Finite Group Databasebeta

← #10592763⋊C3 #13142950

Subgroups — #10592763⋊C3 #13142950

Order 1728.

371 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 100 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3#63635⋊C3C212yesmaximal
4#63634⋊C3C212yesmaximal
5#63630⋊C3C211yesmaximal, characteristic
6#10592763C311yesmaximal, characteristic, complemented
7C3×#5064—31nomaximal
8C2×#634⋊C3—21no—
9C4⋊C4×H(3,𝔽3)—21no—
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×#125—21no—
18C32×#125D311yescharacteristic
19C32×#125C2×C311yescharacteristic
20#64078C2×C311yescharacteristic
21C3×#869—31no—
22#63635C2×C312yes—
23#63634C2×C312yes—
24#63630C2×C311yescharacteristic
25C3×#808—32no—
26C3×#807—32no—
27C3×#803—31no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—42no—
35(C3⋊(C3⋊C8))⋊C3Q814yescomplemented
36C3×#501—91no—
37#5064—31no—
38#5064—91no—
39C2×#634—21no—
40C2×C3×Dic6—61no—
41C32×C4⋊C4—22no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×D311yescharacteristic
50C32×C4⋊C4C2×D311yescharacteristic
51C32×C4⋊C4C22×C311yescharacteristic
52C32×C4⋊C4C22×C311yescharacteristic
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C22×C311yescharacteristic
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)C22×C312yes—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—61no—
61C2×C3×C3⋊C8—32no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×Q812yes—
68(C3⋊Dic3)⋊C3—42no—
69C3×C4⋊Q8—91no—
70C3×#125—31no—
71C3×#125—31no—
72C3×#125—61no—
73C3×#125C3×D311yescharacteristic
74#869—31no—
75#869—91no—
76C3×#107—92no—
77C3×#106—92no—
78C3×#97—91no—
79#808—32no—
80#808—92no—
81#807—32no—
82#807—92no—
83#803—31no—
84#803—91no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4#6711yescharacteristic
95C2×C32×C4#6711yescharacteristic
96C2×C32×C4C3×D411yescharacteristic
97C2×C32×C4C3×D411yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3—22no—
101#634—42no—
102C2×C3×Dic3—62no—
103C3×Dic6—122no—
104C3⋊(C3⋊C8)C3×Q814yescomplemented
105C3×C3⋊C8—34no—
106#501—91noSylow
107C2×H(3,𝔽3)C4⋊Q811yescharacteristic
108C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
109C2×H(3,𝔽3)#13711yescharacteristic
110C2×C3×Q8—181no—
111C2×Dic6—61no—
112C2×Dic6—181no—
113C2×C3×C8—92no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—31no—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—61no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—62no—
125C3×C4⋊C4—62no—
126C3×C4⋊C4—61no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—92no—
129C3×C4⋊C4—181no—
130C3×C4⋊C4C2×C3×D311yescharacteristic
131C3×C4⋊C4C2×C3×D311yescharacteristic
132C3×C42—31no—
133C3×C42—31no—
134C3×C42—61no—
135C3×C42C2×C3×D311yescharacteristic
136C4⋊Dic3—32no—
137C4⋊Dic3—61no—
138C4⋊Dic3—92no—
139C4⋊Dic3—181no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—92no—
142C22×C32—21no—
143C22×C32C2×#6711yescharacteristic
144C22×C32C2×C3×D411yescharacteristic
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—22no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4—42no—
153C32×C4—42no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4D3×Q812yes—
159C32×C4C2×C3×Q812yes—
160C3⋊Dic3—42no—
161C3×Dic3—122no—
162C4⋊Q8—91no—
163#125(C3⋊D3)⋊C311yescharacteristic
164#107—92no—
165#106—92no—
166#97—91no—
167H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
168C3×Q8—362no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—182no—
187C2×C3×C4C3×#6711yescharacteristic
188C2×C3×C4C3×#6711yescharacteristic
189C2×C3×C4C22×C3×D311yescharacteristic
190C2×Dic3—62no—
191C2×Dic3—182no—
192Dic6—122no—
193Dic6—362no—
194C3×C8—94no—
195C3⋊C8—34no—
196C3⋊C8—94no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32#94411yescharacteristic
201C2×C32#94911yescharacteristic
202C2×C32#95111yescharacteristic
203C2×C32C3×C4⋊Q811yescharacteristic
204C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
205C2×C32C3×#13711yescharacteristic
206C2×Q8—181no—
207C2×C8—92no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—92no—
211C4⋊C4—181no—
212C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
213C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
214C42C2×(C3⋊D3)⋊C311yescharacteristic
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—61no—
218C22×C3C2×C3×#6711yescharacteristic
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—32no—
222C3×C4—32no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4—62no—
230C3×C4—121no—
231C3×C4—121no—
232C3×C4—121no—
233C3×C4—121no—
234C3×C4—121no—
235C3×C4—121no—
236C3×C4—122no—
237C3×C4—122no—
238C3×C4—362no—
239C3×C4C3×D3×Q812yes—
240Dic3—122no—
241Dic3—362no—
242C32—21no—
243C32#506411yescharacteristic, complemented
244C32C3×#50111yescharacteristic, complemented
245Q8—362no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—182no—
249C2×C4#638⋊C311yescharacteristic
250C2×C4#638⋊C311yescharacteristic
251C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
252C8—94no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—31no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—61no—
260C2×C3—61no—
261C2×C3—61no—
262C2×C3C3×#94411yescharacteristic
263C2×C3C3×#94911yescharacteristic
264C2×C3C3×#95111yescharacteristic
265C22C2×#638⋊C311yescharacteristic
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—362no—
271C4Q8×(C3⋊D3)⋊C312yes—
272C3—31no—
273C3—31no—
274C3—61no—
275C3C3×#506411yescharacteristic
276C2#64153⋊C311yescharacteristic
277C2#64160⋊C311yescharacteristic
278C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#501
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
6. #10592763C3semidirect
35. (C3⋊(C3⋊C8))⋊C3Q8semidirect
104. C3⋊(C3⋊C8)C3×Q8semidirect
167. H(3,𝔽3)#501semidirect
243. C32#5064semidirect
244. C32C3×#501semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64078⋊C32
#63635⋊C32
#63634⋊C32
#63630⋊C32
#105927633
C3×#50643