Finite Group Databasebeta

← #13139923

Subgroups — #13139923

Order 1728.

521 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 103 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×C4×#1084C211yesmaximal, characteristic
2#63657⋊C3C212yesmaximal
3#11720304C212yesmaximal, complemented
4#11720302C212yesmaximal, complemented
5#10591053—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C411yescharacteristic
8C22×#1084C411yescharacteristic
9C4×#1084C411yescharacteristic
10C4×#1084C411yescharacteristic
11C4×#1084C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2214yes—
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16#63657D312yes—
17C3×#830—32no—
18Dic3⋊(C3⋊C8)—34no—
19Dic3×C3⋊C8—34no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084—22no—
25C2×#1084C814yescomplemented
26C2×#1084C2×C411yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C2×#1084C2×C411yescharacteristic
29C2×#1084C2×C411yescharacteristic
30(C3⋊(C3⋊C8))⋊C3—24no—
31#4925—91no—
32#4718—92no—
33C22×C32×C4—21no—
34C22×C32×C4C2×D312yes—
35C22×C3×Dic3—32no—
36C22×C3×Dic3—32no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C2×C3⋊(C3⋊C8)C2×D314yes—
40C3×C4×Dic3—32no—
41C3×C4×Dic3—32no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C3×C4×Dic3—61no—
46C3×C4×Dic3—61no—
47C3×C4×Dic3—61no—
48C2×C3×C3⋊C8—34no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)C2×C812yes—
52C22×H(3,𝔽3)C22×C411yescharacteristic
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)#4112yes—
55#1084—21no—
56#1084C2×C812yes—
57#1084C2×C812yes—
58#1084M1612yes—
59C2×C3×C42—91no—
60C2×C4×Dic3—32no—
61C2×C4×Dic3—61no—
62C3×C22⋊C8—92no—
63#830—32no—
64#830—92no—
65Dic3⋊C8—94no—
66C8×Dic3—94no—
67C4⋊(C3⋊C8)—92no—
68C4×C3⋊C8—92no—
69C23×C32—21no—
70C23×C32C4×D312yes—
71C2×C32×C4—24no—
72C2×C32×C4—24no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4C4×D312yes—
76C2×C32×C4C22×D312yes—
77C2×C3×Dic3—38no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—32no—
81C2×C3×Dic3—32no—
82C2×C3×Dic3—64no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—64no—
88C2×C3×Dic3—64no—
89C3⋊(C3⋊C8)—24no—
90C3×C3⋊C8—64no—
91#431—271noSylow
92C2×H(3,𝔽3)—22no—
93C2×H(3,𝔽3)#11711yescharacteristic
94C2×H(3,𝔽3)C22×C811yescharacteristic
95C2×H(3,𝔽3)C2×M1611yescharacteristic
96C22×C3×C4—32no—
97C22×C3×C4—61no—
98C22×C3×C4—91no—
99C22×C3×C4—91no—
100C22×C3×C4D3211yescharacteristic
101C22×Dic3—32no—
102C22×Dic3—32no—
103C22×Dic3—61no—
104C22×Dic3—61no—
105C2×C3×C8—94no—
106C3×C42—91no—
107C3×C42—91no—
108C3×C42—91no—
109C3×C42—91no—
110C4×Dic3—32no—
111C4×Dic3—32no—
112C4×Dic3—32no—
113C4×Dic3—32no—
114C4×Dic3—61no—
115C4×Dic3—61no—
116C4×Dic3—61no—
117C4×Dic3—61no—
118C2×C3⋊C8—34no—
119C2×C3⋊C8—94no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—21no—
126C22×C32D3×C814yes—
127C22×C32C2×C4×D312yes—
128C32×C4—22no—
129C32×C4—22no—
130C32×C4—22no—
131C32×C4#23714yes—
132C3×Dic3—34no—
133C3×Dic3—34no—
134C3×Dic3—34no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C3×Dic3—62no—
138C3×Dic3—62no—
139C3×Dic3—62no—
140C2×C42—91no—
141C4⋊C8—272no—
142C22⋊C8—92no—
143C4×C8—272no—
144H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
145C23×C3—32no—
146C23×C3—61no—
147C23×C3#62411yescharacteristic
148C2×C3×C4—22no—
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—64no—
152C2×C3×C4—64no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—94no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—182no—
161C2×C3×C4#62411yescharacteristic
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—38no—
167C2×Dic3—32no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—64no—
173C2×Dic3—64no—
174C2×Dic3—64no—
175C3×C8—184no—
176C3⋊C8—64no—
177C3⋊C8—184no—
178C2×C32—24no—
179C2×C32—24no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32#87712yes—
184C2×C32C2×D3×C812yes—
185C2×C32D3×M1612yes—
186C22×C4—91no—
187C22×C4—91no—
188C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
189C2×C8—94no—
190C42—91no—
191C42—91no—
192C42—91no—
193C42—91no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—32no—
197C22×C3—34no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3—62no—
202C22×C3—61no—
203C22×C3—62no—
204C22×C3#371012yes—
205C22×C3C2×#62411yescharacteristic
206C3×C4—21no—
207C3×C4—34no—
208C3×C4—62no—
209C3×C4—62no—
210C3×C4—62no—
211C3×C4—92no—
212C3×C4—92no—
213C3×C4—92no—
214C3×C4—181no—
215C3×C4#380512yes—
216Dic3—34no—
217Dic3—34no—
218Dic3—34no—
219Dic3—62no—
220Dic3—62no—
221Dic3—62no—
222Dic3—62no—
223Dic3—62no—
224C32—21no—
225C32#471812yescomplemented
226C23#619911yescharacteristic
227C2×C4—22no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—94no—
232C2×C4—91no—
233C2×C4—182no—
234C2×C4#619911yescharacteristic
235C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C8—184no—
237C2×C3—22no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—64no—
242C2×C3—64no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—61no—
246C2×C3C2×#371011yescharacteristic
247C2×C3#6381811yescharacteristic
248C2×C3#6395711yescharacteristic
249C22—21no—
250C22—21no—
251C22#8843412yes—
252C22C2×#619911yescharacteristic
253C4—21no—
254C4—92no—
255C4—92no—
256C4—92no—
257C4—181no—
258C4#8867212yes—
259C3—32no—
260C3—61no—
261C3#1059106111yescharacteristic
262C2—22no—
263C2C2×#8843411yescharacteristic
264C2#1172106511yescharacteristic
265C2#1172129811yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#431
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720304C2semidirect
4. #11720302C2semidirect
25. C2×#1084C8semidirect
144. H(3,𝔽3)#431semidirect
225. C32#4718semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#63657⋊C32
#117203042
#117203022
#105910533