Finite Group Databasebeta

← #13147383

Subgroups — #13147383

Order 1728.

775 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88626C212yesmaximal, complemented
3C22⋊#6188C212yesmaximal, complemented
4#11720784C212yesmaximal, complemented
5C23×#272C411yescharacteristic
6C22×C4×H(3,𝔽3)C411yescharacteristic
7C22×#1084C2211yescharacteristic
8C2×C4×#272—24no—
9C2×C4×#272C411yescharacteristic
10C2×C4×#272C411yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×#6188C2214yescomplemented
14#88626—24no—
15C22×C3×C4×D3—62no—
16C23×H(3,𝔽3)C2×C411yescharacteristic
17C22×#272—22no—
18C22×#272—22no—
19C22×#272C814yescomplemented
20C22×#272C2×C411yescharacteristic
21C22×#272C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×#1084—22no—
26C2×#1084C2×C411yescharacteristic
27C2×#1084C2311yescharacteristic
28C4×#272—22no—
29C4×#272—22no—
30C4×#272—22no—
31C4×#272D414yes—
32#6188—24no—
33C2×C3×C22⋊C8—91nomaximal
34C23×C3×D3—62no—
35C22×C32×C4—22no—
36C22×C3×Dic3—62no—
37C2×C3×C4×D3—62no—
38C2×C3×C4×D3—62no—
39C2×C3×C4×D3—62no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—616no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)C2×C812yescomplemented
45C22×H(3,𝔽3)C22×C411yescharacteristic
46C2×#272—21no—
47C2×#272—21no—
48C2×#272—28no—
49C2×#272C22⋊C412yes—
50C2×#272C2×C812yes—
51C2×#272C2×C812yescomplemented
52C2×#272M1612yes—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)C22⋊C412yes—
55#1084—21no—
56#1084C2×D412yes—
57C23×C3×C4—91no—
58C22×C4×D3—62no—
59C22×C3×C8—92no—
60C3×C22⋊C8—92no—
61C3×C22⋊C8—92no—
62C23×C32—22no—
63C22×C3×D3—68no—
64C22×C3×D3—62no—
65C22×C3×D3—62no—
66C22×C3×D3—68no—
67C22×C3×D3—68no—
68C2×C32×C4—28no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—68no—
74C3×C4×D3—68no—
75C3×C4×D3—68no—
76C3×C4×D3—68no—
77C3×C4×D3—68no—
78C2×C22⋊C8—91noSylow
79C2×H(3,𝔽3)—22no—
80C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
81C2×H(3,𝔽3)C22×C811yescharacteristic
82C2×H(3,𝔽3)C2×M1611yescharacteristic
83#272—22no—
84#272C22⋊C814yescomplemented
85C24×C3—91no—
86C23×D3—62no—
87C22×C3×C4—62no—
88C22×C3×C4—91no—
89C22×C3×C4—91no—
90C22×C3×C4—91no—
91C22×C3×C4—91no—
92C22×C3×C4—91no—
93C22×C3×C4—184no—
94C22×C3×C4#15711yescharacteristic
95C22×Dic3—62no—
96C2×C4×D3—62no—
97C2×C4×D3—62no—
98C2×C4×D3—62no—
99C2×C4×D3—62no—
100C2×C4×D3—616no—
101C2×C3×C8—94no—
102C2×C3×C8—184no—
103C22×C32—24no—
104C22×C32—24no—
105C22×C32—22no—
106C22×C32—24no—
107C2×C3×D3—64no—
108C2×C3×D3—64no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C2×C3×D3—64no—
112C2×C3×D3—632no—
113C2×C3×D3—64no—
114C32×C4—24no—
115C32×C4—24no—
116C3×Dic3—64no—
117C3×Dic3—64no—
118C23×C4—91no—
119C22×C8—92no—
120C22⋊C8—92no—
121C22⋊C8—92no—
122H(3,𝔽3)C2×C22⋊C811yescharacteristic, complemented, Sylow
123C23×C3—62no—
124C23×C3—91no—
125C23×C3—91no—
126C23×C3—94no—
127C23×C3—182no—
128C23×C3—182no—
129C23×C3C2×#15711yescharacteristic
130C22×D3—62no—
131C22×D3—62no—
132C22×D3—68no—
133C22×D3—68no—
134C22×D3—68no—
135C2×C3×C4—22no—
136C2×C3×C4—68no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—91no—
140C2×C3×C4—94no—
141C2×C3×C4—91no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4—182no—
146C2×C3×C4C2×#15711yescharacteristic
147C2×C3×C4C2×#15711yescharacteristic
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—68no—
151C4×D3—68no—
152C4×D3—68no—
153C4×D3—68no—
154C4×D3—68no—
155C3×C8—184no—
156C2×C32—28no—
157C2×C32—22no—
158C2×C32—22no—
159C2×C32—22no—
160C3×D3—68no—
161C3×D3—68no—
162C24—91no—
163C22×C4—91no—
164C22×C4—91no—
165C22×C4—91no—
166C22×C4—91no—
167C22×C4—91no—
168C22×C4—184no—
169C22×C4#108811yescharacteristic
170C2×C8—94no—
171C2×C8—184no—
172C22×C3—21no—
173C22×C3—21no—
174C22×C3—64no—
175C22×C3—64no—
176C22×C3—64no—
177C22×C3—62no—
178C22×C3—92no—
179C22×C3—92no—
180C22×C3—92no—
181C22×C3—92no—
182C22×C3—188no—
183C22×C3—181no—
184C22×C3—181no—
185C22×C3#378712yes—
186C22×C3C22×#15711yescharacteristic
187C2×D3—64no—
188C2×D3—64no—
189C2×D3—64no—
190C2×D3—64no—
191C2×D3—64no—
192C2×D3—632no—
193C2×D3—64no—
194C3×C4—21no—
195C3×C4—64no—
196C3×C4—64no—
197C3×C4—92no—
198C3×C4—181no—
199C3×C4C22⋊#15712yes—
200Dic3—64no—
201Dic3—64no—
202C32—22no—
203C23—91no—
204C23—94no—
205C23—91no—
206C23—182no—
207C23—182no—
208C23C2×#108811yescharacteristic
209C2×C4—22no—
210C2×C4—91no—
211C2×C4—94no—
212C2×C4—91no—
213C2×C4—182no—
214C2×C4—182no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4C2×#108811yescharacteristic
218C2×C4C2×#108811yescharacteristic
219C8—184no—
220C2×C3—22no—
221C2×C3—68no—
222C2×C3—62no—
223C2×C3—62no—
224C2×C3—62no—
225C2×C3—94no—
226C2×C3—182no—
227C2×C3C2×#378711yescharacteristic
228C2×C3#6428311yescharacteristic
229C2×C3C2×C22⋊#15711yescharacteristic
230D3—68no—
231D3—68no—
232C22—21no—
233C22—21no—
234C22—92no—
235C22—92no—
236C22—92no—
237C22—92no—
238C22—188no—
239C22—181no—
240C22—181no—
241C22#8862612yescomplemented
242C22C22×#108811yescharacteristic
243C4—21no—
244C4—92no—
245C4—181no—
246C4C22⋊#108812yes—
247C3—62no—
248C3#1059380011yescharacteristic
249C2—22no—
250C2—94no—
251C2—182no—
252C2C2×#8862611yescharacteristic
253C2#1172244511yescharacteristic
254C2C2×C22⋊#108811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C22×C8
Frattini, socleC22×C3C22×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C4
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C2×C22⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88626C2semidirect
3. C22⋊#6188C2semidirect
4. #11720784C2semidirect
13. C2×#6188C22semidirect
19. C22×#272C8semidirect
44. C22×H(3,𝔽3)C2×C8semidirect
51. C2×#272C2×C8semidirect
84. #272C22⋊C8semidirect
122. H(3,𝔽3)C2×C22⋊C8semidirect
241. C22#88626semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886262
C22⋊#61882
#117207842
C2×C3×C22⋊C89