Finite Group Databasebeta

← #13135220

Subgroups — #13135220

Order 1728.

629 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 111 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×#88552C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3#10589186—32nomaximal
4C24×H(3,𝔽3)C2211yescharacteristic
5C22×#1084C412yescomplemented
6C22×(C3⋊Dic3)⋊C3—22no—
7#88552—22no—
8#3757⋊C3—22no—
9#3757⋊C3C414yescomplemented
10C2×#3757D312yes—
11C2×C3×#217—32no—
12C2×C3×#217—32no—
13C2×C3×#217—61no—
14C23×H(3,𝔽3)—42no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C23×H(3,𝔽3)C2×C412yes—
17C23×H(3,𝔽3)D412yes—
18C23×H(3,𝔽3)D411yescharacteristic
19C23×H(3,𝔽3)Q811yescharacteristic
20C2×#1084—22no—
21C2×#1084—42no—
22C2×(C3⋊Dic3)⋊C3—24no—
23C2×(C3⋊Dic3)⋊C3—44no—
24#4527—92no—
25#4526—91no—
26C24×C32—21no—
27C24×C32C2×D312yes—
28C22×C3⋊Dic3—22no—
29C22×C3×Dic3—34no—
30C22×C3×Dic3—62no—
31C22×C3×Dic3—62no—
32#3757—22no—
33#3757Dic314yescomplemented
34C3×#217—34no—
35C3×#217—64no—
36C3×#217—62no—
37C3×#217—62no—
38C3×#217—62no—
39C22×H(3,𝔽3)—22no—
40C22×H(3,𝔽3)—24no—
41C22×H(3,𝔽3)—42no—
42C22×H(3,𝔽3)—41no—
43C22×H(3,𝔽3)—41no—
44C22×H(3,𝔽3)C4211yescharacteristic
45C22×H(3,𝔽3)C22⋊C412yes—
46C22×H(3,𝔽3)C22⋊C411yescharacteristic
47C22×H(3,𝔽3)C4⋊C411yescharacteristic
48C22×H(3,𝔽3)C4⋊C412yes—
49#1084—42no—
50(C3⋊Dic3)⋊C3—44no—
51C2×C3×C22⋊C4—92no—
52C2×C3×C22⋊C4—91no—
53C2×#217—32no—
54C2×#217—32no—
55C2×#217—61no—
56C2×#217—92no—
57C23×C32—22no—
58C23×C32—21no—
59C23×C32—21no—
60C23×C32—22no—
61C23×C32—21no—
62C23×C32—44no—
63C23×C32—44no—
64C23×C32Dic612yes—
65C23×C32C4×D312yes—
66C23×C32C4×D312yes—
67C23×C32D1212yes—
68C23×C32C2×Dic312yes—
69C23×C32#6712yes—
70C23×C32#6712yes—
71C2×C3⋊Dic3—24no—
72C2×C3⋊Dic3—44no—
73C2×C3×Dic3—64no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—64no—
76C2×C3×Dic3—124no—
77C2×C3×Dic3—124no—
78C2×C3×Dic3—124no—
79#342—271noSylow
80C2×H(3,𝔽3)—21no—
81C2×H(3,𝔽3)—21no—
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)—22no—
84C2×H(3,𝔽3)#9511yescharacteristic
85C2×H(3,𝔽3)#9912yes—
86C24×C3—32no—
87C24×C3—61no—
88C24×C3D3211yescharacteristic
89C22×C3×C4—92no—
90C22×C3×C4—182no—
91C22×Dic3—34no—
92C22×Dic3—62no—
93C22×Dic3—62no—
94C22×Dic3—182no—
95C3×C22⋊C4—94no—
96C3×C22⋊C4—182no—
97C3×C22⋊C4—182no—
98#217—34no—
99#217—62no—
100#217—62no—
101#217—64no—
102#217—62no—
103#217—94no—
104#217—182no—
105C22×C32—24no—
106C22×C32—24no—
107C22×C32—24no—
108C22×C32—24no—
109C22×C32—21no—
110C22×C32—22no—
111C22×C32—21no—
112C22×C32—22no—
113C22×C32—21no—
114C22×C32—44no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—42no—
118C22×C32—42no—
119C22×C32—44no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32C4×Dic312yes—
123C22×C32Dic3⋊C412yes—
124C22×C32Dic3⋊C412yes—
125C22×C32C4⋊Dic312yes—
126C22×C32(C2×D3)⋊C412yes—
127C22×C32(C2×D3)⋊C412yes—
128C22×C32#21712yes—
129C3⋊Dic3—44no—
130C3×Dic3—124no—
131C3×Dic3—124no—
132C3×Dic3—124no—
133C2×C22⋊C4—92no—
134C2×C22⋊C4—91no—
135H(3,𝔽3)#34211yescharacteristic, complemented, Sylow
136C23×C3—32no—
137C23×C3—32no—
138C23×C3—32no—
139C23×C3—32no—
140C23×C3—32no—
141C23×C3—32no—
142C23×C3—32no—
143C23×C3—42no—
144C23×C3—61no—
145C23×C3—62no—
146C23×C3—61no—
147C23×C3—61no—
148C23×C3—62no—
149C23×C3—124no—
150C23×C3—124no—
151C23×C3D3×Dic312yes—
152C23×C3#62411yescharacteristic
153C23×C3#62511yescharacteristic
154C23×C3#62612yes—
155C23×C3#62711yescharacteristic
156C2×C3×C4—184no—
157C2×C3×C4—182no—
158C2×C3×C4—364no—
159C2×C3×C4—362no—
160C2×Dic3—64no—
161C2×Dic3—64no—
162C2×Dic3—64no—
163C2×Dic3—124no—
164C2×Dic3—124no—
165C2×Dic3—124no—
166C2×Dic3—184no—
167C2×Dic3—364no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—22no—
174C2×C32—22no—
175C2×C32—22no—
176C2×C32—21no—
177C2×C32—22no—
178C2×C32—42no—
179C2×C32—42no—
180C2×C32#80614yes—
181C2×C32C3⋊#9512yes—
182C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
183C22×C4—92no—
184C22×C4—182no—
185C22⋊C4—94no—
186C22⋊C4—182no—
187C22⋊C4—182no—
188C22×C3—24no—
189C22×C3—22no—
190C22×C3—32no—
191C22×C3—32no—
192C22×C3—32no—
193C22×C3—32no—
194C22×C3—32no—
195C22×C3—32no—
196C22×C3—32no—
197C22×C3—42no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—64no—
201C22×C3—64no—
202C22×C3—64no—
203C22×C3—64no—
204C22×C3—61no—
205C22×C3—62no—
206C22×C3—61no—
207C22×C3—62no—
208C22×C3—61no—
209C22×C3—124no—
210C22×C3—122no—
211C22×C3—122no—
212C22×C3—122no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3—124no—
216C22×C3—122no—
217C22×C3Dic3211yescharacteristic
218C22×C3(C2×D3)⋊Dic312yes—
219C22×C3#372211yescharacteristic
220C22×C3Dic3⋊Dic312yes—
221C22×C3#372411yescharacteristic
222C3×C4—364no—
223C3×C4—362no—
224Dic3—124no—
225Dic3—124no—
226Dic3—124no—
227Dic3—364no—
228C32—21no—
229C32#452712yescomplemented
230C23—42no—
231C23#619611yescharacteristic
232C23((C3⋊D3)⋊C3)⋊C412yes—
233C23#619811yescharacteristic
234C23#619911yescharacteristic
235C23#620012yes—
236C2×C4—184no—
237C2×C4—182no—
238C2×C4—364no—
239C2×C4—362no—
240C2×C3—22no—
241C2×C3—21no—
242C2×C3—22no—
243C2×C3—21no—
244C2×C3—32no—
245C2×C3—34no—
246C2×C3—62no—
247C2×C3—62no—
248C2×C3—62no—
249C2×C3—62no—
250C2×C3—62no—
251C2×C3—62no—
252C2×C3—62no—
253C2×C3—62no—
254C2×C3—61no—
255C2×C3—62no—
256C2×C3—122no—
257C2×C3—122no—
258C2×C3#6357911yescharacteristic
259C2×C3#6358112yes—
260C22—22no—
261C22—24no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22#8844711yescharacteristic
266C22#8844812yes—
267C22#8844911yescharacteristic
268C22#8845012yes—
269C22#8845111yescharacteristic
270C4—364no—
271C4—362no—
272C3—32no—
273C3—61no—
274C3#1058918711yescharacteristic
275C2—22no—
276C2—21no—
277C2—22no—
278C2—21no—
279C2#1172033411yescharacteristic
280C2#1172033512yes—
subgroupG/H
centerC22#88451
commutatorC22×H(3,𝔽3)C42
FrattiniC24×C3D32
Fitting, prime radical, cosocleC24×H(3,𝔽3)C22
socleC22×C3#3722
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#342
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
5. C22×#1084C4semidirect
9. #3757⋊C3C4semidirect
33. #3757Dic3semidirect
135. H(3,𝔽3)#342semidirect
229. C32#4527semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C2×#3757⋊C32
#105891863