Finite Group Databasebeta

← #13166017

Subgroups — #13166017

Order 1728.

489 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 62 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⋊#226)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4C4×C3⋊#226—31nomaximal
5#10593193—33nomaximal
6#10591537—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊#226)C412yescomplemented
9Dic3×C3⋊Dic3—41nomaximal
10C2×C3⋊#226—31no—
11C2×C3⋊#226—33no—
12C2×C3⋊#226—61no—
13C2×C3⋊#226—63no—
14C2×#3756—31no—
15C3×Q8×Dic3D311yescharacteristic
16C3×C4×SL(2,𝔽3)—31no—
17#63844—31no—
18Dic3×SL(2,𝔽3)D313yes—
19C3⋊#859D311yescharacteristic
20C3×#859—33no—
21C3×#835—31no—
22#63576—31no—
23#63567—31no—
24Dic3×C3⋊C8—31no—
25C2×C3⋊(C3⋊Dic3)—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C4×#226—93no—
30C4×#216—91no—
31#4790—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×#634—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)C2×D311yescharacteristic
38C3⋊#226—32no—
39C3⋊#226—36no—
40C3⋊#226—62no—
41C3⋊#226—66no—
42#3756—32no—
43#3756—32no—
44C4×C3⋊Dic3—121no—
45C2×C3⋊(C3⋊C8)—31no—
46C3×C4⋊Dic3—31no—
47C3×C4⋊Dic3—31no—
48C3×C4×Dic3—31no—
49C2×C3×C3⋊C8—31no—
50#3724—61no—
51Dic32—61no—
52Dic32—123no—
53C22×C33—41no—
54C3⋊(C3⋊Dic3)—42no—
55C3×C3⋊Dic3—42no—
56C32×Dic3—42no—
57C2×#226—93no—
58C2×#226—183no—
59C3×C4×Q8—31no—
60Q8×Dic3C3⋊D311yescharacteristic
61C2×#216—91no—
62C2×#216—91no—
63C2×#216—181no—
64#887—91no—
65C4×Dic6—91no—
66C4×SL(2,𝔽3)—33no—
67#859—33no—
68C3×#103—91no—
69#835—31no—
70#816—91no—
71C8×Dic3—91no—
72#810—91no—
73#807—91no—
74#807—91no—
75C4×C3⋊C8—91no—
76C32×Q8—31no—
77C32×Q8C4×D311yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—121no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—61no—
82C2×C3⋊Dic3—121no—
83C2×C3⋊Dic3—123no—
84C2×C3⋊Dic3—241no—
85C2×C3⋊Dic3—243no—
86#634—32no—
87#634—61no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—43no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—123no—
92C2×C3×Dic3S411yescharacteristic
93C3×SL(2,𝔽3)—21no—
94C3×SL(2,𝔽3)—23no—
95C3×SL(2,𝔽3)C4×D313yes—
96C3×SL(2,𝔽3)C4×D311yescharacteristic
97C3⋊(C3⋊C8)—32no—
98C3×C3⋊C8—61no—
99C4×Q16—271noSylow
100C2×C33—41no—
101C2×C33—41no—
102C2×C33—41no—
103C2×C3×Q8—21no—
104C2×C3×Q8D3211yescharacteristic
105C2×C3×Q8C2×C3⋊D311yescharacteristic
106C2×Dic6—91no—
107C2×Dic6—91no—
108C2×Dic6—181no—
109C2×SL(2,𝔽3)—23no—
110C2×SL(2,𝔽3)D3213yes—
111#226—96no—
112#226—186no—
113C2×C3×C8—91no—
114C3×C4⋊C4—91no—
115C3×C4⋊C4—91no—
116C3×C42—91no—
117#216—92no—
118#216—92no—
119#216—92no—
120#216—92no—
121#216—182no—
122#216—182no—
123C4⋊Dic3—31no—
124C4⋊Dic3—31no—
125Dic3⋊C4—181no—
126Dic3⋊C4—181no—
127C4×Dic3—31no—
128C4×Dic3—181no—
129C4×Dic3—181no—
130C4×Dic3—363no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—91no—
133C2×C3⋊C8—91no—
134C2×C3⋊C8—181no—
135C22×C32—41no—
136C22×C32—43no—
137C22×C32—83no—
138C22×C32—81no—
139C22×C32C2×S411yescharacteristic
140C32×C4—31no—
141C32×C4—31no—
142C32×C4—122no—
143C3⋊Dic3—42no—
144C3⋊Dic3—62no—
145C3⋊Dic3—122no—
146C3⋊Dic3—126no—
147C3⋊Dic3—242no—
148C3⋊Dic3—246no—
149C3×Dic3—46no—
150C3×Dic3—61no—
151C3×Dic3—126no—
152C3×Dic3—121no—
153C3×Dic3#22612yescomplemented
154C2×Q16—271no—
155C4×Q8—31no—
156C4×Q8—271no—
157#107—271no—
158#103—91no—
159#103—271no—
160C4×C8—271no—
161C33—41noSylow
162C3×Q8—21no—
163C3×Q8—31no—
164C3×Q8—31no—
165C3×Q8—61no—
166C3×Q8#62411yescharacteristic
167C3×Q8C4×C3⋊D311yescharacteristic
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—61no—
172C2×C3×C4—91no—
173C2×C3×C4—123no—
174C2×C3×C4—181no—
175C2×Dic3—31no—
176C2×Dic3—61no—
177C2×Dic3—123no—
178C2×Dic3—181no—
179C2×Dic3—181no—
180C2×Dic3—361no—
181C2×Dic3—363no—
182C2×Dic3—723no—
183C2×Dic3C3⋊S411yescharacteristic
184Dic6—92no—
185Dic6—92no—
186Dic6—182no—
187Dic6—181no—
188Dic6—181no—
189Dic6—361no—
190SL(2,𝔽3)—23no—
191SL(2,𝔽3)#62413yes—
192C3×C8—181no—
193C3⋊C8—61no—
194C3⋊C8—92no—
195C3⋊C8—92no—
196C3⋊C8—182no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—43no—
201C2×C32—43no—
202C2×C32—43no—
203C2×C32—83no—
204C2×C32—83no—
205C2×C32—83no—
206C2×C32—81no—
207C2×C32—81no—
208C2×C32—81no—
209C2×C32C4×S411yescharacteristic
210C2×C32C2×#22611yescharacteristic
211C2×C32#98511yescharacteristic
212C2×Q8—271no—
213C2×Q8D3×C3⋊D311yescharacteristic
214Q16—272no—
215Q16—272no—
216C2×C8—91no—
217C2×C8—271no—
218C4⋊C4—91no—
219C4⋊C4—91no—
220C4⋊C4—541no—
221C42—91no—
222C42—541no—
223C22×C3—21no—
224C22×C3—43no—
225C22×C3—83no—
226C22×C3D3×S411yescharacteristic
227C22×C3C2×C3⋊S411yescharacteristic
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—32no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—126no—
236C3×C4—181no—
237C3×C4—361no—
238Dic3—61no—
239Dic3—126no—
240Dic3—121no—
241Dic3—182no—
242Dic3—182no—
243Dic3—362no—
244Dic3—366no—
245Dic3—726no—
246Dic3C3⋊#22612yescomplemented
247C32—41no—
248C32—43no—
249C32—83no—
250C32—81no—
251C32C4×#22611yescharacteristic, complemented
252Q8—31no—
253Q8—272no—
254Q8—541no—
255Q8#628911yescharacteristic
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—541no—
261C8—181no—
262C8—272no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—43no—
267C2×C3—43no—
268C2×C3—43no—
269C2×C3—83no—
270C2×C3—83no—
271C2×C3—83no—
272C2×C3#6419711yescharacteristic
273C2×C3D3×#22611yescharacteristic
274C2×C3(C3⋊S4)⋊C411yescharacteristic
275C2×C3C4×C3⋊S411yescharacteristic
276C2×C3C2×C3⋊#22611yescharacteristic
277C2×C3#6426611yescharacteristic
278C22D3×C3⋊S411yescharacteristic
279C4—32no—
280C4—31no—
281C4—31no—
282C4—181no—
283C4—361no—
284C4—542no—
285C3—21no—
286C3—43no—
287C3—83no—
288C3#1059319311yescharacteristic, complemented
289C3C4×C3⋊#22611yescharacteristic, complemented
290C2#1172419211yescharacteristic
291C2D3×C3⋊#22611yescharacteristic
292C2#1172420111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radical, cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C3⋊(C3⋊#226)C4semidirect
153. C3×Dic3#226semidirect
246. Dic3C3⋊#226semidirect
251. C32C4×#226semidirect
288. C3#10593193semidirect
289. C3C4×C3⋊#226semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊#226)2
C3×Dic3×SL(2,𝔽3)2
C3×C3⋊#8592
C4×C3⋊#2263
#105931933
#105915373
Dic3×C3⋊Dic34