Finite Group Databasebeta

← #13144507

Subgroups — #13144507

Order 1728.

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

normal subgroups only

This group has 616 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)×#124C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88546C211yesmaximal, characteristic
4C4×#6229C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720578C211yesmaximal, characteristic
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#5054—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C22⋊C4×H(3,𝔽3)—21no—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88550—22no—
16#88546—22no—
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19C4×#1084—21no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×#124D314yes—
23C2×C3×#216—34no—
24C2×C3×#214—34no—
25C3×C4×Dic6—34no—
26C3×#835—34no—
27C3×#832—34no—
28C3×C4⋊(C3⋊C8)—34no—
29C23×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229D412yes—
42#6180—21no—
43#6180—21no—
44C3×#464—91no—
45#5054—34no—
46C2×C32×Q8C2×D314yes—
47C2×C32×D4C2×D314yes—
48C2×C3×Dic6—34no—
49C32×C22⋊C4—24no—
50C32×C22⋊C4—24no—
51C32×C42C2×D314yes—
52C3×#216—68no—
53C3×#214—68no—
54C3×C4⋊Dic3—34no—
55C3×Dic3⋊C4—64no—
56C3×C4×Dic3—64no—
57C2×C3×C3⋊C8—34no—
58C2×C3×C3⋊C8—34no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—22no—
69#1084—41no—
70C2×C3×Q16—91no—
71C2×C3×SD16—91no—
72C3×#124—34no—
73C3×#124C3⋊D311yescharacteristic
74C3×C4×Q8—91no—
75C2×#216—34no—
76C2×#214—34no—
77C4×Dic6—34no—
78C3×C4⋊C8—91no—
79C3×#103—91no—
80C3×#102—91no—
81#835—34no—
82#832—34no—
83C4⋊(C3⋊C8)—34no—
84C23×C32—24no—
85C32×Q8—24no—
86C32×Q8—24no—
87C32×D4—24no—
88C32×D4—24no—
89C2×C32×C4—24no—
90C2×C32×C4#6714yes—
91C2×C32×C4#6714yes—
92C2×C32×C4C22×D314yes—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—64no—
95C3×Dic6—38no—
96C3×Dic6—64no—
97C3×C3⋊C8—64no—
98C3×C3⋊C8—64no—
99#464—91noSylow
100C2×H(3,𝔽3)—81no—
101C2×H(3,𝔽3)#12111yescharacteristic
102C2×H(3,𝔽3)#13511yescharacteristic
103C2×H(3,𝔽3)#13711yescharacteristic
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×D4—34no—
108C2×C3×D4C2×C3⋊D311yescharacteristic
109C2×Dic6—34no—
110C3×Q16—182no—
111C3×SD16—182no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C4⋊C4—91no—
115C3×C4⋊C4—181no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C22⋊C4—64no—
119C3×C22⋊C4—64no—
120C3×C42—34no—
121C3×C42—181no—
122C3×C42C2×C3⋊D311yescharacteristic
123#216—68no—
124#214—68no—
125C4⋊Dic3—34no—
126Dic3⋊C4—64no—
127C4×Dic3—64no—
128C2×C3⋊C8—34no—
129C2×C3⋊C8—34no—
130C22×C32—44no—
131C22×C32—44no—
132C22×C32—44no—
133C22×C32C2×#6714yes—
134C32×C4—24no—
135C32×C4—24no—
136C32×C4—44no—
137C32×C4D3×D414yes—
138C32×C4#23714yes—
139C3×Dic3—68no—
140C3×Dic3—124no—
141C2×Q16—91no—
142C2×SD16—91no—
143#124#27211yescharacteristic
144C4×Q8—91no—
145C4⋊C8—91no—
146#103—91no—
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—21no—
150C23×C3—64no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8—92no—
156C3×Q8—181no—
157C3×D4—21no—
158C3×D4—21no—
159C3×D4—64no—
160C3×D4—64no—
161C2×C3×C4—21no—
162C2×C3×C4—34no—
163C2×C3×C4—34no—
164C2×C3×C4—34no—
165C2×C3×C4—64no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4#63811yescharacteristic
169C2×C3×C4#63811yescharacteristic
170C2×C3×C4C22×C3⋊D311yescharacteristic
171C2×Dic3—64no—
172C2×Dic3—64no—
173Dic6—38no—
174Dic6—64no—
175C3×C8—181no—
176C3×C8—181no—
177C3⋊C8—64no—
178C3⋊C8—64no—
179C2×C32—84no—
180C2×C32#93814yes—
181C2×C32#95014yes—
182C2×C32#95114yes—
183C2×Q8—91no—
184C2×Q8C2×#27211yescharacteristic
185C2×D4C2×#27211yescharacteristic
186Q16—182no—
187SD16—182no—
188C2×C8—91no—
189C2×C8—91no—
190C4⋊C4—91no—
191C4⋊C4—181no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C42—181no—
195C42C2×#27211yescharacteristic
196C22×C3—34no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—124no—
201C22×C3—124no—
202C22×C3—124no—
203C22×C3C2×#63811yescharacteristic
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—34no—
207C3×C4—34no—
208C3×C4—41no—
209C3×C4—64no—
210C3×C4—64no—
211C3×C4—124no—
212C3×C4—182no—
213C3×C4—361no—
214C3×C4D4×C3⋊D311yescharacteristic
215C3×C4#383011yescharacteristic
216Dic3—68no—
217Dic3—124no—
218C32#505414yescomplemented
219C23—21no—
220Q8—21no—
221Q8—21no—
222Q8—92no—
223Q8—181no—
224D4—21no—
225D4—21no—
226C2×C4—21no—
227C2×C4—181no—
228C2×C4—181no—
229C2×C4#623511yescharacteristic
230C2×C4#623511yescharacteristic
231C2×C4C22×#27211yescharacteristic
232C8—181no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—81no—
238C2×C3—244no—
239C2×C3#6414711yescharacteristic
240C2×C3#6415911yescharacteristic
241C2×C3#6416011yescharacteristic
242C22—41no—
243C22—41no—
244C22—41no—
245C22C2×#623511yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—182no—
250C4—361no—
251C4D4×#27211yescharacteristic
252C4#8874411yescharacteristic
253C3—34no—
254C3#1059275311yescharacteristic
255C2—81no—
256C2#1172184811yescharacteristic
257C2#1172187111yescharacteristic
258C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
4. C4×#6229C2semidirect
5. #11720583C2semidirect
7. C4⋊#6180C2semidirect
148. H(3,𝔽3)#464semidirect
218. C32#5054semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#885502
C2×#885462
C4×#62292
#117205832
#117205782
C4⋊#61802
C3×#50543