Finite Group Databasebeta

← #13144825

Subgroups — #13144825

Order 1728.

1034 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88546C212yesmaximal, complemented
3#11721824C211yesmaximal, characteristic, complemented
4#11720578C212yesmaximal, complemented
5#11720573C211yesmaximal, characteristic, complemented
6C3×#5218—34nomaximal
7C24×H(3,𝔽3)—21no—
8C2×D4×H(3,𝔽3)—24no—
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6229C2211yescharacteristic, complemented
13#88552—21no—
14#88546—24no—
15C4⋊#1084C2211yescharacteristic, complemented
16#88540—21no—
17C2×#6180C2212yescomplemented
18C22×C32×D4D314yes—
19C2×C3×#214—38no—
20C3×#924—34no—
21C3×#832—38no—
22C3×#830—34no—
23C23×H(3,𝔽3)—24no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)—42no—
26C23×H(3,𝔽3)—42no—
27C23×H(3,𝔽3)D411yescharacteristic
28D4×H(3,𝔽3)—24no—
29D4×H(3,𝔽3)—22no—
30D4×H(3,𝔽3)D414yes—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36#6229—22no—
37#6180—22no—
38C3×#450—91no—
39#5218—34no—
40C24×C32—24no—
41C2×C32×D4—216no—
42C2×C32×D4—24no—
43C2×C32×D4C2×D318yes—
44C22×C32×C4C2×D314yes—
45C2×C3×Dic6—34no—
46C3×#217—64no—
47C3×#214—616no—
48C3×C4⋊Dic3—34no—
49C3×Dic3⋊C4—64no—
50C2×C3×C3⋊C8—38no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—21no—
53C22×H(3,𝔽3)—24no—
54C22×H(3,𝔽3)—48no—
55C22×H(3,𝔽3)—42no—
56C22×H(3,𝔽3)—42no—
57C22×H(3,𝔽3)SD1612yescomplemented
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)—21no—
60C4×H(3,𝔽3)C2×D412yes—
61#1084—41no—
62#1084—41no—
63C22×C3×D4—34no—
64C22×C3×D4C3⋊D311yescharacteristic
65C2×C3×SD16—92no—
66C3×C22⋊Q8—91no—
67C2×#214—38no—
68#924—34no—
69C3×#102—92no—
70C3×C22⋊C8—91no—
71#832—38no—
72#830—34no—
73C23×C32—216no—
74C23×C32—28no—
75C23×C32—48no—
76C23×C32—48no—
77C23×C32#6714yes—
78C32×D4—216no—
79C32×D4—28no—
80C32×D4#67116yes—
81C2×C32×C4—28no—
82C2×C32×C4#6714yes—
83C2×C32×C4C22×D314yes—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C3×Dic6—68no—
87C3×C3⋊C8—68no—
88#450—91noSylow
89C2×H(3,𝔽3)—22no—
90C2×H(3,𝔽3)—44no—
91C2×H(3,𝔽3)#12011yescharacteristic
92C2×H(3,𝔽3)C2×SD1611yescharacteristic
93C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
94C24×C3—21no—
95C24×C3—64no—
96C2×C3×Q8—91no—
97C2×C3×D4—24no—
98C2×C3×D4—21no—
99C2×C3×D4—38no—
100C2×C3×D4—64no—
101C2×C3×D4—616no—
102C2×C3×D4C2×C3⋊D312yes—
103C22×C3×C4—34no—
104C22×C3×C4C2×C3⋊D311yescharacteristic
105C2×Dic6—34no—
106C3×SD16—184no—
107C2×C3×C8—92no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—181no—
110C3×C22⋊C4—181no—
111#217—64no—
112#214—616no—
113C4⋊Dic3—34no—
114Dic3⋊C4—64no—
115C2×C3⋊C8—38no—
116C22×C32—24no—
117C22×C32—24no—
118C22×C32—216no—
119C22×C32—432no—
120C22×C32—48no—
121C22×C32—48no—
122C22×C32#21418yescomplemented
123C22×C32C2×#6714yes—
124C32×C4—24no—
125C32×C4C2×#6718yes—
126C3×Dic3—124no—
127C3×Dic3—124no—
128C22×D4#27211yescharacteristic
129C2×SD16—92no—
130C22⋊Q8—91no—
131#102—92no—
132C22⋊C8—91no—
133H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
134C23×C3—24no—
135C23×C3—22no—
136C23×C3—34no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3—68no—
140C23×C3—616no—
141C23×C3—128no—
142C23×C3—128no—
143C23×C3#63811yescharacteristic
144C3×Q8—182no—
145C3×D4—24no—
146C3×D4—22no—
147C3×D4—316no—
148C3×D4—68no—
149C3×D4—616no—
150C3×D4#63814yes—
151C2×C3×C4—22no—
152C2×C3×C4—34no—
153C2×C3×C4—34no—
154C2×C3×C4—68no—
155C2×C3×C4—181no—
156C2×C3×C4—181no—
157C2×C3×C4#63811yescharacteristic
158C2×C3×C4C22×C3⋊D311yescharacteristic
159C2×Dic3—64no—
160C2×Dic3—64no—
161Dic6—68no—
162C3×C8—182no—
163C3⋊C8—68no—
164C2×C32—28no—
165C2×C32—416no—
166C2×C32#93214yes—
167C2×C32C2×#21414yes—
168C2×C32#95314yes—
169C24—21no—
170C2×Q8—91no—
171C2×D4—24no—
172C2×D4—21no—
173C2×D4C2×#27212yes—
174C22×C4C2×#27211yescharacteristic
175SD16—184no—
176C2×C8—92no—
177C4⋊C4—91no—
178C4⋊C4—181no—
179C22⋊C4—181no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—24no—
183C22×C3—34no—
184C22×C3—38no—
185C22×C3—48no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—64no—
189C22×C3—64no—
190C22×C3—616no—
191C22×C3—128no—
192C22×C3—128no—
193C22×C3—1232no—
194C22×C3#375412yes—
195C22×C3C2×#63811yescharacteristic
196C3×C4—21no—
197C3×C4—38no—
198C3×C4—64no—
199C3×C4—361no—
200C3×C4—361no—
201C3×C4C2×#63812yes—
202Dic3—124no—
203Dic3—124no—
204C32#521814yescomplemented
205C23—24no—
206C23—22no—
207C23—42no—
208C23—42no—
209C23#623511yescharacteristic
210Q8—182no—
211D4—24no—
212D4—22no—
213D4#623514yes—
214C2×C4—22no—
215C2×C4—181no—
216C2×C4—181no—
217C2×C4#623511yescharacteristic
218C2×C4C22×#27211yescharacteristic
219C8—182no—
220C2×C3—22no—
221C2×C3—34no—
222C2×C3—34no—
223C2×C3—34no—
224C2×C3—44no—
225C2×C3—68no—
226C2×C3—1216no—
227C2×C3#6414111yescharacteristic
228C2×C3C2×#375411yescharacteristic
229C2×C3#6416211yescharacteristic
230C22—21no—
231C22—21no—
232C22—24no—
233C22—48no—
234C22—42no—
235C22—42no—
236C22#8854612yescomplemented
237C22C2×#623511yescharacteristic
238C4—21no—
239C4—361no—
240C4—361no—
241C4C2×#623512yes—
242C3—34no—
243C3#1059291711yescharacteristic
244C2—22no—
245C2—44no—
246C2#1172184211yescharacteristic
247C2C2×#8854611yescharacteristic
248C2#1172187611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalC22×D4×H(3,𝔽3)C2
hypercenterC22×C3×D4C3⋊D3
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. #11721824C2semidirect
4. #11720578C2semidirect
5. #11720573C2semidirect
12. C2×#6229C22semidirect
15. C4⋊#1084C22semidirect
17. C2×#6180C22semidirect
57. C22×H(3,𝔽3)SD16semidirect
122. C22×C32#214semidirect
133. H(3,𝔽3)#450semidirect
204. C32#5218semidirect
236. C22#88546semidirect

Maximal subgroups

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

subgroupindex
C22×D4×H(3,𝔽3)2
C2×#885462
#117218242
#117205782
#117205732
C3×#52183