Finite Group Databasebeta

← #13146670

Subgroups — #13146670

Order 1728.

345 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88631C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3#11721298C211yesmaximal, characteristic
4#11720704C212yesmaximal, complemented
5#11720703C212yesmaximal, complemented
6C22×#1084C2211yescharacteristic
7C22×(C3⋊Dic3)⋊C3—21no—
8#88631C414yescomplemented
9C2×#6188C2212yes—
10#3757⋊C3—21no—
11#88449—21no—
12#88449C2211yescharacteristic
13#88449C2212yes—
14#88449C2211yescharacteristic
15#88447—21no—
16C2×Dic3⋊Dic3—61no—
17#63955—61no—
18C23×H(3,𝔽3)D411yescharacteristic
19C2×#1084C2×C412yes—
20C2×#1084D411yescharacteristic
21C2×#1084Q812yes—
22C2×#1084C2311yescharacteristic
23C2×(C3⋊Dic3)⋊C3—21no—
24C2×(C3⋊Dic3)⋊C3—21no—
25C2×(C3⋊Dic3)⋊C3—22no—
26C2×(C3⋊Dic3)⋊C3—42no—
27#6188C2×C414yescomplemented
28#5121—91nomaximal
29C22×C3⋊Dic3—21no—
30C22×C3×Dic3—61no—
31C22×C3×Dic3—61no—
32C22×C3×Dic3—61no—
33#3757—21no—
34C3×#217—61no—
35Dic3⋊Dic3—61no—
36Dic3⋊Dic3—61no—
37Dic3⋊Dic3—62no—
38Dic3⋊Dic3—62no—
39Dic32—61no—
40Dic32—61no—
41C22×H(3,𝔽3)—21no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)C4⋊C412yes—
44C22×H(3,𝔽3)C2×D411yescharacteristic
45#1084C4⋊C412yes—
46#1084C22×C411yescharacteristic
47#1084C2×Q811yescharacteristic
48(C3⋊Dic3)⋊C3—41no—
49(C3⋊Dic3)⋊C3—42no—
50(C3⋊Dic3)⋊C3—41no—
51C2×C3×M16—91no—
52#926—91no—
53C2×C4⋊Dic3—91no—
54C2×Dic3⋊C4—181no—
55#877—181no—
56#818—92no—
57#817—92no—
58C23×C32—21no—
59C23×C32—21no—
60C2×C3⋊Dic3—21no—
61C2×C3⋊Dic3—21no—
62C2×C3⋊Dic3—22no—
63C2×C3⋊Dic3—42no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—122no—
76#428—271noSylow
77C2×H(3,𝔽3)—22no—
78C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
79C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
80C2×H(3,𝔽3)#13711yescharacteristic
81C22×C3×C4—91no—
82C22×C3×C4—181no—
83C22×Dic3—61no—
84C22×Dic3—61no—
85C22×Dic3—61no—
86C22×Dic3—181no—
87C3×M16—94no—
88C2×C3×C8—92no—
89C3×C22⋊C4—181no—
90#217—61no—
91#217—181no—
92C4⋊Dic3—91no—
93C4⋊Dic3—92no—
94C4⋊Dic3—91no—
95C4⋊Dic3—181no—
96Dic3⋊C4—181no—
97Dic3⋊C4—181no—
98Dic3⋊C4—182no—
99Dic3⋊C4—182no—
100C4×Dic3—181no—
101C4×Dic3—181no—
102C4×Dic3—181no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32—21no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—41no—
110C22×C32—41no—
111C3⋊Dic3—41no—
112C3⋊Dic3—41no—
113C3⋊Dic3—42no—
114C3×Dic3—62no—
115C3×Dic3—62no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C3×Dic3—61no—
119C3×Dic3—61no—
120C3×Dic3—121no—
121C3×Dic3—121no—
122C3×Dic3—122no—
123C2×M16—91no—
124#117—271no—
125C2×C4⋊C4—271no—
126#107—272no—
127#106—272no—
128H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
129C23×C3—61no—
130C23×C3—61no—
131C23×C3GO+(4,𝔽2)11yescharacteristic
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—92no—
135C2×C3×C4—92no—
136C2×C3×C4—181no—
137C2×C3×C4—181no—
138C2×C3×C4—182no—
139C2×C3×C4—362no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—62no—
147C2×Dic3—62no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×Dic3—62no—
151C2×Dic3—122no—
152C2×Dic3—181no—
153C2×Dic3—181no—
154C2×Dic3—182no—
155C2×Dic3—362no—
156C3×C8—94no—
157C2×C32—24no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32—42no—
165C22×C4—91no—
166C22×C4—181no—
167M16—94no—
168C2×C8—92no—
169C4⋊C4—271no—
170C4⋊C4—272no—
171C4⋊C4—271no—
172C4⋊C4—541no—
173C22⋊C4—181no—
174C42—541no—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3—62no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—121no—
184C22×C3—121no—
185C22×C3#377312yes—
186C22×C3C2×GO+(4,𝔽2)11yescharacteristic
187C3×C4—91no—
188C3×C4—92no—
189C3×C4—91no—
190C3×C4—362no—
191C3×C4—361no—
192C3×C4—361no—
193Dic3—61no—
194Dic3—61no—
195Dic3—62no—
196Dic3—62no—
197Dic3—61no—
198Dic3—61no—
199Dic3—122no—
200Dic3—121no—
201Dic3—121no—
202Dic3—362no—
203Dic3—361no—
204Dic3—361no—
205C32—21no—
206C32—21no—
207C23#1088⋊C211yescharacteristic
208C2×C4—92no—
209C2×C4—92no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—181no—
213C2×C4—181no—
214C2×C4—182no—
215C2×C4—362no—
216C8—94no—
217C2×C3—22no—
218C2×C3—64no—
219C2×C3—61no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—61no—
223C2×C3—61no—
224C2×C3—61no—
225C2×C3—122no—
226C2×C3C2×#377311yescharacteristic
227C2×C3#6423611yescharacteristic
228C2×C3#6423911yescharacteristic
229C22—21no—
230C22—21no—
231C22#8859012yes—
232C22C2×#1088⋊C211yescharacteristic
233C4—91no—
234C4—92no—
235C4—91no—
236C4—361no—
237C4—362no—
238C4—361no—
239C3—61no—
240C3—61no—
241C3#1059350711yescharacteristic
242C2—22no—
243C2#1172230711yescharacteristic
244C2#1172230911yescharacteristic
245C2C2×#8859011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
cosocleC2×#1084C23
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#428
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11720704C2semidirect
5. #11720703C2semidirect
8. #88631C4semidirect
27. #6188C2×C4semidirect
128. H(3,𝔽3)#428semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C2×#884492
#117212982
#117207042
#117207032
#51219