Finite Group Databasebeta

← #13166031

Subgroups — #13166031

Order 1728.

585 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 58 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⋊#859C211yesmaximal, characteristic, complemented
4#10593628—31nomaximal
5#10593207—33nomaximal
6#10591540—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))C412yescomplemented
9#88804—41nomaximal
10C2×C3⋊GL(2,𝔽3)—31no—
11C2×C3⋊GL(2,𝔽3)—33no—
12C2×C3⋊GL(2,𝔽3)—61no—
13C2×C3⋊GL(2,𝔽3)—63no—
14C2×#3755—31no—
15C3×Q8×Dic3D311yescharacteristic, complemented
16C3×C4×SL(2,𝔽3)—31no—
17#63894—31no—
18Dic3×SL(2,𝔽3)D313yescomplemented
19C3⋊#859D311yescharacteristic, complemented
20C3×#859—33no—
21C3×#835—31no—
22#63576—31no—
23#63564—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#5026—91no—
31#4817—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C4⋊(C3⋊D3)—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×#624—61no—
39C3⋊GL(2,𝔽3)—32no—
40C3⋊GL(2,𝔽3)—36no—
41C3⋊GL(2,𝔽3)—62no—
42C3⋊GL(2,𝔽3)—66no—
43#3755—32no—
44#3755—32no—
45#3752—121no—
46C2×C3⋊(C3⋊C8)—31no—
47C3×C4⋊Dic3—31no—
48C3×C4⋊Dic3—31no—
49C3×C4×Dic3—31no—
50C2×C3×C3⋊C8—31no—
51#3722—61no—
52#3722—123no—
53C22×C33—41no—
54C2×C3⋊(C3⋊D3)—42no—
55C2×C3⋊(C3⋊D3)—81no—
56C2×C3⋊(C3⋊D3)—81no—
57C3×C3⋊Dic3—81no—
58C32×Dic3—81no—
59C2×GL(2,𝔽3)—93no—
60C2×GL(2,𝔽3)—183no—
61C3×C4×Q8—31no—
62Q8×Dic3C3⋊D311yescharacteristic, complemented
63C2×#215—91no—
64C2×#215—91no—
65C2×#215—181no—
66#893—91no—
67C4×D12—91no—
68C4×SL(2,𝔽3)—33no—
69#859—33no—
70C3×#103—91no—
71#835—31no—
72#821—91no—
73#815—91no—
74#809—91no—
75#807—91no—
76#807—91no—
77#803—91no—
78C22×C3⋊D3—61no—
79C22×C3⋊D3—121no—
80C22×C3⋊D3—123no—
81C22×C3⋊D3—241no—
82C22×C3⋊D3—243no—
83C32×Q8—31no—
84C32×Q8C4×D311yescharacteristic
85C2×C32×C4—31no—
86C2×C32×C4—121no—
87C2×C3⋊Dic3—41no—
88C4⋊(C3⋊D3)—32no—
89C4⋊(C3⋊D3)—61no—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—43no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—123no—
94C2×C3×Dic3S411yescharacteristic
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—23no—
97C3×SL(2,𝔽3)C4×D313yescomplemented
98C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
99#624—122no—
100C3⋊(C3⋊C8)—32no—
101C3×C3⋊C8—61no—
102#440—271noSylow
103C2×C33—41no—
104C2×C33—41no—
105C2×C33—41no—
106C3⋊(C3⋊D3)—82no—
107C2×C3×Q8—21no—
108C2×C3×Q8D3211yescharacteristic
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×D12—91no—
111C2×D12—91no—
112C2×D12—181no—
113C2×C4×D3—181no—
114C2×C4×D3—181no—
115C2×SL(2,𝔽3)—23no—
116C2×SL(2,𝔽3)D3213yes—
117GL(2,𝔽3)—96no—
118GL(2,𝔽3)—186no—
119C2×C3×C8—91no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—91no—
122C3×C42—91no—
123#215—92no—
124#215—92no—
125#215—92no—
126#215—92no—
127#215—182no—
128#215—182no—
129(C2×D3)⋊C4—181no—
130(C2×D3)⋊C4—181no—
131(C2×D3)⋊C4—363no—
132C4⋊Dic3—31no—
133C4⋊Dic3—31no—
134C4×Dic3—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—91no—
137C2×C3⋊C8—91no—
138C2×C3⋊C8—181no—
139C22×C32—41no—
140C22×C32—43no—
141C22×C32—83no—
142C22×C32—81no—
143C22×C32C2×S411yescharacteristic
144C2×C3⋊D3—62no—
145C2×C3⋊D3—122no—
146C2×C3⋊D3—126no—
147C2×C3⋊D3—121no—
148C2×C3⋊D3—121no—
149C2×C3⋊D3—242no—
150C2×C3⋊D3—246no—
151C2×C3⋊D3—243no—
152C2×C3⋊D3—243no—
153C2×C3⋊D3—241no—
154C2×C3⋊D3—241no—
155C2×C3⋊D3—242no—
156C2×C3⋊D3—242no—
157C2×C3⋊D3—246no—
158C2×C3⋊D3—246no—
159C32×C4—31no—
160C32×C4—31no—
161C32×C4—241no—
162C3⋊Dic3—81no—
163C3×Dic3—21no—
164C3×Dic3—61no—
165C3×Dic3—83no—
166C3×Dic3—121no—
167C3×Dic3—243no—
168C2×SD16—271no—
169C4×Q8—31no—
170C4×D4—271no—
171#107—271no—
172#103—91no—
173#102—271no—
174#97—271no—
175C33—41noSylow
176C22×D3—181no—
177C22×D3—181no—
178C22×D3—361no—
179C22×D3—363no—
180C22×D3—723no—
181C3×Q8—21no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8#62411yescharacteristic
186C3×Q8C4×C3⋊D311yescharacteristic
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—91no—
192C2×C3×C4—123no—
193C2×C3×C4—181no—
194C2×Dic3—31no—
195C2×Dic3—61no—
196C2×Dic3—123no—
197C2×Dic3C3⋊S411yescharacteristic
198D12—92no—
199D12—92no—
200D12—182no—
201D12—181no—
202D12—181no—
203D12—361no—
204C4×D3—362no—
205C4×D3—362no—
206SL(2,𝔽3)—23no—
207SL(2,𝔽3)#62413yescomplemented
208C3×C8—181no—
209C3⋊C8—61no—
210C3⋊C8—92no—
211C3⋊C8—92no—
212C3⋊C8—182no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—43no—
217C2×C32—43no—
218C2×C32—43no—
219C2×C32—83no—
220C2×C32—83no—
221C2×C32—83no—
222C2×C32—81no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32C4×S411yescharacteristic
226C2×C32#98311yescharacteristic
227C2×C32#98611yescharacteristic
228C3⋊D3—122no—
229C3⋊D3—246no—
230C3⋊D3—242no—
231C3⋊D3—244no—
232C3⋊D3—2412no—
233C2×Q8D3×C3⋊D311yescharacteristic
234C2×D4—271no—
235C22×C4—541no—
236SD16—272no—
237SD16—272no—
238C2×C8—91no—
239C2×C8—271no—
240C4⋊C4—91no—
241C4⋊C4—91no—
242C22⋊C4—541no—
243C42—91no—
244C22×C3—21no—
245C22×C3—43no—
246C22×C3—83no—
247C22×C3D3×S411yescharacteristic
248C22×C3C2×C3⋊S411yescharacteristic
249C2×D3—182no—
250C2×D3—182no—
251C2×D3—362no—
252C2×D3—366no—
253C2×D3—361no—
254C2×D3—361no—
255C2×D3—361no—
256C2×D3—361no—
257C2×D3—726no—
258C2×D3—721no—
259C2×D3—721no—
260C2×D3—723no—
261C2×D3—723no—
262C2×D3—726no—
263C2×D3—726no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—181no—
272C3×C4—243no—
273C3×C4—361no—
274Dic3—21no—
275Dic3—61no—
276Dic3—121no—
277Dic3—243no—
278C32—41no—
279C32—43no—
280C32—83no—
281C32—81no—
282C32#539411yescharacteristic, complemented
283C23—541no—
284Q8—31no—
285Q8#628911yescharacteristic
286D4—272no—
287D4—541no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—91no—
291C2×C4—181no—
292C2×C4—1082no—
293C8—181no—
294C8—272no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—43no—
299C2×C3—43no—
300C2×C3—43no—
301C2×C3—83no—
302C2×C3—83no—
303C2×C3—83no—
304C2×C3#6419911yescharacteristic
305C2×C3#6420211yescharacteristic
306C2×C3(C3⋊S4)⋊C411yescharacteristic
307C2×C3C4×C3⋊S411yescharacteristic
308C2×C3#6426411yescharacteristic
309C2×C3#6426711yescharacteristic
310D3—362no—
311D3—362no—
312D3—722no—
313D3—726no—
314D3—7212no—
315C22—542no—
316C22—1081no—
317C22—1081no—
318C22D3×C3⋊S411yescharacteristic
319C4—31no—
320C4—31no—
321C4—61no—
322C4—181no—
323C4—361no—
324C3—21no—
325C3—43no—
326C3—83no—
327C3#1059320711yescharacteristic, complemented
328C3#1059362811yescharacteristic, complemented
329C2—1082no—
330C2#1172419411yescharacteristic
331C2#1172419711yescharacteristic
332C2#1172420111yescharacteristic
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⋊#859C2semidirect
8. C3⋊(C3⋊GL(2,𝔽3))C4semidirect
15. C3×Q8×Dic3D3semidirect
18. Dic3×SL(2,𝔽3)D3semidirect
19. C3⋊#859D3semidirect
28. C32×SL(2,𝔽3)C2×C4semidirect
62. Q8×Dic3C3⋊D3semidirect
97. C3×SL(2,𝔽3)C4×D3semidirect
98. C3×SL(2,𝔽3)C4×D3semidirect
207. SL(2,𝔽3)#624semidirect
282. C32#5394semidirect
327. C3#10593207semidirect
328. 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⋊#8592
#105936283
#105932073
#105915403
#888044