Finite Group Databasebeta

← #13144491

Subgroups — #13144491

Order 1728.

880 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 99 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#11722664C211yesmaximal, characteristic
2H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
3#11721829C212yesmaximal, complemented
4#11720580C211yesmaximal, characteristic, complemented
5#11720524C212yesmaximal, complemented
6C3×#5053—34nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
9C2×#6235—21no—
10C2×#6235C2212yes—
11#88744—22no—
12D4×#272—22no—
13#88740—21no—
14C4⋊C4×H(3,𝔽3)—21no—
15C22⋊C4×H(3,𝔽3)—21no—
16C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
17#88552C2212yescomplemented
18#88542—22no—
19#88540—22no—
20C3×#1004—34no—
21C32×#123D314yescomplemented
22C3×#929—38no—
23C3×#834—34no—
24C3×#806—38no—
25C23×H(3,𝔽3)D411yescharacteristic
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×#272—21no—
28C22×#272D412yes—
29D4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34#6235—22no—
35#6235—22no—
36#6235—22no—
37C2×#1084—21no—
38C2×#1084—22no—
39C2×#1084D412yes—
40C4⋊#272—21no—
41C4×#272—22no—
42#6229—21no—
43C3×#458—91no—
44#5053—34no—
45C2×C32×D4C2×D314yes—
46C22×C32×C4C2×D314yescomplemented
47C2×C3×#67—38no—
48C2×C3×#67—64no—
49C3×#237—68no—
50C3×D3×D4—68no—
51C3×#235—64no—
52C32×C4⋊C4—24no—
53C32×C22⋊C4—24no—
54C32×C22⋊C4C2×D314yescomplemented
55C3×#217—38no—
56C3×(C2×D3)⋊C4—68no—
57C3×Dic3⋊C4—68no—
58C22×H(3,𝔽3)—21no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)C2×D412yes—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×#272—22no—
64C2×#272—42no—
65C2×#272—81no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)—41no—
69#1084—22no—
70#1084—42no—
71C3×H(5,𝔽2)—91no—
72#1004—34no—
73C3×#123—34no—
74C3×#123—92no—
75C3×#123C3⋊D311yescharacteristic
76#929—38no—
77C3×#99—91no—
78C3×#99—92no—
79#834—34no—
80#806—38no—
81C23×C32#6714yes—
82C23×C32C22×D314yes—
83C22×C3×D3—38no—
84C22×C3×D3—64no—
85C32×D4—28no—
86C2×C32×C4—28no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4#6714yes—
90C3×#67—68no—
91C3×#67—68no—
92C3×#67—68no—
93C2×C3×Dic3—38no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—68no—
96C3×D12—64no—
97C3×C4×D3—68no—
98C3×Dic6—64no—
99#458—91noSylow
100C2×H(3,𝔽3)—22no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)#12011yescharacteristic
104#272—42no—
105C3×#41—182no—
106C3×#41—181no—
107C2×C3×D4—34no—
108C2×C3×D4—92no—
109C2×C3×D4—182no—
110C2×C3×D4—181no—
111C2×C3×D4C2×C3⋊D311yescharacteristic
112C22×C3×C4—34no—
113C22×C3×C4C2×C3⋊D311yescharacteristic
114C2×#67—38no—
115C2×#67—64no—
116#237—68no—
117D3×D4—68no—
118#235—64no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—64no—
121C3×C4⋊C4—182no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—34no—
124C3×C22⋊C4—64no—
125C3×C22⋊C4—92no—
126C3×C22⋊C4—182no—
127C3×C22⋊C4C2×C3⋊D311yescharacteristic
128#217—38no—
129(C2×D3)⋊C4—68no—
130Dic3⋊C4—68no—
131C22×C32—24no—
132C22×C32—44no—
133C22×C32—44no—
134C22×C32D3×D418yes—
135C22×C32C2×#6714yes—
136C2×C3×D3—68no—
137C2×C3×D3—128no—
138C2×C3×D3—244no—
139C32×C4—24no—
140C32×C4—44no—
141C32×C4—44no—
142C3×Dic3—68no—
143C3×Dic3—128no—
144H(5,𝔽2)—91no—
145#123—92no—
146#123#27211yescharacteristic, complemented
147#99—91no—
148#99—92no—
149H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
150C23×C3—34no—
151C23×C3—34no—
152C23×C3—92no—
153C23×C3—181no—
154C23×C3#63811yescharacteristic
155C23×C3C22×C3⋊D311yescharacteristic
156C22×D3—38no—
157C22×D3—64no—
158C3×Q8—181no—
159C3×D4—22no—
160C3×D4—68no—
161C3×D4—181no—
162C3×D4—182no—
163C3×D4—182no—
164C3×D4—182no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—34no—
169C2×C3×C4—68no—
170C2×C3×C4—64no—
171C2×C3×C4—64no—
172C2×C3×C4—92no—
173C2×C3×C4—182no—
174C2×C3×C4—181no—
175C2×C3×C4—182no—
176C2×C3×C4#63811yescharacteristic
177#67—68no—
178#67—68no—
179#67—68no—
180C2×Dic3—38no—
181C2×Dic3—64no—
182C2×Dic3—68no—
183D12—64no—
184C4×D3—68no—
185Dic6—64no—
186C2×C32—28no—
187C2×C32—24no—
188C2×C32—44no—
189C2×C32#93714yes—
190C3×D3—128no—
191#41—182no—
192#41—181no—
193C2×D4—92no—
194C2×D4—182no—
195C2×D4—181no—
196C2×D4C2×#27211yescharacteristic
197C22×C4C2×#27211yescharacteristic, complemented
198C4⋊C4—21no—
199C4⋊C4—182no—
200C22⋊C4—21no—
201C22⋊C4—92no—
202C22⋊C4—182no—
203C22⋊C4C2×#27211yescharacteristic, complemented
204C22×C3—21no—
205C22×C3—38no—
206C22×C3—34no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—64no—
210C22×C3—124no—
211C22×C3—124no—
212C22×C3—182no—
213C22×C3—362no—
214C22×C3—721no—
215C22×C3D4×C3⋊D312yes—
216C22×C3C2×#63811yescharacteristic
217C2×D3—68no—
218C2×D3—128no—
219C2×D3—244no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—64no—
224C3×C4—124no—
225C3×C4—124no—
226C3×C4—182no—
227C3×C4—362no—
228Dic3—68no—
229Dic3—128no—
230C32#505314yescomplemented
231C23—92no—
232C23—181no—
233C23#623511yescharacteristic
234C23C22×#27211yescharacteristic
235Q8—181no—
236D4—22no—
237D4—181no—
238D4—182no—
239D4—182no—
240D4—182no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—92no—
245C2×C4—182no—
246C2×C4—181no—
247C2×C4—182no—
248C2×C4#623511yescharacteristic
249C2×C3—21no—
250C2×C3—22no—
251C2×C3—34no—
252C2×C3—41no—
253C2×C3—68no—
254C2×C3—64no—
255C2×C3—124no—
256C2×C3—362no—
257C2×C3#6414611yescharacteristic
258D3—128no—
259C22—21no—
260C22—41no—
261C22—41no—
262C22—182no—
263C22—362no—
264C22—721no—
265C22D4×#27212yes—
266C22C2×#623511yescharacteristic
267C4—21no—
268C4—41no—
269C4—41no—
270C4—182no—
271C4—362no—
272C3—34no—
273C3#1059275211yescharacteristic
274C2—21no—
275C2—22no—
276C2—41no—
277C2—362no—
278C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C22×C3⋊D3
FittingH(3,𝔽3)×#123C2
prime radical#11722664C2
hypercenterC3×#123C3⋊D3
hypocenterH(3,𝔽3)#458
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#123C2semidirect
3. #11721829C2semidirect
4. #11720580C2semidirect
5. #11720524C2semidirect
8. C22×C4×H(3,𝔽3)C22semidirect
16. C22⋊C4×H(3,𝔽3)C22semidirect
17. #88552C22semidirect
21. C32×#123D3semidirect
46. C22×C32×C4C2×D3semidirect
54. C32×C22⋊C4C2×D3semidirect
146. #123#272semidirect
149. H(3,𝔽3)#458semidirect
197. C22×C4C2×#272semidirect
203. C22⋊C4C2×#272semidirect
230. C32#5053semidirect

Maximal subgroups

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

subgroupindex
#117226642
H(3,𝔽3)×#1232
#117218292
#117205802
#117205242
C3×#50533