Finite Group Databasebeta

← #13139999

Subgroups — #13139999

Order 1728.

901 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3748⋊C3C212yesmaximal, complemented
3#11720323C214yesmaximal, complemented
4#10591069—32nomaximal
5C23×#272C411yescharacteristic
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C411yescharacteristic
8C2×C4×#272—24no—
9C2×C4×#272C412yes—
10C2×C4×#272C2212yes—
11#3748⋊C3—24no—
12C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
13C22×C3×C4×D3—32no—
14C22×C3×C4×D3—61no—
15C2×#3748D312yescomplemented
16C2×C3×#208—32no—
17(C2×D3)⋊(C3⋊C8)—38no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C22×#272—22no—
20C22×#272—24no—
21C22×#272C2×C412yes—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)D414yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26C2×#1084C2×C412yes—
27C4×#272—24no—
28C4×#272—24no—
29(C3⋊(C3⋊C8))⋊C3—24no—
30#5207—91no—
31#5126—92no—
32C23×C3×D3—32no—
33C23×C3×D3—61no—
34C22×C32×C4—21no—
35C22×C32×C4C2×D312yes—
36C22×C3×Dic3—32no—
37C22×C3×Dic3—61no—
38C2×C3×C4×D3—34no—
39C2×C3×C4×D3—34no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—68no—
43C2×C3×C4×D3—64no—
44C2×C3×C4×D3—64no—
45#3748—24no—
46C2×C3⋊(C3⋊C8)C2×D314yescomplemented
47C3×#208—64no—
48C2×C3×C3⋊C8—34no—
49C22×H(3,𝔽3)—22no—
50C22×H(3,𝔽3)C22⋊C412yes—
51C22×H(3,𝔽3)C22×C411yescharacteristic
52C2×#272—28no—
53C2×#272—22no—
54C2×#272—22no—
55C2×#272M1614yescomplemented
56C4×H(3,𝔽3)C22⋊C412yes—
57C4×H(3,𝔽3)C2×D412yes—
58#1084—22no—
59C23×C3×C4—91no—
60C22×C4×D3—32no—
61C22×C4×D3—61no—
62C2×C3×M16—92no—
63C2×#208—32no—
64C2×#208—92no—
65#830—94no—
66(C2×D3)⋊C8—98no—
67C23×C32—21no—
68C23×C32C4×D312yes—
69C22×C3×D3—34no—
70C22×C3×D3—62no—
71C22×C3×D3—68no—
72C22×C3×D3—68no—
73C22×C3×D3—64no—
74C22×C3×D3—64no—
75C2×C32×C4—21no—
76C2×C32×C4—24no—
77C2×C32×C4—21no—
78C2×C32×C4C4×D312yes—
79C2×C32×C4D1214yes—
80C2×C32×C4#6714yes—
81C2×C32×C4C22×D312yes—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C3×C4×D3—68no—
87C3×C4×D3—68no—
88C3×C4×D3—68no—
89C3×C4×D3—64no—
90C3×C4×D3—64no—
91C3⋊(C3⋊C8)—24no—
92C3×C3⋊C8—64no—
93#407—271noSylow
94C2×H(3,𝔽3)—22no—
95C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
96C2×H(3,𝔽3)C2×M1612yes—
97#272—24no—
98C24×C3—91no—
99C23×D3—32no—
100C23×D3—61no—
101C22×C3×C4—32no—
102C22×C3×C4—61no—
103C22×C3×C4—91no—
104C22×C3×C4—92no—
105C22×C3×C4—92no—
106C22×C3×C4—184no—
107C22×C3×C4D3211yescharacteristic
108C22×Dic3—32no—
109C22×Dic3—61no—
110C2×C4×D3—34no—
111C2×C4×D3—34no—
112C2×C4×D3—62no—
113C2×C4×D3—62no—
114C2×C4×D3—64no—
115C2×C4×D3—64no—
116C2×C4×D3—68no—
117C3×M16—184no—
118C2×C3×C8—94no—
119#208—64no—
120#208—184no—
121C2×C3⋊C8—34no—
122C2×C3⋊C8—94no—
123C22×C32—24no—
124C22×C32—24no—
125C22×C32—22no—
126C22×C32—21no—
127C22×C32(C2×D3)⋊C414yes—
128C22×C32C2×C4×D312yes—
129C2×C3×D3—38no—
130C2×C3×D3—64no—
131C2×C3×D3—64no—
132C2×C3×D3—616no—
133C2×C3×D3—616no—
134C2×C3×D3—64no—
135C2×C3×D3—64no—
136C2×C3×D3—64no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4(C2×D3)⋊C414yes—
140C32×C4C2×D1212yes—
141C32×C4C2×#6712yes—
142C3×Dic3—64no—
143C3×Dic3—64no—
144C23×C4—91no—
145C2×M16—92no—
146C22⋊C8—274no—
147H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
148C23×C3—32no—
149C23×C3—61no—
150C23×C3—92no—
151C23×C3—184no—
152C23×C3—182no—
153C23×C3#62411yescharacteristic
154C22×D3—34no—
155C22×D3—62no—
156C22×D3—68no—
157C22×D3—64no—
158C22×D3—64no—
159C22×D3—68no—
160C2×C3×C4—32no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—32no—
164C2×C3×C4—61no—
165C2×C3×C4—64no—
166C2×C3×C4—61no—
167C2×C3×C4—92no—
168C2×C3×C4—182no—
169C2×C3×C4—184no—
170C2×C3×C4—184no—
171C2×C3×C4#62411yescharacteristic
172C2×C3×C4#62614yes—
173C2×C3×C4C2×D3211yescharacteristic
174C2×Dic3—34no—
175C2×Dic3—62no—
176C2×Dic3—64no—
177C2×Dic3—64no—
178C4×D3—64no—
179C4×D3—64no—
180C4×D3—68no—
181C4×D3—68no—
182C4×D3—68no—
183C3×C8—184no—
184C3⋊C8—64no—
185C3⋊C8—184no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32—22no—
189C2×C32—21no—
190C2×C32D3×M1614yes—
191C2×C32C2×(C2×D3)⋊C412yes—
192C3×D3—68no—
193C3×D3—68no—
194C24—91no—
195C22×C4—91no—
196C22×C4—92no—
197C22×C4—92no—
198C22×C4—184no—
199C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200M16—184no—
201C2×C8—94no—
202C22×C3—22no—
203C22×C3—32no—
204C22×C3—34no—
205C22×C3—64no—
206C22×C3—64no—
207C22×C3—61no—
208C22×C3—62no—
209C22×C3—94no—
210C22×C3—188no—
211C22×C3—182no—
212C22×C3—182no—
213C22×C3#372212yes—
214C22×C3C2×#62411yescharacteristic
215C2×D3—38no—
216C2×D3—64no—
217C2×D3—64no—
218C2×D3—64no—
219C2×D3—64no—
220C2×D3—616no—
221C2×D3—616no—
222C2×D3—64no—
223C3×C4—32no—
224C3×C4—32no—
225C3×C4—34no—
226C3×C4—62no—
227C3×C4—62no—
228C3×C4—182no—
229C3×C4#372212yes—
230C3×C4C2×#62612yes—
231Dic3—64no—
232Dic3—64no—
233C32—21no—
234C32#512612yescomplemented
235C23—92no—
236C23—184no—
237C23—182no—
238C23#619911yescharacteristic
239C2×C4—92no—
240C2×C4—182no—
241C2×C4—184no—
242C2×C4—184no—
243C2×C4#619911yescharacteristic
244C2×C4#620014yes—
245C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C8—184no—
247C2×C3—22no—
248C2×C3—32no—
249C2×C3—34no—
250C2×C3—64no—
251C2×C3—64no—
252C2×C3—61no—
253C2×C3—62no—
254C2×C3—184no—
255C2×C3#6381812yes—
256C2×C3C2×#372211yescharacteristic
257D3—68no—
258D3—68no—
259C22—22no—
260C22—94no—
261C22—188no—
262C22—182no—
263C22—182no—
264C22#8845112yes—
265C22C2×#619911yescharacteristic
266C4—182no—
267C4#8845112yes—
268C4C2×#620012yes—
269C3—32no—
270C3—61no—
271C3#1059107411yescharacteristic
272C2—22no—
273C2—184no—
274C2#1172106512yes—
275C2C2×#8845111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3748⋊C3C2semidirect
3. #11720323C2semidirect
12. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
15. C2×#3748D3semidirect
46. C2×C3⋊(C3⋊C8)C2×D3semidirect
55. C2×#272M16semidirect
147. H(3,𝔽3)#407semidirect
234. C32#5126semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#3748⋊C32
#117203232
#105910693