Finite Group Databasebeta

← #13140523

Subgroups — #13140523

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721846C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#63659⋊C3C212yesmaximal
4#11720326C212yesmaximal, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10591478—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9#88552—22no—
10C4×#1084C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88448—22no—
14C3×#936—32no—
15C3×#936—61no—
16#63869—32no—
17#63659D312yes—
18C3×#832—32no—
19#63566—32no—
20#63565—32no—
21Dic3×C3⋊C8—32no—
22C23×H(3,𝔽3)—21no—
23D4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—21no—
26C2×#1084D412yes—
27#6229—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29(C3⋊(C3⋊C8))⋊C3—22no—
30#5056—91no—
31#4768—92no—
32C2×C32×D4—21no—
33C2×C32×D4C2×D312yes—
34C2×C3×Dic6—32no—
35C2×C3×Dic6—61no—
36C4⋊(C3⋊Dic3)C2×D312yes—
37C2×C3⋊(C3⋊C8)C2×D312yes—
38C3×#217—62no—
39C3×#217—62no—
40C3×#217—64no—
41C3×C4⋊Dic3—32no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—61no—
44C2×C3×C3⋊C8—32no—
45#3724—62no—
46Dic3⋊Dic3—62no—
47C22×H(3,𝔽3)—41no—
48C22×H(3,𝔽3)—42no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C4×H(3,𝔽3)#4112yes—
51#1084—22no—
52#1084—41no—
53(C3⋊Dic3)⋊C3—42no—
54C3×#124—91no—
55#936—32no—
56#936—61no—
57#890—92no—
58#870—91no—
59C3×#102—92no—
60#835—92no—
61#832—32no—
62#832—92no—
63#816—92no—
64C8×Dic3—92no—
65#810—92no—
66C4×C3⋊C8—91no—
67C23×C32—22no—
68C23×C32—22no—
69C32×D4—22no—
70C32×D4—22no—
71C32×D4—42no—
72C2×C32×C4—21no—
73C2×C32×C4C22×D312yes—
74C2×C3⋊Dic3—22no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—32no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×Dic6—62no—
82C3×Dic6—62no—
83C3×Dic6—122no—
84C3⋊(C3⋊C8)—22no—
85C3×C3⋊C8—62no—
86#488—271noSylow
87C2×H(3,𝔽3)—81no—
88C2×H(3,𝔽3)#12411yescharacteristic
89C2×H(3,𝔽3)#13512yes—
90C2×C3×Q8—91no—
91C2×C3×D4—32no—
92C2×C3×D4—61no—
93C2×C3×D4D3211yescharacteristic
94C2×Dic6—32no—
95C2×Dic6—61no—
96C2×C3×C8—92no—
97C3×C4⋊C4—92no—
98C3×C22⋊C4—182no—
99C3×C42—91no—
100#217—62no—
101#217—62no—
102#217—64no—
103C4⋊Dic3—32no—
104C4⋊Dic3—92no—
105C4⋊Dic3—182no—
106Dic3⋊C4—182no—
107Dic3⋊C4—182no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—92no—
112C22×C32—21no—
113C22×C32—42no—
114C22×C32—44no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—42no—
118C22×C32D3×D412yes—
119C32×C4—22no—
120C32×C4#23512yes—
121C32×C4#23712yes—
122C3⋊Dic3—42no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—122no—
126C3×Dic3—122no—
127C3×Dic3—122no—
128C3×Dic3—122no—
129#125—271no—
130#124—91no—
131#103—272no—
132#102—92no—
133C4×C8—271no—
134H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
135C23×C3—21no—
136C23×C3—62no—
137C23×C3—62no—
138C3×Q8—182no—
139C3×D4—22no—
140C3×D4—62no—
141C3×D4—62no—
142C3×D4—122no—
143C2×C3×C4—32no—
144C2×C3×C4—61no—
145C2×C3×C4—92no—
146C2×C3×C4—182no—
147C2×C3×C4—181no—
148C2×C3×C4C2×D3211yescharacteristic
149C2×Dic3—32no—
150C2×Dic3—32no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—182no—
156Dic6—62no—
157Dic6—62no—
158Dic6—122no—
159C3×C8—182no—
160C3⋊C8—62no—
161C3⋊C8—182no—
162C2×C32—22no—
163C2×C32—21no—
164C2×C32—82no—
165C2×C32—82no—
166C2×C32#88512yes—
167C2×C32#91212yes—
168C2×C32#91612yes—
169C2×Q8—91no—
170C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
171C2×C8—92no—
172C4⋊C4—92no—
173C4⋊C4—542no—
174C22⋊C4—182no—
175C42—91no—
176C22×C3—32no—
177C22×C3—42no—
178C22×C3—41no—
179C22×C3—61no—
180C22×C3—122no—
181C22×C3—122no—
182C22×C3—122no—
183C22×C3—122no—
184C22×C3—124no—
185C22×C3#381111yescharacteristic
186C3×C4—32no—
187C3×C4—32no—
188C3×C4—62no—
189C3×C4—182no—
190C3×C4—362no—
191C3×C4—361no—
192C3×C4#380412yes—
193Dic3—62no—
194Dic3—62no—
195Dic3—122no—
196Dic3—122no—
197Dic3—122no—
198Dic3—122no—
199Dic3—362no—
200C32—21no—
201C32#476812yescomplemented
202C23—21no—
203Q8—182no—
204D4—22no—
205C2×C4—92no—
206C2×C4—181no—
207C2×C4—182no—
208C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
209C8—182no—
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—32no—
213C2×C3—61no—
214C2×C3—62no—
215C2×C3—81no—
216C2×C3—242no—
217C2×C3—242no—
218C2×C3#6393612yes—
219C2×C3#6395311yescharacteristic
220C22—41no—
221C22—42no—
222C22#8867711yescharacteristic
223C4—182no—
224C4—361no—
225C4—362no—
226C4#8867112yes—
227C3—32no—
228C3—61no—
229C3#1059140311yescharacteristic
230C2—81no—
231C2#1172119212yes—
232C2#1172129411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721095C2semidirect
4. #11720326C2semidirect
5. #11720302C2semidirect
134. H(3,𝔽3)#488semidirect
201. C32#4768semidirect

Maximal subgroups

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

subgroupindex
#117218462
#117210952
#63659⋊C32
#117203262
#117203022
#105914783