Finite Group Databasebeta

← #13147328

Subgroups — #13147328

Order 1728.

371 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 55 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, complemented
2C2×#88627C211yesmaximal, characteristic, complemented
3#11721737C211yesmaximal, characteristic
4C22⋊#6188C211yesmaximal, characteristic, complemented
5#11720785C211yesmaximal, characteristic, complemented
6#11720784C211yesmaximal, characteristic, complemented
7C4×#6188C211yesmaximal, characteristic, complemented
8C2×#6235C411yescharacteristic
9C22×#1084C2211yescharacteristic
10C2×C4×#272C2211yescharacteristic
11#88631—22no—
12C2×#6188C2211yescharacteristic
13C2×#6188C2211yescharacteristic
14C2×#6188C2211yescharacteristic
15C2×#6188C2211yescharacteristic
16#88627—22no—
17C22⋊C4×H(3,𝔽3)C411yescharacteristic
18#88542C411yescharacteristic
19#88540C411yescharacteristic
20C4×#1084C2211yescharacteristic
21C3×#67⋊C4—62no—
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C22×#272C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26#6235—22no—
27C2×#1084—21no—
28C2×#1084—21no—
29C2×#1084C2×C411yescharacteristic
30C2×#1084C2×C411yescharacteristic
31C2×#1084C2311yescharacteristic
32C4×#272—21no—
33C4×#272—21no—
34#6188—21no—
35#6188—21no—
36#6188—21no—
37#6188D412yescomplemented
38C3×#436—91nomaximal
39C2×C3×#67—62no—
40C22×C3×Dic3—62no—
41C2×C3×C4×D3—62no—
42C32×C22⋊C4—22no—
43C3×(C2×D3)⋊C4—62no—
44C3×Dic3⋊C4—62no—
45C3×C4×Dic3—62no—
46C22×H(3,𝔽3)—21no—
47C22×H(3,𝔽3)—21no—
48C22×H(3,𝔽3)—21no—
49C22×H(3,𝔽3)C22×C411yescharacteristic
50C2×#272—21no—
51C2×#272—21no—
52C2×#272—21no—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—21no—
55#1084—21no—
56#1084M1612yescomplemented
57#1084C2×D411yescharacteristic
58#1084#4111yescharacteristic
59C2×C3×M16—91no—
60C2×C3×M16—91no—
61C3×C4×D4—91no—
62#67⋊C4—62no—
63C3×C4⋊C8—91no—
64C3×C22⋊C8—91no—
65C3×C22⋊C8—91no—
66C3×C4×C8—91no—
67C23×C32—22no—
68C22×C3×D3—62no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C3×#67—68no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—122no—
76C2×C3×Dic3—122no—
77C3×C4×D3—122no—
78C3×C4×D3—122no—
79#436—91noSylow
80C2×H(3,𝔽3)—41no—
81C2×H(3,𝔽3)C4×D411yescharacteristic
82C2×H(3,𝔽3)C2×M1611yescharacteristic
83C2×H(3,𝔽3)#13111yescharacteristic
84#272—41no—
85C2×C3×D4—91no—
86C22×C3×C4—91no—
87C22×C3×C4—91no—
88C2×#67—62no—
89C22×Dic3—62no—
90C2×C4×D3—62no—
91C3×M16—182no—
92C3×M16—182no—
93C2×C3×C8—91no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C2×C3×C8—91no—
97C3×C4⋊C4—91no—
98C3×C22⋊C4—62no—
99C3×C22⋊C4—91no—
100C3×C22⋊C4#15711yescharacteristic
101C3×C42—91no—
102(C2×D3)⋊C4—62no—
103Dic3⋊C4—62no—
104C4×Dic3—62no—
105C22×C32—24no—
106C22×C32—22no—
107C22×C32—42no—
108C22×C32—42no—
109C2×C3×D3—64no—
110C2×C3×D3—122no—
111C2×C3×D3—122no—
112C32×C4—42no—
113C32×C4—42no—
114C3×Dic3—64no—
115C3×Dic3—62no—
116C3×Dic3—62no—
117C3×Dic3—122no—
118C2×M16—91no—
119C2×M16—91no—
120C4×D4—91no—
121C4⋊C8—91no—
122C22⋊C8—91no—
123C22⋊C8—91no—
124C4×C8—91no—
125H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
126C23×C3—62no—
127C23×C3—91no—
128C23×C3C2×#15711yescharacteristic
129C22×D3—62no—
130C3×D4—182no—
131C2×C3×C4—62no—
132C2×C3×C4—62no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—181no—
137C2×C3×C4—181no—
138C2×C3×C4—181no—
139C2×C3×C4—181no—
140C2×C3×C4C2×#15711yescharacteristic
141C2×C3×C4C2×#15711yescharacteristic
142#67—68no—
143C2×Dic3—62no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—122no—
147C2×Dic3—122no—
148C4×D3—122no—
149C4×D3—122no—
150C3×C8—92no—
151C3×C8—181no—
152C3×C8—181no—
153C3×C8—181no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—22no—
157C2×C32—44no—
158C3×D3—124no—
159C2×D4—91no—
160C22×C4—91no—
161C22×C4—91no—
162M16—182no—
163M16—182no—
164C2×C8—91no—
165C2×C8—91no—
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—91no—
169C22⋊C4—91no—
170C22⋊C4#108811yescharacteristic
171C42—91no—
172C22×C3—21no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3—64no—
176C22×C3—62no—
177C22×C3—122no—
178C22×C3—122no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3—181no—
182C22×C3C22×#15711yescharacteristic
183C2×D3—64no—
184C2×D3—122no—
185C2×D3—122no—
186C3×C4—21no—
187C3×C4—21no—
188C3×C4—92no—
189C3×C4—91no—
190C3×C4—91no—
191C3×C4—122no—
192C3×C4—122no—
193C3×C4—181no—
194Dic3—62no—
195Dic3—62no—
196Dic3—64no—
197Dic3—122no—
198C32—22no—
199C23—91no—
200C23C2×#108811yescharacteristic
201D4—182no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—181no—
206C2×C4—181no—
207C2×C4—181no—
208C2×C4—181no—
209C2×C4C2×#108811yescharacteristic
210C2×C4C2×#108811yescharacteristic
211C8—92no—
212C8—181no—
213C8—181no—
214C8—181no—
215C2×C3—41no—
216C2×C3—62no—
217C2×C3—62no—
218C2×C3—62no—
219C2×C3—124no—
220C2×C3—361no—
221C2×C3#6428311yescharacteristic
222C2×C3#6428711yescharacteristic
223C2×C3D4×#15711yescharacteristic
224D3—124no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—181no—
229C22—181no—
230C22—181no—
231C22C22×#108811yescharacteristic
232C4—21no—
233C4—21no—
234C4—92no—
235C4—91no—
236C4—91no—
237C4—181no—
238C3—62no—
239C3#1059374211yescharacteristic
240C2—41no—
241C2—361no—
242C2#1172244511yescharacteristic
243C2#1172244911yescharacteristic
244C2D4×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22⋊C4×H(3,𝔽3)C4
prime radicalC2×#6235C4
cosocleC2×#1084C23
hypercenterC3×C22⋊C4#157
hypocenterH(3,𝔽3)#436
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88631C2semidirect
2. C2×#88627C2semidirect
4. C22⋊#6188C2semidirect
5. #11720785C2semidirect
6. #11720784C2semidirect
7. C4×#6188C2semidirect
37. #6188D4semidirect
56. #1084M16semidirect
125. H(3,𝔽3)#436semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C2×#886272
#117217372
C22⋊#61882
#117207852
#117207842
C4×#61882
C3×#4369