Finite Group Databasebeta

← #13157207

Subgroups — #13157207

Order 1728.

356 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#11724514C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#10590191—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88994C412yes—
7C2×C3⋊#622C412yescomplemented
8C4⋊(C3⋊#157)—21no—
9C4×C3⋊#157—21no—
10#88989—21no—
11#88989—21no—
12C33×M16C411yescharacteristic, complemented
13C3×#3748C411yescharacteristic, complemented
14C3×#3743—21no—
15C3×C8×C3⋊D3—21no—
16#64286—31no—
17C2×#3788—31no—
18M16×C3⋊D3D311yescharacteristic
19C3×D3×M16—62no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C3×C4×C3⋊D3D411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3Q811yescharacteristic
28C3⋊#622—22no—
29C33×C8—21no—
30C3×C3⋊(C3⋊C8)—21no—
31#4559—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—62no—
34C22⋊#157—32no—
35C2×#622—32no—
36C4⋊#157—61no—
37C4×#157—61no—
38#3788—61no—
39#3788—61no—
40C32×M16—22no—
41C32×M16—42no—
42C32×M16Dic311yescharacteristic, complemented
43#3748Dic311yescharacteristic, complemented
44#3743—21no—
45C8×C3⋊D3—21no—
46C3×#208—62no—
47C3×ΓL(1,𝔽25)—64no—
48C3×D3×C8—64no—
49C22×C33C4211yescharacteristic
50C2×C3×C3⋊D3—41no—
51C2×C3×C3⋊D3C22⋊C411yescharacteristic
52C2×C3×C3⋊D3C4⋊C411yescharacteristic
53C3⋊#157—42no—
54C33×C4C22⋊C411yescharacteristic
55C33×C4C4⋊C411yescharacteristic
56C3×C3⋊Dic3C22⋊C411yescharacteristic
57C3×C3⋊Dic3C4⋊C411yescharacteristic
58C2×C3×M16—91no—
59#926—91no—
60C2×#208—91no—
61D3×M16—62no—
62C22×C3⋊D3C4×D311yescharacteristic
63C22×C3×D3—62no—
64C2×#157—62no—
65C2×C32×C4—22no—
66C2×C32×C4—42no—
67C2×C32×C4C2×Dic311yescharacteristic
68C2×C3⋊Dic3C4×D311yescharacteristic
69C4×C3⋊D3Dic611yescharacteristic
70C4×C3⋊D3D1211yescharacteristic
71C4×C3⋊D3#6711yescharacteristic
72C4×C3⋊D3#6711yescharacteristic
73C2×C3×Dic3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78#622—62no—
79C32×C8—21no—
80C32×C8—24no—
81C32×C8—44no—
82C3⋊(C3⋊C8)—21no—
83C3×C3⋊C8—64no—
84#344—271noSylow
85C2×C33—21no—
86C2×C33#9511yescharacteristic
87C3×C3⋊D3—21no—
88C3×C3⋊D3—21no—
89C22×C3×C4—91no—
90C2×C4×D3—62no—
91C3×M16—22no—
92C3×M16—42no—
93C3×M16—91no—
94C3×M16—181no—
95C3×M16#15711yescharacteristic, complemented
96C2×C3×C8—181no—
97#217—92no—
98C4⋊Dic3—181no—
99C4×Dic3—181no—
100#208—62no—
101#208—181no—
102#208—181no—
103C2×C3⋊C8—92no—
104ΓL(1,𝔽25)—64no—
105D3×C8—64no—
106C22×C32—22no—
107C22×C32—42no—
108C22×C32C4×Dic311yescharacteristic
109C2×C3⋊D3—41no—
110C2×C3⋊D3Dic3⋊C411yescharacteristic
111C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—124no—
115#157—122no—
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—42no—
119C32×C4—42no—
120C32×C4C4⋊Dic311yescharacteristic
121C32×C4#21711yescharacteristic
122C3⋊Dic3Dic3⋊C411yescharacteristic
123C3⋊Dic3(C2×D3)⋊C411yescharacteristic
124C3×Dic3—62no—
125C3×Dic3—62no—
126C2×M16—91no—
127C2×M16—271no—
128#117—271no—
129C33#34411yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C22×D3—62no—
132C2×C3×C4—22no—
133C2×C3×C4—42no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4C2×#15711yescharacteristic
140C2×Dic3—62no—
141C2×Dic3—182no—
142C4×D3—62no—
143C4×D3—62no—
144C4×D3—62no—
145C4×D3—62no—
146C3×C8—24no—
147C3×C8—21no—
148C3×C8—44no—
149C3×C8—181no—
150C3⋊C8—64no—
151C3⋊C8—182no—
152C2×C32—22no—
153C2×C32—21no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—82no—
157C2×C32C3⋊#9511yescharacteristic
158C3⋊D3—21no—
159C3⋊D3—21no—
160C3×D3—64no—
161C3×D3—122no—
162C22×C4—91no—
163M16—91no—
164M16—181no—
165M16—541no—
166M16—541no—
167M16C3⋊#15711yescharacteristic, complemented
168C2×C8—181no—
169C2×C8—272no—
170C4⋊C4—541no—
171C22⋊C4—272no—
172C42—541no—
173C22×C3—22no—
174C22×C3—42no—
175C22×C3—91no—
176C22×C3—91no—
177C22×C3—361no—
178C22×C3C4×#15711yescharacteristic
179C2×D3—62no—
180C2×D3—62no—
181C2×D3—124no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4C4⋊#15711yescharacteristic
189C3×C4C22⋊#15711yescharacteristic
190Dic3—62no—
191Dic3—62no—
192Dic3—362no—
193C32—22no—
194C32—42no—
195C32#455911yescharacteristic, complemented
196C23—91no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—542no—
203C2×C4C2×C3⋊#15711yescharacteristic
204C8—21no—
205C8—181no—
206C8—542no—
207C2×C3—21no—
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3—82no—
212C2×C3—181no—
213C2×C3—181no—
214C2×C3#6378111yescharacteristic
215D3—64no—
216D3—122no—
217C22—91no—
218C22—91no—
219C22—361no—
220C22C4×C3⋊#15711yescharacteristic
221C4—91no—
222C4—91no—
223C4—1082no—
224C4C4⋊(C3⋊#157)11yescharacteristic
225C4#8899411yescharacteristic
226C3—22no—
227C3—42no—
228C3#1059019111yescharacteristic, complemented
229C2—21no—
230C2—181no—
231C2—181no—
232C2#1172310611yescharacteristic
subgroupG/H
centerC2#11723106
commutatorC22×C33C42
FrattiniC2×C4C2×C3⋊#157
FittingC33×M16C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypercenterM16C3⋊#157
hypocenterC33#344
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
7. C2×C3⋊#622C4semidirect
12. C33×M16C4semidirect
13. C3×#3748C4semidirect
42. C32×M16Dic3semidirect
43. #3748Dic3semidirect
95. C3×M16#157semidirect
129. C33#344semidirect
167. M16C3⋊#157semidirect
195. C32#4559semidirect
228. C3#10590191semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×#889892
C3×M16×C3⋊D32
#105901913
#45599