Finite Group Databasebeta

← #26111368

Subgroups — #26111368

Order 12032.

528 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#20251952C211yesmaximal, characteristic
2#20241817C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic
4#20241291C212yesmaximal, complemented
5C47×#2396C211yesmaximal, characteristic
6C22×#10625329—21no—
7#15689176C2212yes—
8#15689150—24no—
9#15689147—21no—
10#15689147C2212yes—
11Q8⋊(C47⋊C8)—24no—
12#15688849—21no—
13#15688847—21no—
14C2×C4×Q8×C47C2211yescharacteristic
15C2×#117×C47C2211yescharacteristic
16C47×#400C411yescharacteristic
17C47×#399C411yescharacteristic
18C47×#398—21no—
19C47×#396—21no—
20C47×#395C411yescharacteristic
21C47×#393C411yescharacteristic
22C4×C4⋊C4×C47—21no—
23C4×C4⋊C4×C47C2211yescharacteristic
24C4×C22⋊C4×C47—21no—
25C2×#10625329—22no—
26C2×#10625329—22no—
27C2×#10625329—42no—
28C2×#10625329—42no—
29C22×C47⋊C8—22no—
30C4⋊(C47⋊C8)—24no—
31C4⋊(C47⋊C8)—22no—
32C4⋊(C47⋊C8)—22no—
33C4⋊(C47⋊C8)D414yes—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8—22no—
36C22×Q8×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C4×Q8×C47—22no—
39C4×Q8×C47—22no—
40#117×C47—22no—
41#117×C47—22no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47—21no—
54C2×C42×C47—21no—
55C2×C42×C47D411yescharacteristic
56C2×C42×C47C2311yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47C2×C411yescharacteristic
62#95×C47C2×C411yescharacteristic
63C24×C47C22⋊C411yescharacteristic
64C2×Q8×C47—22no—
65C2×Q8×C47—42no—
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—21no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47SD1612yes—
78C22×C4×C47Q1612yes—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C4⋊C4×C47—22no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—42no—
87C4⋊C4×C47—42no—
88C4⋊C4×C47—42no—
89C22⋊C4×C47—22no—
90C22⋊C4×C47—22no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—42no—
93C42×C47—21no—
94C42×C47—21no—
95C42×C47—41no—
96C42×C47—41no—
97C42×C47—41no—
98C42×C47—41no—
99C42×C47C2×D412yes—
100C42×C47#4112yes—
101#10625329—44no—
102#10625329—42no—
103#10625329—42no—
104C2×C47⋊C8—24no—
105C2×C47⋊C8—44no—
106C2×C47⋊C8—42no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47#10312yes—
112C23×C47C2×C22⋊C411yescharacteristic
113Q8×C47—24no—
114Q8×C47—42no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
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—41no—
138C2×C4×C47—41no—
139C2×C4×C47#10312yes—
140C2×C4×C47C4×D412yes—
141C2×C4×C47#12011yescharacteristic
142C2×C4×C47#12112yes—
143C2×C4×C47#12311yescharacteristic
144C2×C4×C47C2×SD1611yescharacteristic
145C2×C4×C47C2×Q1611yescharacteristic
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47#13711yescharacteristic
148C47⋊C8—44no—
149C47⋊C8—82no—
150#9033—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—22no—
154C22×C47—42no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—81no—
158C22×C47#38611yescharacteristic
159C22×C47C2×#10311yescharacteristic
160C22×C47#41711yescharacteristic
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—42no—
165C4×C47—41no—
166C4×C47—41no—
167C4×C47—42no—
168C4×C47—81no—
169C4×C47—81no—
170C4×C47#45011yescharacteristic
171C4×C47#45111yescharacteristic
172C4×C47#48311yescharacteristic
173C4×C47#48411yescharacteristic
174#2905—471no—
175#2396D4711yescharacteristic
176#1487—472no—
177#1458—471no—
178#1303—472no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—42no—
182C2×C47#187211yescharacteristic
183C2×C47#188411yescharacteristic
184C2×C47#189111yescharacteristic
185C22×M16—941no—
186C2×C4×Q8C2×D4711yescharacteristic
187C2×#117C2×D4711yescharacteristic
188#431—472no—
189#424—944no—
190#423—472no—
191#423—941no—
192#400Dic4711yescharacteristic
193#399Dic4711yescharacteristic
194#398—21no—
195#396—21no—
196#395Dic4711yescharacteristic
197#393Dic4711yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2×D4711yescharacteristic
200C4×C22⋊C4—21no—
201Q8⋊C8—944no—
202#324—941no—
203#323—941no—
204C47#903311yescharacteristic, complemented, Sylow
205C22×Q8C2×Dic4711yescharacteristic
206C23×C4C22⋊D4711yescharacteristic
207C2×M16—942no—
208C2×M16—942no—
209C2×M16—1882no—
210C2×M16—1882no—
211C22×C8—942no—
212C4×Q8—22no—
213C4×Q8—22no—
214#117—22no—
215#117—22no—
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⋊C4C22⋊D4711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic4711yescharacteristic
226C2×C22⋊C4C22⋊D4711yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22⋊D4711yescharacteristic
230C2×C42C22×D4711yescharacteristic
231C4⋊C8—474no—
232C4⋊C8—944no—
233C4⋊C8—942no—
234C4⋊C8—942no—
235C22⋊C8—942no—
236C4×C8—942no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95C2×Dic4711yescharacteristic
242#95C2×Dic4711yescharacteristic
243C24C22⋊Dic4711yescharacteristic
244C2×Q8—22no—
245C2×Q8—42no—
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#1062533612yes—
255C22×C4#1062533712yes—
256C22×C4C22⋊Dic4711yescharacteristic
257C22×C4C22⋊Dic4711yescharacteristic
258C22×C4C22⋊Dic4711yescharacteristic
259C22×C4C22×Dic4711yescharacteristic
260C22×C4C2×C22⋊D4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262M16—1884no—
263M16—1882no—
264M16—1882no—
265C2×C8—944no—
266C2×C8—1882no—
267C2×C8—1884no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42D4×D4712yes—
286C42#1062535212yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1309394812yes—
292C23C2×C22⋊Dic4711yescharacteristic
293Q8—24no—
294Q8—42no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
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—41no—
319C2×C4#1309394812yes—
320C2×C4D4×Dic4712yes—
321C2×C4#1309399211yescharacteristic
322C2×C4#1309399612yes—
323C2×C4#1309400511yescharacteristic
324C2×C4C2×#1062533611yescharacteristic
325C2×C4C2×#1062533711yescharacteristic
326C2×C4#1309404911yescharacteristic
327C2×C4#1309405411yescharacteristic
328C8—3762no—
329C8—1884no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1568900311yescharacteristic
338C22C2×#1309394811yescharacteristic
339C22#1568912411yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4#1568927311yescharacteristic
350C4#1568927811yescharacteristic
351C4#1568945111yescharacteristic
352C4#1568945611yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2024369611yescharacteristic
357C2#2024377211yescharacteristic
358C2#2024380711yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2396C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2396D47
hypocenterC47#9033
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #20241817C2semidirect
4. #20241291C2semidirect
204. C47#9033semidirect

Maximal subgroups

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

subgroupindex
#202519522
#202418172
#202417022
#202412912
C47×#23962
#903347