Finite Group Databasebeta

← #13147061

Subgroups — #13147061

Order 1728.

418 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 62 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×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720738C212yesmaximal, complemented
3C3×#5420—33nomaximal
4C3×#5225—31nomaximal
5C2×Q8⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#88552—41nomaximal
8C22×C32×Q8D311yescharacteristic
9C22×C3×SL(2,𝔽3)D313yes—
10C3×#926—31no—
11C2×C3×#208—31no—
12C3×#859—36no—
13C3×#835—32no—
14C3×#835—32no—
15C23×H(3,𝔽3)—41no—
16C2×#1084—42no—
17Q8⋊H(3,𝔽3)C2×C411yescharacteristic
18#5420—33no—
19C3×#419—91no—
20#5225—31no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8Dic312yescomplemented
24C2×C32×Q8C2×D311yescharacteristic
25C22×C32×C4—31no—
26C2×C3×SL(2,𝔽3)Dic316yescomplemented
27C2×C3×SL(2,𝔽3)C2×D313yes—
28C3×#217—61no—
29C3×#217—123no—
30C3×C4⋊Dic3—32no—
31C3×C4×Dic3—61no—
32C3×#208—62no—
33C2×C3×C3⋊C8—32no—
34C22×H(3,𝔽3)—41no—
35C22×H(3,𝔽3)—42no—
36C22×H(3,𝔽3)—82no—
37#1084—82no—
38C22×C3×Q8—31no—
39C22×C3×Q8C3⋊D311yescharacteristic
40C22×SL(2,𝔽3)—33no—
41C2×C3×M16—91no—
42C3×#117—91no—
43#926—31no—
44C2×#208—31no—
45#859—36no—
46C3×#103—92no—
47C3×#103—92no—
48#835—32no—
49#835—32no—
50C23×C32—43no—
51C23×C32S411yescharacteristic
52C32×Q8—32no—
53C32×Q8—31no—
54C32×Q8—61no—
55C32×Q8C2×Dic311yescharacteristic
56C2×C32×C4—32no—
57C2×C32×C4—32no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—126no—
62C3×SL(2,𝔽3)C2×Dic313yes—
63C3×C3⋊C8—62no—
64#419—91noSylow
65C2×H(3,𝔽3)—42no—
66C2×H(3,𝔽3)—41no—
67C2×H(3,𝔽3)—82no—
68C2×C3×Q8—31no—
69C2×C3×Q8—32no—
70C2×C3×Q8—31no—
71C2×C3×Q8—32no—
72C2×C3×Q8—32no—
73C2×C3×Q8—31no—
74C2×C3×Q8—61no—
75C2×C3×Q8—62no—
76C2×C3×Q8C3⋊Dic312yes—
77C2×C3×Q8C2×C3⋊D311yescharacteristic
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—61no—
81C2×SL(2,𝔽3)—36no—
82C2×SL(2,𝔽3)—33no—
83C3×M16—182no—
84C2×C3×C8—92no—
85C3×C4⋊C4—92no—
86C3×C22⋊C4—181no—
87C3×C42—181no—
88#217—61no—
89#217—123no—
90C4⋊Dic3—32no—
91C4×Dic3—61no—
92#208—62no—
93C2×C3⋊C8—32no—
94C22×C32—22no—
95C22×C32—43no—
96C22×C32—46no—
97C22×C32—86no—
98C22×C32SL(2,ℤ/4)12yes—
99C22×C32C2×S411yescharacteristic
100C32×C4—31no—
101C32×C4—31no—
102C32×C4—32no—
103C3×Dic3—122no—
104C3×Dic3—246no—
105C22×Q8#27211yescharacteristic
106C2×M16—91no—
107#117—91no—
108#103—92no—
109#103—92no—
110H(3,𝔽3)—41noSylow
111C23×C3—31no—
112C23×C3—123no—
113C23×C3C3⋊S411yescharacteristic
114C3×Q8—32no—
115C3×Q8—31no—
116C3×Q8—32no—
117C3×Q8—31no—
118C3×Q8—31no—
119C3×Q8—62no—
120C3×Q8—61no—
121C3×Q8—61no—
122C3×Q8—61no—
123C3×Q8—62no—
124C3×Q8C2×C3⋊Dic311yescharacteristic
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—31no—
128C2×C3×C4—32no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—32no—
132C2×C3×C4—32no—
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—61no—
136C2×C3×C4—61no—
137C2×C3×C4—182no—
138C2×Dic3—62no—
139C2×Dic3—126no—
140SL(2,𝔽3)—33no—
141C3×C8—182no—
142C3⋊C8—62no—
143C2×C32—22no—
144C2×C32—46no—
145C2×C32—43no—
146C2×C32—86no—
147C2×C32#98312yes—
148C2×C32C2×SL(2,ℤ/4)11yescharacteristic
149C2×Q8—31no—
150C2×Q8—32no—
151C2×Q8#108412yescomplemented
152C2×Q8C2×#27211yescharacteristic
153C22×C4—31no—
154M16—182no—
155C2×C8—92no—
156C4⋊C4—92no—
157C22⋊C4—181no—
158C42—181no—
159C22×C3—22no—
160C22×C3—32no—
161C22×C3—31no—
162C22×C3—62no—
163C22×C3—126no—
164C22×C3—123no—
165C22×C3—246no—
166C22×C3C3⋊SL(2,ℤ/4)12yes—
167C22×C3C2×C3⋊S411yescharacteristic
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—32no—
171C3×C4—31no—
172C3×C4—31no—
173C3×C4—32no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—62no—
177C3×C4—362no—
178Dic3—122no—
179Dic3—246no—
180C32—43no—
181C32#542011yescharacteristic, complemented
182C23#625811yescharacteristic
183Q8—32no—
184Q8—31no—
185Q8—61no—
186Q8C2×#108411yescharacteristic
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—182no—
192C8—182no—
193C2×C3—22no—
194C2×C3—32no—
195C2×C3—31no—
196C2×C3—62no—
197C2×C3—126no—
198C2×C3—123no—
199C2×C3—246no—
200C2×C3#6426412yes—
201C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
202C22—22no—
203C22#8861312yes—
204C22C2×#625811yescharacteristic
205C4—31no—
206C4—31no—
207C4—32no—
208C4—362no—
209C3—31no—
210C3—123no—
211C3#1059365411yescharacteristic
212C2—22no—
213C2#1172238812yes—
214C2C2×#8861311yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FittingC22×C32×Q8D3
prime radicalC22×Q8⋊H(3,𝔽3)C2
cosocleC2×Q8⋊H(3,𝔽3)C22
hypercenterC23×C3C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. #11720738C2semidirect
5. C2×Q8⋊H(3,𝔽3)C4semidirect
23. C2×C32×Q8Dic3semidirect
26. C2×C3×SL(2,𝔽3)Dic3semidirect
151. C2×Q8#1084semidirect
181. C32#5420semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
#117207382
C3×#54203
C3×#52253
#885524