Finite Group Databasebeta

← #13166043

Subgroups — #13166043

Order 1728.

398 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C3×#64266C211yesmaximal, characteristic
2C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593638—31nomaximal
5#10593219—33nomaximal
6#10591673—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
9C3×C3⋊#226C411yescharacteristic, complemented
10C3⋊D3×C3⋊C8—41nomaximal
11#64266D311yescharacteristic
12C3×#985—33no—
13C3×#950—31no—
14C3×#948D311yescharacteristic, complemented
15C3×#131⋊C3—31no—
16#63814—31no—
17C3⋊(#131⋊C3)D313yescomplemented
18C3⋊GU(2,𝔽3)—31no—
19C3⋊GU(2,𝔽3)—33no—
20C3⋊GU(2,𝔽3)—61no—
21C3⋊GU(2,𝔽3)—63no—
22#63664—31no—
23#63577—31no—
24#63568—31no—
25#63553—31no—
26C3×C4×C3⋊D3—41no—
27C32×SL(2,𝔽3)C2×C411yescharacteristic
28C3⋊(C3⋊(C3⋊C8))—41no—
29C32×C3⋊C8—41no—
30#5404—93no—
31#5196—91no—
32#5144—91no—
33C32×#41C2×D311yescharacteristic
34C3×#235—31no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3—23no—
37C3×#41⋊C3C2×D313yes—
38C3×#41⋊C3C2×D311yescharacteristic
39C3⋊GL(2,𝔽3)Dic311yescharacteristic, complemented
40C3⋊#226Dic311yescharacteristic, complemented
41C3×GL(2,𝔽3)—33no—
42C3×#226—33no—
43C4×C3⋊Dic3—31no—
44#3748—31no—
45C8×C3⋊D3—121no—
46C3×#216—31no—
47C3×#215—31no—
48C3×#214—31no—
49C3×#213—31no—
50C3×#208—31no—
51C2×C3×C3⋊C8—31no—
52C2×C3×C3⋊C8—31no—
53#3711—61no—
54D3×C3⋊C8—61no—
55D3×C3⋊C8—123no—
56C2×C3×C3⋊D3—41no—
57C33×C4—41no—
58C3×C3⋊Dic3—41no—
59#985—33no—
60C3×#135—91no—
61C3×#131—31no—
62#950—31no—
63#948—91no—
64#948C3⋊D311yescharacteristic, complemented
65#901—91no—
66#131⋊C3—33no—
67GU(2,𝔽3)—93no—
68GU(2,𝔽3)—183no—
69#837—91no—
70#837—91no—
71#837—91no—
72#837—181no—
73#825—91no—
74#822—91no—
75#819—91no—
76C8×Dic3—91no—
77C8×Dic3—91no—
78C32×Q8C4×D311yescharacteristic
79C32×D4—31no—
80C2×C32×C4—31no—
81C2×C3⋊Dic3—61no—
82C4×C3⋊D3—41no—
83C3×#67—61no—
84C3×D12—31no—
85C3×C4×D3—61no—
86C3×C4×D3—123no—
87C3×Dic6—31no—
88C3×SL(2,𝔽3)—21no—
89C3×SL(2,𝔽3)—23no—
90C3×SL(2,𝔽3)C4×D313yes—
91C3×SL(2,𝔽3)C2×Dic311yescharacteristic
92C32×C8—121no—
93C3⋊(C3⋊C8)—61no—
94C3⋊(C3⋊C8)—121no—
95C3⋊(C3⋊C8)—123no—
96C3⋊(C3⋊C8)—241no—
97C3⋊(C3⋊C8)—243no—
98C3×C3⋊C8—31no—
99C3×C3⋊C8—31no—
100C3×C3⋊C8—31no—
101C3×C3⋊C8—43no—
102C3×C3⋊C8S411yescharacteristic
103#443—271noSylow
104C2×C33—41no—
105C3×C3⋊D3—42no—
106C3×#41—21no—
107C3×#41—91no—
108C3×#41D3211yescharacteristic
109C3×#41C2×C3⋊D311yescharacteristic
110#235—31no—
111#41⋊C3—23no—
112#41⋊C3D3213yes—
113GL(2,𝔽3)—33no—
114#226—33no—
115C3×Q16—91no—
116C3×SD16—91no—
117C3×SD16—91no—
118C3×D8—91no—
119C3×M16—91no—
120C2×C3×C8—91no—
121C2×C3×C8—91no—
122#216—31no—
123#215—31no—
124#214—31no—
125#213—31no—
126C4×Dic3—91no—
127C4×Dic3—91no—
128C4×Dic3—181no—
129#208—31no—
130#208—91no—
131#208—91no—
132#208—181no—
133#208—181no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—181no—
137ΓL(1,𝔽25)—181no—
138D3×C8—181no—
139D3×C8—363no—
140C22×C32—31no—
141C2×C3⋊D3—41no—
142C2×C3×D3—61no—
143C2×C3×D3—123no—
144C32×C4—31no—
145C32×C4—41no—
146C32×C4—43no—
147C32×C4—83no—
148C32×C4—81no—
149C32×C4C2×S411yescharacteristic
150C3⋊Dic3—41no—
151C3⋊Dic3—62no—
152C3×Dic3—61no—
153C3×Dic3—123no—
154#135—91no—
155#131—31no—
156#131—271no—
157#108—271no—
158#104—271no—
159#104—271no—
160C4×C8—271no—
161C33—41noSylow
162C3×Q8—21no—
163C3×Q8—91no—
164C3×Q8D3×Dic311yescharacteristic
165C3×Q8C4×C3⋊D311yescharacteristic
166C3×D4—31no—
167C3×D4—31no—
168C3×D4—61no—
169C3×D4—91no—
170C3×D4—181no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—61no—
174C2×C3×C4—181no—
175#67—61no—
176C2×Dic3—181no—
177C2×Dic3—181no—
178C2×Dic3—361no—
179D12—31no—
180C4×D3—61no—
181C4×D3—123no—
182Dic6—31no—
183SL(2,𝔽3)—23no—
184SL(2,𝔽3)D3×Dic313yes—
185C3×C8—31no—
186C3×C8—91no—
187C3×C8—91no—
188C3×C8—91no—
189C3×C8—123no—
190C3⋊C8—31no—
191C3⋊C8—31no—
192C3⋊C8—31no—
193C3⋊C8—181no—
194C3⋊C8—181no—
195C3⋊C8—361no—
196C3⋊C8—363no—
197C3⋊C8—723no—
198C3⋊C8C3⋊S411yescharacteristic
199C2×C32—41no—
200C2×C32—43no—
201C2×C32—61no—
202C2×C32—83no—
203C2×C32—81no—
204C2×C32C4×S411yescharacteristic
205C3⋊D3—42no—
206C3×D3—126no—
207C3×D3—121no—
208#41—91no—
209#41D3×C3⋊D311yescharacteristic
210Q16—91no—
211SD16—91no—
212SD16—91no—
213D8—91no—
214M16—91no—
215M16—271no—
216M16—541no—
217C2×C8—91no—
218C2×C8—91no—
219C2×C8—541no—
220C42—271no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—61no—
224C22×C3—181no—
225C2×D3—61no—
226C2×D3—123no—
227C3×C4—21no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—43no—
231C3×C4—61no—
232C3×C4—83no—
233C3×C4—181no—
234C3×C4D3×S411yescharacteristic
235C3×C4C2×C3⋊S411yescharacteristic
236Dic3—61no—
237Dic3—123no—
238Dic3—182no—
239Dic3—182no—
240Dic3—362no—
241C32—41no—
242C32—43no—
243C32—83no—
244C32—81no—
245C32#540411yescharacteristic, complemented
246Q8—91no—
247Q8Dic3×C3⋊D311yescharacteristic
248D4—31no—
249D4—91no—
250D4—181no—
251C2×C4—31no—
252C2×C4—181no—
253C2×C4—541no—
254C8—31no—
255C8—91no—
256C8—91no—
257C8—91no—
258C8—541no—
259C2×C3—21no—
260C2×C3—43no—
261C2×C3—61no—
262C2×C3—61no—
263C2×C3—83no—
264C2×C3—121no—
265C2×C3—361no—
266C2×C3Dic3×S411yescharacteristic
267C2×C3C4×C3⋊S411yescharacteristic
268D3—126no—
269D3—121no—
270C22—31no—
271C22—181no—
272C4—31no—
273C4—181no—
274C4—542no—
275C4D3×C3⋊S411yescharacteristic
276C3—21no—
277C3—43no—
278C3—83no—
279C3#1059321911yescharacteristic, complemented
280C3#1059363811yescharacteristic, complemented
281C2—61no—
282C2—361no—
283C2Dic3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC3×#64266C2
cosocleC32×#41⋊C3C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊(#131⋊C3)C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
8. C3×C3⋊GL(2,𝔽3)C4semidirect
9. C3×C3⋊#226C4semidirect
14. C3×#948D3semidirect
17. C3⋊(#131⋊C3)D3semidirect
39. C3⋊GL(2,𝔽3)Dic3semidirect
40. C3⋊#226Dic3semidirect
64. #948C3⋊D3semidirect
245. C32#5404semidirect
279. C3#10593219semidirect
280. C3#10593638semidirect

Maximal subgroups

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

subgroupindex
C3×#642662
C3×C3⋊(#131⋊C3)2
C3⋊(C3⋊GU(2,𝔽3))2
#105936383
#105932193
#105916733
C3⋊D3×C3⋊C84