Finite Group Databasebeta

← #13140043

Subgroups — #13140043

Order 1728.

523 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88538C211yesmaximal, characteristic, complemented
2#64139⋊C3C212yesmaximal, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4#11720318C212yesmaximal, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#10591165—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10#88538—22no—
11C2×#6180C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#88432—22no—
16#64139D312yescomplemented
17C3×#930—32no—
18C2×C3×#208—32no—
19C2×C3×#208—61no—
20C2×#3714—32no—
21#63573—32no—
22#63564—32no—
23#63563—32no—
24C23×H(3,𝔽3)D411yescharacteristic
25C22×(C3⋊D3)⋊C3—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30#638⋊C3—44no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#6180—21no—
35#6180D412yescomplemented
36#5135—91no—
37#5039—92no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C2×#638—22no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C3×#67—62no—
43C2×C3×D12—32no—
44#3752—22no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C3×(C2×D3)⋊C4—62no—
47C3×C4⋊Dic3—32no—
48C3×#208—64no—
49C3×#208—64no—
50C2×C3×C3⋊C8—32no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—61no—
53C2×C3×C3⋊C8—61no—
54#3714—64no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)—22no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×(C3⋊D3)⋊C3—42no—
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C4×H(3,𝔽3)—21no—
62C4×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64(C3⋊Dic3)⋊C3—42no—
65C2×C3×M16—91no—
66C3×#121—92no—
67C2×#213—92no—
68#930—32no—
69#930—92no—
70C2×#208—32no—
71C2×#208—61no—
72C2×D24—91no—
73#832—92no—
74#821—92no—
75#817—91no—
76#809—92no—
77#808—92no—
78C23×C32—21no—
79C23×C32#6712yes—
80C22×C3⋊D3—22no—
81C22×C3×D3—62no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4#6712yes—
89C2×C32×C4C22×D312yes—
90#638—44no—
91C2×C3⋊Dic3—22no—
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—22no—
94C3×#67—124no—
95C2×C3×Dic3—62no—
96C3×D12—62no—
97C3×D12—62no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—62no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—62no—
102#469—271noSylow
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)#12111yescharacteristic
105C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
106(C3⋊D3)⋊C3—82no—
107C2×C3×D4—92no—
108C2×C3×D4—182no—
109C22×C3×C4—32no—
110C22×C3×C4—61no—
111C22×C3×C4D3211yescharacteristic
112C2×#67—62no—
113C2×#67—182no—
114C2×D12—32no—
115C2×D12—92no—
116C3×M16—182no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C3×C4⋊C4—92no—
120C3×C22⋊C4—182no—
121#213—184no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—182no—
124C4⋊Dic3—32no—
125C4⋊Dic3—92no—
126#208—64no—
127#208—64no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—32no—
130C2×C3⋊C8—61no—
131C2×C3⋊C8—61no—
132D24—182no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—22no—
137C22×C32—21no—
138C22×C32—42no—
139C22×C32C2×#6712yes—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—22no—
147C32×C4—22no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4D3×D412yes—
151C32×C4#23712yes—
152C3⋊Dic3—42no—
153C3×Dic3—122no—
154C2×D8—271no—
155C2×M16—91no—
156#121—92no—
157#106—271no—
158#102—272no—
159H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
160C23×C3—32no—
161C23×C3—61no—
162C23×C3—182no—
163C23×C3#62511yescharacteristic
164C22×D3—62no—
165C22×D3—182no—
166C3×D4—182no—
167C3×D4—182no—
168C3×D4—364no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—182no—
180C2×C3×C4#62511yescharacteristic
181C2×C3×C4C2×D3211yescharacteristic
182#67—124no—
183#67—364no—
184C2×Dic3—62no—
185C2×Dic3—182no—
186D12—62no—
187D12—62no—
188D12—182no—
189D12—182no—
190C3×C8—92no—
191C3×C8—181no—
192C3⋊C8—34no—
193C3⋊C8—62no—
194C3⋊C8—62no—
195C3⋊C8—62no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32—42no—
199C2×C32—42no—
200C2×C32#93212yes—
201C2×C32#93912yes—
202C2×C32#94912yes—
203C3⋊D3—82no—
204C3×D3—242no—
205C2×D4—92no—
206C2×D4—182no—
207C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
208D8—542no—
209M16—182no—
210C2×C8—91no—
211C2×C8—91no—
212C4⋊C4—92no—
213C22⋊C4—182no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—32no—
217C22×C3—62no—
218C22×C3—62no—
219C22×C3—62no—
220C22×C3—62no—
221C22×C3—61no—
222C22×C3—122no—
223C22×C3—362no—
224C22×C3—362no—
225C22×C3—362no—
226C22×C3C2×#62511yescharacteristic
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—362no—
231C2×D3—362no—
232C2×D3—362no—
233C3×C4—21no—
234C3×C4—32no—
235C3×C4—32no—
236C3×C4—62no—
237C3×C4—62no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—362no—
241C3×C4#380511yescharacteristic
242C3×C4#381111yescharacteristic
243Dic3—122no—
244Dic3—362no—
245C32—21no—
246C32#503912yescomplemented
247C23—182no—
248C23#619811yescharacteristic
249D4—182no—
250D4—182no—
251D4—364no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—182no—
255C2×C4#619811yescharacteristic
256C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C8—92no—
258C8—181no—
259C2×C3—32no—
260C2×C3—32no—
261C2×C3—32no—
262C2×C3—41no—
263C2×C3—62no—
264C2×C3—61no—
265C2×C3—122no—
266C2×C3—122no—
267C2×C3—722no—
268C2×C3#6382312yes—
269C2×C3#6398011yescharacteristic
270D3—242no—
271D3—722no—
272C22—22no—
273C22—21no—
274C22—362no—
275C22—362no—
276C22—362no—
277C22C2×#619811yescharacteristic
278C4—21no—
279C4—362no—
280C4#8867211yescharacteristic
281C4#8867711yescharacteristic
282C3—32no—
283C3—61no—
284C3#1059116711yescharacteristic
285C2—41no—
286C2—722no—
287C2#1172106112yes—
288C2#1172131811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#469
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88538C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#88432C2semidirect
4. #11720318C2semidirect
5. #11720314C2semidirect
16. #64139D3semidirect
35. #6180D4semidirect
159. H(3,𝔽3)#469semidirect
246. C32#5039semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64139⋊C32
C2×#884322
#117203182
#117203142
#105911653