Finite Group Databasebeta

← #13146542

Subgroups — #13146542

Order 1728.

329 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 37 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#11721144C211yesmaximal, characteristic
2#11721098C211yesmaximal, characteristic, complemented
3#11720783C211yesmaximal, characteristic, complemented
4#11720705C212yesmaximal, complemented
5#11720705C212yesmaximal, complemented
6C2×C4⋊((C3⋊D3)⋊C3)—21no—
7C2×#6198—21no—
8C2×#6198C2212yes—
9C2×#6188C2212yes—
10C4×#1084C2211yescharacteristic
11#3750⋊C3—21no—
12#88449C2211yescharacteristic
13#88449C2211yescharacteristic
14#88448—21no—
15#63911—61no—
16#63867—61no—
17C22×(C3⋊D3)⋊C3—22no—
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084D411yescharacteristic
20C2×#1084C2311yescharacteristic
21C2×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊Dic3)⋊C3—21no—
23C4⋊((C3⋊D3)⋊C3)—42no—
24#6198—24no—
25#6198—42no—
26#6188—22no—
27#4713—91nomaximal
28C2×C4⋊(C3⋊D3)—21no—
29C2×C3×D12—61no—
30C2×#626—62no—
31C2×#626—62no—
32#3750—21no—
33C3×Dic3⋊C4—61no—
34C3×C4×Dic3—61no—
35C3×C4×Dic3—61no—
36#3724—61no—
37Dic3⋊Dic3—61no—
38Dic3⋊Dic3—61no—
39Dic3⋊Dic3—61no—
40C22×H(3,𝔽3)C2×D411yescharacteristic
41C2×(C3⋊D3)⋊C3—42no—
42C2×(C3⋊D3)⋊C3—42no—
43C2×(C3⋊D3)⋊C3—42no—
44C4×H(3,𝔽3)—21no—
45#1084—21no—
46#1084#4112yes—
47(C3⋊Dic3)⋊C3—41no—
48(C3⋊Dic3)⋊C3—41no—
49#940—181no—
50#889—181no—
51#874—91no—
52#870—91no—
53C3×#97—91no—
54#821—92no—
55#821—92no—
56C22×C3⋊D3—22no—
57C22×C3×D3—62no—
58C2×C32×C4—21no—
59C2×C32×C4—21no—
60C2×C3⋊Dic3—21no—
61C2×C3⋊Dic3—21no—
62C4⋊(C3⋊D3)—42no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C3×D12—122no—
70#626—124no—
71#626—124no—
72#490—271noSylow
73C2×H(3,𝔽3)#12411yescharacteristic
74C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
75C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
76(C3⋊D3)⋊C3—82no—
77C2×C3×D4—181no—
78C2×#67—182no—
79C2×#67—182no—
80C2×D12—61no—
81C2×D12—92no—
82C2×D12—181no—
83C2×D12—181no—
84C2×C3×C8—92no—
85C3×C4⋊C4—181no—
86C3×C42—91no—
87C4⋊Dic3—91no—
88C4⋊Dic3—91no—
89C4⋊Dic3—181no—
90Dic3⋊C4—61no—
91Dic3⋊C4—181no—
92Dic3⋊C4—181no—
93Dic3⋊C4—181no—
94Dic3⋊C4—181no—
95Dic3⋊C4—181no—
96C4×Dic3—61no—
97C4×Dic3—61no—
98C22×C32—21no—
99C22×C32—21no—
100C2×C3⋊D3—42no—
101C2×C3⋊D3—42no—
102C2×C3⋊D3—42no—
103C2×C3×D3—122no—
104C2×C3×D3—122no—
105C2×C3×D3—122no—
106C32×C4—22no—
107C32×C4—41no—
108C3⋊Dic3—41no—
109C3⋊Dic3—41no—
110C3×Dic3—62no—
111C3×Dic3—62no—
112C3×Dic3—62no—
113C3×Dic3—121no—
114C3×Dic3—121no—
115C3×Dic3—121no—
116#127—271no—
117#125—271no—
118#102—272no—
119#102—272no—
120#97—91no—
121H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
122C23×C3—182no—
123C22×D3—62no—
124C22×D3—182no—
125C3×D4—362no—
126C2×C3×C4—61no—
127C2×C3×C4—61no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—181no—
131C2×C3×C4—181no—
132C2×C3×C4GO+(4,𝔽2)11yescharacteristic
133#67—364no—
134#67—364no—
135C2×Dic3—61no—
136C2×Dic3—61no—
137C2×Dic3—61no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140C2×Dic3—61no—
141C2×Dic3—181no—
142C2×Dic3—181no—
143D12—122no—
144D12—184no—
145D12—362no—
146D12—362no—
147C3×C8—182no—
148C2×C32—21no—
149C2×C32—21no—
150C2×C32—21no—
151C2×C32—21no—
152C2×C32—21no—
153C2×C32—21no—
154C3⋊D3—82no—
155C3×D3—242no—
156C2×D4—181no—
157C2×D4—272no—
158C2×D4—541no—
159C2×C8—92no—
160C4⋊C4—181no—
161C4⋊C4—271no—
162C4⋊C4—271no—
163C4⋊C4—541no—
164C42—91no—
165C22×C3—61no—
166C22×C3—61no—
167C22×C3—362no—
168C22×C3—362no—
169C22×C3—362no—
170C22×C3C2×GO+(4,𝔽2)11yescharacteristic
171C2×D3—122no—
172C2×D3—122no—
173C2×D3—122no—
174C2×D3—362no—
175C2×D3—362no—
176C2×D3—362no—
177C3×C4—21no—
178C3×C4—62no—
179C3×C4—92no—
180C3×C4—121no—
181C3×C4—181no—
182C3×C4—361no—
183C3×C4—361no—
184Dic3—62no—
185Dic3—62no—
186Dic3—62no—
187Dic3—121no—
188Dic3—121no—
189Dic3—121no—
190Dic3—361no—
191Dic3—361no—
192C32—21no—
193C32—21no—
194C23—182no—
195D4—362no—
196D4—544no—
197D4—1082no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4#1088⋊C211yescharacteristic
203C8—182no—
204C2×C3—61no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—722no—
211C2×C3#6422111yescharacteristic
212C2×C3#6422411yescharacteristic
213C2×C3#6423611yescharacteristic
214D3—242no—
215D3—722no—
216C22—362no—
217C22—362no—
218C22—362no—
219C22C2×#1088⋊C211yescharacteristic
220C4—21no—
221C4—92no—
222C4—181no—
223C4—361no—
224C4—361no—
225C3—61no—
226C3—61no—
227C3#1059335611yescharacteristic
228C2—722no—
229C2#1172229711yescharacteristic
230C2#1172230211yescharacteristic
231C2#1172230911yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical#11721144C2
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11721098C2semidirect
3. #11720783C2semidirect
4. #11720705C2semidirect
5. #11720705C2semidirect
121. H(3,𝔽3)#490semidirect

Maximal subgroups

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

subgroupindex
#117211442
#117210982
#117207832
#117207052
#47139