Finite Group Databasebeta

← #26110969

Subgroups — #26110969

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252412C211yesmaximal, characteristic
2#20243456C212yesmaximal, complemented
3#20241706C211yesmaximal, characteristic
4#20241236C212yesmaximal, complemented
5C4×#123×C47C211yesmaximal, characteristic
6D4×C47⋊C8—24no—
7#15689176—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15688878—24no—
12#15688849—21no—
13#15688847—21no—
14C2×#123×C47C411yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C22×C42×C47C2211yescharacteristic
17C47×#388—21no—
18C47×#387—21no—
19C47×#386C411yescharacteristic
20C47×#385C411yescharacteristic
21C47×#381C411yescharacteristic
22C4×C4⋊C4×C47—21no—
23C4×C22⋊C4×C47—21no—
24C4×C22⋊C4×C47C2211yescharacteristic
25C22×C47⋊C8—22no—
26C22×C47⋊C8—22no—
27C22×C47⋊C8D414yes—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—22no—
35C22×D4×C47C2×C411yescharacteristic
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40#123×C47—22no—
41#123×C47C814yescomplemented
42C4×D4×C47—22no—
43C4×D4×C47—22no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47—22no—
54C2×C42×C47—21no—
55C2×C42×C47—21no—
56C2×C42×C47—21no—
57C2×C42×C47—22no—
58C2×C42×C47D411yescharacteristic
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63#95×C47—21no—
64C24×C47C22⋊C411yescharacteristic
65C24×C47C22×C411yescharacteristic
66C2×D4×C47—22no—
67C2×D4×C47—22no—
68C2×D4×C47C2×C812yes—
69C2×D4×C47M1612yes—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—21no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—42no—
81C22×C4×C47—42no—
82C22×C4×C47—42no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C2×C812yescomplemented
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4112yes—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—42no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—21no—
97C22⋊C4×C47—22no—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47C2×C812yescomplemented
101C42×C47—21no—
102C42×C47—21no—
103C42×C47—42no—
104C42×C47—41no—
105C42×C47—42no—
106C42×C47—41no—
107C2×C47⋊C8—24no—
108C2×C47⋊C8—44no—
109C2×C47⋊C8—44no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—82no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47C22⋊C812yes—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47C4×D412yes—
121C23×C47C22×C811yescharacteristic
122C23×C47C2×M1611yescharacteristic
123C23×C47#13112yes—
124D4×C47—24no—
125D4×C47—24no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—24no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—24no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—24no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
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—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—81no—
154C2×C4×C47C22⋊C812yes—
155C2×C4×C47#12011yescharacteristic
156C2×C4×C47#12112yes—
157C2×C4×C47#12311yescharacteristic
158C47⋊C8—44no—
159C47⋊C8—82no—
160#8966—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—22no—
163C22×C47—21no—
164C22×C47—22no—
165C22×C47—22no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—81no—
173C22×C47#38611yescharacteristic
174C22×C47C2×C22⋊C811yescharacteristic
175C22×C47#40811yescharacteristic
176C22×C47C8×D412yes—
177C22×C47#43512yes—
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—42no—
182C4×C47—42no—
183C4×C47—41no—
184C4×C47—42no—
185C4×C47—41no—
186C4×C47—81no—
187C4×C47#45812yes—
188#2972—471no—
189C4×#123D4711yescharacteristic
190#1817—472no—
191#1459—471no—
192#1282—472no—
193C2×C47—21no—
194C2×C47—21no—
195C2×C47—22no—
196C2×C47—22no—
197C2×C47—42no—
198C2×C47#186911yescharacteristic
199C2×C47#189011yescharacteristic
200C2×C47#189111yescharacteristic
201C2×#123Dic4711yescharacteristic
202C2×C4×D4C2×D4711yescharacteristic
203C22×C42C2×D4711yescharacteristic
204C8×D4—944no—
205#431—941no—
206C2×C22⋊C8—472no—
207C2×C22⋊C8—472no—
208C2×C22⋊C8—941no—
209#388—21no—
210#387—21no—
211#386Dic4711yescharacteristic
212#385Dic4711yescharacteristic
213#381Dic4711yescharacteristic
214C4×C4⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4C2×D4711yescharacteristic
217#336—944no—
218#324—941no—
219#323—941no—
220C47#896611yescharacteristic, complemented, Sylow
221C22×D4C2×Dic4711yescharacteristic
222C23×C4C2×Dic4711yescharacteristic
223C23×C4C2×Dic4711yescharacteristic
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22×D4711yescharacteristic
226C22×C8—474no—
227C22×C8—942no—
228C22×C8—942no—
229#123—22no—
230#123C47⋊C814yescomplemented
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22⋊D4711yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4C2×Dic4711yescharacteristic
239C2×C22⋊C4C2×Dic4711yescharacteristic
240C2×C22⋊C4C2×Dic4711yescharacteristic
241C2×C22⋊C4C22⋊D4711yescharacteristic
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22⋊D4711yescharacteristic
248C4⋊C8—942no—
249C22⋊C8—942no—
250C22⋊C8—942no—
251C22⋊C8—942no—
252C22⋊C8—942no—
253C22⋊C8—942no—
254C4×C8—942no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24C22⋊Dic4711yescharacteristic
261C24C22×Dic4711yescharacteristic
262C2×D4—22no—
263C2×D4—22no—
264C2×D4C2×C47⋊C812yes—
265C2×D4#1062532912yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C2×C47⋊C812yescomplemented
280C22×C4C22⋊Dic4711yescharacteristic
281C22×C4C22⋊Dic4711yescharacteristic
282C22×C4C22⋊Dic4711yescharacteristic
283C22×C4D4×D4712yes—
284C22×C4#1062535212yes—
285C22×C4C2×C22⋊D4711yescharacteristic
286C22×C4C2×C22⋊D4711yescharacteristic
287C2×C8—944no—
288C2×C8—3762no—
289C2×C8—1884no—
290C2×C8—1882no—
291C2×C8—1884no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—21no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—42no—
301C22⋊C4C2×C47⋊C812yescomplemented
302C42—21no—
303C42—21no—
304C42—42no—
305C42—41no—
306C42—42no—
307C42—41no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C22⋊(C47⋊C8)12yes—
315C23C2×C22⋊Dic4711yescharacteristic
316C23D4×Dic4712yes—
317C23C22×C47⋊C811yescharacteristic
318C23C2×#1062532911yescharacteristic
319C23#1309403112yes—
320D4—24no—
321D4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4C22⋊(C47⋊C8)12yes—
351C2×C4#1309399211yescharacteristic
352C2×C4#1309399612yes—
353C2×C4#1309400511yescharacteristic
354C8—3762no—
355C8—1884no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1568900311yescharacteristic
369C22C2×C22⋊(C47⋊C8)11yescharacteristic
370C22#1568909011yescharacteristic
371C22D4×C47⋊C812yes—
372C22#1568919512yes—
373C4—22no—
374C4—21no—
375C4—22no—
376C4—42no—
377C4—41no—
378C4—42no—
379C4—41no—
380C4—42no—
381C4—81no—
382C4#1568931212yes—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—42no—
388C2#2024368111yescharacteristic
389C2#2024380211yescharacteristic
390C2#2024380711yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC4×#123×C47C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC4×#123D47
hypocenterC47#8966
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
2. #20243456C2semidirect
4. #20241236C2semidirect
41. #123×C47C8semidirect
86. C22×C4×C47C2×C8semidirect
100. C22⋊C4×C47C2×C8semidirect
220. C47#8966semidirect
230. #123C47⋊C8semidirect
279. C22×C4C2×C47⋊C8semidirect
301. C22⋊C4C2×C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202524122
#202434562
#202417062
#202412362
C4×#123×C472
#896647