Finite Group Databasebeta

← #26104441

Subgroups — #26104441

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243368C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241211C212yesmaximal, complemented
4#20241211C212yesmaximal, complemented
5C47×#2296C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×#13093935—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×#117×C47C411yescharacteristic
13C2×#117×C47C411yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#389C411yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#389C2211yescharacteristic
19C47×#385—21no—
20C47×#385—21no—
21C4×C4⋊C4×C47—21no—
22C4×C4⋊C4×C47—21no—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—42no—
26C22⋊(C47⋊C8)—24no—
27C22⋊(C47⋊C8)—42no—
28#13093935—42no—
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C23×C4×C47D411yescharacteristic
32#117×C47—22no—
33#117×C47—22no—
34#117×C47—42no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C42×C47—21no—
46C2×C42×C47—21no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—21no—
79C4⋊C4×C47—21no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47M1612yes—
88C4⋊C4×C47M1612yes—
89C22⋊C4×C47—24no—
90C22⋊C4×C47—42no—
91C42×C47—24no—
92C42×C47—22no—
93C42×C47—22no—
94C42×C47—41no—
95C42×C47—41no—
96C42×C47—42no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47#10112yes—
110C23×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—44no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
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—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47#9912yes—
142C2×C4×C47#10412yes—
143C2×C4×C47C2×M1611yescharacteristic
144C2×C4×C47C2×M1611yescharacteristic
145C2×C4×C47#13512yes—
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47#13711yescharacteristic
148C47⋊C8—44no—
149C47⋊C8—82no—
150#7613—471nomaximal, Sylow
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47#40711yescharacteristic
159C22×C47C2×#9911yescharacteristic
160C22×C47C2×#10111yescharacteristic
161C22×C47#41611yescharacteristic
162C22×C47#41711yescharacteristic
163C22×C47C2×#10411yescharacteristic
164C22×C47#42111yescharacteristic
165C4×C47—22no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—42no—
173C4×C47—42no—
174C4×C47—41no—
175C4×C47—81no—
176C4×C47—81no—
177#2296D4711yescharacteristic
178#1797—471no—
179#1749—471no—
180#1271—472no—
181#1271—472no—
182C2×C47—42no—
183C2×C47#146311yescharacteristic
184C2×C47#148811yescharacteristic
185C2×C47#148911yescharacteristic
186C2×C47#150111yescharacteristic
187C2×C47#151311yescharacteristic
188C2×C47#153111yescharacteristic
189C2×C47#155211yescharacteristic
190C2×#117Dic4711yescharacteristic
191C2×#117Dic4711yescharacteristic
192C22×C42C2×D4711yescharacteristic
193C2×C4⋊C8—941no—
194C2×C22⋊C8—941no—
195C2×C22⋊C8—941no—
196C2×#97—941no—
197#389Dic4711yescharacteristic
198#389Dic4711yescharacteristic
199#389C2×D4711yescharacteristic
200#389C2×D4711yescharacteristic
201#385—21no—
202#385—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205#336—472no—
206#336—472no—
207C47#761311yescharacteristic, complemented, Sylow
208C23×C4C2×Dic4711yescharacteristic
209C23×C4C22⋊D4711yescharacteristic
210C23×C4C22⋊D4711yescharacteristic
211C22×C8—942no—
212C22×C8—942no—
213#117—22no—
214#117—22no—
215#117—42no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic4711yescharacteristic
221C2×C4⋊C4C2×Dic4711yescharacteristic
222C2×C4⋊C4C2×Dic4711yescharacteristic
223C2×C4⋊C4C2×Dic4711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42C2×Dic4711yescharacteristic
230C2×C42C22⋊D4711yescharacteristic
231C2×C42C22⋊D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233C4⋊C8—1882no—
234C22⋊C8—944no—
235C22⋊C8—1882no—
236#97—1882no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24C22⋊Dic4711yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C22⋊Dic4711yescharacteristic
258C22×C4C22⋊Dic4711yescharacteristic
259C22×C4C22⋊Dic4711yescharacteristic
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263C2×C8—944no—
264C2×C8—1882no—
265C2×C8—1882no—
266C2×C8—1882no—
267C2×C8—1882no—
268C2×C8—1882no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4#1062532912yes—
279C4⋊C4#1062532912yes—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C42—24no—
283C42—22no—
284C42—22no—
285C42—41no—
286C42—41no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1309394312yes—
295C23C2×C22⋊Dic4711yescharacteristic
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4#1309393912yes—
327C2×C4#1309395112yes—
328C2×C4C2×#1062532911yescharacteristic
329C2×C4C2×#1062532911yescharacteristic
330C2×C4#1309404412yes—
331C2×C4#1309404911yescharacteristic
332C2×C4#1309405411yescharacteristic
333C8—3762no—
334C8—1884no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22#1568908711yescharacteristic
343C22C2×#1309393911yescharacteristic
344C22C2×#1309394311yescharacteristic
345C22#1568911911yescharacteristic
346C22#1568912411yescharacteristic
347C22C2×#1309395111yescharacteristic
348C22#1568914011yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—42no—
358C4—42no—
359C4—81no—
360C4—81no—
361C2—42no—
362C2#2024171811yescharacteristic
363C2#2024182211yescharacteristic
364C2#2024182711yescharacteristic
365C2#2024187911yescharacteristic
366C2#2024193111yescharacteristic
367C2#2024200311yescharacteristic
368C2#2024208811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2296C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2296D47
hypocenterC47#7613
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
3. #20241211C2semidirect
4. #20241211C2semidirect
207. C47#7613semidirect

Maximal subgroups

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

subgroupindex
#202433682
#202431502
#202412112
C47×#22962
#761347