Finite Group Databasebeta

← #13166247

Subgroups — #13166247

Order 1728.

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

normal subgroups only

This group has 424 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
1#11724247C212yesmaximal
2D3×C8×C3⋊D3C211yesmaximal, characteristic
3#11723093C211yesmaximal, characteristic, complemented
4C3×#63768C211yesmaximal, characteristic, complemented
5#11722715C212yesmaximal, complemented
6#10593768—31nomaximal
7C4×D3×C3⋊D3C2211yescharacteristic
8C4⋊(C3⋊#157)C2212yes—
9C3×C4⋊#157C2212yes—
10(C2×D3)⋊#157—22no—
11Dic3×#157—22no—
12C8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×D3×C8C411yescharacteristic, complemented
15#88785C411yescharacteristic, complemented
16C3⋊D3×C3⋊C8C2211yescharacteristic
17D3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
18#64286—32no—
19C2×C8×C3⋊D3—31no—
20D32×C8—62no—
21#63769—32no—
22#63768—31no—
23#63768D311yescharacteristic, complemented
24C2×D3×C3⋊D3D411yescharacteristic
25C2×C3⋊#157—22no—
26C2×C3×#157—22no—
27C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
28C3×C4×C3⋊D3C2311yescharacteristic
29C32×C4×D3C2×C411yescharacteristic
30#6289Q811yescharacteristic
31Dic3×C3⋊D3D411yescharacteristic
32D3×C3⋊Dic3Q811yescharacteristic
33C33×C8C2×C411yescharacteristic
34C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
35C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
36C32×C3⋊C8C2×C411yescharacteristic
37#4860—91nomaximal
38C2×C4×C3⋊D3—31no—
39C4×D32—62no—
40C22⋊#157—62no—
41C4⋊#157—32no—
42C4⋊#157C2×D312yes—
43C4×#157—62no—
44C2×C32×C8—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C8×C3⋊D3—31no—
47C8×C3⋊D3—31no—
48C8×C3⋊D3—31no—
49C8×C3⋊D3—62no—
50C8×C3⋊D3—122no—
51C8×C3⋊D3C2×D311yescharacteristic
52C3×D3×C8—22no—
53C3×D3×C8—62no—
54#3710—62no—
55D3×C3⋊C8—62no—
56D3×C3⋊C8—62no—
57C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
58C2×C3×C3⋊D3C2×D411yescharacteristic
59C2×C32×D3C4⋊C411yescharacteristic
60D3×C3⋊D3—22no—
61C3⋊#157—42no—
62C3×#157—42no—
63C33×C4C22×C411yescharacteristic
64C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C32×Dic3C4⋊C411yescharacteristic
67C2×D3×C8—91no—
68C2×D3×C8—182no—
69#892—92no—
70C3×#106—91no—
71#817—91no—
72#807—92no—
73C22×C3⋊D3—31no—
74C2×D32—62no—
75C2×#157—22no—
76C2×#157—62no—
77C2×C32×C4—31no—
78C2×C3⋊Dic3—31no—
79C4×C3⋊D3—31no—
80C4×C3⋊D3—31no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—62no—
83C4×C3⋊D3—122no—
84C4×C3⋊D3C22×D311yescharacteristic
85C3×C4×D3—22no—
86C3×C4×D3—62no—
87#624—62no—
88D3×Dic3—62no—
89D3×Dic3—62no—
90C32×C8—22no—
91C32×C8—31no—
92C32×C8—42no—
93C32×C8C4×D311yescharacteristic
94C3⋊(C3⋊C8)—31no—
95C3⋊(C3⋊C8)—62no—
96C3⋊(C3⋊C8)—122no—
97C3⋊(C3⋊C8)C4×D311yescharacteristic
98C3×C3⋊C8—22no—
99C3×C3⋊C8—62no—
100#427—271noSylow
101C2×C33C2×C4⋊C411yescharacteristic
102C3⋊(C3⋊D3)—21no—
103C3×C3⋊D3#13512yes—
104C32×D3—21no—
105C2×C4×D3—91no—
106C2×C4×D3—182no—
107C2×C3×C8—62no—
108C2×C3×C8—91no—
109C3×C4⋊C4—92no—
110(C2×D3)⋊C4—182no—
111C4⋊Dic3—92no—
112C4×Dic3—182no—
113C2×C3⋊C8—91no—
114C2×C3⋊C8—182no—
115D3×C8—62no—
116D3×C8—91no—
117D3×C8—91no—
118D3×C8—91no—
119D3×C8—182no—
120D3×C8—182no—
121D3×C8—182no—
122D3×C8—362no—
123D3×C8#15711yescharacteristic, complemented
124C22×C32—31no—
125C2×C3⋊D3—31no—
126C2×C3⋊D3—62no—
127C2×C3⋊D3—62no—
128C2×C3⋊D3—122no—
129C2×C3⋊D3D3×D411yescharacteristic
130C2×C3×D3—22no—
131C2×C3×D3—62no—
132D32—68no—
133#157—42no—
134#157—122no—
135C32×C4—22no—
136C32×C4—31no—
137C32×C4—42no—
138C32×C4C2×C4×D311yescharacteristic
139C3⋊Dic3—31no—
140C3⋊Dic3—62no—
141C3⋊Dic3—122no—
142C3⋊Dic3D3×Q811yescharacteristic
143C3×Dic3—22no—
144C3×Dic3—62no—
145C22×C8—271no—
146#117—272no—
147#107—272no—
148#106—91no—
149#106—271no—
150C33#42711yescharacteristic, complemented, Sylow
151C22×D3—91no—
152C22×D3—182no—
153C2×C3×C4—62no—
154C2×C3×C4—91no—
155C2×C3×C4—182no—
156C2×Dic3—91no—
157C2×Dic3—182no—
158C2×Dic3—182no—
159C4×D3—62no—
160C4×D3—91no—
161C4×D3—91no—
162C4×D3—91no—
163C4×D3—182no—
164C4×D3—182no—
165C4×D3—182no—
166C4×D3—362no—
167C4×D3C2×#15711yescharacteristic
168C3×C8—22no—
169C3×C8—42no—
170C3×C8—62no—
171C3×C8—91no—
172C3×C8C2×#15711yescharacteristic
173C3⋊C8—62no—
174C3⋊C8—91no—
175C3⋊C8—182no—
176C3⋊C8—362no—
177C3⋊C8C2×#15711yescharacteristic
178C2×C32—22no—
179C2×C32—42no—
180C2×C32—61no—
181C2×C32D3×C4⋊C411yescharacteristic
182C3⋊D3—61no—
183C3⋊D3—64no—
184C3⋊D3—124no—
185C3⋊D3#91612yes—
186C3×D3—24no—
187C3×D3—64no—
188C22×C4—271no—
189C2×C8—31no—
190C2×C8—91no—
191C2×C8—271no—
192C2×C8—271no—
193C2×C8—271no—
194C2×C8—271no—
195C4⋊C4—92no—
196C4⋊C4—272no—
197C22⋊C4—542no—
198C42—542no—
199C22×C3—62no—
200C22×C3—91no—
201C2×D3—62no—
202C2×D3—91no—
203C2×D3—182no—
204C2×D3—182no—
205C2×D3—188no—
206C2×D3—362no—
207C2×D3C4⋊#15711yescharacteristic
208C3×C4—22no—
209C3×C4—42no—
210C3×C4—62no—
211C3×C4—91no—
212C3×C4—362no—
213C3×C4C22×#15711yescharacteristic
214Dic3—62no—
215Dic3—91no—
216Dic3—182no—
217Dic3—362no—
218Dic3—362no—
219Dic3C4⋊#15711yescharacteristic
220C32—22no—
221C32—42no—
222C32#486011yescharacteristic, complemented
223C23—271no—
224C2×C4—31no—
225C2×C4—91no—
226C2×C4—182no—
227C2×C4—271no—
228C2×C4—271no—
229C2×C4—271no—
230C2×C4—271no—
231C2×C4—542no—
232C8—31no—
233C8—91no—
234C8—271no—
235C8D3×#15711yescharacteristic
236C2×C3—22no—
237C2×C3—42no—
238C2×C3—64no—
239C2×C3—92no—
240C2×C3C2×C4⋊#15711yescharacteristic
241D3—21no—
242D3—64no—
243D3—184no—
244D3—181no—
245D3—364no—
246C22—31no—
247C22—91no—
248C22—271no—
249C22—542no—
250C4—31no—
251C4—91no—
252C4—271no—
253C4—362no—
254C4—1082no—
255C4C2×D3×#15711yescharacteristic
256C3—22no—
257C3—42no—
258C3#1059376811yescharacteristic, complemented
259C2—61no—
260C2—92no—
261C2—541no—
262C2D3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C4⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
3. #11723093C2semidirect
4. C3×#63768C2semidirect
5. #11722715C2semidirect
12. C8×C3⋊(C3⋊D3)C4semidirect
14. C32×D3×C8C4semidirect
15. #88785C4semidirect
17. D3×C3⋊(C3⋊C8)C4semidirect
23. #63768D3semidirect
123. D3×C8#157semidirect
150. C33#427semidirect
222. C32#4860semidirect
258. C3#10593768semidirect

Maximal subgroups

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

subgroupindex
#117242472
D3×C8×C3⋊D32
#117230932
C3×#637682
#117227152
#105937683
#48609