Finite Group Databasebeta

← C2×(C3⋊#859)⋊C3 #13146904

Subgroups — C2×(C3⋊#859)⋊C3 #13146904

Order 1728.

414 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊#859)⋊C3C212yesmaximal, direct product factor
3C2×C3⋊#859C311yesmaximal, characteristic, complemented
4C2×C3×#835—31nomaximal
5C22×(C3⋊Dic3)⋊C3—41nomaximal
6C2×Q8⋊H(3,𝔽3)C412yescomplemented
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8C22×C32×Q8D311yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
11C2×C3×C4⋊Dic3—31no—
12C22×C3×C3⋊C8—31no—
13C3⋊#859C2×C312yescomplemented
14C3×#835—32no—
15C3×#835—32no—
16C23×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18C2×(C3⋊Dic3)⋊C3—44no—
19Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
20C2×#859—91no—
21C2×C3×#103—91no—
22C2×#835—31no—
23C2×C32×Q8—32no—
24C2×C32×Q8—31no—
25C2×C32×Q8Dic312yescomplemented
26C2×C32×Q8C2×D311yescharacteristic
27C22×C32×C4—31no—
28C22×C3⋊Dic3—41no—
29C22×C3×Dic3—61no—
30C2×C3×SL(2,𝔽3)—22no—
31C2×C3×SL(2,𝔽3)—21no—
32C2×C3×SL(2,𝔽3)C3×C412yescomplemented
33C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
34C3×C4⋊Dic3—32no—
35C3×C4⋊Dic3—61no—
36C2×C3×C3⋊C8—32no—
37C2×C3×C3⋊C8—62no—
38C22×H(3,𝔽3)—42no—
39C22×H(3,𝔽3)—42no—
40C22×H(3,𝔽3)—42no—
41C22×H(3,𝔽3)—41no—
42(C3⋊Dic3)⋊C3—44no—
43C22×C3×Q8—31no—
44C22×C3×Q8C3×D311yescharacteristic
45C22×SL(2,𝔽3)—31no—
46C22×SL(2,𝔽3)—61no—
47C22×C3×C8—91no—
48C2×C3×C4⋊C4—91no—
49C2×C4⋊Dic3—31no—
50C22×C3⋊C8—31no—
51#859—92no—
52C3×#103—92no—
53C3×#103—92no—
54#835—32no—
55#835—32no—
56C23×C32—41no—
57C23×C32—81no—
58C23×C32S411yescharacteristic
59C32×Q8—32no—
60C32×Q8—31no—
61C32×Q8—61no—
62C32×Q8C2×Dic311yescharacteristic, complemented
63C2×C32×C4—32no—
64C2×C32×C4—32no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C3⋊Dic3—42no—
68C2×C3⋊Dic3—44no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—122no—
71C3×SL(2,𝔽3)—21no—
72C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
73C3×C3⋊C8—62no—
74C2×#103—91noSylow
75C2×H(3,𝔽3)—44no—
76C2×H(3,𝔽3)—41no—
77C2×H(3,𝔽3)—41no—
78C2×H(3,𝔽3)—41no—
79C2×C3×Q8—31no—
80C2×C3×Q8—32no—
81C2×C3×Q8—31no—
82C2×C3×Q8—32no—
83C2×C3×Q8—32no—
84C2×C3×Q8—31no—
85C2×C3×Q8—61no—
86C2×C3×Q8—62no—
87C2×C3×Q8C3×Dic312yes—
88C2×C3×Q8C2×C3×D311yescharacteristic
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—61no—
92C22×C3×C4—181no—
93C22×Dic3—61no—
94C22×Dic3—361no—
95C2×SL(2,𝔽3)—32no—
96C2×SL(2,𝔽3)—31no—
97C2×SL(2,𝔽3)—62no—
98C2×SL(2,𝔽3)—61no—
99C2×C3×C8—92no—
100C2×C3×C8—182no—
101C3×C4⋊C4—92no—
102C3×C4⋊C4—181no—
103C4⋊Dic3—32no—
104C4⋊Dic3—61no—
105C2×C3⋊C8—32no—
106C2×C3⋊C8—62no—
107C22×C32—41no—
108C22×C32—42no—
109C22×C32—42no—
110C22×C32—42no—
111C22×C32—81no—
112C22×C32—82no—
113C22×C32—82no—
114C22×C32—82no—
115C22×C32#22612yes—
116C22×C32GL(2,𝔽3)12yes—
117C22×C32SL(2,ℤ/4)12yes—
118C22×C32C2×S411yescharacteristic
119C32×C4—31no—
120C32×C4—31no—
121C32×C4—32no—
122C3⋊Dic3—44no—
123C3×Dic3—122no—
124C22×Q8(C3⋊D3)⋊C311yescharacteristic
125C22×C8—91no—
126C2×C4⋊C4—91no—
127#103—92no—
128#103—92no—
129H(3,𝔽3)—41noSylow
130C23×C3—31no—
131C23×C3—121no—
132C23×C3—241no—
133C23×C3C3×S411yescharacteristic
134C3×Q8—32no—
135C3×Q8—31no—
136C3×Q8—32no—
137C3×Q8—31no—
138C3×Q8—31no—
139C3×Q8—62no—
140C3×Q8—61no—
141C3×Q8—61no—
142C3×Q8—61no—
143C3×Q8—62no—
144C3×Q8C2×C3×Dic311yescharacteristic
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—62no—
154C2×C3×C4—62no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—182no—
158C2×C3×C4—362no—
159C2×Dic3—62no—
160C2×Dic3—122no—
161C2×Dic3—362no—
162C2×Dic3—364no—
163SL(2,𝔽3)—31no—
164SL(2,𝔽3)—61no—
165C3×C8—182no—
166C3⋊C8—62no—
167C2×C32—41no—
168C2×C32—44no—
169C2×C32—41no—
170C2×C32—41no—
171C2×C32—81no—
172C2×C32—84no—
173C2×C32—81no—
174C2×C32—81no—
175C2×C32#85914yescomplemented
176C2×C32C2×#22611yescharacteristic
177C2×C32C2×GL(2,𝔽3)11yescharacteristic
178C2×C32C2×SL(2,ℤ/4)11yescharacteristic
179C2×Q8—31no—
180C2×Q8—32no—
181C2×Q8(C3⋊Dic3)⋊C312yescomplemented
182C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
183C22×C4—31no—
184C22×C4—181no—
185C2×C8—92no—
186C2×C8—182no—
187C4⋊C4—92no—
188C4⋊C4—181no—
189C22×C3—32no—
190C22×C3—32no—
191C22×C3—32no—
192C22×C3—31no—
193C22×C3—122no—
194C22×C3—122no—
195C22×C3—122no—
196C22×C3—121no—
197C22×C3—242no—
198C22×C3—242no—
199C22×C3—242no—
200C22×C3—241no—
201C22×C3C3×#22612yes—
202C22×C3C3×GL(2,𝔽3)12yes—
203C22×C3C3×SL(2,ℤ/4)12yes—
204C22×C3C2×C3×S411yescharacteristic
205C3×C4—31no—
206C3×C4—31no—
207C3×C4—32no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—32no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—62no—
214C3×C4—362no—
215Dic3—122no—
216Dic3—364no—
217C32—41no—
218C32—81no—
219C32C2×#85911yescharacteristic, complemented
220C23(C3⋊S4)⋊C311yescharacteristic
221Q8—32no—
222Q8—31no—
223Q8—61no—
224Q8C2×(C3⋊Dic3)⋊C311yescharacteristic, complemented
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—182no—
230C2×C4—362no—
231C8—182no—
232C2×C3—34no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—124no—
237C2×C3—121no—
238C2×C3—121no—
239C2×C3—121no—
240C2×C3—244no—
241C2×C3—241no—
242C2×C3—241no—
243C2×C3—241no—
244C2×C3C3×#85914yes—
245C2×C3C2×C3×#22611yescharacteristic
246C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
247C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
248C22(C3⋊#226)⋊C312yes—
249C22(C3⋊GL(2,𝔽3))⋊C312yes—
250C22(C3⋊SL(2,ℤ/4))⋊C312yes—
251C22C2×(C3⋊S4)⋊C311yescharacteristic
252C4—31no—
253C4—31no—
254C4—32no—
255C4—362no—
256C3—31no—
257C3—121no—
258C3—241no—
259C3C2×C3×#85911yescharacteristic
260C2(C3⋊#859)⋊C314yesdirect product factor
261C2C2×(C3⋊#226)⋊C311yescharacteristic
262C2C2×(C3⋊GL(2,𝔽3))⋊C311yescharacteristic
263C2C2×(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22×C3C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×S4
prime radicalC22×Q8⋊H(3,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. (C3⋊#859)⋊C3C2direct (N × H)
3. C2×C3⋊#859C3semidirect
6. C2×Q8⋊H(3,𝔽3)C4semidirect
13. C3⋊#859C2×C3semidirect
19. Q8⋊H(3,𝔽3)C2×C4semidirect
25. C2×C32×Q8Dic3semidirect
32. C2×C3×SL(2,𝔽3)C3×C4semidirect
62. C32×Q8C2×Dic3semidirect
72. C3×SL(2,𝔽3)C2×C3×C4semidirect
175. C2×C32#859semidirect
181. C2×Q8(C3⋊Dic3)⋊C3semidirect
219. C32C2×#859semidirect
224. Q8C2×(C3⋊Dic3)⋊C3semidirect
260. C2(C3⋊#859)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
C2×C3⋊#8593
C2×C3×#8353
C22×(C3⋊Dic3)⋊C34