Finite Group Databasebeta

← #13166507

Subgroups — #13166507

Order 1728.

528 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 80 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×#88925C212yesmaximal
2C4×D3×C3⋊Dic3C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic, complemented
4C3×#63779C211yesmaximal, characteristic, complemented
5Dic3⋊#622C212yesmaximal, complemented
6#10593805—31nomaximal
7C2×C32×C4×D3C411yescharacteristic
8C2×D3×C3⋊Dic3C411yescharacteristic
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12#88925—24no—
13C4×C3⋊(C3⋊Dic3)C411yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3792—32no—
18C2×C4×C3⋊Dic3—31no—
19C4×D3×Dic3—62no—
20#63779—31no—
21#63779D311yescharacteristic, complemented
22C4⋊#622—32no—
23C22×C32×D3C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30C32×C4×D3—22no—
31D3×C3⋊Dic3—22no—
32D3×C3⋊Dic3D412yes—
33D3×C3⋊Dic3Q812yes—
34C3⋊#622—22no—
35C3×#622—22no—
36#4837—91nomaximal
37C22×C32×C4—31no—
38C22×C3⋊Dic3—31no—
39C22×C3⋊Dic3—31no—
40C2×C3×C4×D3—22no—
41C2×D3×Dic3—62no—
42C2×D3×Dic3—62no—
43#3792—64no—
44C2×#622—32no—
45C2×#622C2×D312yes—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3—62no—
50C4×C3⋊Dic3—122no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C3×C4×Dic3—62no—
53Dic32—62no—
54Dic32—62no—
55C22×C33C22×C411yescharacteristic
56C2×C32×D3—21no—
57C2×C32×D3—21no—
58C2×C32×D3C4⋊C412yes—
59C33×C4—21no—
60C3⋊(C3⋊Dic3)—21no—
61C3⋊(C3⋊Dic3)C4⋊C412yes—
62C3×C3⋊Dic3M1612yescomplemented
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3C2×Q811yescharacteristic
65C32×Dic3M1612yescomplemented
66C2×C4×Dic3—182no—
67C2×ΓL(1,𝔽25)—92no—
68C42×D3—91no—
69C3×C4⋊C8—91no—
70Dic3⋊C8—92no—
71C4⋊(C3⋊C8)—91no—
72C23×C32—31no—
73C22×C3×D3—22no—
74C2×C32×C4—22no—
75C2×C32×C4—31no—
76C2×C32×C4—42no—
77C2×C32×C4—62no—
78C2×C32×C4C4×D311yescharacteristic
79C2×C3⋊Dic3—31no—
80C2×C3⋊Dic3—32no—
81C2×C3⋊Dic3—32no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—62no—
86C2×C3⋊Dic3—122no—
87C2×C3⋊Dic3—122no—
88C2×C3⋊Dic3C4×D311yescharacteristic
89C2×C3⋊Dic3C22×D311yescharacteristic
90C2×C3×Dic3—22no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C3×C4×D3—28no—
94D3×Dic3—64no—
95D3×Dic3—64no—
96D3×Dic3—68no—
97#622—22no—
98#622—62no—
99#423—271noSylow
100C2×C33C2×C4⋊C411yescharacteristic
101C2×C33C2×M1611yescharacteristic
102C2×C33C2×M1611yescharacteristic
103C32×D3—22no—
104C22×C3×C4—62no—
105C22×Dic3—182no—
106C22×Dic3—182no—
107C2×C4×D3—91no—
108C2×C4×D3—91no—
109C2×C4×D3#15711yescharacteristic
110C2×C3×C8—92no—
111C3×C42—91no—
112C4×Dic3—62no—
113C4×Dic3—91no—
114C4×Dic3—91no—
115C4×Dic3—91no—
116C4×Dic3—182no—
117C4×Dic3—182no—
118C4×Dic3—182no—
119C4×Dic3—362no—
120C2×C3⋊C8—92no—
121ΓL(1,𝔽25)—184no—
122C22×C32—22no—
123C22×C32—32no—
124C22×C32—42no—
125C22×C32—61no—
126C22×C32—61no—
127C22×C32C2×C4×D311yescharacteristic
128C2×C3×D3—24no—
129C2×C3×D3—24no—
130C2×C3×D3—24no—
131C32×C4—21no—
132C32×C4—24no—
133C32×C4—32no—
134C32×C4—44no—
135C3⋊Dic3—32no—
136C3⋊Dic3—64no—
137C3⋊Dic3—61no—
138C3⋊Dic3—64no—
139C3⋊Dic3—124no—
140C3⋊Dic3—124no—
141C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
142C3⋊Dic3D3×D411yescharacteristic
143C3⋊Dic3D3×Q811yescharacteristic
144C3×Dic3—24no—
145C3×Dic3—64no—
146C3×Dic3—62no—
147C3×Dic3—62no—
148C2×M16—272no—
149C2×C42—271no—
150C4⋊C8—91no—
151C4⋊C8—271no—
152C4⋊C8—272no—
153C33#42311yescharacteristic, complemented, Sylow
154C23×C3—62no—
155C22×D3C2×#15711yescharacteristic
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—62no—
159C2×C3×C4—68no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4C2×#15711yescharacteristic
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—91no—
166C2×Dic3—91no—
167C2×Dic3—184no—
168C2×Dic3—184no—
169C2×Dic3—182no—
170C2×Dic3—182no—
171C2×Dic3—188no—
172C2×Dic3—362no—
173C2×Dic3—362no—
174C2×Dic3C2×#15711yescharacteristic
175C4×D3—22no—
176C4×D3—92no—
177C4×D3—92no—
178C4×D3—182no—
179C3×C8—182no—
180C3⋊C8—182no—
181C2×C32—22no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—42no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32—62no—
188C2×C32D3×C4⋊C411yescharacteristic
189C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
190C2×C32D3×M1611yescharacteristic
191C3×D3—28no—
192C22×C4—31no—
193C22×C4—271no—
194C22×C4—271no—
195M16—544no—
196C2×C8—92no—
197C2×C8—272no—
198C42—91no—
199C42—271no—
200C42—271no—
201C42—271no—
202C22×C3—22no—
203C22×C3—42no—
204C22×C3—64no—
205C22×C3—64no—
206C22×C3—64no—
207C22×C3C22×#15711yescharacteristic
208C2×D3—21no—
209C2×D3—21no—
210C2×D3C4⋊#15712yes—
211C3×C4—24no—
212C3×C4—21no—
213C3×C4—44no—
214C3×C4—64no—
215C3×C4—92no—
216C3×C4—91no—
217C3×C4—91no—
218Dic3—64no—
219Dic3—62no—
220Dic3—62no—
221Dic3—92no—
222Dic3—184no—
223Dic3—184no—
224Dic3—181no—
225Dic3—364no—
226Dic3—364no—
227Dic3#379212yescomplemented
228C32—22no—
229C32—42no—
230C32#483711yescharacteristic, complemented
231C23—31no—
232C2×C4—31no—
233C2×C4—62no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—272no—
237C2×C4—272no—
238C2×C4—271no—
239C2×C4—271no—
240C2×C4—542no—
241C2×C4D3×#15711yescharacteristic
242C8—182no—
243C8—542no—
244C2×C3—22no—
245C2×C3—22no—
246C2×C3—22no—
247C2×C3—42no—
248C2×C3—42no—
249C2×C3—42no—
250C2×C3—68no—
251C2×C3#6428311yescharacteristic
252C2×C3C2×C4⋊#15711yescharacteristic
253C2×C3C2×#379211yescharacteristic
254D3—22no—
255C22—32no—
256C22—61no—
257C22—61no—
258C22C2×D3×#15711yescharacteristic
259C4—21no—
260C4—32no—
261C4—92no—
262C4—91no—
263C4—91no—
264C4—272no—
265C4—541no—
266C3—22no—
267C3—42no—
268C3#1059380511yescharacteristic, complemented
269C2—62no—
270C2#1172425411yescharacteristic
271C2D3×C4⋊#15711yescharacteristic
272C2D3×#379211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#423
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723104C2semidirect
4. C3×#63779C2semidirect
5. Dic3⋊#622C2semidirect
21. #63779D3semidirect
62. C3×C3⋊Dic3M16semidirect
65. C32×Dic3M16semidirect
141. C3⋊Dic3ΓL(1,𝔽25)semidirect
153. C33#423semidirect
227. Dic3#3792semidirect
230. C32#4837semidirect
268. C3#10593805semidirect

Maximal subgroups

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

subgroupindex
C2×#889252
C4×D3×C3⋊Dic32
#117231042
C3×#637792
Dic3⋊#6222
#105938053
#48379