Finite Group Databasebeta

← #13144413

Subgroups — #13144413

Order 1728.

561 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2#11721716C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic
4#11720578C211yesmaximal, characteristic
5#11720530C211yesmaximal, characteristic, complemented
6#11720529C211yesmaximal, characteristic, complemented
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#5009—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17#88540—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×C4×D4D314yes—
21C3×#870—34no—
22C3×#832—34no—
23C3×#832—34no—
24C3×#808—34no—
25C3×#807—34no—
26C3×C4⋊(C3⋊C8)—34no—
27C23×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)Q812yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—42no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084—21no—
37#6180—21no—
38#6180—21no—
39C3×#478—91no—
40#5009—34no—
41C2×C32×D4C2×D314yes—
42C22×C32×C4—24no—
43C32×C4⋊C4C2×D314yes—
44C32×C22⋊C4—24no—
45C32×C42C2×D314yes—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—34no—
48C3×Dic3⋊C4—64no—
49C3×Dic3⋊C4—64no—
50C2×C3×C3⋊C8—34no—
51C2×C3×C3⋊C8—34no—
52C22×H(3,𝔽3)—22no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)C2×Q811yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61#1084—41no—
62#1084—41no—
63C3×#125—91no—
64C3×C4×D4—34no—
65C3×C4×D4C3⋊D311yescharacteristic
66#870—34no—
67C3×#107—91no—
68C3×#106—91no—
69C3×C4⋊C8—91no—
70C3×#102—91no—
71C3×#102—91no—
72#832—34no—
73#832—34no—
74#808—34no—
75#807—34no—
76C4⋊(C3⋊C8)—34no—
77C23×C32—24no—
78C32×D4—24no—
79C32×D4Dic618yes—
80C2×C32×C4—24no—
81C2×C32×C4—48no—
82C2×C32×C4#6714yes—
83C2×C32×C4#6714yes—
84C2×C32×C4C22×D314yes—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×C3⋊C8—64no—
88C3×C3⋊C8—64no—
89#478—91noSylow
90C2×H(3,𝔽3)—42no—
91C2×H(3,𝔽3)C22⋊Q811yescharacteristic
92C2×H(3,𝔽3)#13511yescharacteristic
93C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
94C2×C3×D4—34no—
95C2×C3×D4C2×C3⋊D311yescharacteristic
96C22×C3×C4—21no—
97C22×C3×C4—64no—
98C2×C3×C8—91no—
99C2×C3×C8—91no—
100C3×C4⋊C4—34no—
101C3×C4⋊C4—91no—
102C3×C4⋊C4—91no—
103C3×C4⋊C4—181no—
104C3×C4⋊C4—181no—
105C3×C4⋊C4C2×C3⋊D311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—64no—
108C3×C42—34no—
109C3×C42C2×C3⋊D311yescharacteristic
110C4⋊Dic3—34no—
111C4⋊Dic3—34no—
112Dic3⋊C4—64no—
113Dic3⋊C4—64no—
114C2×C3⋊C8—34no—
115C2×C3⋊C8—34no—
116C22×C32—28no—
117C22×C32—44no—
118C22×C32—44no—
119C22×C32C2×#6714yes—
120C32×C4—24no—
121C32×C4—24no—
122C32×C4—44no—
123C32×C4C2×Dic614yes—
124C32×C4#23514yes—
125C3×Dic3—124no—
126C3×Dic3—124no—
127#125—91no—
128C4×D4#27211yescharacteristic
129#107—91no—
130#106—91no—
131C4⋊C8—91no—
132#102—91no—
133#102—91no—
134H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
135C23×C3—21no—
136C23×C3—64no—
137C3×D4—21no—
138C3×D4—38no—
139C3×D4—64no—
140C3×D4#63412yes—
141C2×C3×C4—21no—
142C2×C3×C4—34no—
143C2×C3×C4—34no—
144C2×C3×C4—34no—
145C2×C3×C4—42no—
146C2×C3×C4—64no—
147C2×C3×C4—128no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4#63811yescharacteristic
151C2×C3×C4#63811yescharacteristic
152C2×C3×C4C22×C3⋊D311yescharacteristic
153C2×Dic3—64no—
154C2×Dic3—64no—
155C3×C8—181no—
156C3×C8—181no—
157C3⋊C8—64no—
158C3⋊C8—64no—
159C2×C32—48no—
160C2×C32#92414yes—
161C2×C32#93214yes—
162C2×C32#95014yes—
163C2×D4C2×#27211yescharacteristic
164C22×C4—21no—
165C2×C8—91no—
166C2×C8—91no—
167C4⋊C4—91no—
168C4⋊C4—91no—
169C4⋊C4—181no—
170C4⋊C4—181no—
171C4⋊C4C2×#27211yescharacteristic
172C22⋊C4—21no—
173C42C2×#27211yescharacteristic
174C22×C3—22no—
175C22×C3—34no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—68no—
179C22×C3—124no—
180C22×C3—124no—
181C22×C3C2×#63811yescharacteristic
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—34no—
185C3×C4—34no—
186C3×C4—41no—
187C3×C4—64no—
188C3×C4—64no—
189C3×C4—124no—
190C3×C4—361no—
191C3×C4—361no—
192C3×C4C2×#63411yescharacteristic
193C3×C4#382811yescharacteristic
194Dic3—124no—
195Dic3—124no—
196C32#500914yescomplemented
197C23—21no—
198D4—21no—
199D4#622912yes—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C4#623511yescharacteristic
205C2×C4#623511yescharacteristic
206C2×C4C22×#27211yescharacteristic
207C8—181no—
208C8—181no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3—42no—
213C2×C3—128no—
214C2×C3#6413311yescharacteristic
215C2×C3#6414111yescharacteristic
216C2×C3#6415911yescharacteristic
217C22—22no—
218C22—41no—
219C22—41no—
220C22C2×#623511yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—361no—
225C4—361no—
226C4C2×#622911yescharacteristic
227C4#8874011yescharacteristic
228C3—34no—
229C3#1059270811yescharacteristic
230C2—42no—
231C2#1172182411yescharacteristic
232C2#1172184211yescharacteristic
233C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×D4×H(3,𝔽3)C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC3×C4×D4C3⋊D3
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. #11721716C2semidirect
5. #11720530C2semidirect
6. #11720529C2semidirect
7. C4⋊#6180C2semidirect
134. H(3,𝔽3)#478semidirect
196. C32#5009semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
#117217162
#117205782
#117205302
#117205292
C4⋊#61802
C3×#50093