Finite Group Databasebeta

← #26115631

Subgroups — #26115631

Order 12032.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×C4⋊Dic47C211yesmaximal, characteristic, complemented
2#20242136C211yesmaximal, characteristic, complemented
3#20241887C211yesmaximal, characteristic, complemented
4#20241883C211yesmaximal, characteristic
5#20241791C211yesmaximal, characteristic, complemented
6#20241787C211yesmaximal, characteristic
7C47×#3131C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689176C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×C8×Dic47—21no—
12C4×C4⋊Dic47C2211yescharacteristic
13#15688877—21no—
14#15688876—21no—
15#15688874—21no—
16#15688860—22no—
17#15688857—22no—
18Q8⋊(C47⋊C8)—22no—
19D4⋊(C47⋊C8)—22no—
20C47×#533C2211yescharacteristic
21C47×#532C2211yescharacteristic
22C47×#488—21no—
23C47×#488—21no—
24C47×#487C411yescharacteristic, complemented
25C47×#487C411yescharacteristic, complemented
26C47×#486C411yescharacteristic, complemented
27C47×#486C411yescharacteristic, complemented
28C47×#416—21no—
29C47×#416—21no—
30C2×C4×C8×C47C2211yescharacteristic
31C22×C47⋊C8—21no—
32C2×C4⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)D412yes—
38C22⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)—21no—
40C4×C47⋊C8—21no—
41C4×C47⋊C8—21no—
42C8×Dic47—42no—
43C47⋊#95D412yes—
44C2×#41×C47—21no—
45C22×C8×C47D411yescharacteristic
46C22×C8×C47D411yescharacteristic
47C4⋊Q8×C47C2×C411yescharacteristic
48C4⋊Q8×C47C2×C411yescharacteristic
49C4⋊Q8×C47C2×C411yescharacteristic
50#127×C47C2×C411yescharacteristic
51#125×C47—21no—
52#124×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47—21no—
55#121×C47—21no—
56#121×C47—21no—
57C4×Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47—21no—
61#117×C47—21no—
62C2×C42×C47C2311yescharacteristic
63#103×C47—21no—
64#103×C47—21no—
65#103×C47—21no—
66#103×C47—21no—
67#102×C47—21no—
68#102×C47—21no—
69#102×C47—21no—
70#102×C47—21no—
71C4×C8×C47C2×C411yescharacteristic
72C4×C8×C47C2×C411yescharacteristic
73C4×C8×C47D411yescharacteristic
74C4×C8×C47D411yescharacteristic
75#41×C47—41no—
76#41×C47—41no—
77#41×C47—41no—
78#41×C47—41no—
79C2×Q8×C47—21no—
80C2×Q8×C47—21no—
81C2×Q8×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C22×Dic47—22no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C2×C8×C47C22⋊C411yescharacteristic
98C2×C8×C47C22⋊C411yescharacteristic
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—41no—
110C42×C47—41no—
111C42×C47C22×C411yescharacteristic
112C42×C47C2×D411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47#4111yescharacteristic
115C4⋊Dic47—24no—
116C4×Dic47—41no—
117C4×Dic47—41no—
118C2×C47⋊C8—22no—
119C2×C47⋊C8—41no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C23×C47—41no—
124C23×C47#12111yescharacteristic
125Q8×C47—41no—
126Q8×C47—41no—
127Q8×C47—42no—
128Q8×C47—81no—
129D4×C47—41no—
130D4×C47—41no—
131D4×C47—42no—
132D4×C47—81no—
133D4×C47—81no—
134D4×C47—81no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—81no—
141C2×C4×C47—81no—
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47C4×D411yescharacteristic
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12411yescharacteristic
147C2×C4×C47#12711yescharacteristic
148C2×C4×C47#13511yescharacteristic
149C2×C4×C47#13511yescharacteristic
150C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C47#13711yescharacteristic
152C2×Dic47—24no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C8×C47—21no—
156C8×C47—21no—
157C8×C47—21no—
158C8×C47—21no—
159C47⋊C8—42no—
160C47⋊C8—81no—
161C47⋊C8—81no—
162#9751—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47#39011yescharacteristic
169C22×C47#46311yescharacteristic
170C22×C47#46411yescharacteristic
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—81no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47#41611yescharacteristic
178C4×C47#41711yescharacteristic
179C4×C47#44312yes—
180C4×C47#46311yescharacteristic
181C4×C47#46411yescharacteristic
182Dic47—44no—
183#3131D4711yescharacteristic
184C8×C4⋊C4—471no—
185#1564—471no—
186#1503—471no—
187#1502—471no—
188#1481—471no—
189#1480—471no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—161no—
193C2×C47#197411yescharacteristic
194C2×C47#197611yescharacteristic
195C2×C47#198311yescharacteristic
196#533C2×D4711yescharacteristic
197#532C2×D4711yescharacteristic
198#488—21no—
199#488—21no—
200#487Dic4711yescharacteristic, complemented
201#487Dic4711yescharacteristic, complemented
202#486Dic4711yescharacteristic, complemented
203#486Dic4711yescharacteristic, complemented
204#431—471no—
205#431—471no—
206C2×C4⋊C8—471no—
207#416—21no—
208#416—21no—
209C2×C4×C8—941no—
210C2×C4×C8C2×D4711yescharacteristic
211C4×C4⋊C4—471no—
212#336—941no—
213#335—941no—
214#334—941no—
215#328—942no—
216#327—942no—
217Q8⋊C8—942no—
218D4⋊C8—942no—
219C47#975111yescharacteristic, complemented, Sylow
220C2×#41—21no—
221C22×C8—941no—
222C22×C8C22⋊D4711yescharacteristic
223C22×C8C22⋊D4711yescharacteristic
224C4⋊Q8C2×Dic4711yescharacteristic
225C4⋊Q8C2×Dic4711yescharacteristic
226C4⋊Q8C2×Dic4711yescharacteristic
227#127C2×Dic4711yescharacteristic
228#125—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117—21no—
239C2×C4⋊C4—941no—
240C2×C42—941no—
241C2×C42C22×D4711yescharacteristic
242C4⋊C8—472no—
243C4⋊C8—941no—
244C4⋊C8—941no—
245C4⋊C8—941no—
246#103—21no—
247#103—21no—
248#103—21no—
249#103—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254C22⋊C8—941no—
255C22⋊C8—941no—
256C4×C8—941no—
257C4×C8—941no—
258C4×C8—1882no—
259C4×C8C2×Dic4711yescharacteristic
260C4×C8C2×Dic4711yescharacteristic
261C4×C8C22⋊D4711yescharacteristic
262C4×C8C22⋊D4711yescharacteristic
263#95—472no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—942no—
277C22×C4D4×D4711yescharacteristic
278C22×C4#1062535211yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—942no—
285C2×C8—1881no—
286C2×C8—1881no—
287C2×C8—1881no—
288C2×C8—1881no—
289C2×C8C22⋊Dic4711yescharacteristic
290C2×C8C22⋊Dic4711yescharacteristic
291C2×C8C22⋊Dic4711yescharacteristic
292C2×C8C22⋊Dic4711yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—944no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—1881no—
307C42—1881no—
308C42D4×D4711yescharacteristic
309C42#1062535211yescharacteristic
310C42C22×Dic4711yescharacteristic
311C42C2×C22⋊D4711yescharacteristic
312C23—41no—
313C23#1309399511yescharacteristic
314Q8—42no—
315Q8—41no—
316Q8—41no—
317Q8—81no—
318D4—42no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—944no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4C2×C22⋊Dic4711yescharacteristic
335C2×C4D4×Dic4711yescharacteristic
336C2×C4D4×Dic4711yescharacteristic
337C2×C4#1309399511yescharacteristic
338C2×C4#1309401111yescharacteristic
339C2×C4#1309402011yescharacteristic
340C2×C4#1309404411yescharacteristic
341C2×C4#1309404411yescharacteristic
342C2×C4#1309404911yescharacteristic
343C2×C4#1309405411yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—3761no—
349C8—3761no—
350C8—1882no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1568902611yescharacteristic
357C22#1568933511yescharacteristic
358C22#1568934211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—1884no—
366C4#1568911911yescharacteristic
367C4#1568912411yescharacteristic
368C4#1568924012yes—
369C4#1568933511yescharacteristic
370C4#1568934211yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2024427711yescharacteristic
375C2#2024428711yescharacteristic
376C2#2024432111yescharacteristic
subgroupG/H
centerC2×C4#13093995
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3131C2
socleC22×C47#390
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3131D47
hypocenterC47#9751
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. C8×C4⋊Dic47C2semidirect
2. #20242136C2semidirect
3. #20241887C2semidirect
5. #20241791C2semidirect
24. C47×#487C4semidirect
25. C47×#487C4semidirect
26. C47×#486C4semidirect
27. C47×#486C4semidirect
200. #487Dic47semidirect
201. #487Dic47semidirect
202. #486Dic47semidirect
203. #486Dic47semidirect
219. C47#9751semidirect

Maximal subgroups

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

subgroupindex
C8×C4⋊Dic472
#202421362
#202418872
#202418832
#202417912
#202417872
C47×#31312
#975147