Finite Group Databasebeta

← C3×#10593622 #13172990

Subgroups — C3×#10593622 #13172990

Order 1728.

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 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×C3×C3⋊#226C211yesmaximal, characteristic
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593622C311yesmaximal, characteristic, direct product factor
5C3×#5388—33nomaximal
6C3×#5030—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊#226C412yescomplemented
9C3×#3750—41nomaximal
10C2×C3⋊#226C2×C311yescharacteristic
11C2×C3×#226—33no—
12C32×C4×Q8D311yescharacteristic
13C2×C3×#216—31no—
14C3×C4×Dic6—31no—
15C3×C4×SL(2,𝔽3)—21no—
16C3×C4×SL(2,𝔽3)—23no—
17C3×C4×SL(2,𝔽3)D313yes—
18C3×C4×SL(2,𝔽3)C2×C311yescharacteristic
19C3⋊#859C2×C311yescharacteristic
20C3×#859—33no—
21C3×#835—31no—
22C3×#810—31no—
23C3×#808—31no—
24C3×#803—31no—
25C2×C33×C4—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C3×C3⋊Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29#5388—33no—
30C3×#441—91no—
31#5030—31no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C3×Dic6—31no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)—23no—
36C2×C3×SL(2,𝔽3)C2×D313yes—
37C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
38C3⋊#226C3×C412yescomplemented
39C3×#226—36no—
40C32×C4⋊C4—31no—
41C32×C42—31no—
42#3750—41no—
43C3×#216—32no—
44C3×#216—32no—
45C3×C4⋊Dic3—31no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—123no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—31no—
50C2×C3×C3⋊C8—31no—
51C22×C33—41no—
52C33×C4—81no—
53C3×C3⋊Dic3—42no—
54C3×C3⋊Dic3—81no—
55C2×#226—33no—
56C2×C3×Q16—91no—
57C3×C4×Q8—21no—
58C3×C4×Q8—91no—
59C3×C4×Q8C3×D311yescharacteristic
60C3×C4×Q8C3⋊D311yescharacteristic
61C2×#216—31no—
62C4×Dic6—31no—
63C4×SL(2,𝔽3)—23no—
64C4×SL(2,𝔽3)C3×D313yes—
65#859—33no—
66C3×#106—91no—
67C3×#103—91no—
68C3×#103—91no—
69C3×#97—91no—
70#835—31no—
71#810—31no—
72#808—31no—
73#803—31no—
74C32×Q8—31no—
75C32×Q8C4×D311yescharacteristic
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C2×C32×C4—41no—
79C2×C32×C4—43no—
80C2×C32×C4—83no—
81C2×C32×C4—81no—
82C2×C32×C4S411yescharacteristic
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—41no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—123no—
88C2×C3×Dic3—123no—
89C3×Dic6—32no—
90C3×Dic6—61no—
91C3×SL(2,𝔽3)—21no—
92C3×SL(2,𝔽3)—23no—
93C3×SL(2,𝔽3)C4×D313yes—
94C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
95C3×C3⋊C8—32no—
96C3×C3⋊C8—61no—
97#441—91noSylow
98C2×C33—41no—
99C2×C33—41no—
100C2×C33—41no—
101C2×C3×Q8—21no—
102C2×C3×Q8—91no—
103C2×C3×Q8C2×C3×D311yescharacteristic
104C2×C3×Q8C2×C3⋊D311yescharacteristic
105C2×Dic6—31no—
106C2×SL(2,𝔽3)—23no—
107C2×SL(2,𝔽3)C2×C3×D313yes—
108#226—36no—
109C3×Q16—92no—
110C3×Q16—92no—
111C2×C3×C8—91no—
112C2×C3×C8—91no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C42—31no—
119C3×C42—31no—
120C3×C42—61no—
121C3×C42—181no—
122#216—32no—
123#216—32no—
124C4⋊Dic3—31no—
125Dic3⋊C4—61no—
126Dic3⋊C4—123no—
127C4×Dic3—61no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—31no—
130C22×C32—41no—
131C22×C32—43no—
132C22×C32—83no—
133C22×C32—81no—
134C22×C32C2×S411yescharacteristic
135C32×C4—21no—
136C32×C4—31no—
137C32×C4—31no—
138C32×C4—61no—
139C32×C4—83no—
140C32×C4—81no—
141C32×C4—86no—
142C32×C4—82no—
143C3⋊Dic3—42no—
144C3⋊Dic3—81no—
145C3×Dic3—62no—
146C3×Dic3—126no—
147C3×Dic3—121no—
148C3×Dic3—243no—
149C2×Q16—91no—
150C4×Q8—91no—
151C4×Q8C3×C3⋊D311yescharacteristic
152#106—91no—
153#103—91no—
154#103—91no—
155#97—91no—
156C33—41noSylow
157C3×Q8—21no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—61no—
161C3×Q8—92no—
162C3×Q8—181no—
163C3×Q8C3×C4×D311yescharacteristic
164C3×Q8C4×C3⋊D311yescharacteristic
165C2×C3×C4—21no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—43no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—83no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4C3×S411yescharacteristic
177C2×C3×C4C3⋊S411yescharacteristic
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—123no—
181C2×Dic3—123no—
182Dic6—32no—
183Dic6—61no—
184SL(2,𝔽3)—23no—
185SL(2,𝔽3)C3×C4×D313yes—
186C3×C8—92no—
187C3×C8—181no—
188C3⋊C8—32no—
189C3⋊C8—61no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—43no—
194C2×C32—43no—
195C2×C32—43no—
196C2×C32—83no—
197C2×C32—83no—
198C2×C32—83no—
199C2×C32—81no—
200C2×C32—81no—
201C2×C32—81no—
202C2×C32C4×S411yescharacteristic
203C2×C32#98311yescharacteristic
204C2×C32#98411yescharacteristic
205C2×Q8—91no—
206C2×Q8C2×C3×C3⋊D311yescharacteristic
207Q16—92no—
208Q16—92no—
209C2×C8—91no—
210C2×C8—91no—
211C4⋊C4—31no—
212C4⋊C4—91no—
213C4⋊C4—181no—
214C42—31no—
215C42—181no—
216C22×C3—21no—
217C22×C3—43no—
218C22×C3—83no—
219C22×C3C2×C3×S411yescharacteristic
220C22×C3C2×C3⋊S411yescharacteristic
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—61no—
229C3×C4—61no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—83no—
233C3×C4—86no—
234C3×C4—121no—
235C3×C4—182no—
236C3×C4—361no—
237Dic3—62no—
238Dic3—126no—
239Dic3—121no—
240Dic3—243no—
241C32—41no—
242C32—43no—
243C32—83no—
244C32—81no—
245C32#538811yescharacteristic, complemented
246Q8—31no—
247Q8—92no—
248Q8—181no—
249Q8C3×C4×C3⋊D311yescharacteristic
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—181no—
253C2×C4—181no—
254C2×C4C3×C3⋊S411yescharacteristic
255C8—92no—
256C8—181no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—43no—
261C2×C3—43no—
262C2×C3—43no—
263C2×C3—83no—
264C2×C3—83no—
265C2×C3—83no—
266C2×C3C3×C4×S411yescharacteristic
267C2×C3C3×#98311yescharacteristic
268C2×C3C3×#98411yescharacteristic
269C2×C3C4×C3⋊S411yescharacteristic
270C2×C3#6426411yescharacteristic
271C2×C3#6426511yescharacteristic
272C22C2×C3×C3⋊S411yescharacteristic
273C4—21no—
274C4—31no—
275C4—31no—
276C4—61no—
277C4—182no—
278C4—361no—
279C3—21no—
280C3—43no—
281C3—83no—
282C3C3×#538811yescharacteristic, complemented
283C3#1059362211yescharacteristic, direct product factor
284C2C3×C4×C3⋊S411yescharacteristic
285C2C3×#6426411yescharacteristic
286C2C3×#6426511yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C32×SL(2,𝔽3)C22
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ 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. #10593622C3direct (N × H)
8. C3×C3⋊#226C4semidirect
38. C3⋊#226C3×C4semidirect
245. C32#5388semidirect
282. C3C3×#5388semidirect
283. C3#10593622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2262
C32×C4×SL(2,𝔽3)2
C3×C3⋊#8592
#105936223
C3×#53883
C3×#50303
C3×#37504