Finite Group Databasebeta

← #13166038

Subgroups — #13166038

Order 1728.

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

normal subgroups only

This group has 669 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⋊GL(2,𝔽3))C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)—31nomaximal
5#10593214—33nomaximal
6#10591536—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6289—41nomaximal
9C3⋊(C3⋊GL(2,𝔽3))C412yescomplemented
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#63574—31no—
23#63564—31no—
24Dic3×C3⋊C8—31no—
25C22×C3⋊(C3⋊D3)—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
29#6289—44no—
30C4×GL(2,𝔽3)—93no—
31C4×#215—91no—
32#4788—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4⋊(C3⋊D3)—31no—
35C2×C4×C3⋊D3—121no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—23no—
38C2×C3×SL(2,𝔽3)C2×D313yes—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40C2×#624—61no—
41C2×#624—123no—
42C3⋊GL(2,𝔽3)—32no—
43C3⋊GL(2,𝔽3)—36no—
44C3⋊GL(2,𝔽3)—62no—
45C3⋊GL(2,𝔽3)—66no—
46#3755—32no—
47#3755—32no—
48C2×C3⋊(C3⋊C8)—31no—
49C3×C4⋊Dic3—31no—
50C3×C4⋊Dic3—31no—
51C3×C4×Dic3—31no—
52C2×C3×C3⋊C8—31no—
53#3722—61no—
54C22×C33—41no—
55C2×C3⋊(C3⋊D3)—42no—
56C2×C3⋊(C3⋊D3)—42no—
57C2×C3⋊(C3⋊D3)—42no—
58C3×C3⋊Dic3—42no—
59C32×Dic3—42no—
60C2×GL(2,𝔽3)—93no—
61C2×GL(2,𝔽3)—183no—
62C3×C4×Q8—31no—
63Q8×Dic3C3⋊D311yescharacteristic, complemented
64C2×#215—91no—
65C2×#215—91no—
66C2×#215—181no—
67#893—91no—
68C4×D12—91no—
69C4×SL(2,𝔽3)—33no—
70#859—33no—
71C3×#103—91no—
72#835—31no—
73#821—91no—
74C8×Dic3—91no—
75#809—91no—
76#808—91no—
77#808—91no—
78C4×C3⋊C8—91no—
79C22×C3⋊D3—61no—
80C22×C3⋊D3—121no—
81C22×C3⋊D3—123no—
82C22×C3⋊D3—241no—
83C22×C3⋊D3—243no—
84C32×Q8—31no—
85C32×Q8C4×D311yescharacteristic, complemented
86C2×C32×C4—31no—
87C2×C32×C4—121no—
88C2×C3⋊Dic3—41no—
89C4⋊(C3⋊D3)—32no—
90C4⋊(C3⋊D3)—61no—
91C4×C3⋊D3—124no—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—43no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—123no—
96C2×C3×Dic3S411yescharacteristic
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—23no—
99C3×SL(2,𝔽3)C4×D313yescomplemented
100C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
101#624—1212no—
102#624—122no—
103C3⋊(C3⋊C8)—32no—
104C3×C3⋊C8—61no—
105C4×SD16—271noSylow
106C2×C33—41no—
107C2×C33—41no—
108C2×C33—41no—
109C3⋊(C3⋊D3)—44no—
110C2×C3×Q8—21no—
111C2×C3×Q8D3211yescharacteristic
112C2×C3×Q8C2×C3⋊D311yescharacteristic
113C2×D12—91no—
114C2×D12—91no—
115C2×D12—181no—
116C2×C4×D3—181no—
117C2×C4×D3—181no—
118C2×C4×D3—363no—
119C2×SL(2,𝔽3)—23no—
120C2×SL(2,𝔽3)D3213yes—
121GL(2,𝔽3)—96no—
122GL(2,𝔽3)—186no—
123C2×C3×C8—91no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—91no—
126C3×C42—91no—
127#215—92no—
128#215—92no—
129#215—92no—
130#215—92no—
131#215—182no—
132#215—182no—
133(C2×D3)⋊C4—181no—
134(C2×D3)⋊C4—181no—
135C4⋊Dic3—31no—
136C4⋊Dic3—31no—
137C4×Dic3—31no—
138C2×C3⋊C8—31no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—181no—
142C22×C32—41no—
143C22×C32—43no—
144C22×C32—83no—
145C22×C32—81no—
146C22×C32C2×S411yescharacteristic
147C2×C3⋊D3—62no—
148C2×C3⋊D3—122no—
149C2×C3⋊D3—126no—
150C2×C3⋊D3—122no—
151C2×C3⋊D3—122no—
152C2×C3⋊D3—126no—
153C2×C3⋊D3—126no—
154C2×C3⋊D3—121no—
155C2×C3⋊D3—121no—
156C2×C3⋊D3—242no—
157C2×C3⋊D3—246no—
158C2×C3⋊D3—246no—
159C2×C3⋊D3—246no—
160C2×C3⋊D3—242no—
161C2×C3⋊D3—242no—
162C32×C4—31no—
163C32×C4—31no—
164C32×C4—122no—
165C3⋊Dic3—42no—
166C3×Dic3—46no—
167C3×Dic3—61no—
168C3×Dic3—126no—
169C3×Dic3—121no—
170C3×Dic3GL(2,𝔽3)12yescomplemented
171C2×SD16—271no—
172C4×Q8—31no—
173C4×D4—271no—
174#106—271no—
175#103—91no—
176#102—271no—
177C4×C8—271no—
178C33—41noSylow
179C22×D3—181no—
180C22×D3—181no—
181C22×D3—363no—
182C22×D3—361no—
183C22×D3—723no—
184C3×Q8—21no—
185C3×Q8—31no—
186C3×Q8—31no—
187C3×Q8—61no—
188C3×Q8#62411yescharacteristic, complemented
189C3×Q8C4×C3⋊D311yescharacteristic, complemented
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—123no—
196C2×C3×C4—181no—
197C2×Dic3—31no—
198C2×Dic3—61no—
199C2×Dic3—123no—
200C2×Dic3C3⋊S411yescharacteristic
201D12—92no—
202D12—92no—
203D12—182no—
204D12—181no—
205D12—181no—
206D12—361no—
207C4×D3—362no—
208C4×D3—3612no—
209C4×D3—362no—
210SL(2,𝔽3)—23no—
211SL(2,𝔽3)#62413yescomplemented
212C3×C8—181no—
213C3⋊C8—61no—
214C3⋊C8—92no—
215C3⋊C8—92no—
216C3⋊C8—182no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—43no—
221C2×C32—43no—
222C2×C32—43no—
223C2×C32—83no—
224C2×C32—83no—
225C2×C32—83no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32—81no—
229C2×C32C4×S411yescharacteristic
230C2×C32C2×GL(2,𝔽3)11yescharacteristic
231C2×C32#98511yescharacteristic
232C3⋊D3—124no—
233C3⋊D3—1212no—
234C3⋊D3—122no—
235C3⋊D3—2412no—
236C3⋊D3—244no—
237C2×Q8D3×C3⋊D311yescharacteristic
238C2×D4—271no—
239C22×C4—541no—
240SD16—272no—
241SD16—272no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—91no—
245C4⋊C4—91no—
246C22⋊C4—541no—
247C42—91no—
248C22×C3—21no—
249C22×C3—43no—
250C22×C3—83no—
251C22×C3D3×S411yescharacteristic
252C22×C3C2×C3⋊S411yescharacteristic
253C2×D3—182no—
254C2×D3—182no—
255C2×D3—366no—
256C2×D3—362no—
257C2×D3—366no—
258C2×D3—366no—
259C2×D3—361no—
260C2×D3—361no—
261C2×D3—361no—
262C2×D3—361no—
263C2×D3—726no—
264C2×D3—721no—
265C2×D3—721no—
266C2×D3—726no—
267C2×D3—726no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—32no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—126no—
276C3×C4—181no—
277C3×C4—361no—
278Dic3—61no—
279Dic3—126no—
280Dic3—121no—
281Dic3C3⋊GL(2,𝔽3)12yescomplemented
282C32—41no—
283C32—43no—
284C32—83no—
285C32—81no—
286C32C4×GL(2,𝔽3)11yescharacteristic, complemented
287C23—541no—
288Q8—31no—
289Q8#628911yescharacteristic, complemented
290D4—272no—
291D4—541no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—91no—
295C2×C4—181no—
296C2×C4—1082no—
297C8—181no—
298C8—272no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—43no—
303C2×C3—43no—
304C2×C3—43no—
305C2×C3—83no—
306C2×C3—83no—
307C2×C3—83no—
308C2×C3#6419811yescharacteristic
309C2×C3D3×GL(2,𝔽3)11yescharacteristic
310C2×C3(C3⋊S4)⋊C411yescharacteristic
311C2×C3C4×C3⋊S411yescharacteristic
312C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
313C2×C3#6426611yescharacteristic
314D3—3612no—
315D3—362no—
316D3—362no—
317D3—722no—
318D3—7212no—
319C22—542no—
320C22—1081no—
321C22—1081no—
322C22D3×C3⋊S411yescharacteristic
323C4—32no—
324C4—31no—
325C4—31no—
326C4—181no—
327C4—361no—
328C3—21no—
329C3—43no—
330C3—83no—
331C3#1059321411yescharacteristic, complemented
332C3C4×C3⋊GL(2,𝔽3)11yescharacteristic, complemented
333C2—1082no—
334C2#1172419311yescharacteristic
335C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
336C2#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
9. 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
63. Q8×Dic3C3⋊D3semidirect
85. C32×Q8C4×D3semidirect
99. C3×SL(2,𝔽3)C4×D3semidirect
100. C3×SL(2,𝔽3)C4×D3semidirect
170. C3×Dic3GL(2,𝔽3)semidirect
188. C3×Q8#624semidirect
189. C3×Q8C4×C3⋊D3semidirect
211. SL(2,𝔽3)#624semidirect
281. Dic3C3⋊GL(2,𝔽3)semidirect
286. C32C4×GL(2,𝔽3)semidirect
289. Q8#6289semidirect
331. C3#10593214semidirect
332. C3C4×C3⋊GL(2,𝔽3)semidirect

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
C4×C3⋊GL(2,𝔽3)3
#105932143
#105915363
C2×#62894