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—81no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—42no—
93C22⋊C4×C47—81no—
94C42×C47—21no—
95C42×C47—21no—
96#10625329—42no—
97#10625329—42no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—82no—
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
112Q8×C47—82no—
113D4×C47—42no—
114D4×C47—42no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47—84no—
141C2×C4×C47#9911yescharacteristic
142C2×C4×C47#9911yescharacteristic
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47C2×Q1611yescharacteristic
145C47⋊C8—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—42no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—41no—
171C4×C47—81no—
172C4×C47—81no—
173C4×C47—81no—
174#2564D4711yescharacteristic
175#1800—471no—
176#1459—471no—
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⋊Q8Dic4711yescharacteristic
187C2×C4×D4C2×D4711yescharacteristic
188#423—941no—
189C2×#100—941no—
190C2×C22⋊C8—941no—
191#400C2×D4711yescharacteristic
192#399Dic4711yescharacteristic
193#395C2×D4711yescharacteristic
194#393Dic4711yescharacteristic
195#387—21no—
196#386—21no—
197#385—21no—
198#380—21no—
199C2×#95Dic4711yescharacteristic
200#343—472no—
201#324—472no—
202#323—941no—
203C47#1190611yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic4711yescharacteristic
205C22×D4C22⋊D4711yescharacteristic
206C23×C4—21no—
207C23×C4C2×Dic4711yescharacteristic
208C23×C4C22⋊D4711yescharacteristic
209C2×M16—942no—
210C22×C8—1881no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic4711yescharacteristic
219C2×C4⋊C4C22×D4711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4C2×Dic4711yescharacteristic
224C2×C22⋊C4C22⋊D4711yescharacteristic
225C2×C42C22⋊D4711yescharacteristic
226C4⋊C8—1881no—
227C4⋊C8—1881no—
228#100—1882no—
229C22⋊C8—942no—
230C22⋊C8—1881no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C2×Dic4711yescharacteristic
237#95C2×Dic4711yescharacteristic
238C24C22⋊Dic4711yescharacteristic
239C24C22⋊Dic4711yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1062533612yes—
245C2×D4#1062533712yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C22⋊Dic4711yescharacteristic
260C22×C4C22⋊Dic4711yescharacteristic
261C22×C4C22×Dic4711yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263C22×C4C2×C22⋊D4711yescharacteristic
264M16—1882no—
265M16—1882no—
266C2×C8—3762no—
267C2×C8—1882no—
268C2×C8—1882no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1309394812yes—
291C23#1309394812yes—
292C23C2×C22⋊Dic4711yescharacteristic
293Q8—82no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—84no—
322C2×C4#1309393911yescharacteristic
323C2×C4#1309393911yescharacteristic
324C2×C4C2×#1062533611yescharacteristic
325C2×C4C2×#1062533711yescharacteristic
326C8—3762no—
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×#1309393911yescharacteristic
341C22C2×#1309394811yescharacteristic
342C22#1568914011yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2024196011yescharacteristic
359C2#2024503411yescharacteristic
360C2#2024507311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2564C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#2564D47
hypocenterC47#11906
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #20241418C2semidirect
4. #20241415C2semidirect
203. C47#11906semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202417062
#202414182
#202414152
C47×#25642
#1190647