Finite Group Databasebeta

← #26104563

Subgroups — #26104563

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243531C211yesmaximal, characteristic, complemented
2#20243505C211yesmaximal, characteristic, complemented
3#20241211C212yesmaximal, complemented
4#20241209C212yesmaximal
5C47×#2293C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7#15688878—21no—
8#15688878—21no—
9#15688878—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×C4×D4×C47C411yescharacteristic
13C2×#117×C47—21no—
14C47×#390C411yescharacteristic
15C47×#390C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#388—21no—
19C47×#388—21no—
20C4×C4⋊C4×C47C411yescharacteristic
21C4×C22⋊C4×C47—21no—
22C43×C47C2211yescharacteristic
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—42no—
26C22×D4×C47C2×C411yescharacteristic
27C23×C4×C47—21no—
28C4×D4×C47—22no—
29C4×D4×C47—22no—
30#117×C47—42no—
31#117×C47—42no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47C2×C411yescharacteristic
35C2×C4⋊C4×C47C2×C411yescharacteristic
36C2×C4⋊C4×C47C2×C411yescharacteristic
37C2×C22⋊C4×C47—21no—
38C2×C22⋊C4×C47—21no—
39C2×C22⋊C4×C47—21no—
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C42×C47—21no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47C2311yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53C24×C47—21no—
54C2×D4×C47—21no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47M1612yes—
58C22×C4×C47—24no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C4⋊C4×C47—21no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47M1612yes—
78C22⋊C4×C47—24no—
79C22⋊C4×C47—42no—
80C22⋊C4×C47—42no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—42no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—22no—
87C42×C47—41no—
88C42×C47—41no—
89C42×C47—41no—
90C42×C47—41no—
91C42×C47—42no—
92C42×C47D812yes—
93C42×C47SD1612yes—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47C2×C22⋊C411yescharacteristic
108D4×C47—42no—
109D4×C47—42no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
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—22no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—48no—
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#10212yes—
137C2×C4×C47#10212yes—
138C2×C4×C47C2×M1611yescharacteristic
139C2×C4×C47C2×M1611yescharacteristic
140C2×C4×C47C2×D811yescharacteristic
141C2×C4×C47C2×SD1611yescharacteristic
142C2×C4×C47#13512yes—
143C47⋊C8—82no—
144C47⋊C8—82no—
145#7642—471nomaximal, Sylow
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47—81no—
153C22×C47#40711yescharacteristic
154C22×C47#41011yescharacteristic
155C22×C47#41311yescharacteristic
156C22×C47C2×#10211yescharacteristic
157C22×C47#41611yescharacteristic
158C22×C47#42111yescharacteristic
159C22×C47#42111yescharacteristic
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—42no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—81no—
171C4×C47—81no—
172#2293D4711yescharacteristic
173#1839—471no—
174#1831—471no—
175#1271—472no—
176#1270—472no—
177C2×C47—82no—
178C2×C47#146411yescharacteristic
179C2×C47#148611yescharacteristic
180C2×C47#148811yescharacteristic
181C2×C47#151711yescharacteristic
182C2×C47#152111yescharacteristic
183C2×C47#153511yescharacteristic
184C2×C47#154011yescharacteristic
185C2×C4×D4Dic4711yescharacteristic
186C2×#117—21no—
187C2×C4⋊C8—941no—
188#390Dic4711yescharacteristic
189#390C2×D4711yescharacteristic
190#389Dic4711yescharacteristic
191#389C2×D4711yescharacteristic
192#388—21no—
193#388—21no—
194C4×C4⋊C4Dic4711yescharacteristic
195C4×C22⋊C4—21no—
196C43C2×D4711yescharacteristic
197#336—472no—
198#336—472no—
199#336—941no—
200#336—941no—
201#336—941no—
202C47#764211yescharacteristic, complemented, Sylow
203C22×D4C2×Dic4711yescharacteristic
204C23×C4—21no—
205C22×C8—942no—
206C22×C8—942no—
207C4×D4—22no—
208C4×D4—22no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×Dic4711yescharacteristic
214C2×C4⋊C4C2×Dic4711yescharacteristic
215C2×C4⋊C4C2×Dic4711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42C2×Dic4711yescharacteristic
223C2×C42C2×Dic4711yescharacteristic
224C2×C42C22⋊D4711yescharacteristic
225C2×C42C22⋊D4711yescharacteristic
226C2×C42C22⋊D4711yescharacteristic
227C2×C42C22⋊D4711yescharacteristic
228C2×C42C22×D4711yescharacteristic
229C4⋊C8—1882no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#1062532912yes—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊Dic4711yescharacteristic
245C22×C4C22⋊Dic4711yescharacteristic
246C22×C4C22⋊Dic4711yescharacteristic
247C22×C4C22⋊Dic4711yescharacteristic
248C22×C4C22×Dic4711yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250C22×C4C2×C22⋊D4711yescharacteristic
251C2×C8—1882no—
252C2×C8—1882no—
253C2×C8—1882no—
254C2×C8—1882no—
255C2×C8—1882no—
256C2×C8—1882no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4#1062532912yes—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42#1062533412yes—
279C42#1062533512yes—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊Dic4711yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
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#1309394512yes—
317C2×C4#1309394512yes—
318C2×C4C2×#1062532911yescharacteristic
319C2×C4C2×#1062532911yescharacteristic
320C2×C4C2×#1062533411yescharacteristic
321C2×C4C2×#1062533511yescharacteristic
322C2×C4#1309404412yes—
323C8—3762no—
324C8—3762no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#1568908711yescharacteristic
333C22#1568909711yescharacteristic
334C22#1568910711yescharacteristic
335C22C2×#1309394511yescharacteristic
336C22#1568911911yescharacteristic
337C22#1568914011yescharacteristic
338C22#1568914011yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—81no—
350C4—81no—
351C2—82no—
352C2#2024172211yescharacteristic
353C2#2024181311yescharacteristic
354C2#2024182211yescharacteristic
355C2#2024194811yescharacteristic
356C2#2024196511yescharacteristic
357C2#2024201911yescharacteristic
358C2#2024203711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2293C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2293D47
hypocenterC47#7642
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #20243531C2semidirect
2. #20243505C2semidirect
3. #20241211C2semidirect
202. C47#7642semidirect

Maximal subgroups

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

subgroupindex
#202435312
#202435052
#202412112
#202412092
C47×#22932
#764247