Finite Group Databasebeta

← #13147411

Subgroups — #13147411

Order 1728.

751 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×D4×#272C211yesmaximal, characteristic
2#11722445C212yesmaximal, complemented
3#11720794C214yesmaximal, complemented
4C23×#272C412yes—
5C2×D4×H(3,𝔽3)C411yescharacteristic
6C2×#6235C2212yes—
7D4×#272—22no—
8D4×#272—22no—
9C2×C4⋊#272C411yescharacteristic
10C2×C4×#272C2211yescharacteristic
11#88631C2214yescomplemented
12#88627—22no—
13#88626—22no—
14C2×C3×D3×D4—62no—
15C23×H(3,𝔽3)C2×C412yes—
16C22×#272—22no—
17C22×#272—22no—
18C22×#272—44no—
19C22×#272C2×C411yescharacteristic
20C22×#272C2×C412yes—
21D4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23#6235—24no—
24C2×#1084C2311yescharacteristic
25C4⋊#272—22no—
26C4×#272D412yes—
27C4×#272D412yes—
28#6188—24no—
29C2×C3×#100—91nomaximal
30C23×C3×D3—64no—
31C2×C32×D4—22no—
32C2×C3×#67—64no—
33C3×D3×D4—68no—
34C3×D3×D4—68no—
35C2×C3×D12—62no—
36C2×C3×C4×D3—62no—
37C22×H(3,𝔽3)—22no—
38C22×H(3,𝔽3)—42no—
39C22×H(3,𝔽3)C22×C411yescharacteristic
40C2×#272—22no—
41C2×#272—22no—
42C2×#272—44no—
43C2×#272—44no—
44C2×#272—42no—
45C2×#272C22⋊C411yescharacteristic
46C2×#272C22⋊C411yescharacteristic
47C4×H(3,𝔽3)C22⋊C412yes—
48#1084C2×D412yes—
49C22×C3×D4—91no—
50C2×D3×D4—62no—
51C2×C3×M16—92no—
52C3×#100—94no—
53C23×C32—24no—
54C22×C3×D3—62no—
55C22×C3×D3—64no—
56C22×C3×D3—68no—
57C22×C3×D3—68no—
58C22×C3×D3—1216no—
59C32×D4—28no—
60C2×C32×C4—22no—
61C3×#67—616no—
62C2×C3×Dic3—62no—
63C3×D12—68no—
64C3×C4×D3—64no—
65C3×C4×D3—64no—
66C2×#100—91noSylow
67C2×H(3,𝔽3)—21no—
68C2×H(3,𝔽3)—42no—
69C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
70#272—21no—
71#272—42no—
72#272#10012yescomplemented
73C24×C3—92no—
74C23×D3—64no—
75C2×C3×D4—62no—
76C2×C3×D4—92no—
77C2×C3×D4—91no—
78C2×C3×D4—182no—
79C2×C3×D4—182no—
80C2×C3×D4#15711yescharacteristic
81C22×C3×C4—91no—
82C2×#67—64no—
83D3×D4—68no—
84D3×D4—68no—
85C2×D12—62no—
86C2×C4×D3—62no—
87C3×M16—94no—
88C3×M16—182no—
89C2×C3×C8—182no—
90C22×C32—28no—
91C22×C32—22no—
92C22×C32—48no—
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—68no—
96C2×C3×D3—68no—
97C2×C3×D3—1216no—
98C2×C3×D3—1216no—
99C2×C3×D3—128no—
100C32×C4—24no—
101C3×Dic3—64no—
102C22×D4—91no—
103C2×M16—92no—
104#100—94no—
105H(3,𝔽3)C2×#10011yescharacteristic, complemented, Sylow
106C23×C3—64no—
107C23×C3—91no—
108C23×C3—92no—
109C23×C3—182no—
110C23×C3—182no—
111C23×C3—364no—
112C23×C3C2×#15712yes—
113C22×D3—62no—
114C22×D3—64no—
115C22×D3—68no—
116C22×D3—68no—
117C22×D3—1216no—
118C3×D4—22no—
119C3×D4—68no—
120C3×D4—184no—
121C3×D4—182no—
122C2×C3×C4—62no—
123C2×C3×C4—91no—
124C2×C3×C4—92no—
125C2×C3×C4—92no—
126C2×C3×C4C2×#15711yescharacteristic
127#67—616no—
128C2×Dic3—62no—
129D12—68no—
130C4×D3—64no—
131C4×D3—64no—
132C3×C8—184no—
133C2×C32—24no—
134C2×C32—22no—
135C2×C32—48no—
136C3×D3—64no—
137C3×D3—64no—
138C3×D3—128no—
139C24—92no—
140C2×D4—92no—
141C2×D4—91no—
142C2×D4—182no—
143C2×D4—182no—
144C2×D4#108811yescharacteristic
145C22×C4—91no—
146M16—94no—
147M16—182no—
148C2×C8—182no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3—68no—
152C22×C3—62no—
153C22×C3—91no—
154C22×C3—91no—
155C22×C3—128no—
156C22×C3—182no—
157C22×C3—182no—
158C22×C3—362no—
159C22×C3—364no—
160C22×C3—364no—
161C22×C3C22×#15711yescharacteristic
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—68no—
165C2×D3—68no—
166C2×D3—1216no—
167C2×D3—1216no—
168C2×D3—128no—
169C3×C4—64no—
170C3×C4—92no—
171C3×C4C22⋊#15712yes—
172Dic3—64no—
173C32—22no—
174C23—91no—
175C23—92no—
176C23—182no—
177C23—182no—
178C23—364no—
179C23C2×#108812yes—
180D4—22no—
181D4—184no—
182D4—182no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—92no—
186C2×C4C2×#108811yescharacteristic
187C8—184no—
188C2×C3—21no—
189C2×C3—42no—
190C2×C3—64no—
191C2×C3—62no—
192C2×C3—92no—
193C2×C3—128no—
194C2×C3—181no—
195C2×C3—362no—
196C2×C3C2×C22⋊#15711yescharacteristic
197D3—64no—
198D3—64no—
199D3—128no—
200C22—22no—
201C22—42no—
202C22—91no—
203C22—91no—
204C22—182no—
205C22—182no—
206C22—362no—
207C22—364no—
208C22—364no—
209C22C22×#108811yescharacteristic
210C4—92no—
211C4C22⋊#108812yes—
212C3—62no—
213C3#1059382511yescharacteristic
214C2—21no—
215C2—42no—
216C2—92no—
217C2—181no—
218C2—362no—
219C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC2×#1084C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)C2×#100
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. #11722445C2semidirect
3. #11720794C2semidirect
11. #88631C22semidirect
72. #272#100semidirect
105. H(3,𝔽3)C2×#100semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
#117224452
#117207942
C2×C3×#1009