Finite Group Databasebeta

← #26104692

Subgroups — #26104692

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243531C211yesmaximal, characteristic, complemented
2C4×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3#20241211C212yesmaximal, complemented
4#20241209C212yesmaximal
5C47×#2860C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C4×C47⋊C8—21no—
8#15688878—21no—
9#15688878—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×#126×C47—21no—
13C47×#390C411yescharacteristic
14C47×#390C411yescharacteristic
15C47×#390C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C411yescharacteristic
18C47×#389C2211yescharacteristic
19C47×#388—21no—
20C47×#388—21no—
21C47×#388—21no—
22C43×C47C2211yescharacteristic
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—24no—
26C4×C47⋊C8—42no—
27C22×D4×C47C2×C411yescharacteristic
28#126×C47—42no—
29#126×C47—42no—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C4⋊C4×C47C2×C411yescharacteristic
35C2×C4⋊C4×C47C2×C411yescharacteristic
36C2×C22⋊C4×C47—21no—
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×C47C2×C411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47C2311yescharacteristic
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52C24×C47—21no—
53C2×D4×C47—22no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C4⋊C4×C47—22no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—42no—
74C22⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—42no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47—41no—
84C42×C47—41no—
85C42×C47—41no—
86C42×C47D812yes—
87C42×C47SD1612yes—
88C2×C47⋊C8—24no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47C2×C22⋊C411yescharacteristic
102D4×C47—82no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
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—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47#10212yes—
128C2×C4×C47#10212yes—
129C2×C4×C47#10412yes—
130C2×C4×C47#13112yes—
131C2×C4×C47C2×D811yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C2×C4×C47#13512yes—
134C47⋊C8—44no—
135C47⋊C8—82no—
136#7672—471nomaximal, Sylow
137C22×C47—81no—
138C22×C47—81no—
139C22×C47—81no—
140C22×C47—81no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47—81no—
144C22×C47#40811yescharacteristic
145C22×C47#41011yescharacteristic
146C22×C47#41311yescharacteristic
147C22×C47C2×#10211yescharacteristic
148C22×C47#41611yescharacteristic
149C22×C47C2×#10411yescharacteristic
150C22×C47#42111yescharacteristic
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—22no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—22no—
158C4×C47—41no—
159C4×C47—81no—
160C4×C47—81no—
161C4×C47—81no—
162#2860D4711yescharacteristic
163#1839—471no—
164C4×C4⋊C8—471no—
165#1271—472no—
166#1270—472no—
167C2×C47—161no—
168C2×C47#147311yescharacteristic
169C2×C47#148011yescharacteristic
170C2×C47#149411yescharacteristic
171C2×C47#150211yescharacteristic
172C2×C47#152111yescharacteristic
173C2×C47#152911yescharacteristic
174C2×C47#154011yescharacteristic
175C2×#126—21no—
176C2×C4⋊C8—941no—
177C2×C4×C8—941no—
178#390Dic4711yescharacteristic
179#390Dic4711yescharacteristic
180#390C2×D4711yescharacteristic
181#389Dic4711yescharacteristic
182#389Dic4711yescharacteristic
183#389C2×D4711yescharacteristic
184#388—21no—
185#388—21no—
186#388—21no—
187C43C2×D4711yescharacteristic
188#336—472no—
189#336—472no—
190#336—941no—
191#336—941no—
192C47#767211yescharacteristic, complemented, Sylow
193C22×D4C2×Dic4711yescharacteristic
194C22×C8—942no—
195C22×C8—942no—
196#126—42no—
197#126—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic4711yescharacteristic
202C2×C4⋊C4C2×Dic4711yescharacteristic
203C2×C4⋊C4C2×Dic4711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42C2×Dic4711yescharacteristic
211C2×C42C2×Dic4711yescharacteristic
212C2×C42C22⋊D4711yescharacteristic
213C2×C42C22⋊D4711yescharacteristic
214C2×C42C22⋊D4711yescharacteristic
215C2×C42C22⋊D4711yescharacteristic
216C2×C42C22×D4711yescharacteristic
217C4⋊C8—944no—
218C4×C8—1882no—
219#95—21no—
220#95—21no—
221#95—21no—
222C24—21no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22⋊Dic4711yescharacteristic
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22×Dic4711yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236C2×C8—944no—
237C2×C8—1882no—
238C2×C8—1882no—
239C2×C8—1882no—
240C2×C8—1882no—
241C2×C8—1882no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42#1062533412yes—
263C42#1062533512yes—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊Dic4711yescharacteristic
272D4—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4#1309394512yes—
298C2×C4#1309394512yes—
299C2×C4#1309395112yes—
300C2×C4#1309403112yes—
301C2×C4C2×#1062533411yescharacteristic
302C2×C4C2×#1062533511yescharacteristic
303C2×C4#1309404412yes—
304C8—3762no—
305C8—1884no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1568909011yescharacteristic
314C22#1568909711yescharacteristic
315C22#1568910711yescharacteristic
316C22C2×#1309394511yescharacteristic
317C22#1568911911yescharacteristic
318C22C2×#1309395111yescharacteristic
319C22#1568914011yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C2—161no—
332C2#2024175711yescharacteristic
333C2#2024178711yescharacteristic
334C2#2024184811yescharacteristic
335C2#2024188311yescharacteristic
336C2#2024196511yescharacteristic
337C2#2024199511yescharacteristic
338C2#2024203711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2860C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2860D47
hypocenterC47#7672
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #20243531C2semidirect
2. C4×C4⋊(C47⋊C8)C2semidirect
3. #20241211C2semidirect
192. C47#7672semidirect

Maximal subgroups

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

subgroupindex
#202435312
C4×C4⋊(C47⋊C8)2
#202412112
#202412092
C47×#28602
#767247