Finite Group Databasebeta

← C2×#11722408 #13164578

Subgroups — C2×#11722408 #13164578

Order 1728.

633 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C22×C9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#11722408C214yesmaximal, direct product factor
3C2×D9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88614C211yesmaximal, characteristic, complemented
5C2×#1007⋊C3—31nomaximal
6C2×#63713C311yesmaximal, characteristic, complemented
7C2×#611⋊C3—41nomaximal
8C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
9C2×C9⋊SL(2,𝔽3)C2212yescomplemented
10D9⋊SL(2,𝔽3)C2212yescomplemented
11#88614C2212yescomplemented
12C22×C3×SL(2,𝔽3)D311yescharacteristic
13#1007⋊C3—34no—
14C2×D3×SL(2,𝔽3)—31no—
15C2×C3⋊(#41⋊C3)—31no—
16C22×Q8×C9C2×C311yescharacteristic, complemented
17#63713—34no—
18#63713C2×C314yescomplemented
19C2×#3701—31no—
20C2×#3701C2×C311yescharacteristic, complemented
21C2×Q8×D9—31no—
22C2×Q8×D9C2×C311yescharacteristic, complemented
23C2×#3697—31no—
24C2×#3697—31no—
25C22×Q8⋊C9—21no—
26C23×3−1+2—41no—
27C22×AGL(1,ℤ/9)—41no—
28#611⋊C3—44no—
29C2×3−1+2⋊C4—41no—
30C9⋊SL(2,𝔽3)C2311yescharacteristic
31C2×#1007—31no—
32C2×2−1+4⋊C3—91no—
33C2×C3×#67—121no—
34C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
35C2×C3×SL(2,𝔽3)C2×D312yes—
36D3×SL(2,𝔽3)—32no—
37C3⋊(#41⋊C3)—32no—
38C2×Q8×C9—31no—
39C2×Q8×C9—32no—
40C2×Q8×C9C22×C311yescharacteristic
41C2×Q8×C9C22×C312yescomplemented
42C22×C4×C9—31no—
43C2×#611—31no—
44C2×#611A411yescharacteristic
45#3701—34no—
46#3701—34no—
47#3701—32no—
48#3701C22×C312yescomplemented
49Q8×D9—34no—
50Q8×D9—32no—
51Q8×D9—34no—
52Q8×D9C22×C312yescomplemented
53#3697—34no—
54#3697—34no—
55#3697—34no—
56#3697—34no—
57C2×D36—31no—
58C2×D36—31no—
59C2×C4×D9—32no—
60C2×C4×D9—31no—
61C2×C4×D9—31no—
62C2×Dic18—31no—
63C2×Dic18—31no—
64C2×Q8⋊C9—21no—
65C2×Q8⋊C9—22no—
66C22×3−1+2—41no—
67C22×3−1+2—42no—
68C22×3−1+2—82no—
69C2×AGL(1,ℤ/9)—42no—
70C2×AGL(1,ℤ/9)—82no—
713−1+2⋊C4—42no—
72C22×C3×Q8C3×D311yescharacteristic
73#1007—34no—
74#1007—94no—
75C2×#239—31no—
76C2×#239—91no—
77C2×D3×Q8—31no—
78C2×D3×Q8—91no—
79C2×#235—91no—
80C2×#235—91no—
812−1+4⋊C3—94no—
82C2×#41⋊C3—91no—
83C22×SL(2,𝔽3)—31no—
84C22×SL(2,𝔽3)—91no—
85C23×C32—41no—
86C22×C3×D3—121no—
87C3×#67—124no—
88C2×C3×Dic3—121no—
89C3×SL(2,𝔽3)C22×D311yescharacteristic
90C23×C9—81no—
91C23×C9C2×A411yescharacteristic
92C22×D9—31no—
93C22×D9C2×A411yescharacteristic
94Q8×C9—31no—
95Q8×C9—32no—
96Q8×C9—32no—
97Q8×C9C23×C311yescharacteristic
98C2×C4×C9—31no—
99C2×C4×C9—31no—
100C2×C4×C9—32no—
101C2×C4×C9—32no—
102#611—34no—
103#611C2×A414yes—
104C2×Dic9—31no—
105C2×Dic9C2×A411yescharacteristic
106D36—32no—
107D36—32no—
108D36—34no—
109C4×D9—38no—
110C4×D9—32no—
111C4×D9—32no—
112C4×D9—32no—
113C4×D9—32no—
114Dic18—32no—
115Dic18—34no—
116Dic18—32no—
117Q8⋊C9—21no—
118C2×2−1+4—91noSylow
119C2×3−1+2—42no—
120C2×3−1+2—41no—
121C2×3−1+2—82no—
122AGL(1,ℤ/9)—82no—
123C2×C3×Q8—31no—
124C2×C3×Q8—32no—
125C2×C3×Q8C2×C3×D311yescharacteristic
126C2×C3×Q8C2×C3×D312yes—
127C2×C3×D4—361no—
128C22×C3×C4—31no—
129C2×#67—31no—
130C2×#67—91no—
131#239—32no—
132#239—94no—
133#239—94no—
134#239—92no—
135D3×Q8—32no—
136D3×Q8—94no—
137D3×Q8—92no—
138D3×Q8—94no—
139#235—94no—
140#235—94no—
141#235—94no—
142#235—94no—
143C2×D12—91no—
144C2×D12—91no—
145C2×C4×D3—91no—
146C2×C4×D3—91no—
147C2×C4×D3—92no—
148C2×Dic6—91no—
149C2×Dic6—91no—
150#41⋊C3—92no—
151C2×SL(2,𝔽3)—31no—
152C2×SL(2,𝔽3)—32no—
153C2×SL(2,𝔽3)—92no—
154C22×C32—41no—
155C22×C32—42no—
156C22×C32—82no—
157C2×C3×D3—122no—
158C2×C3×D3—242no—
159C3×Dic3—122no—
160C22×C9—22no—
161C22×C9—84no—
162C22×C9—82no—
163C22×C9—81no—
164C22×C9C22×A412yes—
165C22×C9C22×A411yescharacteristic
166C2×D9—22no—
167C2×D9—32no—
168C2×D9—62no—
169C2×D9C22×A412yes—
170C4×C9—31no—
171C4×C9—32no—
172C4×C9—31no—
173Dic9—32no—
174Dic9C22×A412yes—
1752−1+4—94no—
1762−1+4—274no—
177C2×#41—91no—
178C2×#41—271no—
179C2×#41—271no—
180C2×#41—271no—
181C22×Q8—91no—
182C22×Q8—271no—
183C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
1843−1+2—41noSylow
185C23×C3—121no—
186C23×C3—361no—
187C23×C3D3×A411yescharacteristic
188C22×D3—31no—
189C22×D3—91no—
190C3×Q8—31no—
191C3×Q8—32no—
192C3×Q8—32no—
193C3×Q8C22×C3×D311yescharacteristic
194C3×D4—364no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—361no—
200#67—34no—
201#67—94no—
202C2×Dic3—31no—
203C2×Dic3—91no—
204D12—94no—
205D12—92no—
206D12—92no—
207C4×D3—92no—
208C4×D3—92no—
209C4×D3—92no—
210C4×D3—92no—
211C4×D3—98no—
212Dic6—94no—
213Dic6—92no—
214Dic6—92no—
215SL(2,𝔽3)—31no—
216C2×C32—42no—
217C2×C32—41no—
218C2×C32—82no—
219C3×D3—242no—
220C2×C9—22no—
221C2×C9—84no—
222C2×C9—82no—
223C2×C9—81no—
224C2×C92−1+4⋊C312yescomplemented
225C2×C9C23×A411yescharacteristic
226D9—22no—
227D9—62no—
228#41—92no—
229#41—274no—
230#41—274no—
231#41—274no—
232#41—274no—
233#41—274no—
234#41—274no—
235#41—272no—
236C2×Q8—31no—
237C2×Q8—32no—
238C2×Q8—92no—
239C2×Q8—271no—
240C2×Q8—271no—
241C2×Q8—274no—
242C2×Q8—274no—
243C2×Q8—272no—
244C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
245C2×Q8C2×AGL(1,ℤ/9)12yescomplemented
246C2×D4—91no—
247C2×D4—271no—
248C2×D4—271no—
249C2×D4—271no—
250C22×C4—31no—
251C22×C4—271no—
252C22×C4—271no—
253C22×C4—272no—
254C22×C3—22no—
255C22×C3—122no—
256C22×C3—121no—
257C22×C3—242no—
258C22×C3—362no—
259C22×C3—722no—
260C22×C3C2×D3×A412yes—
261C22×C3C2×D3×A411yescharacteristic
262C2×D3—32no—
263C2×D3—62no—
264C2×D3—92no—
265C2×D3—182no—
266C3×C4—31no—
267C3×C4—32no—
268C3×C4—31no—
269C3×C4—362no—
270Dic3—32no—
271Dic3—92no—
272C32—41no—
273C9—81no—
274C9C2×2−1+4⋊C311yescharacteristic, complemented
275C23—91no—
276C23—271no—
277C23D9⋊A411yescharacteristic
278Q8—31no—
279Q8—32no—
280Q8—32no—
281Q8—274no—
282Q8—272no—
283Q8—272no—
284Q8C22×AGL(1,ℤ/9)11yescharacteristic
285D4—94no—
286D4—274no—
287D4—272no—
288D4—272no—
289D4—274no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—91no—
295C2×C4—272no—
296C2×C4—272no—
297C2×C4—272no—
298C2×C4—272no—
299C2×C4—278no—
300C2×C4—271no—
301C2×C3—22no—
302C2×C3—121no—
303C2×C3—122no—
304C2×C3—242no—
305C2×C3—722no—
306C2×C3#1007⋊C312yes—
307C2×C3C22×D3×A411yescharacteristic
308D3—62no—
309D3—182no—
310C22—22no—
311C22—92no—
312C22—182no—
313C22—272no—
314C22—542no—
315C22C2×D9⋊A412yes—
316C22C2×D9⋊A411yescharacteristic
317C4—31no—
318C4—32no—
319C4—31no—
320C4—92no—
321C4—272no—
322C3—121no—
323C3C2×#1007⋊C311yescharacteristic
324C2—22no—
325C2—182no—
326C2—542no—
327C2#1172240812yesdirect product factor
328C2C22×D9⋊A411yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocle, hypocenterQ8×C9C23×C3
FrattiniC2×C3C22×D3×A4
FittingC22×Q8×C9C2×C3
socleC22×C3C2×D3×A4
hypercenterC23D9⋊A4
LCSG ▷ Q8×C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C9⋊SL(2,𝔽3)C2semidirect
2. #11722408C2direct (N × H)
3. C2×D9⋊SL(2,𝔽3)C2semidirect
4. C2×#88614C2semidirect
6. C2×#63713C3semidirect
9. C2×C9⋊SL(2,𝔽3)C22semidirect
10. D9⋊SL(2,𝔽3)C22semidirect
11. #88614C22semidirect
16. C22×Q8×C9C2×C3semidirect
18. #63713C2×C3semidirect
20. C2×#3701C2×C3semidirect
22. C2×Q8×D9C2×C3semidirect
41. C2×Q8×C9C22×C3semidirect
48. #3701C22×C3semidirect
52. Q8×D9C22×C3semidirect
183. C22×Q8AGL(1,ℤ/9)semidirect
224. C2×C92−1+4⋊C3semidirect
245. C2×Q8C2×AGL(1,ℤ/9)semidirect
274. C9C2×2−1+4⋊C3semidirect
327. C2#11722408direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C9⋊SL(2,𝔽3)2
#117224082
C2×D9⋊SL(2,𝔽3)2
C2×#886142
C2×#1007⋊C33
C2×#637133
C2×#611⋊C34