Finite Group Databasebeta

← #26131525

Subgroups — #26131525

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252159C211yesmaximal, characteristic
2C2×#15688934C211yesmaximal, characteristic
3C2×#15688932C211yesmaximal, characteristic
4M32×Dic47C211yesmaximal, characteristic, complemented
5#106×Dic47C211yesmaximal, characteristic, complemented
6#20241493C211yesmaximal, characteristic, complemented
7C47×#1386C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic
9#15689591—21no—
10C2×#13093964C2211yescharacteristic
11#15689217C411yescharacteristic, complemented
12#15689217C411yescharacteristic, complemented
13#15689210—21no—
14#15689210—21no—
15#15689205C411yescharacteristic, complemented
16#15689205C411yescharacteristic, complemented
17C2×#13093956C2211yescharacteristic
18#15689117—21no—
19C2×C8×Dic47C2211yescharacteristic
20C4⋊C4×Dic47—21no—
21#15688934C414yescomplemented
22#15688932C414yescomplemented
23#15688898—21no—
24C16×Dic47—21no—
25#15688885—21no—
26C2×M32×C47C2211yescharacteristic
27C47×#427C2211yescharacteristic
28C2×#106×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094046—21no—
31#13094046—21no—
32#13094046C2×C412yes—
33#13094046C2×C412yes—
34C2×Dic188C2×C411yescharacteristic
35C2×#10625325—21no—
36C2×D376C2×C411yescharacteristic
37C22×C47⋊C8D411yescharacteristic
38#13093989—21no—
39#13093989—21no—
40#13093986—21no—
41#13093986—21no—
42(C22⋊D47)⋊C4—41no—
43(C22⋊D47)⋊C4—41no—
44#13093981—41no—
45C2×C4⋊Dic47—21no—
46C2×C4×Dic47—41no—
47C2×C4×Dic47D411yescharacteristic
48C42×D47—41no—
49#13093964—21no—
50#13093964C2×C412yescomplemented
51C2×C47⋊C16—21no—
52#13093956—21no—
53#13093956C2×C412yescomplemented
54#13093947—21no—
55#13093947—21no—
56#13093944—21no—
57#13093944—21no—
58C8×Dic47C2×C411yescharacteristic
59C8×Dic47C2×C411yescharacteristic
60C8×Dic47D411yescharacteristic
61C8×Dic47D411yescharacteristic
62C47⋊#95—41no—
63C22×C8×C47C2311yescharacteristic
64#117×C47—21no—
65C2×C4⋊C4×C47—21no—
66M32×C47—21no—
67M32×C47C2×C412yescomplemented
68C2×C16×C47—21no—
69#107×C47C2×C411yescharacteristic
70#107×C47C2×C411yescharacteristic
71#106×C47—21no—
72#106×C47—21no—
73#106×C47C2×C412yescomplemented
74C22×C4×C47—41no—
75C22×C4×C47C2×D411yescharacteristic
76C2×C22⋊D47—41no—
77C22×Dic47—41no—
78C22×Dic47D811yescharacteristic
79C22×Dic47D811yescharacteristic
80#10625350—22no—
81#10625350—22no—
82#10625350—41no—
83#10625350—41no—
84C2×D188—21no—
85C2×C4×D47—41no—
86C2×C4×D47—41no—
87C2×C4×D47—41no—
88C2×Dic94—21no—
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C22×C411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C42×C47—41no—
101(C2×D47)⋊C4—41no—
102(C2×D47)⋊C4—41no—
103C4⋊Dic47—22no—
104C4⋊Dic47—41no—
105Dic47⋊C4—41no—
106Dic47⋊C4—41no—
107C4×Dic47—41no—
108C4×Dic47—41no—
109C4×Dic47—82no—
110C4×Dic47C22⋊C411yescharacteristic
111C4×Dic47C22⋊C411yescharacteristic
112C4×Dic47SD1611yescharacteristic
113C4×Dic47SD1611yescharacteristic
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8C22⋊C411yescharacteristic
117C2×C47⋊C8C22⋊C411yescharacteristic
118Dic188—21no—
119Dic188C4212yescomplemented
120D376—21no—
121D376C4212yescomplemented
122#10625325—22no—
123#10625325—22no—
124C16×C47—22no—
125C47⋊C16—22no—
126C23×C47C2×D811yescharacteristic
127C22×D47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47C2×SD1611yescharacteristic
137C2×C4×C47#13511yescharacteristic
138C2×C4×C47#13511yescharacteristic
139C22⋊D47—42no—
140C22⋊D47—42no—
141C2×Dic47—22no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—42no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47#10211yescharacteristic
148C2×Dic47#10211yescharacteristic
149C2×Dic47#10211yescharacteristic
150C2×Dic47#10211yescharacteristic
151D188—22no—
152D188—41no—
153C4×D47—42no—
154C4×D47—42no—
155C4×D47—82no—
156C4×D47—82no—
157Dic94—22no—
158Dic94—41no—
159C8×C47C2×C4211yescharacteristic
160C8×C47#11711yescharacteristic
161C8×C47C4×D411yescharacteristic
162C8×C47C4×D411yescharacteristic
163C47⋊C8—21no—
164C47⋊C8—21no—
165C4×#360—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47C2×#10211yescharacteristic
169C22×C47C4×D811yescharacteristic
170C22×C47C4×D811yescharacteristic
171C2×D47—42no—
172C2×D47—81no—
173C2×D47—81no—
174C4×C47—82no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47C4×C22⋊C411yescharacteristic
178C4×C47#41611yescharacteristic
179C4×C47C4×SD1611yescharacteristic
180C4×C47C4×SD1611yescharacteristic
181Dic47—21no—
182Dic47—21no—
183Dic47—21no—
184Dic47—42no—
185Dic47—82no—
186Dic47#36012yescomplemented
187C4×#135—471no—
188C2×#360—471no—
189C2×#360—471no—
190C4×M32—471no—
191C4×#106—471no—
192#1386—471no—
193#1386D4711yescharacteristic, complemented
194C2×C47—21no—
195C2×C47—21no—
196C2×C47C4×#10211yescharacteristic
197C2×C47C2×#36011yescharacteristic
198C2×C47#214511yescharacteristic
199D47—82no—
200C2×#135—471no—
201C4×#41—941no—
202C2×M32—471no—
203C2×M32C2×D4711yescharacteristic
204C4×Q16—471no—
205C4×Q16—471no—
206C4×SD16—941no—
207C4×SD16—941no—
208C4×D8—471no—
209C4×D8—471no—
210#427C2×D4711yescharacteristic
211C2×#106—471no—
212C2×#106C2×D4711yescharacteristic
213#416—941no—
214C2×C4×C8—471no—
215C4×C4⋊C4—941no—
216#360—474no—
217#360—474no—
218#346—941no—
219C4×C16—941no—
220#340—941no—
221C47C4×#36011yescharacteristic, complemented, Sylow
222C2×#41—941no—
223#135—472no—
224#135—472no—
225#135—941no—
226#135—941no—
227C2×Q16—471no—
228C2×SD16—941no—
229C2×D8—471no—
230C22×C8—471no—
231C22×C8C22×D4711yescharacteristic
232C4×Q8—941no—
233C4×Q8—941no—
234C4×D4—941no—
235C4×D4—941no—
236C4×D4—1881no—
237C4×D4—1881no—
238#117—21no—
239#117—1881no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—941no—
242C2×C42—471no—
243C2×C42—1881no—
244C2×C42—1881no—
245M32—21no—
246M32—472no—
247M32—941no—
248M32C4×D4712yescomplemented
249C2×C16—21no—
250C2×C16—941no—
251#107C4×D4711yescharacteristic
252#107C4×D4711yescharacteristic
253#106—21no—
254#106—21no—
255#106—472no—
256#106—941no—
257#106C4×D4712yescomplemented
258#103—941no—
259#103—941no—
260#102—941no—
261#102—941no—
262C4×C8—471no—
263C4×C8—471no—
264C4×C8—471no—
265C4×C8—471no—
266#95—1881no—
267#41—942no—
268#41—942no—
269#41—1881no—
270#41—1881no—
271C2×Q8—941no—
272C2×D4—941no—
273C2×D4—1881no—
274C22×C4—41no—
275C22×C4—471no—
276C22×C4—471no—
277C22×C4—1881no—
278C22×C4—1881no—
279C22×C4—1881no—
280C22×C4—1881no—
281C22×C4D4×D4711yescharacteristic
282Q16—472no—
283Q16—941no—
284SD16—942no—
285SD16—942no—
286D8—472no—
287D8—941no—
288C2×C8—471no—
289C2×C8—471no—
290C2×C8—941no—
291C2×C8—941no—
292C2×C8C2×C4×D4711yescharacteristic
293C2×C8C2×C4×D4711yescharacteristic
294C2×C8C2×C4×D4711yescharacteristic
295C2×C8D4×D4711yescharacteristic
296C2×C8#1062535211yescharacteristic
297C2×C8#1062535411yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—942no—
303C4⋊C4—1881no—
304C4⋊C4—1881no—
305C4⋊C4—1881no—
306C22⋊C4—41no—
307C22⋊C4—1881no—
308C22⋊C4—1881no—
309C42—41no—
310C42—471no—
311C42—471no—
312C42—471no—
313C42—471no—
314C42—3762no—
315C42—1881no—
316C42—1881no—
317C16—22no—
318C16—942no—
319C23—1881no—
320C23D8×D4711yescharacteristic
321Q8—942no—
322Q8—1881no—
323D4—942no—
324D4—1882no—
325D4—1882no—
326D4—1881no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—471no—
333C2×C4—471no—
334C2×C4—471no—
335C2×C4—471no—
336C2×C4—942no—
337C2×C4—3762no—
338C2×C4—3762no—
339C2×C4—3761no—
340C2×C4—3761no—
341C2×C4—1881no—
342C2×C4—1882no—
343C2×C4—1881no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4C22⋊C4×D4711yescharacteristic
347C2×C4(C22⋊D47)⋊C411yescharacteristic
348C2×C4#1309398611yescharacteristic
349C2×C4SD16×D4711yescharacteristic
350C2×C4#1309404511yescharacteristic
351C2×C4#1309404711yescharacteristic
352C8—941no—
353C8—941no—
354C8C42×D4711yescharacteristic
355C8#1309398211yescharacteristic
356C8(C22⋊D47)⋊C411yescharacteristic
357C8#1309398611yescharacteristic
358C22—21no—
359C22—21no—
360C22—3761no—
361C22—3761no—
362C22—1882no—
363C22#102×D4711yescharacteristic
364C22#1568920311yescharacteristic
365C22#1568920511yescharacteristic
366C4—82no—
367C4—81no—
368C4—81no—
369C4—472no—
370C4—941no—
371C4—941no—
372C4—941no—
373C4—3762no—
374C4—1882no—
375C4#1568897011yescharacteristic
376C4#1568911811yescharacteristic
377C4#1568920811yescharacteristic
378C4#1568921011yescharacteristic
379C2—21no—
380C2—21no—
381C2—3762no—
382C2#2024320211yescharacteristic
383C2#360×D4711yescharacteristic
384C2#2024512211yescharacteristic
subgroupG/H
centerC22#102×D47
commutatorC8×C47C2×C42
FrattiniC22×C8C22×D47
FittingC47×#1386C2
socleC22×C47C2×#102
prime radicalC2×#13094046C22
cosocleC22×C8×C47C23
hypercenter#1386D47
hypocenterC47C4×#360
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. M32×Dic47C2semidirect
5. #106×Dic47C2semidirect
6. #20241493C2semidirect
7. C47×#1386C2semidirect
11. #15689217C4semidirect
12. #15689217C4semidirect
15. #15689205C4semidirect
16. #15689205C4semidirect
21. #15688934C4semidirect
22. #15688932C4semidirect
50. #13093964C2×C4semidirect
53. #13093956C2×C4semidirect
67. M32×C47C2×C4semidirect
73. #106×C47C2×C4semidirect
119. Dic188C42semidirect
121. D376C42semidirect
186. Dic47#360semidirect
193. #1386D47semidirect
221. C47C4×#360semidirect
248. M32C4×D47semidirect
257. #106C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202521592
C2×#156889342
C2×#156889322
M32×Dic472
#106×Dic472
#202414932
C47×#13862
C4×#36047