Finite Group Databasebeta

← #26127324

Subgroups — #26127324

Order 12032.

764 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 107 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, complemented
2#20241706C211yesmaximal, characteristic, complemented
3#20241391C214yesmaximal, complemented
4C47×#2408C211yesmaximal, characteristic
5#15689147—21no—
6C2×#13093941—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688892C2212yescomplemented
9#15688849C2212yescomplemented
10#15688847—21no—
11C23×D4×C47C411yescharacteristic
12C2×#121×C47—21no—
13C2×#120×C47—21no—
14C2×C4×D4×C47C2211yescharacteristic
15C47×#394C2212yes—
16C47×#392C412yes—
17C47×#390—21no—
18C47×#380C411yescharacteristic
19C47×#379—21no—
20C2×#10625329—22no—
21C22×C47⋊C8—41no—
22C4⋊(C47⋊C8)—41no—
23C4⋊(C47⋊C8)—41no—
24#13093941—42no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—41no—
27C25×C47—21no—
28C22×D4×C47—22no—
29C22×D4×C47—21no—
30C22×D4×C47—41no—
31C22×D4×C47—41no—
32C22×D4×C47—44no—
33C22×D4×C47C2×C412yes—
34C22×D4×C47D411yescharacteristic
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37#121×C47—44no—
38#120×C47—42no—
39#120×C47—42no—
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51#95×C47C2×C412yes—
52C24×C47—22no—
53C24×C47—42no—
54C24×C47—44no—
55C24×C47—42no—
56C24×C47—42no—
57C24×C47—82no—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C22⋊C411yescharacteristic
60C2×D4×C47—24no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47—42no—
64C2×D4×C47—42no—
65C2×D4×C47—48no—
66C2×D4×C47—44no—
67C2×D4×C47—44no—
68C2×D4×C47—44no—
69C2×D4×C47—44no—
70C2×D4×C47—44no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—82no—
74C2×D4×C47D812yes—
75C2×D4×C47SD1612yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—42no—
82C22×C4×C47—42no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C4⋊C4×C47—41no—
89C4⋊C4×C47—42no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—44no—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—82no—
97C42×C47—21no—
98C42×C47—21no—
99#10625329—42no—
100#10625329—42no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—82no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—24no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—48no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—84no—
121C23×C47—88no—
122C23×C47—88no—
123C23×C47—84no—
124C23×C47#10212yes—
125C23×C47#10212yes—
126C23×C47C2×C22⋊C411yescharacteristic
127D4×C47—42no—
128D4×C47—42no—
129D4×C47—44no—
130D4×C47—44no—
131D4×C47—44no—
132D4×C47—44no—
133D4×C47—88no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—22no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—81no—
150C2×C4×C47#9911yescharacteristic
151C2×C4×C47#9911yescharacteristic
152C2×C4×C47C2×D811yescharacteristic
153C2×C4×C47C2×SD1611yescharacteristic
154C47⋊C8—82no—
155C47⋊C8—82no—
156#11880—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—44no—
165C22×C47—44no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—44no—
171C22×C47—81no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47—816no—
175C22×C47—82no—
176C22×C47—82no—
177C22×C47C2×#9911yescharacteristic
178C22×C47C2×#10211yescharacteristic
179C22×C47#42111yescharacteristic
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—82no—
187C4×C47AΣL(1,𝔽16)12yes—
188#2408D4711yescharacteristic
189#1800—471no—
190#1459—471no—
191#1345—474no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—42no—
195C2×C47—42no—
196C2×C47—84no—
197C2×C47#151911yescharacteristic
198C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
199C2×C47#213411yescharacteristic
200C23×D4Dic4711yescharacteristic
201C2×#121—21no—
202C2×#120—21no—
203C2×C4×D4C2×D4711yescharacteristic
204#423—941no—
205C2×#100—941no—
206C2×C22⋊C8—941no—
207#394C2×D4712yes—
208#392Dic4712yes—
209#390—21no—
210#380Dic4711yescharacteristic
211#379—21no—
212#343—472no—
213#324—472no—
214#323—941no—
215C47#1188011yescharacteristic, complemented, Sylow
216C25—21no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—41no—
220C22×D4—41no—
221C22×D4—44no—
222C22×D4C2×Dic4712yes—
223C22×D4C22⋊D4711yescharacteristic
224C23×C4C2×Dic4711yescharacteristic
225C23×C4C22⋊D4711yescharacteristic
226C2×M16—942no—
227C22×C8—1881no—
228#121—44no—
229#120—42no—
230#120—42no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22×D4711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C2×Dic4711yescharacteristic
240C2×C22⋊C4C22⋊D4711yescharacteristic
241C2×C42C22⋊D4711yescharacteristic
242C4⋊C8—1881no—
243C4⋊C8—1881no—
244#100—1882no—
245C22⋊C8—942no—
246C22⋊C8—1881no—
247#95C2×Dic4712yes—
248C24—22no—
249C24—42no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—82no—
254C24C22⋊Dic4711yescharacteristic
255C24C22⋊Dic4711yescharacteristic
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—48no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4#1062533412yes—
271C2×D4#1062533512yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C22⋊Dic4711yescharacteristic
280C22×C4C22⋊Dic4711yescharacteristic
281C22×C4C22×Dic4711yescharacteristic
282C22×C4C2×C22⋊D4711yescharacteristic
283C22×C4C2×C22⋊D4711yescharacteristic
284M16—1882no—
285M16—1882no—
286C2×C8—3762no—
287C2×C8—1882no—
288C2×C8—1882no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—82no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—24no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—48no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—84no—
317C23—88no—
318C23—88no—
319C23—84no—
320C23#1309394512yes—
321C23#1309394512yes—
322C23C2×C22⋊Dic4711yescharacteristic
323D4—42no—
324D4—42no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—44no—
329D4—88no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4#1309393911yescharacteristic
347C2×C4#1309393911yescharacteristic
348C2×C4C2×#1062533411yescharacteristic
349C2×C4C2×#1062533511yescharacteristic
350C8—3762no—
351C8—3762no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—81no—
367C22—82no—
368C22—82no—
369C22—816no—
370C22—82no—
371C22—82no—
372C22C2×#1309393911yescharacteristic
373C22C2×#1309394511yescharacteristic
374C22#1568914011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4#1568891012yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—84no—
388C2#2024195611yescharacteristic
389C2C2×#1568891011yescharacteristic
390C2#2024508011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
Fitting, prime radicalC47×#2408C2
socleC22×C47C2×#99
cosocleC2×C4⋊C4×C47C23
hypercenter#2408D47
hypocenterC47#11880
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #20243380C2semidirect
2. #20241706C2semidirect
3. #20241391C2semidirect
8. #15688892C22semidirect
9. #15688849C22semidirect
215. C47#11880semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202417062
#202413912
C47×#24082
#1188047