Finite Group Databasebeta

← #26112979

Subgroups — #26112979

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246114C211yesmaximal, characteristic
2#20244370C211yesmaximal, characteristic, complemented
3#20242056C211yesmaximal, characteristic
4#20241952C211yesmaximal, characteristic, complemented
5#20241247C211yesmaximal, characteristic, complemented
6#20241216C211yesmaximal, characteristic, complemented
7C47×#3208C211yesmaximal, characteristic
8C2×D4×Dic47—21no—
9#15689176C2211yescharacteristic
10#15689147C2211yescharacteristic
11#15689026—21no—
12#15689003—22no—
13#15689003—22no—
14#15688989C2211yescharacteristic
15#15688969C2211yescharacteristic
16#15688882—21no—
17#15688882—21no—
18#15688867—22no—
19#15688860—22no—
20C47×#533C2211yescharacteristic
21C2×#127×C47C2211yescharacteristic
22C47×#492C412yescomplemented
23C47×#486C412yescomplemented
24C47×#460—22no—
25C47×#450—22no—
26C47×#431C2211yescharacteristic
27C23×Dic47—22no—
28C2×#10625329—21no—
29D4×Dic47—42no—
30D4×Dic47—42no—
31C2×C4⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—22no—
35C2×C4×Dic47D411yescharacteristic
36C2×C4×Dic47D411yescharacteristic
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C22⋊(C47⋊C8)—21no—
41C4×C47⋊C8—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95D412yes—
46C22×D4×C47—21no—
47C22×D4×C47—21no—
48C22×D4×C47D412yes—
49C2×SD16×C47—24no—
50C4⋊Q8×C47C2×C411yescharacteristic, complemented
51C4⋊Q8×C47C2×C411yescharacteristic, complemented
52#127×C47—21no—
53#127×C47—21no—
54#127×C47C2×C412yes—
55#127×C47D412yes—
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61#117×C47—21no—
62C2×C42×C47C2311yescharacteristic
63C4⋊C8×C47—21no—
64#102×C47—24no—
65C22⋊C8×C47—21no—
66C4×C8×C47C2×C412yescomplemented
67C24×C47—22no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×D4×C47—22no—
71C2×D4×C47—22no—
72C2×D4×C47—22no—
73C2×D4×C47—22no—
74C2×D4×C47—22no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—82no—
80C2×D4×C47—82no—
81C2×D4×C47C22⋊C414yes—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—22no—
88C22×Dic47—44no—
89C22×Dic47—82no—
90C22×Dic47—82no—
91SD16×C47—84no—
92C2×C8×C47—22no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C42×C47—41no—
101C42×C47C22×C411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C42×C47#4111yescharacteristic
105C22⋊Dic47—44no—
106C22⋊Dic47—82no—
107C22⋊Dic47—82no—
108C4⋊Dic47—41no—
109C4⋊Dic47—41no—
110C4×Dic47—21no—
111C4×Dic47—21no—
112C4×Dic47—21no—
113C4×Dic47SD1612yescomplemented
114#10625329—41no—
115#10625329—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C23×C47—24no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—42no—
124C23×C47—42no—
125C23×C47#12011yescharacteristic
126Q8×C47—42no—
127Q8×C47—81no—
128D4×C47—44no—
129D4×C47—42no—
130D4×C47—42no—
131D4×C47—42no—
132D4×C47—42no—
133D4×C47—82no—
134D4×C47—82no—
135D4×C47—82no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47C4×D411yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12311yescharacteristic
146C2×C4×C47C2×SD1611yescharacteristic
147C2×C4×C47#13511yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C47#13711yescharacteristic
150C2×Dic47—22no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—41no—
159C2×Dic47—88no—
160C8×C47—42no—
161C47⋊C8—81no—
162C47⋊C8—81no—
163#9295—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—44no—
167C22×C47—44no—
168C22×C47—44no—
169C22×C47—82no—
170C22×C47—82no—
171C22×C47—82no—
172C22×C47—82no—
173C22×C47#38611yescharacteristic
174C22×C47#45011yescharacteristic
175C22×C47#45211yescharacteristic
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—81no—
181C4×C47—81no—
182C4×C47C4×SD1611yescharacteristic
183C4×C47#44011yescharacteristic
184C4×C47#46111yescharacteristic
185C4×C47#46311yescharacteristic
186Dic47—42no—
187Dic47—81no—
188Dic47—82no—
189#3208D4711yescharacteristic
190#2328—471no—
191#1995—471no—
192#1545—471no—
193#1518—471no—
194#1286—471no—
195#1274—471no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—84no—
199C2×C47#187711yescharacteristic
200C2×C47#190411yescharacteristic
201C2×C47#191111yescharacteristic
202#533C2×D4711yescharacteristic
203C2×#127C2×D4711yescharacteristic
204C2×C4×D4—941no—
205#492Dic4712yescomplemented
206#486Dic4712yescomplemented
207#460—22no—
208#450—22no—
209#431—471no—
210#431C2×D4711yescharacteristic
211#423—471no—
212#390—941no—
213#386—942no—
214#386—942no—
215#382—471no—
216#376—471no—
217#339—941no—
218#339—941no—
219#331—942no—
220#328—942no—
221C47#929511yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D4712yes—
225C23×C4—942no—
226C2×SD16—24no—
227C2×M16—941no—
228C4⋊Q8C2×Dic4711yescharacteristic, complemented
229C4⋊Q8C2×Dic4711yescharacteristic, complemented
230#127—21no—
231#127—21no—
232#127C2×Dic4712yes—
233#127C22⋊D4712yes—
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—1882no—
240C4×D4—1882no—
241#117—21no—
242C2×C4⋊C4—941no—
243C2×C22⋊C4—942no—
244C2×C22⋊C4—942no—
245C2×C22⋊C4—942no—
246C2×C42—471no—
247C2×C42—471no—
248C2×C42C22×D4711yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—941no—
251C4⋊C8—941no—
252C4⋊C8—941no—
253#102—24no—
254C22⋊C8—21no—
255C22⋊C8—941no—
256C4×C8—941no—
257C4×C8C2×Dic4712yescomplemented
258#95—472no—
259#95—941no—
260#95—941no—
261#95—941no—
262C24—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4C22⋊Dic4714yes—
277C22×C4—941no—
278C22×C4—942no—
279C22×C4—941no—
280C22×C4—3762no—
281C22×C4—3762no—
282C22×C4—1884no—
283C22×C4D4×D4711yescharacteristic
284C22×C4D4×D4711yescharacteristic
285C22×C4C2×C22⋊D4711yescharacteristic
286SD16—84no—
287M16—1881no—
288M16—1881no—
289C2×C8—22no—
290C2×C8—1881no—
291C2×C8—1881no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1881no—
299C4⋊C4—1881no—
300C22⋊C4—41no—
301C22⋊C4—3762no—
302C22⋊C4—3762no—
303C22⋊C4—1884no—
304C42—41no—
305C42—472no—
306C42—941no—
307C42—941no—
308C42—941no—
309C42#1062535211yescharacteristic
310C42#1062535211yescharacteristic
311C42C22×Dic4711yescharacteristic
312C42C2×C22⋊D4711yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1309399211yescharacteristic
321Q8—42no—
322Q8—81no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—942no—
336C2×C4—3768no—
337C2×C4—1881no—
338C2×C4—1881no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4C2×C22⋊Dic4711yescharacteristic
346C2×C4D4×Dic4711yescharacteristic
347C2×C4D4×Dic4711yescharacteristic
348C2×C4#1309399611yescharacteristic
349C2×C4#1309399611yescharacteristic
350C2×C4#1309400511yescharacteristic
351C2×C4SD16×D4711yescharacteristic
352C2×C4#1309404511yescharacteristic
353C2×C4#1309405111yescharacteristic
354C2×C4#1309405311yescharacteristic
355C8—42no—
356C8—3761no—
357C8—3761no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1568900311yescharacteristic
368C22#1568927211yescharacteristic
369C22#1568928211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—3762no—
377C4—3761no—
378C4—1882no—
379C4SD16×Dic4711yescharacteristic
380C4#1568922411yescharacteristic
381C4#1568932611yescharacteristic
382C4#1568934011yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2024372411yescharacteristic
387C2#2024387711yescharacteristic
388C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3208C2
socleC22×C47#386
prime radicalC2×#127×C47C22
cosocleC2×C42×C47C23
hypercenter#3208D47
hypocenterC47#9295
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
2. #20244370C2semidirect
4. #20241952C2semidirect
5. #20241247C2semidirect
6. #20241216C2semidirect
22. C47×#492C4semidirect
23. C47×#486C4semidirect
50. C4⋊Q8×C47C2×C4semidirect
51. C4⋊Q8×C47C2×C4semidirect
66. C4×C8×C47C2×C4semidirect
113. C4×Dic47SD16semidirect
205. #492Dic47semidirect
206. #486Dic47semidirect
221. C47#9295semidirect
228. C4⋊Q8C2×Dic47semidirect
229. C4⋊Q8C2×Dic47semidirect
257. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202461142
#202443702
#202420562
#202419522
#202412472
#202412162
C47×#32082
#929547