Finite Group Databasebeta

← #13166406

Subgroups — #13166406

Order 1728.

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

normal subgroups only

This group has 440 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
1C2×#88785C211yesmaximal, characteristic
2#11723098C211yesmaximal, characteristic, complemented
3C3×#63773C211yesmaximal, characteristic, complemented
4#11722721C212yesmaximal, complemented
5#11722719C212yesmaximal, complemented
6#10593762—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C2×C32×C3⋊C8C411yescharacteristic
10#88785C411yescharacteristic
11#88785C411yescharacteristic
12#88785C2211yescharacteristic
13#88785C2211yescharacteristic
14C3⋊#3708C2212yescomplemented
15C3×#3708C2212yescomplemented
16C2×C8×C3⋊D3—31no—
17C2×#3710—62no—
18#63773—31no—
19#63773D311yescharacteristic, complemented
20#63765—32no—
21#63764—32no—
22C22×C3⋊(C3⋊D3)C811yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C811yescharacteristic
25C4×C3⋊(C3⋊D3)C811yescharacteristic
26C4×C3⋊(C3⋊D3)C811yescharacteristic
27C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
28C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
29C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
30C3×C3⋊(C3⋊C8)C2311yescharacteristic
31C32×C3⋊C8C2×C411yescharacteristic
32C32×C3⋊C8C2×C411yescharacteristic
33D3×M32—91nomaximal
34C2×C4×C3⋊D3—31no—
35C2×C4×C3⋊D3—62no—
36C2×C4×C3⋊D3—122no—
37C2×C32×C8—31no—
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C8×C3⋊D3—31no—
40C8×C3⋊D3—31no—
41C8×C3⋊D3—31no—
42C8×C3⋊D3—31no—
43C2×C3×C3⋊C8—22no—
44C2×C3×C3⋊C8—62no—
45#3710—62no—
46#3710—62no—
47#3710—62no—
48#3710—62no—
49#3708—32no—
50#3708C2×D312yescomplemented
51C22×C33C2×C811yescharacteristic
52C2×C3⋊(C3⋊D3)—22no—
53C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
54C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
55C33×C4C2×C811yescharacteristic
56C33×C4C22×C411yescharacteristic
57C3⋊(C3⋊Dic3)C2×C811yescharacteristic
58C3⋊(C3⋊Dic3)C2×C811yescharacteristic
59C2×D3×C8—91no—
60C2×D3×C8—182no—
61C3×M32—91no—
62#812—91no—
63#798—92no—
64D3×C16—92no—
65C22×C3⋊D3—31no—
66C22×C3⋊D3—62no—
67C22×C3⋊D3—122no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4C4×D311yescharacteristic
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—62no—
73C2×C3⋊Dic3—122no—
74C4×C3⋊D3—31no—
75C4×C3⋊D3—31no—
76C4×C3⋊D3—31no—
77C4×C3⋊D3—31no—
78C4×C3⋊D3—62no—
79C4×C3⋊D3—62no—
80C4×C3⋊D3—62no—
81C4×C3⋊D3—62no—
82C4×C3⋊D3—122no—
83C4×C3⋊D3—122no—
84C4×C3⋊D3—122no—
85C4×C3⋊D3—122no—
86C32×C8—31no—
87C32×C8—31no—
88C3⋊(C3⋊C8)C4×D311yescharacteristic
89C3⋊(C3⋊C8)C22×D311yescharacteristic
90C3×C3⋊C8—22no—
91C3×C3⋊C8—22no—
92C3×C3⋊C8—62no—
93C3×C3⋊C8—62no—
94C2×M32—271noSylow
95C2×C33—21no—
96C2×C33C22×C811yescharacteristic
97C3⋊(C3⋊D3)—21no—
98C3⋊(C3⋊D3)M3212yescomplemented
99C2×C4×D3—91no—
100C2×C4×D3—182no—
101C2×C4×D3—362no—
102C2×C3×C8—62no—
103C2×C3×C8—91no—
104C2×C3⋊C8—62no—
105C2×C3⋊C8#15711yescharacteristic
106D3×C8—91no—
107D3×C8—91no—
108D3×C8—91no—
109D3×C8—91no—
110D3×C8—182no—
111D3×C8—182no—
112D3×C8—182no—
113D3×C8—182no—
114C3×C16—92no—
115C3⋊C16—92no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32D3×C811yescharacteristic
119C2×C3⋊D3—31no—
120C2×C3⋊D3—31no—
121C2×C3⋊D3—62no—
122C2×C3⋊D3—62no—
123C2×C3⋊D3—62no—
124C2×C3⋊D3—68no—
125C2×C3⋊D3—128no—
126C2×C3⋊D3—122no—
127C2×C3⋊D3—122no—
128C32×C4—22no—
129C32×C4—22no—
130C32×C4—42no—
131C32×C4—42no—
132C32×C4D3×C811yescharacteristic
133C32×C4C2×C4×D311yescharacteristic
134C3⋊Dic3—31no—
135C3⋊Dic3—31no—
136C3⋊Dic3—62no—
137C3⋊Dic3—62no—
138C3⋊Dic3—122no—
139C3⋊Dic3—122no—
140C22×C8—271no—
141M32—91no—
142M32—272no—
143M32—271no—
144C2×C16—272no—
145C33C2×M3211yescharacteristic, complemented, Sylow
146C22×D3—91no—
147C22×D3—182no—
148C22×D3—362no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4C2×#15711yescharacteristic
152C2×Dic3—91no—
153C2×Dic3—182no—
154C2×Dic3—362no—
155C4×D3—91no—
156C4×D3—91no—
157C4×D3—91no—
158C4×D3—91no—
159C4×D3—182no—
160C4×D3—182no—
161C4×D3—182no—
162C4×D3—182no—
163C4×D3—362no—
164C4×D3—362no—
165C4×D3—362no—
166C4×D3—362no—
167C3×C8—62no—
168C3×C8—62no—
169C3×C8—91no—
170C3×C8—91no—
171C3⋊C8—62no—
172C3⋊C8—62no—
173C3⋊C8C2×#15711yescharacteristic
174C3⋊C8C2×#15711yescharacteristic
175C2×C32—21no—
176C2×C32—24no—
177C2×C32—22no—
178C2×C32—44no—
179C2×C32—42no—
180C2×C32C2×D3×C811yescharacteristic
181C3⋊D3—32no—
182C3⋊D3—64no—
183C3⋊D3—61no—
184C3⋊D3—64no—
185C3⋊D3—124no—
186C3⋊D3—124no—
187C22×C4—271no—
188C2×C8—31no—
189C2×C8—91no—
190C2×C8—271no—
191C2×C8—271no—
192C2×C8—271no—
193C2×C8—271no—
194C16—92no—
195C16—272no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3#378711yescharacteristic
199C2×D3—91no—
200C2×D3—91no—
201C2×D3—182no—
202C2×D3—182no—
203C2×D3—182no—
204C2×D3—188no—
205C2×D3—362no—
206C2×D3—362no—
207C2×D3—368no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—42no—
211C3×C4—42no—
212C3×C4#378711yescharacteristic
213C3×C4C22×#15711yescharacteristic
214Dic3—91no—
215Dic3—91no—
216Dic3—182no—
217Dic3—182no—
218Dic3—362no—
219Dic3—362no—
220C32—22no—
221C32—42no—
222C32D3×M3211yescharacteristic, complemented
223C23—271no—
224C2×C4—271no—
225C2×C4—271no—
226C2×C4—271no—
227C2×C4—271no—
228C2×C4—271no—
229C2×C4D3×#15711yescharacteristic
230C8—31no—
231C8—31no—
232C8—91no—
233C8—91no—
234C2×C3—24no—
235C2×C3—21no—
236C2×C3—22no—
237C2×C3—44no—
238C2×C3—42no—
239C2×C3C2×#378711yescharacteristic
240D3—92no—
241D3—184no—
242D3—184no—
243D3—181no—
244D3—364no—
245D3—364no—
246C22—271no—
247C22—271no—
248C22—542no—
249C22#8892411yescharacteristic
250C4—271no—
251C4—271no—
252C4#8892411yescharacteristic
253C4C2×D3×#15711yescharacteristic
254C3—22no—
255C3—42no—
256C3#1059376211yescharacteristic, complemented
257C2—21no—
258C2—272no—
259C2—541no—
260C2C2×#8892411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C22×C8
FittingC2×C33×C4C2×C4
prime radicalC22×C3⋊(C3⋊D3)C8
cosocleC3×C3⋊(C3⋊C8)C23
hypercenterC2×C4D3×#157
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
3. C3×#63773C2semidirect
4. #11722721C2semidirect
5. #11722719C2semidirect
14. C3⋊#3708C22semidirect
15. C3×#3708C22semidirect
19. #63773D3semidirect
50. #3708C2×D3semidirect
98. C3⋊(C3⋊D3)M32semidirect
145. C33C2×M32semidirect
222. C32D3×M32semidirect
256. C3#10593762semidirect

Maximal subgroups

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

subgroupindex
C2×#887852
#117230982
C3×#637732
#117227212
#117227192
#105937623
D3×M329