Finite Group Databasebeta

← #13146606

Subgroups — #13146606

Order 1728.

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

normal subgroups only

This group has 319 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×#88626C211yesmaximal, characteristic
2C2×#88589C211yesmaximal, characteristic, complemented
3#11721096C212yesmaximal
4#11720704C211yesmaximal, characteristic, complemented
5#11720702C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C2×#6196C2212yes—
8C2×#6188C2211yescharacteristic
9C2×#6188C2211yescharacteristic
10#88626—22no—
11#88589—22no—
12#3750⋊C3—22no—
13#88451—22no—
14#88449C2212yes—
15#63851—62no—
16C22×#272D411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084C2311yescharacteristic
19C4×#272—21no—
20C4×#272—21no—
21C2×(C3⋊Dic3)⋊C3—22no—
22C2×(C3⋊Dic3)⋊C3—22no—
23#6196—22no—
24#6196—22no—
25#6188—21no—
26#6188D412yes—
27#5116—91nomaximal
28C2×C3×C4×D3—62no—
29C2×#627—62no—
30#3750—22no—
31C3×Dic3⋊C4—62no—
32Dic3⋊Dic3—62no—
33(C2×D3)⋊Dic3—62no—
34(C2×D3)⋊Dic3—62no—
35C22×H(3,𝔽3)C2×D411yescharacteristic
36C2×#272—21no—
37C2×#272—21no—
38C2×#272Q1612yescomplemented
39C4×H(3,𝔽3)—21no—
40#1084C2×D411yescharacteristic
41#1084#4111yescharacteristic
42(C3⋊Dic3)⋊C3—42no—
43(C3⋊Dic3)⋊C3—42no—
44C22×C3×C8—91no—
45#924—92no—
46C2×Dic12—91no—
47#896—182no—
48#818—91no—
49#816—92no—
50C22×C3×D3—62no—
51C2×C32×C4—22no—
52C2×C3⋊Dic3—22no—
53C2×C3⋊Dic3—22no—
54C2×C3×Dic3—62no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—62no—
57C3×C4×D3—122no—
58C3×C4×D3—122no—
59#627—122no—
60#627—122no—
61#467—271noSylow
62C2×H(3,𝔽3)#12111yescharacteristic
63C2×H(3,𝔽3)C2×Q1611yescharacteristic
64C2×H(3,𝔽3)#13511yescharacteristic
65#272—22no—
66C22×C3×C4—91no—
67C2×C4×D3—62no—
68C2×Dic6—92no—
69C2×Dic6—182no—
70C2×C3×C8—91no—
71C2×C3×C8—91no—
72C2×C3×C8—182no—
73C3×C4⋊C4—182no—
74#217—182no—
75(C2×D3)⋊C4—182no—
76(C2×D3)⋊C4—182no—
77C4⋊Dic3—92no—
78Dic3⋊C4—62no—
79Dic3⋊C4—182no—
80Dic3⋊C4—182no—
81Dic12—182no—
82C22×C32—22no—
83C2×C3×D3—64no—
84C2×C3×D3—122no—
85C2×C3×D3—122no—
86C32×C4—42no—
87C3⋊Dic3—42no—
88C3⋊Dic3—42no—
89C3×Dic3—62no—
90C3×Dic3—62no—
91C3×Dic3—122no—
92C3×Dic3—122no—
93C2×Q16—271no—
94C22×C8—91no—
95C22⋊Q8—272no—
96#107—271no—
97#103—272no—
98H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
99C23×C3—91no—
100C22×D3—62no—
101C2×C3×C4—62no—
102C2×C3×C4—91no—
103C2×C3×C4—181no—
104C2×C3×C4—181no—
105C2×C3×C4—182no—
106C2×C3×C4—182no—
107C2×C3×C4GO+(4,𝔽2)11yescharacteristic
108C2×Dic3—62no—
109C2×Dic3—62no—
110C2×Dic3—62no—
111C2×Dic3—182no—
112C2×Dic3—182no—
113C4×D3—122no—
114C4×D3—122no—
115Dic6—182no—
116Dic6—182no—
117Dic6—362no—
118Dic6—362no—
119C3×C8—92no—
120C3×C8—181no—
121C2×C32—22no—
122C2×C32—22no—
123C2×C32—22no—
124C3×D3—124no—
125C2×Q8—272no—
126C22×C4—91no—
127Q16—542no—
128C2×C8—91no—
129C2×C8—91no—
130C2×C8—182no—
131C4⋊C4—182no—
132C4⋊C4—272no—
133C22⋊C4—542no—
134C22×C3—62no—
135C22×C3—92no—
136C22×C3—181no—
137C22×C3—181no—
138C22×C3C2×GO+(4,𝔽2)11yescharacteristic
139C2×D3—64no—
140C2×D3—122no—
141C2×D3—122no—
142C3×C4—21no—
143C3×C4—91no—
144C3×C4—91no—
145C3×C4—122no—
146C3×C4—362no—
147C3×C4—362no—
148Dic3—62no—
149Dic3—62no—
150Dic3—122no—
151Dic3—122no—
152Dic3—362no—
153Dic3—362no—
154C32—22no—
155C23—91no—
156Q8—542no—
157Q8—542no—
158C2×C4—91no—
159C2×C4—181no—
160C2×C4—181no—
161C2×C4—182no—
162C2×C4—182no—
163C2×C4#1088⋊C211yescharacteristic
164C8—92no—
165C8—181no—
166C2×C3—62no—
167C2×C3—62no—
168C2×C3—62no—
169C2×C3—182no—
170C2×C3#6422311yescharacteristic
171C2×C3#6422811yescharacteristic
172C2×C3#6424211yescharacteristic
173D3—124no—
174C22—92no—
175C22—181no—
176C22—181no—
177C22C2×#1088⋊C211yescharacteristic
178C4—21no—
179C4—91no—
180C4—91no—
181C4—362no—
182C4—362no—
183C3—62no—
184C3#1059344611yescharacteristic
185C2—182no—
186C2#1172229611yescharacteristic
187C2#1172229911yescharacteristic
188C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC22×#272D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#467
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
2. C2×#88589C2semidirect
4. #11720704C2semidirect
5. #11720702C2semidirect
38. C2×#272Q16semidirect
98. H(3,𝔽3)#467semidirect

Maximal subgroups

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

subgroupindex
C2×#886262
C2×#885892
#117210962
#117207042
#117207022
#51169