Finite Group Databasebeta

← #13144466

Subgroups — #13144466

Order 1728.

660 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 111 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)×#121C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic, complemented
3#11721824C211yesmaximal, characteristic, complemented
4C2×#88538C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720532C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#5042—34nomaximal
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15#88552—21no—
16#88546—22no—
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19#88538—22no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×#121D314yes—
23C2×C3×#214—34no—
24C3×#924—34no—
25C2×C3×#208—34no—
26C3×#832—34no—
27C3×#810—34no—
28C3×#807—34no—
29C23×H(3,𝔽3)—21no—
30C23×H(3,𝔽3)D411yescharacteristic
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—42no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41#6229—21no—
42#6229—21no—
43#6180—21no—
44#6180D412yescomplemented
45C3×#468—91no—
46#5042—34no—
47C2×C32×D4—24no—
48C2×C32×D4C2×D314yes—
49C22×C32×C4C2×D314yes—
50C2×C3×Dic6—34no—
51C32×C4⋊C4C2×D314yes—
52C32×C22⋊C4—24no—
53C3×#217—64no—
54C3×#214—68no—
55C3×C4⋊Dic3—34no—
56C3×Dic3⋊C4—64no—
57C3×#208—68no—
58C2×C3×C3⋊C8—34no—
59C2×C3×C3⋊C8—34no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—41no—
72#1084—41no—
73C2×C3×SD16—91no—
74C2×C3×M16—91no—
75C3×C22⋊Q8—91no—
76C3×#121—34no—
77C3×#121C3⋊D311yescharacteristic
78C2×#214—34no—
79#924—34no—
80C2×#208—34no—
81C3×#107—91no—
82C3×#103—91no—
83C3×#102—91no—
84#832—34no—
85#810—34no—
86#807—34no—
87C23×C32—24no—
88C23×C32#6714yes—
89C32×D4—24no—
90C32×D4—24no—
91C32×D4—48no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4#6714yes—
96C2×C32×C4C22×D314yes—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C3×Dic6—64no—
100C3×Dic6—64no—
101C3×C3⋊C8—38no—
102C3×C3⋊C8—64no—
103#468—91noSylow
104C2×H(3,𝔽3)—41no—
105C2×H(3,𝔽3)—81no—
106C2×H(3,𝔽3)#12111yescharacteristic
107C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
108C2×H(3,𝔽3)#13711yescharacteristic
109C2×C3×Q8—91no—
110C2×C3×D4—21no—
111C2×C3×D4—34no—
112C2×C3×D4—64no—
113C2×C3×D4C2×C3⋊D311yescharacteristic
114C22×C3×C4—34no—
115C22×C3×C4C2×C3⋊D311yescharacteristic
116C2×Dic6—34no—
117C3×SD16—182no—
118C3×M16—182no—
119C2×C3×C8—91no—
120C2×C3×C8—91no—
121C3×C4⋊C4—34no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—181no—
124C3×C4⋊C4C2×C3⋊D311yescharacteristic
125C3×C22⋊C4—21no—
126C3×C22⋊C4—64no—
127C3×C22⋊C4—181no—
128#217—64no—
129#214—68no—
130C4⋊Dic3—34no—
131Dic3⋊C4—64no—
132#208—68no—
133C2×C3⋊C8—34no—
134C2×C3⋊C8—34no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—24no—
138C22×C32—44no—
139C22×C32—44no—
140C22×C32—44no—
141C22×C32C2×#6714yes—
142C32×C4—24no—
143C32×C4—44no—
144C32×C4D3×D414yes—
145C32×C4#23714yes—
146C3×Dic3—124no—
147C3×Dic3—124no—
148C2×SD16—91no—
149C2×M16—91no—
150C22⋊Q8—91no—
151#121#27211yescharacteristic
152#107—91no—
153#103—91no—
154#102—91no—
155H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—34no—
158C23×C3—64no—
159C23×C3#63811yescharacteristic
160C3×Q8—181no—
161C3×Q8—181no—
162C3×D4—21no—
163C3×D4—21no—
164C3×D4—42no—
165C3×D4—64no—
166C3×D4—64no—
167C3×D4—128no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—34no—
172C2×C3×C4—34no—
173C2×C3×C4—64no—
174C2×C3×C4—64no—
175C2×C3×C4—64no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4#63811yescharacteristic
179C2×C3×C4C22×C3⋊D311yescharacteristic
180C2×Dic3—64no—
181C2×Dic3—64no—
182Dic6—64no—
183Dic6—64no—
184C3×C8—92no—
185C3×C8—181no—
186C3⋊C8—38no—
187C3⋊C8—64no—
188C2×C32—44no—
189C2×C32—84no—
190C2×C32#93214yes—
191C2×C32#93914yes—
192C2×C32#95114yes—
193C2×Q8—91no—
194C2×D4—21no—
195C2×D4C2×#27211yescharacteristic
196C22×C4C2×#27211yescharacteristic
197SD16—182no—
198M16—182no—
199C2×C8—91no—
200C2×C8—91no—
201C4⋊C4—91no—
202C4⋊C4—181no—
203C4⋊C4C2×#27211yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—181no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—34no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—64no—
214C22×C3—64no—
215C22×C3—64no—
216C22×C3—124no—
217C22×C3—124no—
218C22×C3—124no—
219C22×C3C2×#63811yescharacteristic
220C3×C4—21no—
221C3×C4—34no—
222C3×C4—34no—
223C3×C4—41no—
224C3×C4—64no—
225C3×C4—124no—
226C3×C4—361no—
227C3×C4—361no—
228C3×C4D4×C3⋊D311yescharacteristic
229C3×C4#383011yescharacteristic
230Dic3—124no—
231Dic3—124no—
232C32#504214yescomplemented
233C23—21no—
234C23#623511yescharacteristic
235Q8—181no—
236Q8—181no—
237D4—21no—
238D4—21no—
239D4—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4#623511yescharacteristic
246C2×C4C22×#27211yescharacteristic
247C8—92no—
248C8—181no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—41no—
253C2×C3—81no—
254C2×C3—124no—
255C2×C3—244no—
256C2×C3#6414111yescharacteristic
257C2×C3#6414811yescharacteristic
258C2×C3#6416011yescharacteristic
259C22—21no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22C2×#623511yescharacteristic
266C4—21no—
267C4—41no—
268C4—361no—
269C4—361no—
270C4D4×#27211yescharacteristic
271C4#8874411yescharacteristic
272C3—34no—
273C3#1059274111yescharacteristic
274C2—41no—
275C2—81no—
276C2#1172184211yescharacteristic
277C2#1172184911yescharacteristic
278C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#468
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. #11721824C2semidirect
4. C2×#88538C2semidirect
5. #11720578C2semidirect
6. #11720532C2semidirect
7. #11720529C2semidirect
44. #6180D4semidirect
155. H(3,𝔽3)#468semidirect
232. C32#5042semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1212
C2×#885462
#117218242
C2×#885382
#117205782
#117205322
#117205292
C3×#50423