Finite Group Databasebeta

← #13138885

Subgroups — #13138885

Order 1728.

453 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 69 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×#88527C211yesmaximal, characteristic
2#11721094C212yesmaximal
3C2×#88433C211yesmaximal, characteristic, complemented
4#63643⋊C3C212yesmaximal, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#10590849—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#88527—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88451—22no—
14#88448—22no—
15#88433—22no—
16C2×C3×ΓL(1,𝔽25)—32no—
17C2×C3×ΓL(1,𝔽25)—61no—
18#63857—32no—
19#63856—32no—
20C2×#3719—32no—
21#63643D312yescomplemented
22C3×#816—32no—
23#63573—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#5137—91no—
36#4816—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×ΓL(1,𝔽25)—64no—
48C3×ΓL(1,𝔽25)—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—22no—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—21no—
60(C3⋊Dic3)⋊C3—42no—
61(C3⋊Dic3)⋊C3—42no—
62C2×C3×M16—91no—
63#946—92no—
64C2×#216—92no—
65#924—92no—
66C2×Dic12—91no—
67C2×ΓL(1,𝔽25)—32no—
68C2×ΓL(1,𝔽25)—61no—
69#897—92no—
70C3×#103—92no—
71#817—91no—
72#816—32no—
73#816—92no—
74#808—92no—
75C22×C3×D3—32no—
76C22×C3×D3—61no—
77C2×C32×C4—21no—
78C2×C32×C4C22×D312yes—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3—22no—
81#634—22no—
82#634—22no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C3×Dic6—62no—
92C3×Dic6—62no—
93C32×C8—22no—
94C32×C8—22no—
95C3×C3⋊C8—34no—
96C3×C3⋊C8—62no—
97#470—271noSylow
98C2×H(3,𝔽3)#12111yescharacteristic
99C2×H(3,𝔽3)#13712yes—
100#272—41no—
101C2×C3×Q8—92no—
102C22×C3×C4—91no—
103C2×C4×D3—32no—
104C2×C4×D3—61no—
105C2×Dic6—32no—
106C2×Dic6—92no—
107C3×M16—182no—
108C2×C3×C8—32no—
109C2×C3×C8—61no—
110C2×C3×C8—91no—
111C2×C3×C8D3211yescharacteristic
112C3×C4⋊C4—92no—
113#217—182no—
114#216—184no—
115(C2×D3)⋊C4—182no—
116(C2×D3)⋊C4—182no—
117C4⋊Dic3—32no—
118C4⋊Dic3—92no—
119C4⋊Dic3—182no—
120Dic3⋊C4—182no—
121Dic3⋊C4—182no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—61no—
124Dic12—182no—
125ΓL(1,𝔽25)—64no—
126ΓL(1,𝔽25)—64no—
127C22×C32—21no—
128C22×C32D3×D412yes—
129C2×C3×D3—62no—
130C2×C3×D3—62no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—122no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4#23512yes—
137C32×C4D3×D412yes—
138C3⋊Dic3—42no—
139C3⋊Dic3—42no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144C2×Q16—271no—
145C2×M16—91no—
146C22⋊Q8—272no—
147#106—271no—
148#103—92no—
149H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
150C23×C3—91no—
151C22×D3—32no—
152C22×D3—61no—
153C3×Q8—182no—
154C3×Q8—182no—
155C2×C3×C4—32no—
156C2×C3×C4—61no—
157C2×C3×C4—91no—
158C2×C3×C4—181no—
159C2×C3×C4—181no—
160C2×C3×C4—182no—
161C2×C3×C4—182no—
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—32no—
164C2×Dic3—61no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—182no—
168C2×Dic3—182no—
169C4×D3—62no—
170C4×D3—62no—
171C4×D3—62no—
172C4×D3—62no—
173Dic6—62no—
174Dic6—62no—
175Dic6—182no—
176Dic6—182no—
177C3×C8—21no—
178C3×C8—62no—
179C3×C8—62no—
180C3×C8—92no—
181C3⋊C8—34no—
182C3⋊C8—62no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32#88412yes—
186C2×C32#91512yes—
187C2×C32#91812yes—
188C3×D3—122no—
189C3×D3—122no—
190C2×Q8—92no—
191C22×C4—91no—
192Q16—542no—
193M16—182no—
194C2×C8—91no—
195C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C4⋊C4—92no—
197C4⋊C4—542no—
198C22⋊C4—542no—
199C22×C3—32no—
200C22×C3—61no—
201C22×C3—182no—
202C22×C3—181no—
203C22×C3#380211yescharacteristic
204C2×D3—62no—
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—122no—
209C3×C4—32no—
210C3×C4—32no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—181no—
214C3×C4—362no—
215C3×C4—362no—
216C3×C4#379811yescharacteristic
217C3×C4#381111yescharacteristic
218Dic3—62no—
219Dic3—62no—
220Dic3—122no—
221Dic3—122no—
222Dic3—362no—
223Dic3—362no—
224C32—21no—
225C32#481612yescomplemented
226C23—91no—
227Q8—182no—
228Q8—182no—
229C2×C4—91no—
230C2×C4—181no—
231C2×C4—181no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C8—21no—
236C8—92no—
237C2×C3—32no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—62no—
241C2×C3—61no—
242C2×C3—361no—
243C2×C3#6381112yes—
244C2×C3#6391211yescharacteristic
245D3—122no—
246D3—122no—
247C22—182no—
248C22—181no—
249C22#8865511yescharacteristic
250C4—181no—
251C4—362no—
252C4—362no—
253C4#8865011yescharacteristic
254C4#8867711yescharacteristic
255C3—32no—
256C3—61no—
257C3#1059085611yescharacteristic
258C2—361no—
259C2#1172084512yes—
260C2#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)#470
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. #11720314C2semidirect
21. #63643D3semidirect
149. H(3,𝔽3)#470semidirect
225. C32#4816semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117210942
C2×#884332
#63643⋊C32
#117203142
#105908493