Finite Group Databasebeta

← #13144356

Subgroups — #13144356

Order 1728.

1023 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1023 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
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88542C212yesmaximal
4C2×#88540C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#4993—34nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×#1084C2212yes—
12C4⋊C4×H(3,𝔽3)—22no—
13#88542—22no—
14#88542—24no—
15#88540—24no—
16C2×C32×C4⋊C4D314yescomplemented
17C2×C3×(C2×D3)⋊C4—38no—
18C2×C3×(C2×D3)⋊C4—34no—
19C2×C3×Dic3⋊C4—38no—
20C3×C3⋊#95—34no—
21C23×H(3,𝔽3)C2311yescharacteristic
22C22×#272—22no—
23C22×#272—21no—
24C22×#272—22no—
25C22×#272—21no—
26C22×#272Q812yes—
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—22no—
35C2×#1084—21no—
36C2×#1084D414yes—
37C3×#399—91no—
38#4993—34no—
39C23×C3×D3—34no—
40C22×C32×C4C2×D318yes—
41C22×C32×C4C2×D314yes—
42C22×C3×Dic3—38no—
43C22×C3×Dic3—34no—
44C32×C4⋊C4—28no—
45C3×(C2×D3)⋊C4—616no—
46C3×(C2×D3)⋊C4—68no—
47C3×Dic3⋊C4—616no—
48C22×H(3,𝔽3)C2×D412yes—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C22×H(3,𝔽3)C2×Q811yescharacteristic
51C22×H(3,𝔽3)#4112yes—
52C22×H(3,𝔽3)#4111yescharacteristic
53C2×#272—24no—
54C2×#272—22no—
55C2×#272—42no—
56C2×#272—41no—
57C2×#272—41no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)—42no—
60#1084—24no—
61#1084—41no—
62C2×C3×C4⋊C4—34no—
63C2×C3×C4⋊C4—92no—
64C2×C3×C4⋊C4C3⋊D311yescharacteristic
65C2×C3×C22⋊C4—92no—
66C2×C3×C22⋊C4—91no—
67C2×(C2×D3)⋊C4—38no—
68C2×(C2×D3)⋊C4—34no—
69C2×Dic3⋊C4—38no—
70C3×#95—91no—
71C3⋊#95—34no—
72C23×C32C22×D314yes—
73C22×C3×D3—38no—
74C22×C3×D3—68no—
75C22×C3×D3—68no—
76C22×C3×D3—64no—
77C22×C3×D3—64no—
78C2×C32×C4—28no—
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4—28no—
82C2×C32×C4#6718yes—
83C2×C3×Dic3—316no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—68no—
87C2×C3×Dic3—64no—
88#399—91noSylow
89C2×H(3,𝔽3)C22⋊Q812yes—
90C2×H(3,𝔽3)C22⋊Q811yescharacteristic
91C2×H(3,𝔽3)#12312yes—
92C2×H(3,𝔽3)#12311yescharacteristic
93C2×H(3,𝔽3)#12711yescharacteristic
94#272—42no—
95C24×C3—91no—
96C23×D3—34no—
97C22×C3×C4—34no—
98C22×C3×C4—38no—
99C22×C3×C4—92no—
100C22×C3×C4—91no—
101C22×C3×C4C2×C3⋊D312yes—
102C22×C3×C4C2×C3⋊D311yescharacteristic
103C22×Dic3—38no—
104C22×Dic3—34no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—68no—
107C3×C4⋊C4—184no—
108C3×C22⋊C4—184no—
109C3×C22⋊C4—182no—
110(C2×D3)⋊C4—616no—
111(C2×D3)⋊C4—68no—
112Dic3⋊C4—616no—
113C22×C32#23518yes—
114C22×C32D3×D418yes—
115C22×C32D3×Q814yes—
116C22×C32#23914yes—
117C22×C32C2×#6714yes—
118C2×C3×D3—616no—
119C2×C3×D3—68no—
120C2×C3×D3—128no—
121C2×C3×D3—124no—
122C2×C3×D3—124no—
123C32×C4—28no—
124C32×C4—48no—
125C3×Dic3—616no—
126C3×Dic3—124no—
127C2×C4⋊C4—92no—
128C2×C4⋊C4#27211yescharacteristic, complemented
129C2×C22⋊C4—92no—
130C2×C22⋊C4—91no—
131#95—91no—
132H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
133C23×C3—34no—
134C23×C3—92no—
135C23×C3—182no—
136C23×C3—182no—
137C23×C3—181no—
138C23×C3—181no—
139C23×C3C22×C3⋊D311yescharacteristic
140C22×D3—38no—
141C22×D3—68no—
142C22×D3—64no—
143C22×D3—68no—
144C22×D3—64no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—38no—
150C2×C3×C4—68no—
151C2×C3×C4—68no—
152C2×C3×C4—68no—
153C2×C3×C4—68no—
154C2×C3×C4—94no—
155C2×C3×C4—182no—
156C2×C3×C4—182no—
157C2×C3×C4—182no—
158C2×C3×C4—181no—
159C2×C3×C4#63812yes—
160C2×Dic3—316no—
161C2×Dic3—68no—
162C2×Dic3—68no—
163C2×Dic3—68no—
164C2×Dic3—64no—
165C2×C32#89418yes—
166C2×C32#89618yes—
167C2×C32#92914yes—
168C2×C32#94014yes—
169C2×C32#94614yes—
170C3×D3—128no—
171C24—91no—
172C22×C4—92no—
173C22×C4—91no—
174C22×C4C2×#27212yes—
175C22×C4C2×#27211yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—184no—
178C22⋊C4—184no—
179C22⋊C4—182no—
180C22×C3—38no—
181C22×C3—38no—
182C22×C3—34no—
183C22×C3—34no—
184C22×C3—34no—
185C22×C3—184no—
186C22×C3—182no—
187C22×C3—362no—
188C22×C3—361no—
189C22×C3—361no—
190C22×C3#382812yes—
191C22×C3D4×C3⋊D312yes—
192C22×C3Q8×C3⋊D311yescharacteristic
193C22×C3#383211yescharacteristic
194C22×C3C2×#63811yescharacteristic
195C2×D3—616no—
196C2×D3—68no—
197C2×D3—128no—
198C2×D3—124no—
199C2×D3—124no—
200C3×C4—22no—
201C3×C4—42no—
202C3×C4—68no—
203C3×C4—128no—
204C3×C4—184no—
205C3×C4—361no—
206Dic3—616no—
207Dic3—124no—
208C32#499314yescomplemented
209C23—92no—
210C23—182no—
211C23—182no—
212C23—181no—
213C23—181no—
214C23C22×#27211yescharacteristic
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—94no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4—181no—
224C2×C4#623512yes—
225C2×C3—38no—
226C2×C3—38no—
227C2×C3—34no—
228C2×C3—34no—
229C2×C3—34no—
230C2×C3—362no—
231C2×C3#6410312yes—
232C2×C3#6410512yes—
233C2×C3#6413811yescharacteristic
234C2×C3#6414911yescharacteristic
235C2×C3#6415511yescharacteristic
236D3—128no—
237C22—184no—
238C22—182no—
239C22—362no—
240C22—361no—
241C22—361no—
242C22#8874012yes—
243C22D4×#27212yes—
244C22Q8×#27211yescharacteristic
245C22#8874811yescharacteristic
246C22C2×#623511yescharacteristic
247C4—22no—
248C4—42no—
249C4—184no—
250C4—361no—
251C3—34no—
252C3#1059269211yescharacteristic
253C2—362no—
254C2#1172176212yes—
255C2#1172176412yes—
256C2#1172182911yescharacteristic
257C2#1172185011yescharacteristic
258C2#1172186311yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×#272C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×#88540C2semidirect
5. #11720574C2semidirect
16. C2×C32×C4⋊C4D3semidirect
128. C2×C4⋊C4#272semidirect
132. H(3,𝔽3)#399semidirect
208. C32#4993semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
C2×#885402
#117205742
C3×#49933