Finite Group Databasebeta

← #13166028

Subgroups — #13166028

Order 1728.

526 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 56 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⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593661—31nomaximal
5#10593203—33nomaximal
6#10591586—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)—21no—
9C3×C3⋊GL(2,𝔽3)—21no—
10#88803—41nomaximal
11C2×C3×#239D311yescharacteristic, complemented
12C2×C3×#41⋊C3—31no—
13C2×C3⋊(#41⋊C3)D313yescomplemented
14C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
15C2×C3×GL(2,𝔽3)—33no—
16C2×C3×#215—31no—
17#63908—31no—
18C2×#3713—31no—
19C3⋊#859—31no—
20C3⋊#859—33no—
21C3⋊#859—61no—
22C3⋊#859—63no—
23#63662—31no—
24#63563—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C22×C3×C3⋊D3—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#5427—93no—
31#5238—91no—
32#5172—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C3×#239—21no—
35C3×#239—61no—
36C2×C3×D12—31no—
37C2×C3×D12—31no—
38C2×C3×C4×D3—31no—
39C3×#41⋊C3—61no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—23no—
42C2×C3×SL(2,𝔽3)C2×D313yes—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44C2×#625—61no—
45C3⋊(#41⋊C3)—23no—
46C3⋊GL(2,𝔽3)—21no—
47C3×GL(2,𝔽3)—63no—
48#3752—121no—
49C4⋊(C3⋊Dic3)—31no—
50C2×C3⋊(C3⋊C8)—31no—
51C3×#215—61no—
52C3×#215—61no—
53C2×C3×C3⋊C8—31no—
54(C2×D3)⋊Dic3—61no—
55(C2×D3)⋊Dic3—123no—
56#3713—62no—
57C22×C33—41no—
58C2×C3×C3⋊D3—42no—
59C2×C3×C3⋊D3—81no—
60C2×C3×C3⋊D3—81no—
61C3⋊(C3⋊Dic3)—81no—
62C32×Dic3—81no—
63C2×C3×#41—31no—
64C2×#239C3⋊D311yescharacteristic, complemented
65C2×#41⋊C3—33no—
66C2×GL(2,𝔽3)—33no—
67C2×C3×SD16—91no—
68C2×#215—31no—
69#938—91no—
70C2×#213—91no—
71C2×#213—91no—
72#930—91no—
73#859—93no—
74#859—183no—
75#835—91no—
76#835—91no—
77#835—181no—
78#832—91no—
79#830—91no—
80#821—91no—
81(C2×D3)⋊C8—91no—
82C22×C3⋊D3—41no—
83C22×C3×D3—31no—
84C22×C3×D3—61no—
85C22×C3×D3—123no—
86C32×Q8—31no—
87C32×Q8#6711yescharacteristic
88C2×C32×C4—31no—
89C2×C32×C4—121no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—121no—
92C2×C3⋊Dic3—123no—
93C2×C3⋊Dic3—241no—
94C2×C3⋊Dic3—243no—
95C2×C3×Dic3—43no—
96C2×C3×Dic3S411yescharacteristic
97C3×D12—32no—
98C3×D12—61no—
99C3×D12—61no—
100C3×D12—61no—
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—23no—
105C3×SL(2,𝔽3)#6713yescomplemented
106C3×SL(2,𝔽3)#6711yescharacteristic, complemented
107#625—122no—
108C3⋊(C3⋊C8)—61no—
109C3×C3⋊C8—61no—
110#449—271noSylow
111C2×C33—41no—
112C2×C33—41no—
113C2×C33—41no—
114C3×C3⋊D3—82no—
115C3×#41—61no—
116C3×#41—181no—
117C2×C3×Q8—21no—
118C2×C3×Q8D3211yescharacteristic
119C2×C3×Q8C2×C3⋊D311yescharacteristic
120C2×C3×D4—91no—
121C2×C3×D4—91no—
122C22×C3×C4—91no—
123C2×#67—181no—
124C2×#67—181no—
125#239—21no—
126#239—61no—
127C2×D12—31no—
128C2×D12—31no—
129C2×C4×D3—31no—
130#41⋊C3—63no—
131C2×SL(2,𝔽3)—23no—
132C2×SL(2,𝔽3)D3213yes—
133GL(2,𝔽3)—63no—
134C3×SD16—181no—
135C3×SD16—181no—
136C2×C3×C8—91no—
137#217—181no—
138#215—61no—
139#215—61no—
140#213—182no—
141#213—182no—
142(C2×D3)⋊C4—181no—
143(C2×D3)⋊C4—363no—
144C4⋊Dic3—91no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—181no—
151C22×C32—41no—
152C22×C32—43no—
153C22×C32—83no—
154C22×C32—81no—
155C22×C32C2×S411yescharacteristic
156C2×C3⋊D3—42no—
157C2×C3⋊D3—81no—
158C2×C3⋊D3—81no—
159C2×C3×D3—61no—
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C2×C3×D3—126no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C2×C3×D3—243no—
167C2×C3×D3—243no—
168C32×C4—31no—
169C32×C4—31no—
170C32×C4—241no—
171C3⋊Dic3—121no—
172C3⋊Dic3—243no—
173C3⋊Dic3—241no—
174C3⋊Dic3—242no—
175C3⋊Dic3—246no—
176C3×Dic3—21no—
177C3×Dic3—83no—
178C2×#41—31no—
179C2×SD16—91no—
180C2×D8—271no—
181#121—271no—
182#103—271no—
183#102—271no—
184C22⋊C8—271no—
185C33—41noSylow
186C23×C3—91no—
187C23×C3—181no—
188C22×D3—31no—
189C22×D3—61no—
190C22×D3—123no—
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8#62511yescharacteristic
196C3×Q8#63811yescharacteristic
197C3×D4—92no—
198C3×D4—181no—
199C3×D4—181no—
200C3×D4—181no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—61no—
205C2×C3×C4—123no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208#67—362no—
209#67—362no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—361no—
213C2×Dic3—363no—
214C2×Dic3—723no—
215C2×Dic3C3⋊S411yescharacteristic
216D12—32no—
217D12—61no—
218D12—61no—
219D12—61no—
220C4×D3—61no—
221C4×D3—61no—
222SL(2,𝔽3)—23no—
223SL(2,𝔽3)#62513yescomplemented
224C3×C8—181no—
225C3⋊C8—61no—
226C3⋊C8—181no—
227C3⋊C8—181no—
228C3⋊C8—361no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—43no—
233C2×C32—43no—
234C2×C32—43no—
235C2×C32—83no—
236C2×C32—83no—
237C2×C32—83no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—81no—
241C2×C32#98511yescharacteristic
242C2×C32#98611yescharacteristic
243C2×C32GL(2,ℤ/4)11yescharacteristic
244C3⋊D3—82no—
245C3×D3—121no—
246C3×D3—246no—
247C3×D3—241no—
248#41—61no—
249#41—181no—
250C2×Q8D3×C3⋊D311yescharacteristic
251C2×D4—91no—
252C2×D4—91no—
253C2×D4—541no—
254C22×C4—91no—
255SD16—181no—
256SD16—181no—
257D8—542no—
258C2×C8—91no—
259C2×C8—271no—
260C4⋊C4—271no—
261C22⋊C4—541no—
262C22×C3—21no—
263C22×C3—43no—
264C22×C3—83no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3—181no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3—361no—
271C22×C3D3×S411yescharacteristic
272C22×C3C2×C3⋊S411yescharacteristic
273C2×D3—61no—
274C2×D3—61no—
275C2×D3—61no—
276C2×D3—126no—
277C2×D3—121no—
278C2×D3—121no—
279C2×D3—121no—
280C2×D3—243no—
281C2×D3—243no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4—61no—
289C3×C4—243no—
290Dic3—21no—
291Dic3—361no—
292Dic3—361no—
293Dic3—721no—
294Dic3—723no—
295Dic3—726no—
296C32—41no—
297C32—43no—
298C32—83no—
299C32—81no—
300C32#542711yescharacteristic, complemented
301C23—91no—
302C23—181no—
303Q8—31no—
304Q8#629011yescharacteristic
305D4—92no—
306D4—181no—
307D4—181no—
308D4—181no—
309D4—1082no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—541no—
315C8—181no—
316C8—541no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—43no—
321C2×C3—43no—
322C2×C3—43no—
323C2×C3—83no—
324C2×C3—83no—
325C2×C3—83no—
326C2×C3—361no—
327C2×C3—721no—
328C2×C3#6419711yescharacteristic
329C2×C3#6419911yescharacteristic
330C2×C3#6420911yescharacteristic
331C2×C3#6426611yescharacteristic
332C2×C3#6426711yescharacteristic
333C2×C3#6426911yescharacteristic
334D3—121no—
335D3—246no—
336D3—241no—
337C22—181no—
338C22—181no—
339C22—181no—
340C22—361no—
341C22—361no—
342C22—361no—
343C22D3×C3⋊S411yescharacteristic
344C4—31no—
345C4—31no—
346C4—61no—
347C4—1081no—
348C3—21no—
349C3—43no—
350C3—83no—
351C3#1059320311yescharacteristic, complemented
352C3#1059366111yescharacteristic, complemented
353C2—361no—
354C2—721no—
355C2#1172419211yescharacteristic
356C2#1172419411yescharacteristic
357C2#1172420411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
11. C2×C3×#239D3semidirect
13. C2×C3⋊(#41⋊C3)D3semidirect
14. C2×C3⋊GL(2,𝔽3)D3semidirect
29. C32×SL(2,𝔽3)D4semidirect
64. C2×#239C3⋊D3semidirect
105. C3×SL(2,𝔽3)#67semidirect
106. C3×SL(2,𝔽3)#67semidirect
223. SL(2,𝔽3)#625semidirect
300. C32#5427semidirect
351. C3#10593203semidirect
352. C3#10593661semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3×C3⋊GL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105936613
#105932033
#105915863
#888034