Finite Group Databasebeta

← #13138897

Subgroups — #13138897

Order 1728.

559 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 107 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×C8×#272C211yesmaximal, characteristic
2C4×#6199C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63641⋊C3C212yesmaximal, complemented
5#11720311C212yesmaximal, complemented
6D3×Dic3⋊C8—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#6199C411yescharacteristic
9C2×#6199C411yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C8×#272—22no—
13C4×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88447C412yes—
16#88434—22no—
17C2×C3×D3×C8—32no—
18C2×C3×D3×C8—61no—
19C4×D3×Dic3—32no—
20C2×D3×C3⋊C8—32no—
21#63641D312yescomplemented
22C3×Dic3⋊C8—32no—
23#63572—32no—
24Dic3⋊(C3⋊C8)—32no—
25C22×#272C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C4×#272D412yes—
29C4×#272Q812yes—
30C2×(C3⋊Dic3)⋊C3C2×C412yes—
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32#6199—22no—
33#6199C814yescomplemented
34C8×H(3,𝔽3)—21no—
35#6180—21no—
36(C3⋊(C3⋊C8))⋊C3—22no—
37C2×Dic3⋊C8—91no—
38D3×C4⋊C8—92no—
39C2×C3×C4×D3—32no—
40C2×C3×C4×D3—61no—
41C2×D3×Dic3—32no—
42C2×D3×Dic3—32no—
43C2×C32×C8—21no—
44C2×C32×C8C2×D312yes—
45C4×C3⋊Dic3C2×D312yes—
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C3×C4×Dic3—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—61no—
51C3×D3×C8—64no—
52C3×D3×C8—64no—
53Dic32—32no—
54Dic32—32no—
55D3×C3⋊C8—64no—
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C2×#272C4⋊C412yes—
58C2×#272C2×C812yes—
59C2×#272M1612yes—
60C4×H(3,𝔽3)C2×D411yescharacteristic
61C4×H(3,𝔽3)C2×Q811yescharacteristic
62#1084C4⋊C412yes—
63(C3⋊Dic3)⋊C3—22no—
64(C3⋊Dic3)⋊C3C2×C814yescomplemented
65C22×C3×C8—91no—
66C2×C4×Dic3—91no—
67C22×C3⋊C8—91no—
68C2×D3×C8—32no—
69C2×D3×C8—61no—
70C2×D3×C8—92no—
71C42×D3—92no—
72C3×C4⋊C8—92no—
73Dic3⋊C8—32no—
74Dic3⋊C8—92no—
75Dic3⋊C8—92no—
76Dic3⋊C8—92no—
77C4⋊(C3⋊C8)—92no—
78C22×C3×D3—32no—
79C22×C3×D3—61no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D312yes—
82C2×C3⋊Dic3C4×D312yes—
83C2×C3⋊Dic3C4×D312yes—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—32no—
87C2×C3×Dic3—61no—
88C3×C4×D3—34no—
89C3×C4×D3—34no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92D3×Dic3—38no—
93D3×Dic3—64no—
94C32×C8—22no—
95C32×C8—22no—
96C3⋊(C3⋊C8)—22no—
97C3×C3⋊C8—62no—
98C3×C3⋊C8—62no—
99C3×C3⋊C8—62no—
100C2×C4⋊C8—271noSylow
101C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
102C2×H(3,𝔽3)C22×C811yescharacteristic
103C2×H(3,𝔽3)C2×M1611yescharacteristic
104#272C4⋊C814yescomplemented
105C22×C3×C4—91no—
106C22×Dic3—91no—
107C22×Dic3—91no—
108C2×C4×D3—32no—
109C2×C4×D3—61no—
110C2×C4×D3—92no—
111C2×C4×D3—92no—
112C2×C3×C8—32no—
113C2×C3×C8—61no—
114C2×C3×C8—92no—
115C2×C3×C8—91no—
116C2×C3×C8—182no—
117C2×C3×C8D3211yescharacteristic
118C3×C42—92no—
119C4×Dic3—32no—
120C4×Dic3—92no—
121C4×Dic3—92no—
122C4×Dic3—92no—
123C4×Dic3—92no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—92no—
128C2×C3⋊C8—182no—
129D3×C8—64no—
130D3×C8—64no—
131D3×C8—184no—
132C22×C32—21no—
133C22×C32C2×C4×D312yes—
134C2×C3×D3—34no—
135C2×C3×D3—34no—
136C2×C3×D3—34no—
137C2×C3×D3—62no—
138C2×C3×D3—62no—
139C2×C3×D3—62no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4D3×D412yes—
143C32×C4D3×Q812yes—
144C3⋊Dic3—22no—
145C3⋊Dic3D3×C814yescomplemented
146C3×Dic3—34no—
147C3×Dic3—34no—
148C3×Dic3—62no—
149C3×Dic3—62no—
150C22×C8—91no—
151C22×C8—271no—
152C2×C42—271no—
153C4⋊C8—92no—
154C4⋊C8—272no—
155H(3,𝔽3)C2×C4⋊C811yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C22×D3—32no—
158C22×D3—61no—
159C2×C3×C4—32no—
160C2×C3×C4—61no—
161C2×C3×C4—92no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4C2×D3211yescharacteristic
167C2×Dic3—32no—
168C2×Dic3—32no—
169C2×Dic3—32no—
170C2×Dic3—61no—
171C2×Dic3—92no—
172C2×Dic3—92no—
173C2×Dic3—94no—
174C2×Dic3—182no—
175C4×D3—34no—
176C4×D3—34no—
177C4×D3—62no—
178C4×D3—62no—
179C4×D3—98no—
180C4×D3—184no—
181C3×C8—21no—
182C3×C8—62no—
183C3×C8—62no—
184C3×C8—182no—
185C3×C8—181no—
186C3⋊C8—62no—
187C3⋊C8—62no—
188C3⋊C8—62no—
189C3⋊C8—182no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32D3×C4⋊C412yes—
194C2×C32C2×D3×C812yes—
195C2×C32D3×M1612yes—
196C3×D3—38no—
197C3×D3—64no—
198C22×C4—91no—
199C22×C4—271no—
200C22×C4—271no—
201C2×C8—92no—
202C2×C8—91no—
203C2×C8—182no—
204C2×C8—542no—
205C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
206C42—92no—
207C42—272no—
208C22×C3—32no—
209C22×C3—61no—
210C22×C3—92no—
211C22×C3—92no—
212C22×C3—92no—
213C22×C3C4×D3211yescharacteristic
214C2×D3—34no—
215C2×D3—34no—
216C2×D3—34no—
217C2×D3—62no—
218C2×D3—62no—
219C2×D3—62no—
220C3×C4—32no—
221C3×C4—32no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—94no—
225C3×C4—92no—
226C3×C4—182no—
227C3×C4#379711yescharacteristic
228C3×C4#381111yescharacteristic
229Dic3—34no—
230Dic3—34no—
231Dic3—62no—
232Dic3—62no—
233Dic3—94no—
234Dic3—182no—
235C32—21no—
236C32D3×C4⋊C812yescomplemented
237C23—91no—
238C2×C4—92no—
239C2×C4—91no—
240C2×C4—92no—
241C2×C4—92no—
242C2×C4—92no—
243C2×C4—274no—
244C2×C4—542no—
245C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C8—21no—
247C8—181no—
248C8—182no—
249C2×C3—32no—
250C2×C3—32no—
251C2×C3—32no—
252C2×C3—61no—
253C2×C3—61no—
254C2×C3—61no—
255C2×C3—94no—
256C2×C3D32×C811yescharacteristic
257C2×C3#6379111yescharacteristic
258C2×C3#6388311yescharacteristic
259D3—38no—
260D3—64no—
261C22—92no—
262C22—92no—
263C22—92no—
264C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
265C4—94no—
266C4—92no—
267C4—182no—
268C4#8865111yescharacteristic
269C4#8867711yescharacteristic
270C3—32no—
271C3—61no—
272C3#1059065811yescharacteristic
273C2—94no—
274C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C2#1172083611yescharacteristic
276C2#1172114711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63641⋊C3C2semidirect
5. #11720311C2semidirect
21. #63641D3semidirect
33. #6199C8semidirect
64. (C3⋊Dic3)⋊C3C2×C8semidirect
104. #272C4⋊C8semidirect
145. C3⋊Dic3D3×C8semidirect
155. H(3,𝔽3)C2×C4⋊C8semidirect
236. C32D3×C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
C4×#61992
C2×#884342
#63641⋊C32
#117203112
D3×Dic3⋊C83