Finite Group Databasebeta

← #13153890

Subgroups — #13153890

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)×#117C211yesmaximal, characteristic
2#11721824C212yesmaximal
3#11721757C214yesmaximal, complemented
4#11721735C214yesmaximal
5#11721716C211yesmaximal, characteristic, complemented
6#11721715C211yesmaximal, characteristic, complemented
7C4×#6229C212yesmaximal, complemented
8C3×#5518—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2212yes—
11C4⋊C4×H(3,𝔽3)C2212yes—
12C22⋊C4×H(3,𝔽3)C2212yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88552C2214yes—
16C4⋊#1084C2214yes—
17C4⋊#1084C2212yes—
18#88540C2214yes—
19#88540C2214yes—
20#88540C2214yes—
21C4×#1084C2214yes—
22C32×#117D314yes—
23C3×#924—38no—
24C3×#889—316no—
25C3×#879—316no—
26C3×#870—34no—
27C3×#869—34no—
28C3×C4×Dic6—38no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×C4×H(3,𝔽3)C2312yes—
33C2×C4×H(3,𝔽3)C2312yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084C2314yes—
36C2×#1084C2314yes—
37#6229—24no—
38C3×#541—91no—
39#5518—34no—
40C22×C32×C4C2×D314yes—
41C2×C3×Dic6—38no—
42C32×C4⋊C4C2×D318yes—
43C32×C22⋊C4C2×D318yes—
44C32×C42C2×D314yes—
45C32×C42C2×D314yes—
46C3×#217—316no—
47C3×C4⋊Dic3—316no—
48C3×C4⋊Dic3—38no—
49C3×Dic3⋊C4—316no—
50C3×Dic3⋊C4—316no—
51C3×Dic3⋊C4—316no—
52C3×C4×Dic3—316no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—22no—
55C22×H(3,𝔽3)C2411yescharacteristic
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)#4112yes—
60#1084—24no—
61#1084—24no—
62C3×C4⋊Q8—91no—
63C3×#126—94no—
64C3×#125—94no—
65C3×#125—91no—
66C3×C22⋊Q8—92no—
67C3×C4×Q8—92no—
68C3×#117—34no—
69C3×#117C3⋊D311yescharacteristic
70#924—38no—
71#889—316no—
72#879—316no—
73#870—34no—
74#869—34no—
75C4×Dic6—38no—
76C23×C32C22×D314yes—
77C2×C32×C4—28no—
78C2×C32×C4C22×D318yes—
79C2×C32×C4C22×D314yes—
80C2×C32×C4C22×D318yes—
81C2×C32×C4C22×D314yes—
82C2×C3×Dic3—316no—
83C2×C3×Dic3—316no—
84C3×Dic6—616no—
85#541—91noSylow
86C2×H(3,𝔽3)—41no—
87C2×H(3,𝔽3)C2×#4111yescharacteristic
88C2×H(3,𝔽3)2−1+412yes—
89C2×C3×Q8—92no—
90C22×C3×C4—34no—
91C22×C3×C4C2×C3⋊D311yescharacteristic
92C2×Dic6—38no—
93C3×C4⋊C4—38no—
94C3×C4⋊C4—94no—
95C3×C4⋊C4—94no—
96C3×C4⋊C4—94no—
97C3×C4⋊C4—92no—
98C3×C4⋊C4—94no—
99C3×C4⋊C4C2×C3⋊D312yes—
100C3×C22⋊C4—38no—
101C3×C22⋊C4—94no—
102C3×C22⋊C4C2×C3⋊D312yes—
103C3×C42—34no—
104C3×C42—34no—
105C3×C42—94no—
106C3×C42C2×C3⋊D311yescharacteristic
107C3×C42C2×C3⋊D311yescharacteristic
108#217—316no—
109C4⋊Dic3—38no—
110C4⋊Dic3—316no—
111Dic3⋊C4—316no—
112Dic3⋊C4—316no—
113Dic3⋊C4—316no—
114C4×Dic3—316no—
115C22×C32—24no—
116C22×C32—28no—
117C22×C32C23×D314yes—
118C32×C4—24no—
119C32×C4—28no—
120C32×C4—28no—
121C32×C4#23518yes—
122C3×Dic3—616no—
123C3×Dic3—616no—
124C4⋊Q8—91no—
125#126—94no—
126#125—94no—
127#125—91no—
128C22⋊Q8—92no—
129C4×Q8—92no—
130#117#27211yescharacteristic
131H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
132C23×C3—34no—
133C23×C3C22×C3⋊D311yescharacteristic
134C3×Q8—184no—
135C2×C3×C4—22no—
136C2×C3×C4—34no—
137C2×C3×C4—38no—
138C2×C3×C4—34no—
139C2×C3×C4—38no—
140C2×C3×C4—68no—
141C2×C3×C4—94no—
142C2×C3×C4—94no—
143C2×C3×C4C22×C3⋊D312yes—
144C2×C3×C4C22×C3⋊D312yes—
145C2×C3×C4C22×C3⋊D311yescharacteristic
146C2×C3×C4C22×C3⋊D311yescharacteristic
147C2×Dic3—316no—
148C2×Dic3—316no—
149Dic6—616no—
150C2×C32—44no—
151C2×C32C2×#23514yes—
152C2×C32#101018yes—
153C2×Q8—92no—
154C22×C4C2×#27211yescharacteristic
155C4⋊C4—92no—
156C4⋊C4—94no—
157C4⋊C4—94no—
158C4⋊C4—94no—
159C4⋊C4—94no—
160C4⋊C4C2×#27212yes—
161C22⋊C4—94no—
162C22⋊C4C2×#27212yes—
163C42—94no—
164C42C2×#27211yescharacteristic
165C42C2×#27211yescharacteristic
166C22×C3—21no—
167C22×C3—22no—
168C22×C3—34no—
169C22×C3—64no—
170C22×C3—68no—
171C22×C3C23×C3⋊D311yescharacteristic
172C3×C4—21no—
173C3×C4—22no—
174C3×C4—22no—
175C3×C4—38no—
176C3×C4—64no—
177C3×C4—68no—
178C3×C4—68no—
179C3×C4—184no—
180C3×C4—184no—
181C3×C4#382812yes—
182Dic3—616no—
183Dic3—616no—
184C32#551814yescomplemented
185C23C22×#27211yescharacteristic
186Q8—184no—
187C2×C4—22no—
188C2×C4—94no—
189C2×C4—94no—
190C2×C4C22×#27212yes—
191C2×C4C22×#27212yes—
192C2×C4C22×#27211yescharacteristic
193C2×C4C22×#27211yescharacteristic
194C2×C3—34no—
195C2×C3—38no—
196C2×C3—41no—
197C2×C3—124no—
198C2×C3C2×#382811yescharacteristic
199C2×C3#6436712yes—
200C22—21no—
201C22—22no—
202C22C23×#27211yescharacteristic
203C4—21no—
204C4—22no—
205C4—22no—
206C4—184no—
207C4—184no—
208C4#8874012yes—
209C3—34no—
210C3#1059598911yescharacteristic
211C2—41no—
212C2C2×#8874011yescharacteristic
213C2#1172267612yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
prime radicalC23×H(3,𝔽3)C23
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. #11721757C2semidirect
5. #11721716C2semidirect
6. #11721715C2semidirect
7. C4×#6229C2semidirect
131. H(3,𝔽3)#541semidirect
184. C32#5518semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#117218242
#117217572
#117217352
#117217162
#117217152
C4×#62292
C3×#55183