Finite Group Databasebeta

← #13143719

Subgroups — #13143719

Order 1728.

1111 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×#6229C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88542C212yesmaximal
4H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
5#11720574C212yesmaximal, complemented
6C3×#4673—34nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×#1084C2212yes—
12C2×#6229—22no—
13C2×#6229—24no—
14#88542—22no—
15#88542—24no—
16C22×C3×Dic6—34no—
17C2×C3×(C2×D3)⋊C4—38no—
18C2×C3×(C2×D3)⋊C4—34no—
19C32×#95D314yescomplemented
20C3×C3⋊#95—38no—
21C23×H(3,𝔽3)C2311yescharacteristic
22C22×#272—21no—
23C22×#272—22no—
24C22×#272—22no—
25C22×#272—21no—
26C22×#272Q812yescomplemented
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D412yes—
32C2×#1084—22no—
33C2×#1084—22no—
34C2×#1084—21no—
35C2×#1084—22no—
36C2×#1084D414yes—
37#6229—28no—
38C3×#393—91no—
39#4673—34no—
40C23×C3×D3—34no—
41C22×C32×C4C2×D318yes—
42C22×C32×C4C2×D314yes—
43C22×C3×Dic3—38no—
44C22×C3×Dic3—34no—
45C2×C3×Dic6—616no—
46C2×C3×Dic6—68no—
47C3×(C2×D3)⋊C4—616no—
48C3×(C2×D3)⋊C4—68no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C22×H(3,𝔽3)C2×D412yes—
51C22×H(3,𝔽3)C2×Q811yescharacteristic
52C22×H(3,𝔽3)#4112yes—
53C22×H(3,𝔽3)#4111yescharacteristic
54C2×#272—24no—
55C2×#272—22no—
56C2×#272—41no—
57C2×#272—41no—
58C2×#272—42no—
59C4×H(3,𝔽3)—22no—
60C4×H(3,𝔽3)—42no—
61#1084—24no—
62#1084—41no—
63C22×C3×Q8—91no—
64C22×Dic6—34no—
65C2×C3×C22⋊C4—92no—
66C2×C3×C22⋊C4—91no—
67C2×(C2×D3)⋊C4—38no—
68C2×(C2×D3)⋊C4—34no—
69C3×#95—34no—
70C3×#95—92no—
71C3×#95C3⋊D311yescharacteristic
72C3⋊#95—38no—
73C23×C32C22×D314yes—
74C22×C3×D3—38no—
75C22×C3×D3—64no—
76C22×C3×D3—68no—
77C22×C3×D3—68no—
78C22×C3×D3—64no—
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4—28no—
82C2×C32×C4—28no—
83C2×C32×C4D1218yes—
84C2×C3×Dic3—316no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—68no—
87C2×C3×Dic3—64no—
88C2×C3×Dic3—68no—
89C3×Dic6—632no—
90#393—91noSylow
91C2×H(3,𝔽3)#12011yescharacteristic
92C2×H(3,𝔽3)C22⋊Q812yes—
93C2×H(3,𝔽3)C22⋊Q811yescharacteristic
94C2×H(3,𝔽3)#12411yescharacteristic
95C2×H(3,𝔽3)#12412yes—
96#272—42no—
97C24×C3—91no—
98C23×D3—34no—
99C2×C3×Q8—184no—
100C2×C3×Q8—182no—
101C22×C3×C4—34no—
102C22×C3×C4—38no—
103C22×C3×C4—92no—
104C22×C3×C4—91no—
105C22×C3×C4C2×C3⋊D312yes—
106C22×C3×C4C2×C3⋊D311yescharacteristic
107C22×Dic3—38no—
108C22×Dic3—34no—
109C2×Dic6—616no—
110C2×Dic6—68no—
111C3×C22⋊C4—184no—
112C3×C22⋊C4—182no—
113(C2×D3)⋊C4—616no—
114(C2×D3)⋊C4—68no—
115C22×C32C2×D1214yes—
116C22×C32#23518yes—
117C22×C32D3×D418yes—
118C22×C32#23714yes—
119C22×C32D3×Q814yes—
120C2×C3×D3—616no—
121C2×C3×D3—68no—
122C2×C3×D3—124no—
123C2×C3×D3—124no—
124C2×C3×D3—128no—
125C32×C4—28no—
126C32×C4—48no—
127C3×Dic3—616no—
128C3×Dic3—124no—
129C22×Q8—91no—
130C2×C22⋊C4—92no—
131C2×C22⋊C4—91no—
132#95—92no—
133#95#27211yescharacteristic, complemented
134H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
135C23×C3—34no—
136C23×C3—92no—
137C23×C3—181no—
138C23×C3—182no—
139C23×C3—182no—
140C23×C3—181no—
141C23×C3C22×C3⋊D311yescharacteristic
142C22×D3—38no—
143C22×D3—64no—
144C22×D3—68no—
145C22×D3—68no—
146C22×D3—64no—
147C3×Q8—188no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—38no—
153C2×C3×C4—68no—
154C2×C3×C4—68no—
155C2×C3×C4—68no—
156C2×C3×C4—68no—
157C2×C3×C4—94no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4—181no—
161C2×C3×C4—182no—
162C2×C3×C4C4⋊(C3⋊D3)12yes—
163C2×Dic3—316no—
164C2×Dic3—68no—
165C2×Dic3—68no—
166C2×Dic3—64no—
167C2×Dic3—68no—
168Dic6—632no—
169C2×C32#87514yes—
170C2×C32#88214yes—
171C2×C32#88518yes—
172C2×C32#89618yes—
173C2×C32#89714yes—
174C3×D3—128no—
175C24—91no—
176C2×Q8—184no—
177C2×Q8—182no—
178C22×C4—92no—
179C22×C4—91no—
180C22×C4C2×#27212yes—
181C22×C4C2×#27211yescharacteristic
182C22⋊C4—184no—
183C22⋊C4—182no—
184C22×C3—34no—
185C22×C3—38no—
186C22×C3—38no—
187C22×C3—34no—
188C22×C3—34no—
189C22×C3—184no—
190C22×C3—182no—
191C22×C3—361no—
192C22×C3—361no—
193C22×C3—362no—
194C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
195C22×C3#382812yes—
196C22×C3D4×C3⋊D312yes—
197C22×C3#383011yescharacteristic
198C22×C3Q8×C3⋊D311yescharacteristic
199C2×D3—616no—
200C2×D3—68no—
201C2×D3—124no—
202C2×D3—128no—
203C2×D3—124no—
204C3×C4—22no—
205C3×C4—42no—
206C3×C4—68no—
207C3×C4—128no—
208C3×C4—184no—
209C3×C4—361no—
210Dic3—616no—
211Dic3—124no—
212C32#467314yescomplemented
213C23—92no—
214C23—181no—
215C23—182no—
216C23—182no—
217C23—181no—
218C23C22×#27211yescharacteristic
219Q8—188no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—94no—
225C2×C4—182no—
226C2×C4—182no—
227C2×C4—181no—
228C2×C4—182no—
229C2×C4C4⋊#27212yes—
230C2×C3—34no—
231C2×C3—38no—
232C2×C3—38no—
233C2×C3—34no—
234C2×C3—34no—
235C2×C3—362no—
236C2×C3#6408411yescharacteristic
237C2×C3#6409111yescharacteristic
238C2×C3#6409412yes—
239C2×C3#6410512yes—
240C2×C3#6410611yescharacteristic
241D3—128no—
242C22—184no—
243C22—182no—
244C22—361no—
245C22—362no—
246C22—361no—
247C22C2×C4⋊#27211yescharacteristic
248C22#8874012yes—
249C22D4×#27212yes—
250C22#8874411yescharacteristic
251C22Q8×#27211yescharacteristic
252C4—22no—
253C4—42no—
254C4—184no—
255C4—361no—
256C3—34no—
257C3#1059237211yescharacteristic
258C2—362no—
259C2#1172172111yescharacteristic
260C2#1172173811yescharacteristic
261C2#1172174112yes—
262C2#1172176412yes—
263C2#1172176511yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
prime radicalC23×#272C22
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#393
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
4. H(3,𝔽3)×#95C2semidirect
5. #11720574C2semidirect
19. C32×#95D3semidirect
26. C22×#272Q8semidirect
133. #95#272semidirect
134. H(3,𝔽3)#393semidirect
212. C32#4673semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#885422
H(3,𝔽3)×#952
#117205742
C3×#46733