Finite Group Databasebeta

← #13138884

Subgroups — #13138884

Order 1728.

475 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 71 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×C8×#272C211yesmaximal, characteristic
2#11721094C212yesmaximal
3C2×#88433C211yesmaximal, characteristic, complemented
4#63643⋊C3C212yesmaximal, complemented
5#11720315C211yesmaximal, characteristic, complemented
6#10590848—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C8×#272—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88451—22no—
14#88448—22no—
15#88433—22no—
16C2×C3×D3×C8—32no—
17C2×C3×D3×C8—61no—
18#63857—32no—
19#63856—32no—
20C2×#3719—32no—
21#63643D312yescomplemented
22C3×#816—32no—
23#63575—32no—
24C22×#272D411yescharacteristic
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084D411yescharacteristic
27C4×#272—21no—
28C4×#272—21no—
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3—22no—
33C8×H(3,𝔽3)—21no—
34#6180D412yes—
35#5116—91no—
36#4813—92no—
37C2×#634C2×D312yes—
38C2×C3×C4×D3—32no—
39C2×C3×C4×D3—61no—
40C2×C3×Dic6—32no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D312yes—
43C4⋊(C3⋊Dic3)C2×D312yes—
44C3×C4⋊Dic3—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C3×D3×C8—64no—
48C3×D3×C8—64no—
49#3724—62no—
50Dic3⋊Dic3—62no—
51(C2×D3)⋊Dic3—62no—
52(C2×D3)⋊Dic3—62no—
53#3719—64no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×#272—21no—
56C2×#272—21no—
57C2×#272Q1612yescomplemented
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61(C3⋊Dic3)⋊C3—42no—
62(C3⋊Dic3)⋊C3—42no—
63C22×C3×C8—91no—
64#946—92no—
65C2×#216—92no—
66#924—92no—
67C2×Dic12—91no—
68C2×D3×C8—32no—
69C2×D3×C8—61no—
70#897—92no—
71C3×#103—92no—
72#818—91no—
73#816—32no—
74#816—92no—
75#807—92no—
76C22×C3×D3—32no—
77C22×C3×D3—61no—
78C2×C32×C4—21no—
79C2×C32×C4C22×D312yes—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3—22no—
82#634—22no—
83#634—22no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C32×C8—22no—
95C32×C8—22no—
96C3×C3⋊C8—34no—
97C3×C3⋊C8—62no—
98#467—271noSylow
99C2×H(3,𝔽3)#12111yescharacteristic
100C2×H(3,𝔽3)C2×Q1611yescharacteristic
101C2×H(3,𝔽3)#13511yescharacteristic
102#272—22no—
103C2×C3×Q8—92no—
104C22×C3×C4—91no—
105C2×C4×D3—32no—
106C2×C4×D3—61no—
107C2×Dic6—32no—
108C2×Dic6—92no—
109C2×C3×C8—32no—
110C2×C3×C8—61no—
111C2×C3×C8—91no—
112C2×C3×C8—182no—
113C2×C3×C8D3211yescharacteristic
114C3×C4⋊C4—92no—
115#217—182no—
116#216—184no—
117(C2×D3)⋊C4—182no—
118(C2×D3)⋊C4—182no—
119C4⋊Dic3—32no—
120C4⋊Dic3—92no—
121C4⋊Dic3—182no—
122Dic3⋊C4—182no—
123Dic3⋊C4—182no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—61no—
126Dic12—182no—
127D3×C8—64no—
128D3×C8—64no—
129C22×C32—21no—
130C22×C32D3×D412yes—
131C2×C3×D3—34no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4#23512yes—
140C32×C4D3×D412yes—
141C3⋊Dic3—42no—
142C3⋊Dic3—42no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C2×Q16—271no—
148C22×C8—91no—
149C22⋊Q8—272no—
150#107—271no—
151#103—92no—
152H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
153C23×C3—91no—
154C22×D3—32no—
155C22×D3—61no—
156C3×Q8—182no—
157C3×Q8—182no—
158C2×C3×C4—32no—
159C2×C3×C4—61no—
160C2×C3×C4—91no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163C2×C3×C4—182no—
164C2×C3×C4—182no—
165C2×C3×C4C2×D3211yescharacteristic
166C2×Dic3—32no—
167C2×Dic3—61no—
168C2×Dic3—62no—
169C2×Dic3—62no—
170C2×Dic3—182no—
171C2×Dic3—182no—
172C4×D3—62no—
173C4×D3—62no—
174C4×D3—62no—
175C4×D3—62no—
176Dic6—62no—
177Dic6—62no—
178Dic6—182no—
179Dic6—182no—
180C3×C8—21no—
181C3×C8—62no—
182C3×C8—62no—
183C3×C8—92no—
184C3⋊C8—34no—
185C3⋊C8—62no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32#88412yes—
190C2×C32#91612yes—
191C2×C32D3×Q1612yes—
192C3×D3—64no—
193C3×D3—64no—
194C2×Q8—92no—
195C22×C4—91no—
196Q16—542no—
197C2×C8—91no—
198C2×C8—182no—
199C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C4⋊C4—92no—
201C4⋊C4—542no—
202C22⋊C4—542no—
203C22×C3—32no—
204C22×C3—61no—
205C22×C3—92no—
206C22×C3—181no—
207C22×C3—181no—
208C22×C3#380211yescharacteristic
209C2×D3—34no—
210C2×D3—62no—
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—62no—
214C2×D3—62no—
215C3×C4—32no—
216C3×C4—32no—
217C3×C4—61no—
218C3×C4—61no—
219C3×C4—181no—
220C3×C4—362no—
221C3×C4—362no—
222C3×C4#379811yescharacteristic
223C3×C4#381111yescharacteristic
224Dic3—62no—
225Dic3—62no—
226Dic3—122no—
227Dic3—122no—
228Dic3—362no—
229Dic3—362no—
230C32—21no—
231C32#481312yescomplemented
232C23—91no—
233Q8—182no—
234Q8—182no—
235C2×C4—91no—
236C2×C4—181no—
237C2×C4—181no—
238C2×C4—182no—
239C2×C4—182no—
240C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C8—21no—
242C8—92no—
243C2×C3—32no—
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—182no—
250C2×C3#6380211yescharacteristic
251C2×C3#6380911yescharacteristic
252C2×C3#6391211yescharacteristic
253D3—64no—
254D3—64no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22#8865511yescharacteristic
259C4—181no—
260C4—362no—
261C4—362no—
262C4#8865011yescharacteristic
263C4#8867711yescharacteristic
264C3—32no—
265C3—61no—
266C3#1059085511yescharacteristic
267C2—182no—
268C2#1172083311yescharacteristic
269C2#1172084311yescharacteristic
270C2#1172114211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C2×#88433C2semidirect
4. #63643⋊C3C2semidirect
5. #11720315C2semidirect
21. #63643D3semidirect
57. C2×#272Q16semidirect
152. H(3,𝔽3)#467semidirect
231. C32#4813semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117210942
C2×#884332
#63643⋊C32
#117203152
#105908483