Finite Group Databasebeta

← C2×#127⋊H(3,𝔽3) #13152189

Subgroups — C2×#127⋊H(3,𝔽3) #13152189

Order 1728.

633 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 56 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#127⋊H(3,𝔽3)C212yesmaximal, direct product factor
2C2×C32⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
3C2×C32×#127C311yesmaximal, characteristic, complemented
4C2×C3×#127⋊C3C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)—41nomaximal
6C32⋊(C42⋊C3)C2211yescharacteristic, complemented
7C22×C32×D4—31no—
8C22×C32×D4—31no—
9C32×#127—32no—
10C32×#127C2×C312yescomplemented
11C2×C32×C42C2×C311yescharacteristic, complemented
12C3×#127⋊C3C2×C316yescomplemented
13C2×C3×C42⋊C3C2×C313yescomplemented
14C2×C22⋊H(3,𝔽3)—41no—
15C2×C22⋊H(3,𝔽3)—42no—
16C2×C3×#127—31no—
17C2×C3×#127C3211yescharacteristic
18C2×#127⋊C3—33no—
19C24×C32—31no—
20C24×C32A411yescharacteristic
21C22×C3×A4—43no—
22C2×C32×D4—32no—
23C2×C32×D4—32no—
24C2×C32×D4—32no—
25C2×C32×D4—32no—
26C2×C32×D4—62no—
27C2×C32×D4—62no—
28C2×C32×D4—62no—
29C2×C32×D4—62no—
30C22×C32×C4—31no—
31C32×C42—31no—
32C32×C42C22×C311yescharacteristic, complemented
33C3×C42⋊C3C22×C313yescomplemented
34C22×H(3,𝔽3)—161no—
35C22⋊H(3,𝔽3)—41no—
36C22×C3×D4—31no—
37C22×C3×D4—33no—
38C22×C3×D4—33no—
39C22×C3×D4—31no—
40C3×#127—32no—
41C3×#127—36no—
42C3×#127—32no—
43C3×#127C2×C3212yes—
44C2×C3×C42—31no—
45C2×C3×C42C2×C3211yescharacteristic
46#127⋊C3—36no—
47C2×C42⋊C3—33no—
48C23×C32—32no—
49C23×C32—61no—
50C23×C32—61no—
51C23×C32—61no—
52C23×C32—62no—
53C23×C32—61no—
54C23×C32—62no—
55C23×C32C2×A411yescharacteristic
56C23×C32C2×A412yes—
57C2×C3×A4—43no—
58C2×C3×A4—46no—
59C32×D4—64no—
60C32×D4—64no—
61C32×D4—64no—
62C32×D4—64no—
63C2×C32×C4—32no—
64C2×C32×C4—32no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×#127H(3,𝔽3)11yescharacteristic, complemented, Sylow
68C2×H(3,𝔽3)—162no—
69C2×H(3,𝔽3)—161no—
70C24×C3—33no—
71C24×C3—31no—
72C24×C3—31no—
73C24×C3C3×A411yescharacteristic
74C22×A4—123no—
75C2×C3×D4—32no—
76C2×C3×D4—36no—
77C2×C3×D4—36no—
78C2×C3×D4—32no—
79C2×C3×D4—36no—
80C2×C3×D4—36no—
81C2×C3×D4—32no—
82C2×C3×D4—32no—
83C2×C3×D4—62no—
84C2×C3×D4—62no—
85C2×C3×D4—62no—
86C2×C3×D4—62no—
87C2×C3×D4—66no—
88C2×C3×D4—66no—
89C2×C3×D4—66no—
90C2×C3×D4—66no—
91C22×C3×C4—33no—
92C22×C3×C4—31no—
93C3×C42—31no—
94C3×C42—33no—
95C3×C42—31no—
96C3×C42C22×C3211yescharacteristic
97C42⋊C3—33no—
98C22×C32—31no—
99C22×C32—31no—
100C22×C32—41no—
101C22×C32—62no—
102C22×C32—62no—
103C22×C32—64no—
104C22×C32—121no—
105C22×C32—122no—
106C22×C32—121no—
107C22×C32—121no—
108C22×C32—163no—
109C22×C32C22×A411yescharacteristic
110C3×A4—43no—
111C32×C4—32no—
112C32×C4—32no—
113C22×D4—31no—
114C22×D4—31no—
115#127—32no—
116#127C2×H(3,𝔽3)12yescomplemented
117C2×C42C2×H(3,𝔽3)11yescharacteristic, complemented
118H(3,𝔽3)—161noSylow
119C23×C3—36no—
120C23×C3—32no—
121C23×C3—31no—
122C23×C3—32no—
123C23×C3—61no—
124C23×C3—61no—
125C23×C3—63no—
126C23×C3—63no—
127C23×C3—61no—
128C23×C3—62no—
129C23×C3—63no—
130C23×C3—66no—
131C23×C3—63no—
132C23×C3—66no—
133C23×C3—61no—
134C23×C3—62no—
135C23×C3C2×C3×A411yescharacteristic
136C23×C3C2×C3×A412yes—
137C2×A4—123no—
138C2×A4—126no—
139C3×D4—64no—
140C3×D4—64no—
141C3×D4—64no—
142C3×D4—612no—
143C3×D4—612no—
144C3×D4—612no—
145C3×D4—612no—
146C3×D4—64no—
147C2×C3×C4—33no—
148C2×C3×C4—33no—
149C2×C3×C4—32no—
150C2×C3×C4—36no—
151C2×C3×C4—36no—
152C2×C3×C4—32no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C32—31no—
156C2×C32—31no—
157C2×C32—42no—
158C2×C32—122no—
159C2×C32—166no—
160C2×C32—163no—
161C2×C32#127⋊C311yescharacteristic, complemented
162C24—31no—
163C24C22⋊H(3,𝔽3)11yescharacteristic
164C2×D4—32no—
165C2×D4—32no—
166C2×D4—32no—
167C2×D4—32no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C22×C4—31no—
173C42—31no—
174C42C22×H(3,𝔽3)11yescharacteristic, complemented
175C22×C3—31no—
176C22×C3—33no—
177C22×C3—33no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—41no—
181C22×C3—62no—
182C22×C3—66no—
183C22×C3—66no—
184C22×C3—612no—
185C22×C3—62no—
186C22×C3—64no—
187C22×C3—121no—
188C22×C3—121no—
189C22×C3—123no—
190C22×C3—126no—
191C22×C3—123no—
192C22×C3—122no—
193C22×C3—121no—
194C22×C3—123no—
195C22×C3—121no—
196C22×C3—483no—
197C22×C3C22×C3×A411yescharacteristic
198A4—123no—
199C3×C4—36no—
200C3×C4—36no—
201C3×C4—32no—
202C3×C4—32no—
203C32—163no—
204C32C2×#127⋊C311yescharacteristic, complemented
205C23—32no—
206C23—61no—
207C23—61no—
208C23—61no—
209C23—62no—
210C23—61no—
211C23—62no—
212C23C2×C22⋊H(3,𝔽3)11yescharacteristic
213C23C2×C22⋊H(3,𝔽3)12yes—
214D4—64no—
215D4—64no—
216D4—64no—
217D4—64no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C3—33no—
223C2×C3—33no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—42no—
228C2×C3—122no—
229C2×C3—126no—
230C2×C3—122no—
231C2×C3—486no—
232C2×C3—483no—
233C2×C3C3×#127⋊C311yescharacteristic
234C22—31no—
235C22—31no—
236C22—41no—
237C22—62no—
238C22—62no—
239C22—64no—
240C22—121no—
241C22—122no—
242C22—121no—
243C22—121no—
244C22C22×C22⋊H(3,𝔽3)11yescharacteristic
245C4—32no—
246C4—32no—
247C3—31no—
248C3—483no—
249C3C2×C3×#127⋊C311yescharacteristic
250C2—31no—
251C2—31no—
252C2—42no—
253C2—122no—
254C2#127⋊H(3,𝔽3)11yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
FittingC2×C32×#127C3
socleC23×C3C2×C3×A4
hypercenterC2×C32#127⋊C3
hypocenterC42C22×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #127⋊H(3,𝔽3)C2direct (N × H)
2. C2×C32⋊(C42⋊C3)C2semidirect
3. C2×C32×#127C3semidirect
4. C2×C3×#127⋊C3C3semidirect
6. C32⋊(C42⋊C3)C22semidirect
10. C32×#127C2×C3semidirect
11. C2×C32×C42C2×C3semidirect
12. C3×#127⋊C3C2×C3semidirect
13. C2×C3×C42⋊C3C2×C3semidirect
32. C32×C42C22×C3semidirect
33. C3×C42⋊C3C22×C3semidirect
67. C2×#127H(3,𝔽3)semidirect
116. #127C2×H(3,𝔽3)semidirect
117. C2×C42C2×H(3,𝔽3)semidirect
161. C2×C32#127⋊C3semidirect
174. C42C22×H(3,𝔽3)semidirect
204. C32C2×#127⋊C3semidirect
254. C2#127⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
#127⋊H(3,𝔽3)2
C2×C32⋊(C42⋊C3)2
C2×C32×#1273
C2×C3×#127⋊C33
C22×C22⋊H(3,𝔽3)4