Finite Group Databasebeta

← #26104036

Subgroups — #26104036

Order 12032.

276 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688922C211yesmaximal, characteristic
2#20243782C211yesmaximal, characteristic
3C2×C47×#333C211yesmaximal, characteristic
4#15689601C411yescharacteristic
5#15689132—21no—
6C2×#13093943—21no—
7C2×#13093943C411yescharacteristic
8#15688922—21no—
9#15688922C412yescomplemented
10#15688883—21no—
11C2×C4⋊Q8×C47C2211yescharacteristic
12C2×C4⋊C8×C47—21no—
13C47×#333—21no—
14C47×#333C412yescomplemented
15C2×#10625329—21no—
16C2×#10625329—41no—
17Q8×Dic47—42no—
18#13093983—21no—
19#13093983—41no—
20#13093948—42no—
21#13093943—22no—
22#13093943—41no—
23#13093943—41no—
24C22×Q8×C47C2×C411yescharacteristic
25C22×Q8×C47D411yescharacteristic
26C22×C8×C47—41no—
27C4⋊Q8×C47—41no—
28C4⋊Q8×C47C2×C411yescharacteristic
29C4⋊Q8×C47C2×C411yescharacteristic
30C4⋊Q8×C47D411yescharacteristic
31C4⋊Q8×C47Q811yescharacteristic
32C2×C4⋊C4×C47—41no—
33C2×C42×C47D411yescharacteristic
34C4⋊C8×C47—22no—
35C4⋊C8×C47—41no—
36C2×Q8×C47—21no—
37C2×Q8×C47—21no—
38C2×Q8×C47—41no—
39C2×Q8×C47—41no—
40C2×Q8×C47—41no—
41C2×Q8×C47—41no—
42C2×Q8×C47C4211yescharacteristic
43C2×Q8×C47C22⋊C411yescharacteristic
44C2×Q8×C47C4⋊C411yescharacteristic
45C2×Q8×C47C4⋊C411yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47C22⋊C411yescharacteristic
50C2×C8×C47—42no—
51C2×C8×C47—82no—
52C4⋊C4×C47—41no—
53C4⋊C4×C47—41no—
54C4⋊C4×C47—41no—
55C4⋊C4×C47—41no—
56C42×C47—21no—
57C42×C47C22⋊C411yescharacteristic
58C42×C47C4⋊C411yescharacteristic
59C22⋊Dic47—41no—
60C4⋊Dic47—22no—
61C4⋊Dic47—42no—
62C4×Dic47—41no—
63C4×Dic47—42no—
64#10625329—42no—
65#10625329—41no—
66#10625329—41no—
67#10625329—81no—
68C2×C47⋊C8—41no—
69C2×C47⋊C8—81no—
70C23×C47#9911yescharacteristic
71Q8×C47—41no—
72Q8×C47—41no—
73Q8×C47—41no—
74Q8×C47—41no—
75Q8×C47—42no—
76Q8×C47—81no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47#9511yescharacteristic
92C2×C4×C47#9911yescharacteristic
93C2×Dic47—42no—
94C8×C47—82no—
95C47⋊C8—81no—
96C47⋊C8—82no—
97#7491—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47#34211yescharacteristic
100C22×C47AΣL(1,𝔽16)11yescharacteristic
101C22×C47#35211yescharacteristic
102C4×C47—21no—
103C4×C47—22no—
104C4×C47—21no—
105C4×C47—21no—
106C4×C47—21no—
107C4×C47—41no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111Dic47—82no—
112C2×#356—471no—
113#1886—471no—
114C2×#333D4711yescharacteristic
115C2×C47—21no—
116C2×C47—21no—
117C2×C47#139011yescharacteristic
118C2×C47#140512yes—
119C2×C4⋊Q8C2×D4711yescharacteristic
120#519—471no—
121C2×C4⋊C8—21no—
122#419—941no—
123C2×#101—471no—
124C2×#101—941no—
125#356—472no—
126#356—941no—
127#339—941no—
128#333—21no—
129#333Dic4712yescomplemented
130C47#749111yescharacteristic, complemented, Sylow
131C22×Q8C4×D4711yescharacteristic
132C22×Q8D18811yescharacteristic
133C2×M16—941no—
134C2×M16—1881no—
135C22×C8—41no—
136C4⋊Q8—41no—
137C4⋊Q8Dic9411yescharacteristic
138C4⋊Q8C4×D4711yescharacteristic
139C4⋊Q8C2×Dic4711yescharacteristic
140C4⋊Q8C22⋊D4711yescharacteristic
141C4×Q8—1882no—
142#117—941no—
143#117—1881no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D4711yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—41no—
148#103—1882no—
149#101—942no—
150#101—1881no—
151#101—1881no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C4×Dic4711yescharacteristic
159C2×Q8Dic47⋊C411yescharacteristic
160C2×Q8C4⋊Dic4711yescharacteristic
161C2×Q8(C2×D47)⋊C411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4(C2×D47)⋊C411yescharacteristic
166M16—3761no—
167M16—1882no—
168M16—1881no—
169M16—1881no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8—3761no—
173C2×C8—1881no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—942no—
179C4⋊C4—1882no—
180C22⋊C4—1881no—
181C42—21no—
182C42—1881no—
183C42—1882no—
184C42Dic47⋊C411yescharacteristic
185C42C22⋊Dic4711yescharacteristic
186C23#1309393811yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—1882no—
208C2×C4C47⋊#9511yescharacteristic
209C2×C4#1309393811yescharacteristic
210C8—82no—
211C8—3761no—
212C8—3762no—
213C22—22no—
214C22#1568888911yescharacteristic
215C22#1568891111yescharacteristic
216C22#1568891411yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—3762no—
227C2—21no—
228C2—21no—
229C2#2024150711yescharacteristic
230C2#2024154312yes—
subgroupG/H
centerC22#15688911
commutatorC2×Q8×C47C42
FrattiniC2×C4⋊Q8C2×D47
FittingC2×C47×#333C2
socleC22×C47AΣL(1,𝔽16)
prime radicalC23×C47#99
cosocleC2×C4⋊Q8×C47C22
hypercenterC2×#333D47
hypocenterC47#7491
LCSG ▷ C2×Q8×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
9. #15688922C4semidirect
14. C47×#333C4semidirect
129. #333Dic47semidirect
130. C47#7491semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
#202437822
C2×C47×#3332
#749147