Finite Group Databasebeta

← #13144483

Subgroups — #13144483

Order 1728.

627 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 125 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)×C22⋊Q8C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic, complemented
3#11721824C211yesmaximal, characteristic
4C22×#6180C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720532C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#5051—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15#88552—21no—
16#88550—22no—
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19C2×#6180—22no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×C22⋊Q8D314yes—
23C2×C3×#216—34no—
24C3×#924—34no—
25C22×C3×C3⋊C8—34no—
26C3×#835—34no—
27C3×#810—34no—
28C3×#807—34no—
29C23×H(3,𝔽3)D411yescharacteristic
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229—21no—
42#6180—21no—
43#6180D412yes—
44C3×#467—91no—
45#5051—34no—
46C2×C32×Q8C2×D314yes—
47C22×C32×C4C2×D314yes—
48C2×C3×Dic6—34no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4C2×D314yes—
51C32×C22⋊C4—24no—
52C3×#217—64no—
53C3×#216—68no—
54C3×C4⋊Dic3—34no—
55C3×Dic3⋊C4—64no—
56C2×C3×C3⋊C8—34no—
57C2×C3×C3⋊C8—34no—
58C2×C3×C3⋊C8—68no—
59C22×H(3,𝔽3)—21no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)Q1612yescomplemented
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—41no—
69#1084—41no—
70C2×C3×Q16—91no—
71C22×C3×C8—91no—
72C3×C22⋊Q8—34no—
73C3×C22⋊Q8—91no—
74C3×C22⋊Q8C3⋊D311yescharacteristic
75C2×#216—34no—
76#924—34no—
77C22×C3⋊C8—34no—
78C3×#107—91no—
79C3×#103—91no—
80C3×#103—91no—
81#835—34no—
82#810—34no—
83#807—34no—
84C23×C32#6714yes—
85C32×Q8—24no—
86C32×Q8—24no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4#6714yes—
92C2×C32×C4C22×D314yes—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—64no—
95C3×Dic6—64no—
96C3×Dic6—64no—
97C3×C3⋊C8—38no—
98C3×C3⋊C8—64no—
99#467—91noSylow
100C2×H(3,𝔽3)—22no—
101C2×H(3,𝔽3)#12111yescharacteristic
102C2×H(3,𝔽3)C2×Q1611yescharacteristic
103C2×H(3,𝔽3)#13511yescharacteristic
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C22×C3×C4—34no—
108C22×C3×C4C2×C3⋊D311yescharacteristic
109C2×Dic6—34no—
110C3×Q16—182no—
111C2×C3×C8—91no—
112C2×C3×C8—91no—
113C2×C3×C8—182no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—34no—
116C3×C4⋊C4—64no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4—181no—
119C3×C4⋊C4C2×C3⋊D311yescharacteristic
120C3×C22⋊C4—21no—
121C3×C22⋊C4—64no—
122C3×C22⋊C4—181no—
123#217—64no—
124#216—68no—
125C4⋊Dic3—34no—
126Dic3⋊C4—64no—
127C2×C3⋊C8—34no—
128C2×C3⋊C8—34no—
129C2×C3⋊C8—68no—
130C22×C32—24no—
131C22×C32—24no—
132C22×C32#21618yescomplemented
133C22×C32C2×#6714yes—
134C32×C4—24no—
135C32×C4—44no—
136C32×C4—44no—
137C32×C4D3×D414yes—
138C32×C4#23714yes—
139C3×Dic3—124no—
140C3×Dic3—124no—
141C2×Q16—91no—
142C22×C8—91no—
143C22⋊Q8—91no—
144C22⋊Q8#27211yescharacteristic
145#107—91no—
146#103—91no—
147#103—91no—
148H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
149C23×C3—34no—
150C23×C3#63811yescharacteristic
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8—181no—
156C3×Q8—181no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—64no—
164C2×C3×C4—64no—
165C2×C3×C4—64no—
166C2×C3×C4—64no—
167C2×C3×C4—181no—
168C2×C3×C4—181no—
169C2×C3×C4#63811yescharacteristic
170C2×C3×C4C22×C3⋊D311yescharacteristic
171C2×Dic3—64no—
172C2×Dic3—64no—
173Dic6—64no—
174Dic6—64no—
175C3×C8—92no—
176C3×C8—181no—
177C3⋊C8—38no—
178C3⋊C8—64no—
179C2×C32—28no—
180C2×C32#93914yes—
181C2×C32C2×#21614yes—
182C2×C32#95014yes—
183C2×Q8—91no—
184C2×Q8C2×#27211yescharacteristic
185C22×C4C2×#27211yescharacteristic
186Q16—182no—
187C2×C8—91no—
188C2×C8—91no—
189C2×C8—182no—
190C4⋊C4—21no—
191C4⋊C4—91no—
192C4⋊C4—181no—
193C4⋊C4C2×#27211yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—181no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—34no—
199C22×C3—38no—
200C22×C3—64no—
201C22×C3—64no—
202C22×C3#375612yes—
203C22×C3C2×#63811yescharacteristic
204C3×C4—21no—
205C3×C4—34no—
206C3×C4—34no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—64no—
210C3×C4—124no—
211C3×C4—124no—
212C3×C4—361no—
213C3×C4—361no—
214C3×C4D4×C3⋊D311yescharacteristic
215C3×C4#383011yescharacteristic
216Dic3—124no—
217Dic3—124no—
218C32#505114yescomplemented
219C23#623511yescharacteristic
220Q8—21no—
221Q8—21no—
222Q8—181no—
223Q8—181no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—181no—
229C2×C4—181no—
230C2×C4#623511yescharacteristic
231C2×C4C22×#27211yescharacteristic
232C8—92no—
233C8—181no—
234C2×C3—22no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—34no—
238C2×C3—68no—
239C2×C3#6414811yescharacteristic
240C2×C3C2×#375611yescharacteristic
241C2×C3#6415911yescharacteristic
242C22—21no—
243C22—21no—
244C22#8855012yescomplemented
245C22C2×#623511yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—361no—
250C4—361no—
251C4D4×#27211yescharacteristic
252C4#8874411yescharacteristic
253C3—34no—
254C3#1059275011yescharacteristic
255C2—22no—
256C2#1172184911yescharacteristic
257C2C2×#8855011yescharacteristic
258C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#467
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
5. #11720583C2semidirect
6. #11720532C2semidirect
7. #11720529C2semidirect
61. C22×H(3,𝔽3)Q16semidirect
132. C22×C32#216semidirect
148. H(3,𝔽3)#467semidirect
218. C32#5051semidirect
244. C22#88550semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×#885502
#117218242
C22×#61802
#117205832
#117205322
#117205292
C3×#50513