Finite Group Databasebeta

← #13143848

Subgroups — #13143848

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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#11721826C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
4#11720551C211yesmaximal, characteristic, complemented
5#11720550C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720549C211yesmaximal, characteristic, complemented
8C3×#4723—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#88552—21no—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19C4×#1084—21no—
20C3×#926—34no—
21C3×#924—34no—
22C32×C22⋊C8D314yes—
23C3×#818—34no—
24C3×#817—34no—
25C3×#816—34no—
26C3×#816—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084—21no—
36#6229—21no—
37#6229—21no—
38C8×H(3,𝔽3)—21no—
39C8×H(3,𝔽3)—21no—
40C3×#485—91no—
41#4723—34no—
42C22×C32×C4C2×D314yes—
43C2×C3×Dic6—34no—
44C2×C32×C8C2×D314yes—
45C2×C32×C8C2×D314yes—
46C3×#217—64no—
47C3×#217—64no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—34no—
50C3×C4⋊Dic3—34no—
51C3×Dic3⋊C4—64no—
52C3×C4×Dic3—64no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)#4111yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—41no—
61#1084—41no—
62#1084—41no—
63#1084—41no—
64C3×C22⋊Q8—91no—
65C3×#117—91no—
66#926—34no—
67#924—34no—
68C3×#107—91no—
69C3×#106—91no—
70C3×#103—91no—
71C3×#103—91no—
72C3×C22⋊C8—34no—
73C3×C22⋊C8C3⋊D311yescharacteristic
74#818—34no—
75#817—34no—
76#816—34no—
77#816—34no—
78C23×C32D1214yes—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4D1214yes—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×Dic6—64no—
88C3×Dic6—64no—
89C32×C8—24no—
90C32×C8—24no—
91#485—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)#12311yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×H(3,𝔽3)#13711yescharacteristic
96C2×C3×Q8—91no—
97C22×C3×C4—34no—
98C22×C3×C4C2×C3⋊D311yescharacteristic
99C2×Dic6—34no—
100C2×C3×C8—34no—
101C2×C3×C8—34no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103C2×C3×C8C2×C3⋊D311yescharacteristic
104C3×C4⋊C4—91no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—91no—
107C3×C4⋊C4—181no—
108C3×C22⋊C4—181no—
109C3×C22⋊C4—181no—
110C3×C42—181no—
111#217—64no—
112#217—64no—
113C4⋊Dic3—34no—
114C4⋊Dic3—34no—
115C4⋊Dic3—34no—
116Dic3⋊C4—64no—
117C4×Dic3—64no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32C2×D1214yes—
122C32×C4—24no—
123C32×C4#23714yes—
124C32×C4#23714yes—
125C3×Dic3—124no—
126C3×Dic3—124no—
127C3×Dic3—124no—
128C3×Dic3—124no—
129C22⋊Q8—91no—
130#117—91no—
131#107—91no—
132#106—91no—
133#103—91no—
134#103—91no—
135C22⋊C8#27211yescharacteristic
136H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
137C23×C3—34no—
138C23×C3C4⋊(C3⋊D3)11yescharacteristic
139C3×Q8—181no—
140C3×Q8—181no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—34no—
144C2×C3×C4—34no—
145C2×C3×C4—64no—
146C2×C3×C4—64no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
152C2×C3×C4C22×C3⋊D311yescharacteristic
153C2×Dic3—64no—
154C2×Dic3—64no—
155C2×Dic3—64no—
156C2×Dic3—64no—
157Dic6—64no—
158Dic6—64no—
159C3×C8—21no—
160C3×C8—21no—
161C3×C8—64no—
162C3×C8—64no—
163C2×C32—44no—
164C2×C32#88614yes—
165C2×C32#90414yes—
166C2×C32#90914yes—
167C2×Q8—91no—
168C22×C4C2×#27211yescharacteristic
169C2×C8C2×#27211yescharacteristic
170C2×C8C2×#27211yescharacteristic
171C4⋊C4—91no—
172C4⋊C4—91no—
173C4⋊C4—91no—
174C4⋊C4—181no—
175C22⋊C4—181no—
176C22⋊C4—181no—
177C42—181no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—34no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
186C3×C4—21no—
187C3×C4—34no—
188C3×C4—34no—
189C3×C4—64no—
190C3×C4—361no—
191C3×C4—361no—
192C3×C4—361no—
193C3×C4—361no—
194C3×C4#383011yescharacteristic
195C3×C4#383011yescharacteristic
196Dic3—124no—
197Dic3—124no—
198Dic3—124no—
199Dic3—124no—
200C32#472314yescomplemented
201C23C4⋊#27211yescharacteristic
202Q8—181no—
203Q8—181no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—181no—
207C2×C4—181no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4C4⋊#27211yescharacteristic
211C2×C4C22×#27211yescharacteristic
212C8—21no—
213C8—21no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—41no—
218C2×C3—124no—
219C2×C3#6409511yescharacteristic
220C2×C3#6411311yescharacteristic
221C2×C3#6411811yescharacteristic
222C22—21no—
223C22—21no—
224C22—21no—
225C22C2×C4⋊#27211yescharacteristic
226C4—21no—
227C4—361no—
228C4—361no—
229C4—361no—
230C4—361no—
231C4#8874411yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059242211yescharacteristic
235C2—41no—
236C2#1172174211yescharacteristic
237C2#1172177911yescharacteristic
238C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊C8C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #11720551C2semidirect
5. #11720550C2semidirect
6. #11720549C2semidirect
7. #11720549C2semidirect
136. H(3,𝔽3)#485semidirect
200. C32#4723semidirect

Maximal subgroups

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

subgroupindex
#117218262
#117218242
H(3,𝔽3)×C22⋊C82
#117205512
#117205502
#117205492
C3×#47233