Finite Group Databasebeta

← #13135163

Subgroups — #13135163

Order 1728.

435 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 111 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⋊#1084C211yesmaximal, characteristic
2C2×#3748⋊C3C212yesmaximal
3#10589167—32nomaximal
4C22×C4×H(3,𝔽3)C2211yescharacteristic
5C22×#1084C412yes—
6C4⋊#1084—22no—
7#3748⋊C3—22no—
8#3748⋊C3C414yescomplemented
9C2×(C3⋊(C3⋊C8))⋊C3—22no—
10C2×#3748D312yes—
11C2×C3×C4⋊Dic3—32no—
12C2×C3×C4⋊Dic3—61no—
13C2×C3×#208—32no—
14C23×H(3,𝔽3)C2×C411yescharacteristic
15C2×C4×H(3,𝔽3)C2×C412yes—
16C2×C4×H(3,𝔽3)D411yescharacteristic
17C2×C4×H(3,𝔽3)D412yes—
18C2×C4×H(3,𝔽3)Q811yescharacteristic
19C2×#1084—22no—
20C2×#1084—42no—
21(C3⋊(C3⋊C8))⋊C3—24no—
22#4554—92no—
23#4530—91no—
24C22×C32×C4—21no—
25C22×C32×C4C2×D312yes—
26C22×C3×Dic3—34no—
27C22×C3×Dic3—62no—
28#3748—22no—
29#3748Dic314yescomplemented
30C2×C3⋊(C3⋊C8)—22no—
31C3×C4⋊Dic3—64no—
32C3×C4⋊Dic3—62no—
33C3×C4⋊Dic3—62no—
34C3×#208—34no—
35C3×#208—62no—
36C2×C3×C3⋊C8—62no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)—21no—
39C22×H(3,𝔽3)C4211yescharacteristic
40C22×H(3,𝔽3)C22⋊C411yescharacteristic
41C22×H(3,𝔽3)C4⋊C411yescharacteristic
42C4×H(3,𝔽3)C22⋊C412yes—
43C4×H(3,𝔽3)C4⋊C412yes—
44#1084—42no—
45C2×C3×M16—92no—
46C2×C3×C4⋊C4—91no—
47C2×C4⋊Dic3—32no—
48C2×C4⋊Dic3—61no—
49C2×#208—32no—
50C2×#208—92no—
51C23×C32—21no—
52C23×C32C4×D312yes—
53C2×C32×C4—22no—
54C2×C32×C4—21no—
55C2×C32×C4—22no—
56C2×C32×C4—21no—
57C2×C32×C4Dic612yes—
58C2×C32×C4C4×D312yes—
59C2×C32×C4D1212yes—
60C2×C32×C4C2×Dic312yes—
61C2×C32×C4#6712yes—
62C2×C32×C4#6712yes—
63C2×C3×Dic3—64no—
64C2×C3×Dic3—64no—
65C2×C3×Dic3—124no—
66C2×C3×Dic3—124no—
67C3⋊(C3⋊C8)—24no—
68C3×C3⋊C8—64no—
69#343—271noSylow
70C2×H(3,𝔽3)—21no—
71C2×H(3,𝔽3)—21no—
72C2×H(3,𝔽3)#9511yescharacteristic
73C2×H(3,𝔽3)#10011yescharacteristic
74C2×H(3,𝔽3)#10111yescharacteristic
75C22×C3×C4—32no—
76C22×C3×C4—61no—
77C22×C3×C4—92no—
78C22×C3×C4D3211yescharacteristic
79C22×Dic3—34no—
80C22×Dic3—62no—
81C3×M16—94no—
82C3×M16—182no—
83C2×C3×C8—182no—
84C3×C4⋊C4—182no—
85C4⋊Dic3—62no—
86C4⋊Dic3—62no—
87C4⋊Dic3—64no—
88#208—34no—
89#208—62no—
90#208—94no—
91#208—182no—
92C2×C3⋊C8—62no—
93C2×C3⋊C8—182no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—22no—
97C22×C32—22no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32C4×Dic312yes—
102C22×C32Dic3⋊C412yes—
103C22×C32(C2×D3)⋊C412yes—
104C32×C4—22no—
105C32×C4—22no—
106C32×C4Dic3⋊C412yes—
107C32×C4C4⋊Dic312yes—
108C32×C4(C2×D3)⋊C412yes—
109C32×C4#21712yes—
110C3×Dic3—124no—
111C3×Dic3—124no—
112C2×M16—92no—
113C2×C4⋊C4—91no—
114H(3,𝔽3)#34311yescharacteristic, complemented, Sylow
115C23×C3—32no—
116C23×C3—61no—
117C23×C3#62411yescharacteristic
118C2×C3×C4—32no—
119C2×C3×C4—32no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—32no—
124C2×C3×C4—61no—
125C2×C3×C4—62no—
126C2×C3×C4—61no—
127C2×C3×C4—62no—
128C2×C3×C4—182no—
129C2×C3×C4—362no—
130C2×C3×C4D3×Dic312yes—
131C2×C3×C4#62511yescharacteristic
132C2×C3×C4#62612yes—
133C2×C3×C4#62711yescharacteristic
134C2×Dic3—64no—
135C2×Dic3—64no—
136C2×Dic3—124no—
137C2×Dic3—124no—
138C3×C8—184no—
139C3⋊C8—64no—
140C3⋊C8—184no—
141C2×C32—22no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—22no—
145C2×C32—21no—
146C2×C32—21no—
147C2×C32—21no—
148C2×C32#82312yes—
149C2×C32#82412yes—
150C2×C32C3⋊#9512yes—
151C22×C4—92no—
152C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
153M16—94no—
154M16—182no—
155C2×C8—182no—
156C4⋊C4—182no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—32no—
160C22×C3—32no—
161C22×C3—32no—
162C22×C3—62no—
163C22×C3—62no—
164C22×C3—62no—
165C22×C3—62no—
166C22×C3—61no—
167C22×C3—61no—
168C22×C3—61no—
169C22×C3Dic3211yescharacteristic
170C22×C3#372211yescharacteristic
171C22×C3#372411yescharacteristic
172C3×C4—32no—
173C3×C4—32no—
174C3×C4—32no—
175C3×C4—32no—
176C3×C4—62no—
177C3×C4—62no—
178C3×C4—362no—
179C3×C4(C2×D3)⋊Dic312yes—
180C3×C4Dic3⋊Dic312yes—
181Dic3—124no—
182Dic3—124no—
183C32—21no—
184C32#455412yescomplemented
185C23#619911yescharacteristic
186C2×C4—182no—
187C2×C4—362no—
188C2×C4#619611yescharacteristic
189C2×C4((C3⋊D3)⋊C3)⋊C412yes—
190C2×C4#619811yescharacteristic
191C2×C4#620012yes—
192C8—184no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—32no—
196C2×C3—32no—
197C2×C3—32no—
198C2×C3—62no—
199C2×C3—62no—
200C2×C3—62no—
201C2×C3—62no—
202C2×C3—61no—
203C2×C3—61no—
204C2×C3—61no—
205C2×C3#6355911yescharacteristic
206C2×C3#6356111yescharacteristic
207C2×C3#6357911yescharacteristic
208C22—21no—
209C22—21no—
210C22#8844711yescharacteristic
211C22#8844911yescharacteristic
212C22#8845111yescharacteristic
213C4—362no—
214C4#8844812yes—
215C4#8845012yes—
216C3—32no—
217C3—61no—
218C3#1058916811yescharacteristic
219C2—21no—
220C2—21no—
221C2#1172032511yescharacteristic
222C2#1172032711yescharacteristic
223C2#1172033411yescharacteristic
subgroupG/H
centerC22#88451
commutatorC22×H(3,𝔽3)C42
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC22×C3#3722
prime radicalC23×H(3,𝔽3)C2×C4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#343
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. #3748⋊C3C4semidirect
29. #3748Dic3semidirect
114. H(3,𝔽3)#343semidirect
184. C32#4554semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3748⋊C32
#105891673