Finite Group Databasebeta

← #13146599

Subgroups — #13146599

Order 1728.

347 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 41 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×#88589C211yesmaximal, characteristic, complemented
2C2×#88588C211yesmaximal, characteristic
3C2×#88587C212yesmaximal, complemented
4#11721145C212yesmaximal, complemented
5C4×#6188C211yesmaximal, characteristic, complemented
6C2×#6198C2212yes—
7C2×#6196C2212yes—
8C2×#6188C2211yescharacteristic
9C2×#6188C2211yescharacteristic
10#88589—22no—
11#88588—22no—
12#88587—24no—
13C4×#1084C2211yescharacteristic
14#3752⋊C3—22no—
15#88450—22no—
16#63852—62no—
17C22×(C3⋊D3)⋊C3—22no—
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084D411yescharacteristic
20C2×#1084C2311yescharacteristic
21C2×(C3⋊Dic3)⋊C3—22no—
22#6198—22no—
23#6198—22no—
24#6196—22no—
25#6196—22no—
26#6188D412yes—
27#6188D412yes—
28#4696—91nomaximal
29C2×#627—62no—
30C2×#626—62no—
31#3752—22no—
32C3×(C2×D3)⋊C4—62no—
33C3×C4×Dic3—62no—
34#3722—62no—
35(C2×D3)⋊Dic3—62no—
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C2×(C3⋊D3)⋊C3—42no—
38C2×(C3⋊D3)⋊C3—42no—
39C2×(C3⋊D3)⋊C3—42no—
40C4×H(3,𝔽3)—21no—
41#1084—21no—
42#1084C2×D411yescharacteristic
43#1084C2×D411yescharacteristic
44(C3⋊Dic3)⋊C3—42no—
45C2×Dic12—91no—
46C2×D24—91no—
47C2×#204—92no—
48#885—182no—
49#875—92no—
50C3×C4×C8—91no—
51C22×C3⋊D3—22no—
52C22×C3×D3—62no—
53C2×C32×C4—22no—
54C2×C3⋊Dic3—22no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3—62no—
58#627—122no—
59#627—122no—
60#626—122no—
61#626—122no—
62#495—271noSylow
63C2×H(3,𝔽3)#12711yescharacteristic
64C2×H(3,𝔽3)#13511yescharacteristic
65C2×H(3,𝔽3)#13511yescharacteristic
66(C3⋊D3)⋊C3—82no—
67C2×#67—182no—
68C2×D12—92no—
69C2×Dic6—92no—
70C2×Dic6—182no—
71C2×C3×C8—91no—
72C2×C3×C8—91no—
73C3×C22⋊C4—182no—
74C3×C42—91no—
75#217—182no—
76(C2×D3)⋊C4—62no—
77(C2×D3)⋊C4—182no—
78(C2×D3)⋊C4—182no—
79(C2×D3)⋊C4—182no—
80C4×Dic3—62no—
81Dic12—182no—
82D24—182no—
83#204—184no—
84C22×C32—22no—
85C2×C3⋊D3—42no—
86C2×C3⋊D3—42no—
87C2×C3⋊D3—42no—
88C2×C3×D3—122no—
89C2×C3×D3—122no—
90C2×C3×D3—122no—
91C32×C4—42no—
92C3⋊Dic3—42no—
93C3×Dic3—62no—
94C3×Dic3—62no—
95C3×Dic3—122no—
96C3×Dic3—122no—
97C2×Q16—271no—
98C2×SD16—272no—
99C2×D8—271no—
100#124—272no—
101C4×C8—91no—
102H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
103C23×C3—182no—
104C22×D3—62no—
105C22×D3—182no—
106C2×C3×C4—62no—
107C2×C3×C4—91no—
108C2×C3×C4—91no—
109C2×C3×C4—182no—
110C2×C3×C4GO+(4,𝔽2)11yescharacteristic
111#67—362no—
112#67—362no—
113C2×Dic3—62no—
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—182no—
117D12—182no—
118D12—182no—
119Dic6—182no—
120Dic6—182no—
121Dic6—362no—
122Dic6—362no—
123C3×C8—92no—
124C3×C8—92no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—22no—
128C3⋊D3—82no—
129C3×D3—242no—
130C2×Q8—272no—
131C2×D4—272no—
132Q16—542no—
133SD16—544no—
134D8—542no—
135C2×C8—91no—
136C2×C8—91no—
137C22⋊C4—182no—
138C22⋊C4—542no—
139C42—91no—
140C22×C3—62no—
141C22×C3—362no—
142C22×C3—362no—
143C22×C3—362no—
144C22×C3C2×GO+(4,𝔽2)11yescharacteristic
145C2×D3—122no—
146C2×D3—122no—
147C2×D3—122no—
148C2×D3—362no—
149C2×D3—362no—
150C2×D3—362no—
151C3×C4—21no—
152C3×C4—91no—
153C3×C4—91no—
154C3×C4—122no—
155C3×C4—181no—
156C3×C4—362no—
157Dic3—62no—
158Dic3—62no—
159Dic3—122no—
160Dic3—122no—
161Dic3—362no—
162C32—22no—
163C23—182no—
164Q8—542no—
165Q8—542no—
166D4—542no—
167D4—542no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—182no—
171C2×C4#1088⋊C211yescharacteristic
172C8—92no—
173C8—92no—
174C2×C3—62no—
175C2×C3—62no—
176C2×C3—62no—
177C2×C3—722no—
178C2×C3#6422311yescharacteristic
179C2×C3#6423111yescharacteristic
180C2×C3#6423711yescharacteristic
181D3—242no—
182D3—722no—
183C22—362no—
184C22—362no—
185C22—362no—
186C22C2×#1088⋊C211yescharacteristic
187C4—21no—
188C4—91no—
189C4—91no—
190C4—181no—
191C4—362no—
192C3—62no—
193C3#1059342411yescharacteristic
194C2—722no—
195C2#1172229611yescharacteristic
196C2#1172230511yescharacteristic
197C2#1172231011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC2×#88588C2
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88589C2semidirect
3. C2×#88587C2semidirect
4. #11721145C2semidirect
5. C4×#6188C2semidirect
102. H(3,𝔽3)#495semidirect

Maximal subgroups

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

subgroupindex
C2×#885892
C2×#885882
C2×#885872
#117211452
C4×#61882
#46969