Finite Group Databasebeta

← #13166233

Subgroups — #13166233

Order 1728.

376 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D3×C8×C3⋊D3C211yesmaximal, characteristic
2#11723089C211yesmaximal, characteristic, complemented
3C3×#63764C211yesmaximal, characteristic, complemented
4#11722721C212yesmaximal, complemented
5#11722720C212yesmaximal, complemented
6#10593762—31nomaximal
7C4×D3×C3⋊D3C411yescharacteristic
8C8×C3⋊(C3⋊D3)C411yescharacteristic
9C3×C8×C3⋊D3C2211yescharacteristic
10C32×D3×C8C411yescharacteristic
11#88785C2211yescharacteristic
12C3⋊D3×C3⋊C8C411yescharacteristic
13D3×C3⋊(C3⋊C8)C2211yescharacteristic
14C3⋊#3708C2212yescomplemented
15C3×#3708C2212yescomplemented
16C2×C8×C3⋊D3—31no—
17D32×C8—62no—
18#63773—32no—
19#63765—32no—
20#63764—31no—
21#63764D311yescharacteristic, complemented
22C2×D3×C3⋊D3C811yescharacteristic
23C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C32×C4×D3C2×C411yescharacteristic
26#6289C811yescharacteristic
27Dic3×C3⋊D3C811yescharacteristic
28D3×C3⋊Dic3C811yescharacteristic
29C33×C8C2×C411yescharacteristic
30C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
31C3×C3⋊(C3⋊C8)C2311yescharacteristic
32C32×C3⋊C8C2×C411yescharacteristic
33C2×#798—91nomaximal
34C2×C4×C3⋊D3—31no—
35C4×D32—62no—
36C2×C32×C8—31no—
37C2×C3⋊(C3⋊C8)—31no—
38C8×C3⋊D3—31no—
39C8×C3⋊D3—31no—
40C8×C3⋊D3—31no—
41C8×C3⋊D3—62no—
42C8×C3⋊D3—122no—
43C8×C3⋊D3C2×D311yescharacteristic
44C3×D3×C8—22no—
45C3×D3×C8—62no—
46#3710—62no—
47D3×C3⋊C8—62no—
48D3×C3⋊C8—62no—
49#3708—32no—
50#3708C2×D312yescomplemented
51C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
52C2×C3×C3⋊D3C2×C811yescharacteristic
53C2×C32×D3C2×C811yescharacteristic
54D3×C3⋊D3—22no—
55C33×C4C22×C411yescharacteristic
56C3⋊(C3⋊Dic3)C2×C811yescharacteristic
57C3×C3⋊Dic3C2×C811yescharacteristic
58C32×Dic3C2×C811yescharacteristic
59C2×D3×C8—91no—
60C2×D3×C8—182no—
61C2×C3×C16—91no—
62C2×C3⋊C16—91no—
63#798—92no—
64#798—92no—
65C22×C3⋊D3—31no—
66C2×D32—62no—
67C2×C32×C4—31no—
68C2×C3⋊Dic3—31no—
69C4×C3⋊D3—31no—
70C4×C3⋊D3—31no—
71C4×C3⋊D3—31no—
72C4×C3⋊D3—62no—
73C4×C3⋊D3—122no—
74C4×C3⋊D3C4×D311yescharacteristic
75C3×C4×D3—22no—
76C3×C4×D3—62no—
77#624—62no—
78D3×Dic3—62no—
79D3×Dic3—62no—
80C32×C8—22no—
81C32×C8—31no—
82C32×C8—42no—
83C32×C8C4×D311yescharacteristic
84C3⋊(C3⋊C8)—31no—
85C3⋊(C3⋊C8)—62no—
86C3⋊(C3⋊C8)—122no—
87C3⋊(C3⋊C8)C22×D311yescharacteristic
88C3×C3⋊C8—22no—
89C3×C3⋊C8—62no—
90C2×M32—271noSylow
91C2×C33C22×C811yescharacteristic
92C3⋊(C3⋊D3)—21no—
93C3×C3⋊D3M3212yescomplemented
94C32×D3—21no—
95C2×C4×D3—91no—
96C2×C4×D3—182no—
97C2×C3×C8—62no—
98C2×C3×C8—91no—
99C2×C3⋊C8—91no—
100C2×C3⋊C8—182no—
101D3×C8—62no—
102D3×C8—91no—
103D3×C8—91no—
104D3×C8—91no—
105D3×C8—182no—
106D3×C8—182no—
107D3×C8—182no—
108D3×C8—362no—
109D3×C8#15711yescharacteristic
110C3×C16—92no—
111C3⋊C16—92no—
112C22×C32—31no—
113C2×C3⋊D3—31no—
114C2×C3⋊D3—62no—
115C2×C3⋊D3—62no—
116C2×C3⋊D3—122no—
117C2×C3⋊D3D3×C811yescharacteristic
118C2×C3×D3—22no—
119C2×C3×D3—62no—
120D32—68no—
121C32×C4—22no—
122C32×C4—31no—
123C32×C4—42no—
124C32×C4C2×C4×D311yescharacteristic
125C3⋊Dic3—31no—
126C3⋊Dic3—62no—
127C3⋊Dic3—122no—
128C3⋊Dic3D3×C811yescharacteristic
129C3×Dic3—22no—
130C3×Dic3—62no—
131C22×C8—271no—
132M32—272no—
133M32—272no—
134C2×C16—91no—
135C2×C16—271no—
136C33C2×M3211yescharacteristic, complemented, Sylow
137C22×D3—91no—
138C22×D3—182no—
139C2×C3×C4—62no—
140C2×C3×C4—91no—
141C2×Dic3—91no—
142C2×Dic3—182no—
143C4×D3—62no—
144C4×D3—91no—
145C4×D3—91no—
146C4×D3—91no—
147C4×D3—182no—
148C4×D3—182no—
149C4×D3—182no—
150C4×D3—362no—
151C4×D3C2×#15711yescharacteristic
152C3×C8—22no—
153C3×C8—42no—
154C3×C8—62no—
155C3×C8—91no—
156C3×C8C2×#15711yescharacteristic
157C3⋊C8—62no—
158C3⋊C8—91no—
159C3⋊C8—182no—
160C3⋊C8—362no—
161C3⋊C8C2×#15711yescharacteristic
162C2×C32—22no—
163C2×C32—42no—
164C2×C32—61no—
165C2×C32C2×D3×C811yescharacteristic
166C3⋊D3—61no—
167C3⋊D3—64no—
168C3⋊D3—124no—
169C3⋊D3#79812yescomplemented
170C3×D3—24no—
171C3×D3—64no—
172C22×C4—271no—
173C2×C8—31no—
174C2×C8—91no—
175C2×C8—271no—
176C2×C8—271no—
177C2×C8—271no—
178C2×C8—271no—
179C16—92no—
180C16—272no—
181C22×C3—62no—
182C22×C3—91no—
183C2×D3—62no—
184C2×D3—91no—
185C2×D3—182no—
186C2×D3—182no—
187C2×D3—188no—
188C2×D3—362no—
189C2×D3#378711yescharacteristic
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—62no—
193C3×C4—91no—
194C3×C4C22×#15711yescharacteristic
195Dic3—62no—
196Dic3—91no—
197Dic3—182no—
198Dic3—362no—
199Dic3#378711yescharacteristic
200C32—22no—
201C32—42no—
202C32C2×#79811yescharacteristic, complemented
203C23—271no—
204C2×C4—31no—
205C2×C4—91no—
206C2×C4—271no—
207C2×C4—271no—
208C2×C4—271no—
209C2×C4—271no—
210C8—31no—
211C8—91no—
212C8—271no—
213C8D3×#15711yescharacteristic
214C2×C3—22no—
215C2×C3—42no—
216C2×C3—64no—
217C2×C3—92no—
218C2×C3C2×#378711yescharacteristic
219D3—21no—
220D3—64no—
221D3—184no—
222D3—181no—
223D3—364no—
224C22—31no—
225C22—91no—
226C22—271no—
227C22—542no—
228C4—31no—
229C4—91no—
230C4—271no—
231C4C2×D3×#15711yescharacteristic
232C3—22no—
233C3—42no—
234C3#1059376211yescharacteristic, complemented
235C2—61no—
236C2—92no—
237C2—541no—
238C2D3×#378711yescharacteristic
subgroupG/H
center, hypercenterC8D3×#157
commutator, socleC2×C33C22×C8
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
prime radicalC2×D3×C3⋊D3C8
cosocleC3×C3⋊(C3⋊C8)C23
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723089C2semidirect
3. C3×#63764C2semidirect
4. #11722721C2semidirect
5. #11722720C2semidirect
14. C3⋊#3708C22semidirect
15. C3×#3708C22semidirect
21. #63764D3semidirect
50. #3708C2×D3semidirect
93. C3×C3⋊D3M32semidirect
136. C33C2×M32semidirect
169. C3⋊D3#798semidirect
202. C32C2×#798semidirect
234. C3#10593762semidirect

Maximal subgroups

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

subgroupindex
D3×C8×C3⋊D32
#117230892
C3×#637642
#117227212
#117227202
#105937623
C2×#7989