Finite Group Databasebeta

← #13166019

Subgroups — #13166019

Order 1728.

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

normal subgroups only

This group has 497 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
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593659—31nomaximal
5#10593197—33nomaximal
6#10591599—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)—21no—
9C3⋊(C3⋊#226)—21no—
10#88807—41nomaximal
11C2×C3×#239D311yescharacteristic
12C2×C3×#41⋊C3—31no—
13C2×C3⋊(#41⋊C3)D313yes—
14C2×C3⋊#226—31no—
15C2×C3⋊#226—33no—
16C2×C3⋊#226—61no—
17C2×C3⋊#226—63no—
18C2×#3756—31no—
19#63857—31no—
20C2×#3716—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24#63562—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5425—93no—
31#5239—91no—
32#4808—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×#634—31no—
35C3×#239—21no—
36C3×#239—61no—
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×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C2×D313yes—
44C3⋊(#41⋊C3)—23no—
45C3⋊#226—62no—
46C3⋊#226—66no—
47C3⋊#226—61no—
48C3⋊#226—63no—
49#3756—61no—
50#3756—61no—
51#3750—121no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×C4⋊Dic3—31no—
54C2×C3×C3⋊C8—31no—
55Dic3⋊Dic3—61no—
56Dic3⋊Dic3—123no—
57(C2×D3)⋊Dic3—61no—
58#3716—62no—
59C22×C33—41no—
60C3⋊(C3⋊Dic3)—81no—
61C3×C3⋊Dic3—42no—
62C32×Dic3—81no—
63C2×C3×#41—31no—
64C2×#239C3⋊D311yescharacteristic
65C2×#41⋊C3—33no—
66C2×#226—93no—
67C2×#226—183no—
68C2×#216—91no—
69C2×#216—91no—
70C2×#216—181no—
71C2×#214—91no—
72#924—91no—
73C2×#204—91no—
74#897—91no—
75#859—33no—
76C3×#103—91no—
77#835—31no—
78#832—91no—
79#830—91no—
80(C2×D3)⋊C8—91no—
81#809—91no—
82C22×C3×D3—31no—
83C32×Q8—31no—
84C32×Q8#6711yescharacteristic
85C2×C32×C4—31no—
86C2×C32×C4—121no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—121no—
90C2×C3⋊Dic3—123no—
91C2×C3⋊Dic3—241no—
92C2×C3⋊Dic3—243no—
93#634—61no—
94#634—61no—
95C2×C3×Dic3—43no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—123no—
98C2×C3×Dic3S411yescharacteristic
99C3×D12—32no—
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)D1211yescharacteristic
106C3×SL(2,𝔽3)#6713yes—
107C3⋊(C3⋊C8)—61no—
108C3×C3⋊C8—61no—
109#452—271noSylow
110C2×C33—41no—
111C2×C33—41no—
112C2×C33—41no—
113C3×#41—61no—
114C3×#41—181no—
115C2×C3×Q8—21no—
116C2×C3×Q8D3211yescharacteristic
117C2×C3×Q8C2×C3⋊D311yescharacteristic
118C2×C3×D4—91no—
119C22×C3×C4—91no—
120#239—21no—
121#239—61no—
122C2×D12—31no—
123C2×C4×D3—31no—
124C2×Dic6—91no—
125C2×Dic6—91no—
126C2×Dic6—181no—
127#41⋊C3—63no—
128C2×SL(2,𝔽3)—23no—
129C2×SL(2,𝔽3)D3213yes—
130#226—186no—
131#226—183no—
132C2×C3×C8—91no—
133C3×C4⋊C4—91no—
134#217—181no—
135#216—182no—
136#216—181no—
137#216—181no—
138#216—182no—
139#216—181no—
140#216—181no—
141#214—182no—
142(C2×D3)⋊C4—181no—
143C4⋊Dic3—31no—
144C4⋊Dic3—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—363no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—181no—
151#204—182no—
152C22×C32—41no—
153C22×C32—43no—
154C22×C32—83no—
155C22×C32—81no—
156C22×C32C2×S411yescharacteristic
157C2×C3×D3—61no—
158C2×C3×D3—61no—
159C2×C3×D3—61no—
160C32×C4—31no—
161C32×C4—31no—
162C32×C4—241no—
163C3⋊Dic3—42no—
164C3⋊Dic3—121no—
165C3⋊Dic3—243no—
166C3⋊Dic3—241no—
167C3⋊Dic3—246no—
168C3⋊Dic3—242no—
169C3×Dic3—21no—
170C3×Dic3—83no—
171C3×Dic3—126no—
172C3×Dic3—121no—
173C2×#41—31no—
174C2×Q16—271no—
175C2×SD16—271no—
176C22⋊Q8—271no—
177#103—91no—
178#102—271no—
179C22⋊C8—271no—
180C33—41noSylow
181C23×C3—91no—
182C22×D3—31no—
183C3×Q8—21no—
184C3×Q8—31no—
185C3×Q8—31no—
186C3×Q8—61no—
187C3×Q8#62611yescharacteristic
188C3×Q8#63811yescharacteristic
189C3×D4—92no—
190C3×D4—181no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—61no—
195C2×C3×C4—123no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4—181no—
199C2×Dic3—61no—
200C2×Dic3—123no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203C2×Dic3—361no—
204C2×Dic3—363no—
205C2×Dic3—723no—
206C2×Dic3C3⋊S411yescharacteristic
207D12—32no—
208D12—61no—
209C4×D3—61no—
210C4×D3—61no—
211Dic6—181no—
212Dic6—181no—
213Dic6—181no—
214Dic6—181no—
215Dic6—182no—
216Dic6—361no—
217SL(2,𝔽3)—23no—
218SL(2,𝔽3)#62613yes—
219C3×C8—181no—
220C3⋊C8—61no—
221C3⋊C8—181no—
222C3⋊C8—181no—
223C3⋊C8—182no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—43no—
228C2×C32—43no—
229C2×C32—43no—
230C2×C32—83no—
231C2×C32—83no—
232C2×C32—83no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32#98411yescharacteristic
237C2×C32#98511yescharacteristic
238C2×C32GL(2,ℤ/4)11yescharacteristic
239C3×D3—121no—
240#41—61no—
241#41—181no—
242C2×Q8—271no—
243C2×Q8D3×C3⋊D311yescharacteristic
244C2×D4—91no—
245C22×C4—91no—
246Q16—541no—
247Q16—541no—
248SD16—542no—
249C2×C8—91no—
250C2×C8—271no—
251C4⋊C4—91no—
252C4⋊C4—541no—
253C22⋊C4—541no—
254C22×C3—21no—
255C22×C3—43no—
256C22×C3—83no—
257C22×C3—181no—
258C22×C3—181no—
259C22×C3—181no—
260C22×C3D3×S411yescharacteristic
261C22×C3C2×C3⋊S411yescharacteristic
262C2×D3—61no—
263C2×D3—61no—
264C2×D3—61no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—243no—
273C3×C4—361no—
274Dic3—21no—
275Dic3—126no—
276Dic3—121no—
277Dic3—362no—
278Dic3—361no—
279Dic3—361no—
280Dic3—723no—
281Dic3—726no—
282C32—41no—
283C32—43no—
284C32—83no—
285C32—81no—
286C32#542511yescharacteristic, complemented
287C23—91no—
288Q8—31no—
289Q8—541no—
290Q8—541no—
291Q8#629111yescharacteristic
292D4—92no—
293D4—181no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4—541no—
300C8—181no—
301C8—541no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—43no—
306C2×C3—43no—
307C2×C3—43no—
308C2×C3—83no—
309C2×C3—83no—
310C2×C3—83no—
311C2×C3—361no—
312C2×C3#6419611yescharacteristic
313C2×C3#6419711yescharacteristic
314C2×C3#6421011yescharacteristic
315C2×C3#6426511yescharacteristic
316C2×C3#6426611yescharacteristic
317C2×C3#6426911yescharacteristic
318D3—121no—
319C22—181no—
320C22—181no—
321C22—181no—
322C22D3×C3⋊S411yescharacteristic
323C4—31no—
324C4—31no—
325C4—61no—
326C4—361no—
327C4—1081no—
328C3—21no—
329C3—43no—
330C3—83no—
331C3#1059319711yescharacteristic, complemented
332C3#1059365911yescharacteristic, complemented
333C2—361no—
334C2#1172419111yescharacteristic
335C2#1172419211yescharacteristic
336C2#1172420511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊(#41⋊C3)C2
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
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C3×C3⋊#859C2semidirect
21. C3⋊#859D3semidirect
286. C32#5425semidirect
331. C3#10593197semidirect
332. C3#10593659semidirect

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⋊#226)2
C3×C3⋊#8592
#105936593
#105931973
#105915993
#888074