Finite Group Databasebeta

← #13154300

Subgroups — #13154300

Order 1728.

1056 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 205 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721863C212yesmaximal
3Q8×#1084C212yesmaximal, complemented
4#11721766C214yesmaximal
5#11721757C214yesmaximal, complemented
6#11721718C211yesmaximal, characteristic
7#11721716C211yesmaximal, characteristic, complemented
8C3×#5731—34nomaximal
9C2×Q8×H(3,𝔽3)C2212yes—
10C2×C4×#272C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2214yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#88542C2214yes—
14#88542C2212yes—
15C4⋊#1084C2214yes—
16C4⋊#1084C2212yes—
17#88540C2214yes—
18#88540C2212yes—
19#88540C2214yes—
20C4×#1084C2214yes—
21C4×#1084C2211yescharacteristic
22C32×C4⋊Q8D314yescomplemented
23C3×#946—38no—
24C3×Q8×Dic3—38no—
25C3×#898—316no—
26C3×#889—316no—
27C3×#872—34no—
28C3×#870—34no—
29C22×#272C2311yescharacteristic
30Q8×H(3,𝔽3)—24no—
31C2×C4×H(3,𝔽3)C2314yes—
32C2×C4×H(3,𝔽3)C2312yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084C2311yescharacteristic
35C2×#1084C2312yes—
36C2×#1084C2314yes—
37C4×#272—22no—
38C3×#541—91no—
39#5731—34no—
40C2×C32×Q8C2×D318yes—
41C2×C3×C4×D3—34no—
42C32×C4⋊C4C2×D3116yes—
43C32×C42C2×D314yes—
44C3×(C2×D3)⋊C4—316no—
45C3×(C2×D3)⋊C4—38no—
46C3×C4⋊Dic3—316no—
47C3×C4⋊Dic3—38no—
48C3×Dic3⋊C4—316no—
49C3×Dic3⋊C4—38no—
50C3×Dic3⋊C4—316no—
51C3×C4×Dic3—316no—
52C3×C4×Dic3—34no—
53C22×H(3,𝔽3)C2411yescharacteristic
54C2×#272—21no—
55C2×#272—22no—
56C4×H(3,𝔽3)—24no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)#4112yes—
59#1084—21no—
60#1084—22no—
61#1084—24no—
62C3×C4⋊Q8—34no—
63C3×C4⋊Q8C3⋊D311yescharacteristic
64C3×#126—94no—
65C3×#125—94no—
66C3×#125—91no—
67C3×C22⋊Q8—92no—
68C3×C4×Q8—92no—
69C3×#117—91no—
70#946—38no—
71Q8×Dic3—38no—
72#898—316no—
73#889—316no—
74#872—34no—
75#870—34no—
76C22×C3×D3—34no—
77C32×Q8—216no—
78C2×C32×C4C22×D3116yes—
79C2×C32×C4C22×D318yes—
80C2×C32×C4C22×D314yes—
81C2×C3×Dic3—38no—
82C2×C3×Dic3—316no—
83C2×C3×Dic3—34no—
84C3×C4×D3—68no—
85#541—91noSylow
86C2×H(3,𝔽3)C2×#4111yescharacteristic
87C2×H(3,𝔽3)2−1+412yes—
88#272—41no—
89C2×C3×Q8—38no—
90C2×C3×Q8C2×C3⋊D312yes—
91C22×C3×C4—91no—
92C2×C4×D3—34no—
93C3×C4⋊C4—316no—
94C3×C4⋊C4—92no—
95C3×C4⋊C4—94no—
96C3×C4⋊C4—94no—
97C3×C4⋊C4—92no—
98C3×C4⋊C4—94no—
99C3×C4⋊C4C2×C3⋊D314yes—
100C3×C22⋊C4—92no—
101C3×C22⋊C4—94no—
102C3×C42—34no—
103C3×C42—91no—
104C3×C42—94no—
105C3×C42C2×C3⋊D311yescharacteristic
106(C2×D3)⋊C4—38no—
107(C2×D3)⋊C4—316no—
108C4⋊Dic3—38no—
109C4⋊Dic3—316no—
110Dic3⋊C4—316no—
111Dic3⋊C4—38no—
112Dic3⋊C4—316no—
113C4×Dic3—34no—
114C4×Dic3—316no—
115C22×C32C23×D314yes—
116C2×C3×D3—64no—
117C2×C3×D3—68no—
118C32×C4—216no—
119C32×C4—28no—
120C32×C4#23718yes—
121C3×Dic3—64no—
122C3×Dic3—68no—
123C3×Dic3—616no—
124C4⋊Q8#27211yescharacteristic, complemented
125#126—94no—
126#125—94no—
127#125—91no—
128C22⋊Q8—92no—
129C4×Q8—92no—
130#117—91no—
131H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—34no—
134C3×Q8—24no—
135C3×Q8—616no—
136C2×C3×C4—34no—
137C2×C3×C4—316no—
138C2×C3×C4—38no—
139C2×C3×C4—92no—
140C2×C3×C4—94no—
141C2×C3×C4—91no—
142C2×C3×C4—182no—
143C2×C3×C4C22×C3⋊D314yes—
144C2×C3×C4C22×C3⋊D312yes—
145C2×C3×C4C22×C3⋊D311yescharacteristic
146C2×Dic3—38no—
147C2×Dic3—316no—
148C2×Dic3—34no—
149C4×D3—68no—
150C2×C32C2×#23714yes—
151C2×C32#100718yes—
152C3×D3—124no—
153C2×Q8C2×#27212yes—
154C22×C4—91no—
155C4⋊C4—92no—
156C4⋊C4—92no—
157C4⋊C4—94no—
158C4⋊C4—94no—
159C4⋊C4—94no—
160C4⋊C4C2×#27214yes—
161C22⋊C4—92no—
162C22⋊C4—94no—
163C42—91no—
164C42—94no—
165C42C2×#27211yescharacteristic
166C22×C3—34no—
167C22×C3—181no—
168C22×C3—182no—
169C22×C3C23×C3⋊D311yescharacteristic
170C2×D3—64no—
171C2×D3—68no—
172C3×C4—24no—
173C3×C4—22no—
174C3×C4—38no—
175C3×C4—616no—
176C3×C4—68no—
177C3×C4—181no—
178C3×C4—182no—
179C3×C4—184no—
180C3×C4#383012yes—
181Dic3—64no—
182Dic3—68no—
183Dic3—616no—
184C32#573114yescomplemented
185C23—91no—
186Q8—24no—
187C2×C4—92no—
188C2×C4—94no—
189C2×C4—91no—
190C2×C4—182no—
191C2×C4C22×#27214yes—
192C2×C4C22×#27212yes—
193C2×C4C22×#27211yescharacteristic
194C2×C3—34no—
195C2×C3—38no—
196C2×C3—361no—
197C2×C3C2×#383011yescharacteristic
198C2×C3#6436412yes—
199D3—124no—
200C22—181no—
201C22—182no—
202C22C23×#27211yescharacteristic
203C4—24no—
204C4—22no—
205C4—181no—
206C4—182no—
207C4—184no—
208C4#8874412yes—
209C3—34no—
210C3#1059620211yescharacteristic
211C2—361no—
212C2C2×#8874411yescharacteristic
213C2#1172267012yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×#272C23
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×#1084C2semidirect
5. #11721757C2semidirect
7. #11721716C2semidirect
22. C32×C4⋊Q8D3semidirect
124. C4⋊Q8#272semidirect
131. H(3,𝔽3)#541semidirect
184. C32#5731semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#117218632
Q8×#10842
#117217662
#117217572
#117217182
#117217162
C3×#57313