Finite Group Databasebeta

← #13137366

Subgroups — #13137366

Order 1728.

383 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11721715C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic
4#11720326C214yesmaximal
5#10590291—31nomaximal
6#10590289—31nomaximal
7C2×#6229C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084—22no—
10C4⋊((C3⋊Dic3)⋊C3)C2212yes—
11#3750⋊C3—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#64079D311yescharacteristic
14C3×#870—31no—
15C3×#869—31no—
16C3×#869—31no—
17C3×#869—61no—
18#63630D311yescharacteristic
19C3×#803—31no—
20#63566—34no—
21#63565—34no—
22C2×C4×H(3,𝔽3)D412yes—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084—22no—
25#6229—24no—
26C2×(C3⋊Dic3)⋊C3—22no—
27(C3⋊(C3⋊C8))⋊C3—22no—
28#5082—91no—
29#5069—91no—
30#4703—91no—
31C2×C3×Dic6—32no—
32C2×C3×Dic6—32no—
33C2×C3×Dic6—62no—
34C32×C42—21no—
35C32×C42C2×D311yescharacteristic
36C32×C42C2×D311yescharacteristic
37C4⋊(C3⋊Dic3)C2×D312yes—
38#3750—22no—
39C2×C3⋊(C3⋊C8)C2×D312yes—
40C3×C4⋊Dic3—32no—
41C3×C4⋊Dic3—64no—
42C3×C4⋊Dic3—62no—
43C3×C4⋊Dic3—62no—
44C3×Dic3⋊C4—62no—
45C2×C3×C3⋊C8—32no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)—22no—
48C4×H(3,𝔽3)#4112yes—
49#1084—42no—
50(C3⋊Dic3)⋊C3—42no—
51C3×C4⋊Q8—91no—
52C3×#125—91no—
53#870—31no—
54#870—91no—
55#869—31no—
56#869—31no—
57#869—61no—
58C3×#97—91no—
59#835—94no—
60#816—94no—
61#810—94no—
62#803—31no—
63#803—91no—
64C2×C32×C4—22no—
65C2×C32×C4—21no—
66C2×C32×C4D1212yes—
67C2×C32×C4#6712yes—
68C2×C32×C4C22×D311yescharacteristic
69C2×C32×C4C22×D311yescharacteristic
70C2×C3⋊Dic3—22no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C3×Dic6—64no—
76C3×Dic6—64no—
77C3×Dic6—124no—
78C3⋊(C3⋊C8)—22no—
79C3×C3⋊C8—62no—
80#491—271noSylow
81C2×H(3,𝔽3)#12411yescharacteristic
82C2×H(3,𝔽3)#13712yes—
83C2×C3×Q8—92no—
84C2×Dic6—32no—
85C2×Dic6—32no—
86C2×Dic6—62no—
87C2×C3×C8—92no—
88C3×C4⋊C4—92no—
89C3×C4⋊C4—182no—
90C3×C4⋊C4—182no—
91C3×C42—31no—
92C3×C42—31no—
93C3×C42—61no—
94C3×C42D3211yescharacteristic
95C4⋊Dic3—32no—
96C4⋊Dic3—62no—
97C4⋊Dic3—62no—
98C4⋊Dic3—64no—
99C4⋊Dic3—92no—
100Dic3⋊C4—62no—
101Dic3⋊C4—182no—
102C2×C3⋊C8—32no—
103C2×C3⋊C8—92no—
104C22×C32—21no—
105C22×C32C2×D1211yescharacteristic
106C22×C32C2×#6711yescharacteristic
107C32×C4—24no—
108C32×C4—22no—
109C32×C4—22no—
110C32×C4—22no—
111C32×C4#23512yes—
112C32×C4#23912yes—
113C3⋊Dic3—42no—
114C3×Dic3—124no—
115C3×Dic3—122no—
116C3×Dic3—122no—
117C3×Dic3—122no—
118C4⋊Q8—91no—
119#125—91no—
120#103—274no—
121#97—91no—
122H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
123C3×Q8—184no—
124C2×C3×C4—32no—
125C2×C3×C4—32no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—62no—
129C2×C3×C4—61no—
130C2×C3×C4—182no—
131C2×C3×C4—182no—
132C2×C3×C4#62612yes—
133C2×C3×C4C2×D3211yescharacteristic
134C2×Dic3—62no—
135C2×Dic3—62no—
136C2×Dic3—64no—
137C2×Dic3—62no—
138C2×Dic3—182no—
139Dic6—64no—
140Dic6—64no—
141Dic6—124no—
142C3×C8—182no—
143C3⋊C8—62no—
144C3⋊C8—182no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32#87511yescharacteristic
148C2×C32#90912yes—
149C2×C32#94711yescharacteristic
150C2×C32#95112yes—
151C2×Q8—92no—
152C2×C8—92no—
153C4⋊C4—92no—
154C4⋊C4—182no—
155C4⋊C4—182no—
156C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
157C22×C3—31no—
158C22×C3—31no—
159C22×C3—61no—
160C22×C3C2×#62611yescharacteristic
161C3×C4—22no—
162C3×C4—32no—
163C3×C4—32no—
164C3×C4—62no—
165C3×C4—62no—
166C3×C4—64no—
167C3×C4—62no—
168C3×C4—362no—
169C3×C4—362no—
170C3×C4#379912yes—
171Dic3—122no—
172Dic3—122no—
173Dic3—124no—
174Dic3—122no—
175Dic3—362no—
176C32—21no—
177C32#470311yescharacteristic, complemented
178C32#506911yescharacteristic, complemented
179Q8—184no—
180C2×C4—182no—
181C2×C4—182no—
182C2×C4#620012yes—
183C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
184C8—182no—
185C2×C3—32no—
186C2×C3—32no—
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—62no—
190C2×C3—61no—
191C2×C3#6383112yes—
192C2×C3#6386411yescharacteristic
193C22C2×#620011yescharacteristic
194C4—22no—
195C4—362no—
196C4—362no—
197C4#8865212yes—
198C3—31no—
199C3—31no—
200C3—61no—
201C3#1059029511yescharacteristic
202C2#1172107412yes—
203C2#1172111411yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#491
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
122. H(3,𝔽3)#491semidirect
177. C32#4703semidirect
178. C32#5069semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64079⋊C32
#63630⋊C32
#117203262
#105902913
#105902893