Finite Group Databasebeta

← #13138913

Subgroups — #13138913

Order 1728.

475 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#11721097C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63641⋊C3C212yesmaximal, complemented
5#11720311C212yesmaximal, complemented
6#10590694—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C8×#272—22no—
11C4×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88451C412yes—
14#88449C411yescharacteristic
15#88449C411yescharacteristic
16#88435—22no—
17C2×C3×D3×C8—32no—
18C2×C3×D3×C8—61no—
19#63855—32no—
20C2×#3711—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—
32C8×H(3,𝔽3)—21no—
33#6180—21no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5102—91no—
36#4839—92no—
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—61no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D312yes—
41C4×C3⋊Dic3C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×C4×Dic3—32no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C3×D3×C8—64no—
48C3×D3×C8—64no—
49Dic3⋊Dic3—32no—
50Dic3⋊Dic3—32no—
51(C2×D3)⋊Dic3—32no—
52(C2×D3)⋊Dic3—32no—
53#3711—64no—
54C22×H(3,𝔽3)C22×C411yescharacteristic
55C2×#272—21no—
56C2×#272—21no—
57C2×#272C4⋊C412yes—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)C2×Q811yescharacteristic
60#1084C4⋊C412yes—
61(C3⋊Dic3)⋊C3—22no—
62(C3⋊Dic3)⋊C3—22no—
63C22×C3×C8—91no—
64#926—91no—
65C2×#208—91no—
66C2×ΓL(1,𝔽25)—92no—
67C2×D3×C8—32no—
68C2×D3×C8—61no—
69#872—92no—
70C3×C4⋊C8—92no—
71Dic3⋊C8—32no—
72Dic3⋊C8—92no—
73Dic3⋊C8—92no—
74Dic3⋊C8—92no—
75C4⋊(C3⋊C8)—92no—
76C22×C3×D3—32no—
77C22×C3×D3—61no—
78C2×C32×C4—21no—
79C2×C32×C4C22×D312yes—
80C2×C3⋊Dic3C4×D312yes—
81C2×C3⋊Dic3C4×D312yes—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C3×C4×D3—34no—
87C3×C4×D3—34no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C32×C8—22no—
91C32×C8—22no—
92C3⋊(C3⋊C8)—22no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95C3×C3⋊C8—62no—
96#424—271noSylow
97C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
98C2×H(3,𝔽3)#13111yescharacteristic
99C2×H(3,𝔽3)#13111yescharacteristic
100#272—22no—
101C22×C3×C4—91no—
102C2×C4×D3—32no—
103C2×C4×D3—61no—
104C2×C3×C8—32no—
105C2×C3×C8—61no—
106C2×C3×C8—92no—
107C2×C3×C8—91no—
108C2×C3×C8—182no—
109C2×C3×C8D3211yescharacteristic
110C3×C42—92no—
111#217—92no—
112(C2×D3)⋊C4—92no—
113(C2×D3)⋊C4—92no—
114C4⋊Dic3—91no—
115C4⋊Dic3—91no—
116Dic3⋊C4—92no—
117Dic3⋊C4—92no—
118C4×Dic3—32no—
119C4×Dic3—92no—
120#208—182no—
121C2×C3⋊C8—32no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—92no—
125ΓL(1,𝔽25)—184no—
126D3×C8—64no—
127D3×C8—64no—
128C22×C32—21no—
129C22×C32C2×C4×D312yes—
130C2×C3×D3—34no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4D3×D412yes—
139C32×C4D3×Q812yes—
140C3⋊Dic3—22no—
141C3⋊Dic3—22no—
142C3×Dic3—34no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—62no—
146C2×M16—271no—
147C22×C8—91no—
148#117—271no—
149C4⋊C8—92no—
150C4⋊C8—272no—
151H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
152C23×C3—91no—
153C22×D3—32no—
154C22×D3—61no—
155C2×C3×C4—32no—
156C2×C3×C4—61no—
157C2×C3×C4—91no—
158C2×C3×C4—92no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—92no—
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—61no—
167C2×Dic3—92no—
168C2×Dic3—92no—
169C4×D3—34no—
170C4×D3—34no—
171C4×D3—62no—
172C4×D3—62no—
173C3×C8—21no—
174C3×C8—62no—
175C3×C8—62no—
176C3×C8—182no—
177C3×C8—181no—
178C3⋊C8—62no—
179C3⋊C8—62no—
180C3⋊C8—62no—
181C3⋊C8—182no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32D3×C4⋊C412yes—
186C2×C32#90112yes—
187C2×C32#90712yes—
188C3×D3—64no—
189C3×D3—64no—
190C22×C4—91no—
191M16—542no—
192C2×C8—92no—
193C2×C8—91no—
194C2×C8—182no—
195C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C4⋊C4—271no—
197C4⋊C4—271no—
198C22⋊C4—272no—
199C42—92no—
200C22×C3—32no—
201C22×C3—61no—
202C22×C3—92no—
203C22×C3—181no—
204C22×C3—181no—
205C22×C3C4×D3211yescharacteristic
206C2×D3—34no—
207C2×D3—62no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—62no—
212C3×C4—32no—
213C3×C4—32no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—92no—
217C3×C4—182no—
218C3×C4—182no—
219C3×C4#379711yescharacteristic
220C3×C4#381111yescharacteristic
221Dic3—34no—
222Dic3—62no—
223Dic3—62no—
224Dic3—62no—
225Dic3—182no—
226Dic3—182no—
227C32—21no—
228C32#483912yescomplemented
229C23—91no—
230C2×C4—91no—
231C2×C4—92no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—92no—
235C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C8—21no—
237C8—181no—
238C8—182no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—182no—
246C2×C3#6380311yescharacteristic
247C2×C3#6380511yescharacteristic
248C2×C3#6388311yescharacteristic
249D3—64no—
250D3—64no—
251C22—92no—
252C22—181no—
253C22—181no—
254C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C4—92no—
256C4—182no—
257C4—182no—
258C4#8865111yescharacteristic
259C4#8867711yescharacteristic
260C3—32no—
261C3—61no—
262C3#1059070111yescharacteristic
263C2—182no—
264C2#1172084011yescharacteristic
265C2#1172084211yescharacteristic
266C2#1172114711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
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)#424
LCSG ▷ C22×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
151. H(3,𝔽3)#424semidirect
228. C32#4839semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117210972
C2×#884352
#63641⋊C32
#117203112
#105906943