Finite Group Databasebeta

← #13152058

Subgroups — #13152058

Order 1728.

979 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 979 conjugacy classes of subgroups — too many to draw. Only its 179 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×#88552C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3#11721309C214yesmaximal, complemented
4#11721299C218yesmaximal, complemented
5#10595521—32nomaximal
6C24×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2212yes—
8C22×(C3⋊Dic3)⋊C3C2214yes—
9C2×#6196C2214yescomplemented
10#88552C2214yes—
11#3757⋊C3C2214yes—
12#3757⋊C3C2214yes—
13#88449C2214yescomplemented
14#88448C2218yescomplemented
15C2×#3757D312yes—
16C2×C3×#217—32no—
17C2×C3×#217—32no—
18C2×C3×#217—61no—
19#63982—38no—
20#63964—38no—
21#63964—38no—
22C23×H(3,𝔽3)—42no—
23C23×H(3,𝔽3)Q814yescomplemented
24C23×H(3,𝔽3)C2312yes—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—42no—
27C2×#1084C2314yes—
28C2×(C3⋊Dic3)⋊C3—44no—
29C2×(C3⋊Dic3)⋊C3C2318yes—
30#6196—44no—
31#5483—91no—
32#5483—92no—
33C24×C32—21no—
34C24×C32C2×D312yes—
35C22×C3⋊Dic3C2×D314yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—62no—
39C2×#627—38no—
40#3757C2×D314yes—
41#3757C2×D314yes—
42C3×#217—34no—
43C3×#217—34no—
44C3×#217—34no—
45C3×#217—34no—
46C3×#217—64no—
47#3724—38no—
48Dic3⋊Dic3—38no—
49Dic3⋊Dic3—38no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—24no—
52C22×H(3,𝔽3)—44no—
53C22×H(3,𝔽3)C2×Q812yes—
54C22×H(3,𝔽3)C2×Q814yes—
55C22×H(3,𝔽3)C2411yescharacteristic
56#1084—24no—
57(C3⋊Dic3)⋊C3—28no—
58C2×C3×C22⋊C4—92no—
59C2×C3×C22⋊C4—91no—
60C2×#217—32no—
61C2×#217—32no—
62C2×#217—61no—
63C2×#217—92no—
64#924—94no—
65#924—98no—
66#878—98no—
67#878—98no—
68#878—98no—
69C23×C32—24no—
70C23×C32—21no—
71C23×C32—22no—
72C23×C32—44no—
73C23×C32—44no—
74C23×C32Dic618yescomplemented
75C23×C32C22×D312yes—
76C23×C32C22×D312yes—
77C23×C32C22×D312yes—
78C2×C3⋊Dic3—44no—
79C2×C3⋊Dic3C22×D318yes—
80C2×C3×Dic3—38no—
81C2×C3×Dic3—38no—
82C2×C3×Dic3—64no—
83C2×C3×Dic3—124no—
84C2×C3×Dic3—124no—
85C2×C3×Dic3—124no—
86#627—128no—
87#543—271noSylow
88C2×H(3,𝔽3)—22no—
89C2×H(3,𝔽3)—24no—
90C2×H(3,𝔽3)C22×Q811yescharacteristic
91C2×H(3,𝔽3)H(5,𝔽2)12yes—
92C24×C3—32no—
93C24×C3—61no—
94C24×C3D3211yescharacteristic
95C22×C3×C4—94no—
96C22×C3×C4—92no—
97C22×Dic3—34no—
98C22×Dic3—34no—
99C22×Dic3—62no—
100C22×Dic3—94no—
101C2×Dic6—94no—
102C2×Dic6—98no—
103C3×C22⋊C4—94no—
104C3×C22⋊C4—94no—
105C3×C22⋊C4—94no—
106#217—34no—
107#217—34no—
108#217—34no—
109#217—34no—
110#217—64no—
111#217—94no—
112#217—94no—
113C4⋊Dic3—94no—
114C4⋊Dic3—98no—
115Dic3⋊C4—98no—
116Dic3⋊C4—98no—
117Dic3⋊C4—98no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32—24no—
122C22×C32—21no—
123C22×C32—22no—
124C22×C32—24no—
125C22×C32—44no—
126C22×C32—48no—
127C22×C32—48no—
128C22×C32C2×Dic614yes—
129C22×C32C2×Dic614yes—
130C22×C32C2×Dic614yes—
131C22×C32C23×D312yes—
132C3⋊Dic3—28no—
133C3×Dic3—68no—
134C3×Dic3—68no—
135C3×Dic3—124no—
136C22⋊Q8—278no—
137C22⋊Q8—274no—
138C2×C22⋊C4—91no—
139C2×C22⋊C4—92no—
140H(3,𝔽3)#54311yescharacteristic, complemented, Sylow
141C23×C3—38no—
142C23×C3—32no—
143C23×C3—32no—
144C23×C3—32no—
145C23×C3—42no—
146C23×C3—64no—
147C23×C3—61no—
148C23×C3—62no—
149C23×C3—124no—
150C23×C3—124no—
151C23×C3#62714yes—
152C23×C3C2×D3211yescharacteristic
153C23×C3C2×D3212yes—
154C2×C3×C4—98no—
155C2×C3×C4—94no—
156C2×C3×C4—362no—
157C2×C3×C4—364no—
158C2×Dic3—38no—
159C2×Dic3—38no—
160C2×Dic3—64no—
161C2×Dic3—98no—
162C2×Dic3—124no—
163C2×Dic3—124no—
164C2×Dic3—124no—
165C2×Dic3—364no—
166Dic6—364no—
167Dic6—368no—
168C2×C32—24no—
169C2×C32—24no—
170C2×C32—24no—
171C2×C32—24no—
172C2×C32—21no—
173C2×C32—22no—
174C2×C32—44no—
175C2×C32C22×Dic612yes—
176C2×C32#100414yes—
177C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
178C2×Q8—274no—
179C22×C4—92no—
180C22×C4—94no—
181C4⋊C4—278no—
182C4⋊C4—274no—
183C22⋊C4—94no—
184C22⋊C4—94no—
185C22⋊C4—94no—
186C22×C3—24no—
187C22×C3—22no—
188C22×C3—32no—
189C22×C3—34no—
190C22×C3—34no—
191C22×C3—34no—
192C22×C3—44no—
193C22×C3—64no—
194C22×C3—64no—
195C22×C3—64no—
196C22×C3—64no—
197C22×C3—61no—
198C22×C3—62no—
199C22×C3—64no—
200C22×C3—124no—
201C22×C3—128no—
202C22×C3—128no—
203C22×C3C2×#62712yes—
204C22×C3C2×#62714yes—
205C22×C3C22×D3211yescharacteristic
206C3×C4—188no—
207C3×C4—184no—
208Dic3—68no—
209Dic3—68no—
210Dic3—124no—
211Dic3—188no—
212C32—21no—
213C32#548312yescomplemented
214C23—42no—
215C23#619614yescomplemented
216C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C23C2×((C3⋊D3)⋊C3)⋊C212yes—
218Q8—1084no—
219C2×C4—94no—
220C2×C4—98no—
221C2×C4—362no—
222C2×C4—364no—
223C2×C3—22no—
224C2×C3—24no—
225C2×C3—32no—
226C2×C3—34no—
227C2×C3—64no—
228C2×C3—64no—
229C2×C3—64no—
230C2×C3—64no—
231C2×C3—61no—
232C2×C3—62no—
233C2×C3—124no—
234C2×C3C22×#62711yescharacteristic
235C2×C3#6433012yes—
236C22—22no—
237C22—24no—
238C22—44no—
239C22C2×#619612yes—
240C22C2×#619614yes—
241C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C4—184no—
243C4—188no—
244C3—32no—
245C3—61no—
246C3#1059552111yescharacteristic
247C2—22no—
248C2—24no—
249C2C22×#619611yescharacteristic
250C2#1172254512yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#543
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
3. #11721309C2semidirect
4. #11721299C2semidirect
9. C2×#6196C22semidirect
13. #88449C22semidirect
14. #88448C22semidirect
23. C23×H(3,𝔽3)Q8semidirect
74. C23×C32Dic6semidirect
140. H(3,𝔽3)#543semidirect
213. C32#5483semidirect
215. C23#6196semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C2×#3757⋊C32
#117213092
#117212992
#105955213