Finite Group Databasebeta

← #13166027

Subgroups — #13166027

Order 1728.

502 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 60 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⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593628—31nomaximal
5#10593204—33nomaximal
6#10591566—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C412yescomplemented
9#88803—41nomaximal
10C2×C3⋊GL(2,𝔽3)D311yescharacteristic
11C2×C3×GL(2,𝔽3)—33no—
12C3×Q8×Dic3D311yescharacteristic, complemented
13C2×C3×#215—31no—
14C3×C4×SL(2,𝔽3)—31no—
15Dic3×D12—31no—
16Dic3×SL(2,𝔽3)D313yescomplemented
17C3⋊#859—31no—
18C3⋊#859—33no—
19C3⋊#859—61no—
20C3⋊#859—63no—
21#63662—31no—
22#63575—31no—
23#63562—31no—
24#63554—31no—
25C22×C3×C3⋊D3—41no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
29#5394—93no—
30#5164—91no—
31#5026—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C3×D12—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
38C2×D3×Dic3—61no—
39C3⋊GL(2,𝔽3)Dic312yescomplemented
40C3×GL(2,𝔽3)—36no—
41#3752—121no—
42C4⋊(C3⋊Dic3)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C3×#215—32no—
45C3×#215—32no—
46C3×C4⋊Dic3—31no—
47C3×C4×Dic3—31no—
48C2×C3×C3⋊C8—31no—
49(C2×D3)⋊Dic3—61no—
50(C2×D3)⋊Dic3—123no—
51C22×C33—41no—
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3—81no—
54C2×C3×C3⋊D3—81no—
55C3⋊(C3⋊Dic3)—81no—
56C32×Dic3—81no—
57C2×GL(2,𝔽3)—33no—
58C2×C3×SD16—91no—
59C3×C4×Q8—31no—
60Q8×Dic3C3⋊D311yescharacteristic, complemented
61C2×#215—31no—
62D4×Dic3—91no—
63C4×D12—91no—
64C4×SL(2,𝔽3)—33no—
65#859—93no—
66#859—183no—
67#835—91no—
68#835—91no—
69#835—181no—
70#832—91no—
71#818—91no—
72#815—91no—
73#809—91no—
74#807—91no—
75#803—91no—
76C22×C3⋊D3—41no—
77C22×C3×D3—61no—
78C22×C3×D3—123no—
79C32×Q8—31no—
80C32×Q8C4×D311yescharacteristic
81C2×C32×C4—31no—
82C2×C32×C4—121no—
83C2×C3⋊Dic3—61no—
84C2×C3⋊Dic3—121no—
85C2×C3⋊Dic3—123no—
86C2×C3⋊Dic3—241no—
87C2×C3⋊Dic3—243no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—43no—
90C2×C3×Dic3S411yescharacteristic
91C3×D12—32no—
92C3×D12—61no—
93C3×SL(2,𝔽3)—21no—
94C3×SL(2,𝔽3)—23no—
95C3×SL(2,𝔽3)C4×D313yescomplemented
96C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
97D3×Dic3—122no—
98C3⋊(C3⋊C8)—61no—
99C3×C3⋊C8—32no—
100#440—271noSylow
101C2×C33—41no—
102C2×C33—41no—
103C2×C33—41no—
104C3×C3⋊D3—82no—
105C2×C3×Q8—21no—
106C2×C3×Q8D3211yescharacteristic
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—91no—
109C22×Dic3—181no—
110C2×D12—31no—
111C2×C4×D3—181no—
112C2×SL(2,𝔽3)—23no—
113C2×SL(2,𝔽3)D3213yes—
114GL(2,𝔽3)—36no—
115C3×SD16—92no—
116C3×SD16—92no—
117C2×C3×C8—91no—
118C3×C4⋊C4—91no—
119C3×C42—91no—
120#217—181no—
121#215—32no—
122#215—32no—
123(C2×D3)⋊C4—181no—
124(C2×D3)⋊C4—363no—
125C4⋊Dic3—31no—
126C4⋊Dic3—91no—
127C4⋊Dic3—91no—
128C4⋊Dic3—181no—
129C4×Dic3—31no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—91no—
132C2×C3⋊C8—91no—
133C2×C3⋊C8—181no—
134C22×C32—41no—
135C22×C32—43no—
136C22×C32—83no—
137C22×C32—81no—
138C22×C32C2×S411yescharacteristic
139C2×C3⋊D3—42no—
140C2×C3⋊D3—81no—
141C2×C3⋊D3—81no—
142C2×C3×D3—62no—
143C2×C3×D3—126no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146C2×C3×D3—243no—
147C2×C3×D3—243no—
148C32×C4—31no—
149C32×C4—31no—
150C32×C4—241no—
151C3⋊Dic3—121no—
152C3⋊Dic3—243no—
153C3⋊Dic3—241no—
154C3⋊Dic3—242no—
155C3⋊Dic3—246no—
156C3×Dic3—21no—
157C3×Dic3—61no—
158C3×Dic3—83no—
159C2×SD16—91no—
160C4×Q8—31no—
161C4×D4—271no—
162#107—271no—
163#103—271no—
164#102—271no—
165#97—271no—
166C33—41noSylow
167C23×C3—181no—
168C22×D3—61no—
169C22×D3—123no—
170C3×Q8—21no—
171C3×Q8—31no—
172C3×Q8—31no—
173C3×Q8—61no—
174C3×Q8D3×Dic311yescharacteristic
175C3×Q8C4×C3⋊D311yescharacteristic
176C3×D4—92no—
177C3×D4—181no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—91no—
183C2×C3×C4—123no—
184C2×Dic3—31no—
185C2×Dic3—181no—
186C2×Dic3—181no—
187C2×Dic3—361no—
188C2×Dic3—363no—
189C2×Dic3—362no—
190C2×Dic3—723no—
191C2×Dic3C3⋊S411yescharacteristic
192D12—32no—
193D12—61no—
194C4×D3—362no—
195SL(2,𝔽3)—23no—
196SL(2,𝔽3)D3×Dic313yescomplemented
197C3×C8—92no—
198C3⋊C8—32no—
199C3⋊C8—181no—
200C3⋊C8—181no—
201C3⋊C8—361no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—43no—
206C2×C32—43no—
207C2×C32—43no—
208C2×C32—83no—
209C2×C32—83no—
210C2×C32—83no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32C4×S411yescharacteristic
215C2×C32#98311yescharacteristic
216C2×C32#98611yescharacteristic
217C3⋊D3—82no—
218C3×D3—122no—
219C3×D3—246no—
220C2×Q8D3×C3⋊D311yescharacteristic
221C2×D4—91no—
222C22×C4—541no—
223SD16—92no—
224SD16—92no—
225C2×C8—91no—
226C2×C8—271no—
227C4⋊C4—91no—
228C4⋊C4—271no—
229C22⋊C4—541no—
230C42—91no—
231C22×C3—21no—
232C22×C3—43no—
233C22×C3—83no—
234C22×C3—182no—
235C22×C3—361no—
236C22×C3—361no—
237C22×C3D3×S411yescharacteristic
238C22×C3C2×C3⋊S411yescharacteristic
239C2×D3—62no—
240C2×D3—126no—
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—243no—
244C2×D3—243no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—181no—
253C3×C4—243no—
254Dic3—21no—
255Dic3—61no—
256Dic3—361no—
257Dic3—361no—
258Dic3—721no—
259Dic3—723no—
260Dic3—726no—
261C32—41no—
262C32—43no—
263C32—83no—
264C32—81no—
265C32#539411yescharacteristic, complemented
266C23—181no—
267Q8—31no—
268Q8Dic3×C3⋊D311yescharacteristic
269D4—92no—
270D4—181no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—91no—
274C2×C4—541no—
275C2×C4—1082no—
276C8—92no—
277C8—541no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—43no—
282C2×C3—43no—
283C2×C3—43no—
284C2×C3—83no—
285C2×C3—83no—
286C2×C3—83no—
287C2×C3—362no—
288C2×C3#6419911yescharacteristic
289C2×C3#6420111yescharacteristic
290C2×C3Dic3×S411yescharacteristic
291C2×C3C4×C3⋊S411yescharacteristic
292C2×C3#6426411yescharacteristic
293C2×C3#6426711yescharacteristic
294D3—122no—
295D3—246no—
296C22—182no—
297C22—361no—
298C22—361no—
299C22D3×C3⋊S411yescharacteristic
300C4—31no—
301C4—31no—
302C4—61no—
303C4—181no—
304C4—1081no—
305C3—21no—
306C3—43no—
307C3—83no—
308C3#1059320411yescharacteristic, complemented
309C3#1059362811yescharacteristic, complemented
310C2—362no—
311C2#1172419411yescharacteristic
312C2#1172419611yescharacteristic
313C2Dic3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
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
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
8. C3×C3⋊GL(2,𝔽3)C4semidirect
12. C3×Q8×Dic3D3semidirect
16. Dic3×SL(2,𝔽3)D3semidirect
28. C32×SL(2,𝔽3)C2×C4semidirect
39. C3⋊GL(2,𝔽3)Dic3semidirect
60. Q8×Dic3C3⋊D3semidirect
95. C3×SL(2,𝔽3)C4×D3semidirect
96. C3×SL(2,𝔽3)C2×Dic3semidirect
196. SL(2,𝔽3)D3×Dic3semidirect
265. C32#5394semidirect
308. C3#10593204semidirect
309. C3#10593628semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105936283
#105932043
#105915663
#888034