Finite Group Databasebeta

← #13137409

Subgroups — #13137409

Order 1728.

475 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 91 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#11721718C211yesmaximal, characteristic
2C2×#88435C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4C4⋊((C3⋊(C3⋊C8))⋊C3)C212yesmaximal, complemented
5#11720304C212yesmaximal, complemented
6#10590324—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C42×H(3,𝔽3)C2211yescharacteristic
9#88542C412yes—
10#88540C412yes—
11C4×#1084C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88435—22no—
15#88434—22no—
16C3×#872—32no—
17C3×#872—61no—
18C2×#3711—32no—
19C2×D3×C3⋊C8—32no—
20C4⋊(C3⋊(C3⋊C8))D312yescomplemented
21C3×C4⋊(C3⋊C8)—32no—
22Dic3⋊(C3⋊C8)—32no—
23Dic3⋊(C3⋊C8)—32no—
24C22×#272C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C412yes—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084C2×C412yes—
28C2×#1084C2×C411yescharacteristic
29C4×#272D412yes—
30C4×#272Q812yes—
31(C3⋊(C3⋊C8))⋊C3—22no—
32(C3⋊(C3⋊C8))⋊C3—22no—
33#4999—91no—
34#4839—92no—
35C2×C3×C4×D3—32no—
36C2×C3×C4×D3—61no—
37C32×C42—21no—
38C32×C42C2×D312yes—
39C2×C3⋊(C3⋊C8)C2×D312yes—
40C2×C3⋊(C3⋊C8)C2×D312yes—
41C3×(C2×D3)⋊C4—32no—
42C3×(C2×D3)⋊C4—32no—
43C3×(C2×D3)⋊C4—62no—
44C3×Dic3⋊C4—32no—
45C3×Dic3⋊C4—32no—
46C3×Dic3⋊C4—62no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—32no—
51#3711—64no—
52D3×C3⋊C8—64no—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C2×#272—22no—
55C2×#272C4⋊C412yes—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)C2×Q811yescharacteristic
59#1084—22no—
60#1084C4⋊C412yes—
61C3×#117—91no—
62C2×#208—91no—
63C22×C3⋊C8—91no—
64C2×ΓL(1,𝔽25)—92no—
65C2×D3×C8—92no—
66#872—32no—
67#872—61no—
68C3×C4⋊C8—92no—
69Dic3⋊C8—92no—
70Dic3⋊C8—92no—
71C4⋊(C3⋊C8)—32no—
72C4⋊(C3⋊C8)—92no—
73C4⋊(C3⋊C8)—92no—
74C22×C3×D3—32no—
75C22×C3×D3—61no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4C4×D312yes—
79C2×C32×C4C4×D312yes—
80C2×C32×C4C22×D312yes—
81C2×C3×Dic3—32no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—62no—
86C3×C4×D3—34no—
87C3×C4×D3—34no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3⋊(C3⋊C8)—22no—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—62no—
93C3×C3⋊C8—62no—
94#424—271noSylow
95C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
96C2×H(3,𝔽3)#13112yes—
97#272—22no—
98C22×C3×C4—91no—
99C2×C4×D3—32no—
100C2×C4×D3—61no—
101C2×C3×C8—92no—
102C2×C3×C8—92no—
103C3×C4⋊C4—92no—
104C3×C22⋊C4—92no—
105C3×C42—32no—
106C3×C42—61no—
107C3×C42—91no—
108C3×C42D3211yescharacteristic
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—62no—
112Dic3⋊C4—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—62no—
115C4×Dic3—32no—
116C4×Dic3—61no—
117#208—182no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123ΓL(1,𝔽25)—184no—
124D3×C8—184no—
125C22×C32—21no—
126C22×C32C2×C4×D312yes—
127C2×C3×D3—34no—
128C2×C3×D3—62no—
129C2×C3×D3—62no—
130C2×C3×D3—62no—
131C2×C3×D3—122no—
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—42no—
137C32×C4D3×D412yes—
138C32×C4D3×Q812yes—
139C3×Dic3—34no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—122no—
144C2×M16—271no—
145C22×C8—271no—
146#117—91no—
147C4⋊C8—92no—
148C4⋊C8—272no—
149H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
150C23×C3—91no—
151C22×D3—32no—
152C22×D3—61no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—62no—
157C2×C3×C4—61no—
158C2×C3×C4—91no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—92no—
162C2×C3×C4#62412yes—
163C2×C3×C4C2×D3211yescharacteristic
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—32no—
167C2×Dic3—61no—
168C2×Dic3—62no—
169C4×D3—34no—
170C4×D3—34no—
171C4×D3—62no—
172C4×D3—62no—
173C3×C8—182no—
174C3×C8—182no—
175C3⋊C8—62no—
176C3⋊C8—62no—
177C3⋊C8—182no—
178C3⋊C8—182no—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32D3×C4⋊C412yes—
182C2×C32#90112yes—
183C2×C32#90712yes—
184C3×D3—64no—
185C3×D3—122no—
186C22×C4—91no—
187M16—542no—
188C2×C8—92no—
189C2×C8—92no—
190C2×C8—542no—
191C4⋊C4—92no—
192C22⋊C4—92no—
193C42—91no—
194C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C22×C3—32no—
196C22×C3—61no—
197C22×C3—92no—
198C22×C3—182no—
199C22×C3C2×#62411yescharacteristic
200C2×D3—34no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—122no—
205C3×C4—22no—
206C3×C4—32no—
207C3×C4—32no—
208C3×C4—62no—
209C3×C4—62no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4—92no—
213C3×C4—122no—
214C3×C4—182no—
215C3×C4#379711yescharacteristic
216C3×C4#380211yescharacteristic
217Dic3—34no—
218Dic3—62no—
219Dic3—62no—
220Dic3—62no—
221Dic3—122no—
222C32—21no—
223C32#483912yescomplemented
224C23—91no—
225C2×C4—91no—
226C2×C4—92no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4#619912yes—
230C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
231C8—182no—
232C8—182no—
233C2×C3—32no—
234C2×C3—32no—
235C2×C3—32no—
236C2×C3—62no—
237C2×C3—61no—
238C2×C3—182no—
239C2×C3#6381712yes—
240C2×C3#6390011yescharacteristic
241D3—64no—
242D3—122no—
243C22—92no—
244C22—182no—
245C22C2×#619911yescharacteristic
246C4—22no—
247C4—92no—
248C4—182no—
249C4#8865111yescharacteristic
250C4#8865511yescharacteristic
251C3—32no—
252C3—61no—
253C3#1059033211yescharacteristic
254C2—182no—
255C2#1172106412yes—
256C2#1172115211yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720304C2semidirect
20. C4⋊(C3⋊(C3⋊C8))D3semidirect
149. H(3,𝔽3)#424semidirect
223. C32#4839semidirect

Maximal subgroups

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

subgroupindex
#117217182
C2×#884352
C2×#884342
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203042
#105903243