Finite Group Databasebeta

← C3×#10593623 #13172991

Subgroups — C3×#10593623 #13172991

Order 1728.

396 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593623C311yesmaximal, characteristic, direct product factor
5C3×#5389—33nomaximal
6C3×#5022—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×#3750—41nomaximal
9C32×C4×Q8D311yescharacteristic
10C3×#870—31no—
11C3×C4×SL(2,𝔽3)—21no—
12C3×C4×SL(2,𝔽3)—23no—
13C3×C4×SL(2,𝔽3)D313yes—
14C3×C4×SL(2,𝔽3)C2×C311yescharacteristic
15C3⋊#859C2×C311yescharacteristic
16C3⋊#859C2×C311yescharacteristic
17C3×#859—33no—
18C3×#859—33no—
19C3×#835—31no—
20C3×#835—31no—
21C3×#808—31no—
22C3×#807—31no—
23C3×C4⋊(C3⋊C8)—31no—
24C2×C33×C4—41no—
25C2×C3×C3⋊Dic3—41no—
26C2×C3×C3⋊Dic3—41no—
27C32×SL(2,𝔽3)Q811yescharacteristic
28#5389—33no—
29C3×#479—91no—
30#5022—31no—
31C2×C32×Q8C2×D311yescharacteristic
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)—23no—
34C2×C3×SL(2,𝔽3)C2×D313yes—
35C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
36C32×C4⋊C4—31no—
37C32×C42—31no—
38#3750—41no—
39C3×C4⋊Dic3—31no—
40C3×C4⋊Dic3—31no—
41C3×Dic3⋊C4—61no—
42C3×Dic3⋊C4—61no—
43C3×Dic3⋊C4—123no—
44C2×C3×C3⋊C8—31no—
45C2×C3×C3⋊C8—31no—
46C22×C33—41no—
47C33×C4—81no—
48C3×C3⋊Dic3—42no—
49C3×C3⋊Dic3—81no—
50C3×#125—91no—
51C3×C4×Q8—21no—
52C3×C4×Q8C3×D311yescharacteristic
53C3×C4×Q8C3⋊D311yescharacteristic
54#870—31no—
55C4×SL(2,𝔽3)—23no—
56C4×SL(2,𝔽3)C3×D313yes—
57#859—33no—
58#859—33no—
59C3×#107—91no—
60C3×#106—91no—
61C3×C4⋊C8—91no—
62C3×#103—91no—
63C3×#103—91no—
64#835—31no—
65#835—31no—
66#808—31no—
67#807—31no—
68C4⋊(C3⋊C8)—31no—
69C32×Q8—31no—
70C32×Q8Dic611yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—41no—
74C2×C32×C4—43no—
75C2×C32×C4—83no—
76C2×C32×C4—81no—
77C2×C32×C4S411yescharacteristic
78C2×C3⋊Dic3—41no—
79C2×C3⋊Dic3—41no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—123no—
83C2×C3×Dic3—123no—
84C3×SL(2,𝔽3)—21no—
85C3×SL(2,𝔽3)—23no—
86C3×SL(2,𝔽3)Dic613yes—
87C3×SL(2,𝔽3)C3×Q811yescharacteristic
88C3×C3⋊C8—61no—
89C3×C3⋊C8—61no—
90#479—91noSylow
91C2×C33—41no—
92C2×C33—41no—
93C2×C33—41no—
94C2×C3×Q8—21no—
95C2×C3×Q8C2×C3×D311yescharacteristic
96C2×C3×Q8C2×C3⋊D311yescharacteristic
97C2×SL(2,𝔽3)—23no—
98C2×SL(2,𝔽3)C2×C3×D313yes—
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—61no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4—181no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—61no—
111C4⋊Dic3—31no—
112C4⋊Dic3—31no—
113Dic3⋊C4—61no—
114Dic3⋊C4—61no—
115Dic3⋊C4—123no—
116C2×C3⋊C8—31no—
117C2×C3⋊C8—31no—
118C22×C32—41no—
119C22×C32—43no—
120C22×C32—83no—
121C22×C32—81no—
122C22×C32C2×S411yescharacteristic
123C32×C4—21no—
124C32×C4—31no—
125C32×C4—31no—
126C32×C4—61no—
127C32×C4—83no—
128C32×C4—81no—
129C32×C4—86no—
130C32×C4—82no—
131C3⋊Dic3—42no—
132C3⋊Dic3—81no—
133C3×Dic3—126no—
134C3×Dic3—121no—
135C3×Dic3—121no—
136C3×Dic3—243no—
137#125—91no—
138C4×Q8C3×C3⋊D311yescharacteristic
139#107—91no—
140#106—91no—
141C4⋊C8—91no—
142#103—91no—
143#103—91no—
144C33—41noSylow
145C3×Q8—21no—
146C3×Q8—31no—
147C3×Q8—31no—
148C3×Q8—61no—
149C3×Q8C3×Dic611yescharacteristic
150C3×Q8#63411yescharacteristic
151C2×C3×C4—21no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—43no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—83no—
160C2×C3×C4—181no—
161C2×C3×C4—181no—
162C2×C3×C4C3×S411yescharacteristic
163C2×C3×C4C3⋊S411yescharacteristic
164C2×Dic3—61no—
165C2×Dic3—61no—
166C2×Dic3—123no—
167C2×Dic3—123no—
168SL(2,𝔽3)—23no—
169SL(2,𝔽3)C3×Dic613yes—
170C3×C8—181no—
171C3×C8—181no—
172C3⋊C8—61no—
173C3⋊C8—61no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—41no—
177C2×C32—43no—
178C2×C32—43no—
179C2×C32—43no—
180C2×C32—83no—
181C2×C32—83no—
182C2×C32—83no—
183C2×C32—81no—
184C2×C32—81no—
185C2×C32—81no—
186C2×C32A4⋊Q811yescharacteristic
187C2×C32#98311yescharacteristic
188C2×C32#98511yescharacteristic
189C2×Q8C2×C3×C3⋊D311yescharacteristic
190C2×C8—91no—
191C2×C8—91no—
192C4⋊C4—31no—
193C4⋊C4—91no—
194C4⋊C4—91no—
195C4⋊C4—181no—
196C4⋊C4—181no—
197C42—31no—
198C22×C3—21no—
199C22×C3—43no—
200C22×C3—83no—
201C22×C3C2×C3×S411yescharacteristic
202C22×C3C2×C3⋊S411yescharacteristic
203C3×C4—22no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—61no—
214C3×C4—83no—
215C3×C4—86no—
216C3×C4—121no—
217C3×C4—361no—
218C3×C4—361no—
219Dic3—126no—
220Dic3—121no—
221Dic3—121no—
222Dic3—243no—
223C32—41no—
224C32—43no—
225C32—83no—
226C32—81no—
227C32#538911yescharacteristic, complemented
228Q8—31no—
229Q8C3×#63411yescharacteristic
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—181no—
233C2×C4—181no—
234C2×C4C3×C3⋊S411yescharacteristic
235C8—181no—
236C8—181no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—43no—
241C2×C3—43no—
242C2×C3—43no—
243C2×C3—83no—
244C2×C3—83no—
245C2×C3—83no—
246C2×C3C3×A4⋊Q811yescharacteristic
247C2×C3C3×#98311yescharacteristic
248C2×C3C3×#98511yescharacteristic
249C2×C3#6425911yescharacteristic
250C2×C3#6426411yescharacteristic
251C2×C3#6426611yescharacteristic
252C22C2×C3×C3⋊S411yescharacteristic
253C4—21no—
254C4—31no—
255C4—31no—
256C4—61no—
257C4—361no—
258C4—361no—
259C3—21no—
260C3—43no—
261C3—83no—
262C3C3×#538911yescharacteristic, complemented
263C3#1059362311yescharacteristic, direct product factor
264C2C3×#6425911yescharacteristic
265C2C3×#6426411yescharacteristic
266C2C3×#6426611yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C32×SL(2,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10593623C3direct (N × H)
227. C32#5389semidirect
262. C3C3×#5389semidirect
263. C3#10593623direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C4×SL(2,𝔽3)2
C3×C3⋊#8592
#105936233
C3×#53893
C3×#50223
C3×#37504