Finite Group Databasebeta

← #13143861

Subgroups — #13143861

Order 1728.

627 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 123 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#11721830C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
4#11720554C212yesmaximal, complemented
5#11720551C212yesmaximal, complemented
6C3×#4736—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—21no—
9C22×#1084—21no—
10C2×C4⋊#272C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12#88542—21no—
13C4⋊#1084—21no—
14C4⋊#1084C2212yes—
15C4⋊#1084C2211yescharacteristic
16C3×#930—34no—
17C2×C3×C4⋊Dic3—34no—
18C32×C22⋊C8D314yescomplemented
19C3×#821—38no—
20C3×#818—38no—
21C23×H(3,𝔽3)D411yescharacteristic
22C22×#272—21no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26#6235—42no—
27C2×#1084—22no—
28C2×#1084—21no—
29C2×#1084—42no—
30C4⋊#272—22no—
31C8×H(3,𝔽3)—22no—
32C3×#480—91no—
33#4736—34no—
34C22×C32×C4C2×D314yes—
35C2×C3×#67—64no—
36C22×C3×Dic3—64no—
37C2×C3×D12—34no—
38C2×C32×C8C2×D318yes—
39C3×(C2×D3)⋊C4—64no—
40C3×C4⋊Dic3—38no—
41C3×C4⋊Dic3—34no—
42C3×C4⋊Dic3—64no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)—21no—
45C22×H(3,𝔽3)D812yes—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×#272—41no—
48C2×#272—41no—
49C2×#272—41no—
50C4×H(3,𝔽3)—21no—
51C4×H(3,𝔽3)#4112yes—
52#1084—42no—
53#1084—41no—
54C3×#121—91no—
55C2×C3×C4⋊C4—91no—
56#930—34no—
57C2×C4⋊Dic3—34no—
58C3×#107—92no—
59C3×#102—92no—
60C3×C22⋊C8—34no—
61C3×C22⋊C8C3⋊D311yescharacteristic
62#821—38no—
63#818—38no—
64C23×C32D1214yes—
65C22×C3×D3—64no—
66C2×C32×C4—28no—
67C2×C32×C4D1214yes—
68C2×C32×C4C22×D314yes—
69C3×#67—128no—
70C2×C3×Dic3—68no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—128no—
73C3×D12—68no—
74C32×C8—28no—
75#480—91noSylow
76C2×H(3,𝔽3)—22no—
77C2×H(3,𝔽3)#12311yescharacteristic
78C2×H(3,𝔽3)C2×D811yescharacteristic
79C2×H(3,𝔽3)#13711yescharacteristic
80#272—81no—
81C2×C3×D4—91no—
82C2×C3×D4—181no—
83C22×C3×C4—34no—
84C22×C3×C4—181no—
85C22×C3×C4C2×C3⋊D311yescharacteristic
86C2×#67—64no—
87C22×Dic3—64no—
88C2×D12—34no—
89C2×C3×C8—38no—
90C2×C3×C8C2×C3⋊D312yes—
91C3×C4⋊C4—92no—
92C3×C4⋊C4—91no—
93C3×C4⋊C4—181no—
94C3×C22⋊C4—181no—
95(C2×D3)⋊C4—64no—
96C4⋊Dic3—34no—
97C4⋊Dic3—38no—
98C4⋊Dic3—64no—
99C22×C32—24no—
100C22×C32—24no—
101C22×C32D2418yes—
102C22×C32C2×D1214yes—
103C2×C3×D3—124no—
104C2×C3×D3—124no—
105C2×C3×D3—124no—
106C32×C4—24no—
107C32×C4#23718yes—
108C3×Dic3—128no—
109C3×Dic3—124no—
110#121—91no—
111C2×C4⋊C4—91no—
112#107—92no—
113#102—92no—
114C22⋊C8#27211yescharacteristic, complemented
115H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
116C23×C3—34no—
117C23×C3—181no—
118C23×C3C4⋊(C3⋊D3)11yescharacteristic
119C22×D3—64no—
120C3×D4—182no—
121C3×D4—362no—
122C2×C3×C4—22no—
123C2×C3×C4—34no—
124C2×C3×C4—34no—
125C2×C3×C4—68no—
126C2×C3×C4—181no—
127C2×C3×C4—182no—
128C2×C3×C4—362no—
129C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
130C2×C3×C4C22×C3⋊D311yescharacteristic
131#67—128no—
132C2×Dic3—68no—
133C2×Dic3—64no—
134C2×Dic3—128no—
135D12—68no—
136C3×C8—22no—
137C3×C8—68no—
138C2×C32—28no—
139C2×C32#88614yes—
140C2×C32C2×D2414yes—
141C2×C32#90914yes—
142C3×D3—244no—
143C2×D4—91no—
144C2×D4—181no—
145C22×C4—181no—
146C22×C4C2×#27211yescharacteristic
147C2×C8C2×#27212yes—
148C4⋊C4—92no—
149C4⋊C4—91no—
150C4⋊C4—181no—
151C22⋊C4—181no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—64no—
157C22×C3—64no—
158C22×C3—361no—
159C22×C3—361no—
160C22×C3—361no—
161C22×C3#374512yes—
162C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
163C2×D3—124no—
164C2×D3—124no—
165C2×D3—124no—
166C3×C4—21no—
167C3×C4—38no—
168C3×C4—64no—
169C3×C4—362no—
170C3×C4—361no—
171C3×C4#383012yes—
172Dic3—128no—
173Dic3—124no—
174C32#473614yescomplemented
175C23—181no—
176C23C4⋊#27211yescharacteristic
177D4—182no—
178D4—362no—
179C2×C4—22no—
180C2×C4—181no—
181C2×C4—182no—
182C2×C4—362no—
183C2×C4C4⋊#27211yescharacteristic
184C2×C4C22×#27211yescharacteristic
185C8—22no—
186C2×C3—22no—
187C2×C3—34no—
188C2×C3—34no—
189C2×C3—34no—
190C2×C3—68no—
191C2×C3—721no—
192C2×C3#6409511yescharacteristic
193C2×C3C2×#374511yescharacteristic
194C2×C3#6411811yescharacteristic
195D3—244no—
196C22—21no—
197C22—21no—
198C22—361no—
199C22—361no—
200C22—361no—
201C22#8852912yes—
202C22C2×C4⋊#27211yescharacteristic
203C4—21no—
204C4—362no—
205C4—361no—
206C4#8874412yes—
207C3—34no—
208C3#1059243511yescharacteristic
209C2—22no—
210C2—721no—
211C2#1172174211yescharacteristic
212C2C2×#8852911yescharacteristic
213C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊C8C2
prime radical#11721830C2
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#480
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. H(3,𝔽3)×C22⋊C8C2semidirect
4. #11720554C2semidirect
5. #11720551C2semidirect
18. C32×C22⋊C8D3semidirect
114. C22⋊C8#272semidirect
115. H(3,𝔽3)#480semidirect
174. C32#4736semidirect

Maximal subgroups

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

subgroupindex
#117218302
C2×C4⋊#10842
H(3,𝔽3)×C22⋊C82
#117205542
#117205512
C3×#47363