Finite Group Databasebeta

← #13159029

Subgroups — #13159029

Order 1728.

1048 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 172 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
1#11723916C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C2×#88799C212yesmaximal, complemented
4C2×#88795C212yesmaximal
5C3×#63630C211yesmaximal, characteristic, complemented
6#10592785—31nomaximal
7#10590374—34nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×#6308C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C33×C42C2211yescharacteristic
12#88859—22no—
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4⋊Dic3—22no—
15#88799—24no—
16#88795—24no—
17C32×C4⋊Q8—31no—
18C2×#3756—32no—
19C2×#3755—32no—
20#64084—31no—
21#64084—34no—
22#64084—64no—
23C3×#869D314yescomplemented
24C2×#3719—38no—
25C2×#3717—38no—
26#63630D311yescharacteristic, complemented
27C3×#803—34no—
28C22×C3⋊(C3⋊D3)—21no—
29C2×C33×C4D412yes—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32C4⋊(C3⋊(C3⋊D3))—22no—
33#6308—22no—
34C2×C32×Dic3—22no—
35C32×Dic6—22no—
36C32×Dic6—22no—
37C3×C3⋊(C3⋊C8)D414yes—
38#5086—94no—
39#4715—91no—
40C2×C32×Q8—32no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4⋊(C3⋊D3)—34no—
43C2×C4⋊(C3⋊D3)—64no—
44C2×#634—31no—
45C2×#634—34no—
46C2×#634—64no—
47C2×C3×Dic6C2×D318yes—
48C32×C4⋊C4—62no—
49C32×C42—24no—
50C32×C42C2×D311yescharacteristic
51C32×C42C2×D314yes—
52#3756—64no—
53#3755—64no—
54#3752—616no—
55#3752—62no—
56#3752—68no—
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—28no—
59C2×C3×C3⋊C8—38no—
60#3719—616no—
61#3717—616no—
62C22×C33C2×D411yescharacteristic
63C2×C3⋊(C3⋊D3)—41no—
64C2×C3⋊(C3⋊D3)—42no—
65C33×C4—22no—
66C33×C4C2×D412yes—
67C3⋊(C3⋊Dic3)—41no—
68C32×Dic3—42no—
69C3×C4⋊Q8—34no—
70C2×#216—98no—
71C2×#215—98no—
72C2×Dic12—92no—
73C2×#204—92no—
74#875—94no—
75#875—91no—
76#875—184no—
77#869C3⋊D311yescharacteristic, complemented
78C3×#97—91no—
79#803—34no—
80C22×C3⋊D3—68no—
81C22×C3⋊D3—61no—
82C22×C3⋊D3—64no—
83C32×Q8—62no—
84C32×Q8—62no—
85C2×C32×C4—28no—
86C2×C32×C4—24no—
87C2×C32×C4—62no—
88C2×C32×C4D1212yes—
89C2×C32×C4#6718yes—
90C2×C32×C4C22×D314yes—
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—68no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—64no—
95C4⋊(C3⋊D3)—62no—
96C4⋊(C3⋊D3)—68no—
97C4⋊(C3⋊D3)—128no—
98#634—62no—
99#634—68no—
100#634—128no—
101C2×C3×Dic3—28no—
102C3×Dic6—28no—
103C3×Dic6—28no—
104C3⋊(C3⋊C8)D1214yes—
105C3×C3⋊C8—316no—
106#497—271noSylow
107C2×C33#12711yescharacteristic
108C2×C33#13712yes—
109C3⋊(C3⋊D3)—81no—
110C2×C3×Q8—38no—
111C2×D12—94no—
112C2×D12—91no—
113C2×D12—184no—
114C2×Dic6—94no—
115C2×Dic6—91no—
116C2×Dic6—184no—
117C2×Dic6C2×C3⋊D312yes—
118C2×C3×C8—92no—
119C3×C4⋊C4—68no—
120C3×C42—24no—
121C3×C42D3214yes—
122C3×C42C2×C3⋊D311yescharacteristic
123#216—1816no—
124#215—1816no—
125(C2×D3)⋊C4—1816no—
126(C2×D3)⋊C4—188no—
127(C2×D3)⋊C4—182no—
128C4⋊Dic3—22no—
129C2×C3⋊C8—38no—
130Dic12—184no—
131#204—184no—
132C22×C32—24no—
133C22×C32C2×D1211yescharacteristic
134C22×C32C2×#6714yes—
135C2×C3⋊D3—128no—
136C2×C3⋊D3—1216no—
137C2×C3⋊D3—121no—
138C2×C3⋊D3—122no—
139C2×C3⋊D3—124no—
140C2×C3⋊D3—128no—
141C32×C4—22no—
142C32×C4—28no—
143C32×C4—28no—
144C32×C4—48no—
145C32×C4—122no—
146C32×C4C2×D1212yes—
147C32×C4D3×D418yes—
148C3⋊Dic3—128no—
149C3⋊Dic3—121no—
150C3⋊Dic3—124no—
151C3×Dic3—48no—
152C2×Q16—272no—
153C2×SD16—272no—
154C4⋊Q8—31no—
155#124—271no—
156#97—91no—
157C33#49711yescharacteristic, complemented, Sylow
158C22×D3—188no—
159C22×D3—184no—
160C22×D3—181no—
161C3×Q8—68no—
162C3×Q8—68no—
163C2×C3×C4—28no—
164C2×C3×C4—24no—
165C2×C3×C4—68no—
166C2×C3×C4#62618yes—
167C2×C3×C4#63812yes—
168C2×C3×C4C2×D3214yes—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—22no—
171C2×Dic3—188no—
172C2×Dic3—184no—
173C2×Dic3—181no—
174D12—188no—
175D12—182no—
176D12—368no—
177Dic6—22no—
178Dic6—22no—
179Dic6—188no—
180Dic6—182no—
181Dic6—368no—
182C3×C8—94no—
183C3⋊C8—316no—
184C2×C32—28no—
185C2×C32—24no—
186C2×C32#87411yescharacteristic
187C2×C32#90912yes—
188C2×C32#94014yes—
189C2×C32#94218yes—
190C3⋊D3—248no—
191C3⋊D3—241no—
192C3⋊D3—244no—
193C2×Q8—32no—
194C2×Q8—271no—
195C2×D4—271no—
196Q16—544no—
197SD16—544no—
198C2×C8—92no—
199C4⋊C4—62no—
200C22⋊C4—542no—
201C42D3×C3⋊D311yescharacteristic
202C22×C3—24no—
203C22×C3C2×#62614yes—
204C22×C3C2×#63811yescharacteristic
205C2×D3—368no—
206C2×D3—3616no—
207C2×D3—368no—
208C2×D3—361no—
209C2×D3—362no—
210C2×D3—364no—
211C3×C4—28no—
212C3×C4—22no—
213C3×C4—28no—
214C3×C4—48no—
215C3×C4—128no—
216C3×C4D3×D1218yes—
217C3×C4D4×C3⋊D312yes—
218Dic3—42no—
219Dic3—368no—
220Dic3—361no—
221Dic3—364no—
222C32—24no—
223C32#471511yescharacteristic, complemented
224C32#508614yescomplemented
225C23—541no—
226Q8—62no—
227Q8—62no—
228Q8—542no—
229D4—542no—
230C2×C4—62no—
231C2×C4—541no—
232C2×C4#629112yes—
233C2×C4C2×D3×C3⋊D311yescharacteristic
234C8—94no—
235C2×C3—24no—
236C2×C3—28no—
237C2×C3#6383318yes—
238C2×C3#6391114yes—
239C2×C3#6414911yescharacteristic
240C2×C3#6415112yes—
241D3—728no—
242D3—721no—
243D3—724no—
244C22—1081no—
245C22—1082no—
246C22C2×#629111yescharacteristic
247C4—22no—
248C4—122no—
249C4—1081no—
250C4C3⋊D3×D1212yes—
251C3—24no—
252C3#1059037414yescomplemented
253C3#1059278511yescharacteristic, complemented
254C2—2161no—
255C2#1172338112yes—
256C2#1172352411yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC42D3×C3⋊D3
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×#869C2semidirect
3. C2×#88799C2semidirect
5. C3×#63630C2semidirect
23. C3×#869D3semidirect
26. #63630D3semidirect
77. #869C3⋊D3semidirect
157. C33#497semidirect
223. C32#4715semidirect
224. C32#5086semidirect
252. C3#10590374semidirect
253. C3#10592785semidirect

Maximal subgroups

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

subgroupindex
#117239162
C32×#8692
C2×#887992
C2×#887952
C3×#636302
#105927853
#105903743