Finite Group Databasebeta

← C2×D9⋊(#41⋊C3) #13164586

Subgroups — C2×D9⋊(#41⋊C3) #13164586

Order 1728.

673 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 90 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×#88683C211yesmaximal, characteristic, complemented
2D9⋊(#41⋊C3)C214yesmaximal, direct product factor
3C2×D9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88614C211yesmaximal, characteristic, complemented
5C2×D3×#41⋊C3—31nomaximal
6C2×#41×D9C311yesmaximal, characteristic, complemented
7C2×C4×AGL(1,ℤ/9)—41nomaximal
8#88683C2212yescomplemented
9C2×C9⋊SL(2,𝔽3)C2211yescharacteristic, complemented
10D9⋊SL(2,𝔽3)C2212yescomplemented
11#88614C2212yescomplemented
12C2×C3×#41⋊C3D311yescharacteristic
13D3×#41⋊C3—34no—
14C2×D3×SL(2,𝔽3)—31no—
15C2×C3⋊(#41⋊C3)—31no—
16C2×C9×#41C2×C311yescharacteristic, complemented
17#41×D9—34no—
18#41×D9C2×C314yescomplemented
19C2×#3701C2×C311yescharacteristic, complemented
20C2×Q8×D9C2×C311yescharacteristic
21C2×#3699—31no—
22C2×D4×D9—31no—
23C2×#3697—31no—
24C22×C4×D9—31no—
25C2×#41⋊C9—21no—
26C22×AGL(1,ℤ/9)—41no—
27C2×C4×3−1+2—41no—
28C2×3−1+2⋊C4—41no—
29C4×AGL(1,ℤ/9)—44no—
30C9⋊SL(2,𝔽3)C2311yescharacteristic, complemented
31C2×D3×#41—31no—
32C22×#41⋊C3—91no—
33C2×C3×C4×D3—121no—
34C3×#41⋊C3C2×D312yes—
35C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
36D3×SL(2,𝔽3)—32no—
37C3⋊(#41⋊C3)—32no—
38C23×D9—31no—
39C9×#41—32no—
40C9×#41C22×C312yescomplemented
41C2×Q8×C9C22×C311yescharacteristic
42C2×D4×C9—31no—
43C22×C4×C9—31no—
44C2×#611—32no—
45C22×Dic9—31no—
46#3701—32no—
47#3701C22×C312yescomplemented
48Q8×D9—32no—
49Q8×D9C22×C312yes—
50#3699—34no—
51#3699—34no—
52D4×D9—34no—
53D4×D9—34no—
54#3697—34no—
55#3697—34no—
56C2×D36—31no—
57C2×C4×D9—31no—
58C2×C4×D9—31no—
59C2×C4×D9—31no—
60C2×C4×D9—34no—
61C2×C4×D9—34no—
62C2×C4×D9A411yescharacteristic
63C2×Dic18—31no—
64#41⋊C9—22no—
65C2×Q8⋊C9—21no—
66C22×3−1+2—41no—
67C2×AGL(1,ℤ/9)—44no—
68C2×AGL(1,ℤ/9)—42no—
69C4×3−1+2—42no—
703−1+2⋊C4—42no—
71C2×C3×#41C3×D311yescharacteristic
72D3×#41—34no—
73D3×#41—94no—
74C2×#239—31no—
75C2×D3×Q8—31no—
76C2×#237—91no—
77C2×D3×D4—91no—
78C2×#235—91no—
79C22×C4×D3—91no—
80C2×#41⋊C3—31no—
81C2×#41⋊C3—91no—
82C2×#41⋊C3—94no—
83C22×SL(2,𝔽3)—91no—
84C22×C3×D3—121no—
85C2×C32×C4—41no—
86C2×C3×Dic3—121no—
87C3×C4×D3—124no—
88C3×SL(2,𝔽3)C22×D311yescharacteristic
89C23×C9—31no—
90C22×D9—31no—
91C22×D9—34no—
92C22×D9—64no—
93C22×D9C2×A411yescharacteristic
94Q8×C9—31no—
95Q8×C9C23×C311yescharacteristic
96D4×C9—32no—
97D4×C9—32no—
98C2×C4×C9—31no—
99C2×C4×C9—32no—
100C2×C4×C9—32no—
101C2×C4×C9—81no—
102C2×C4×C9C2×A411yescharacteristic
103#611—38no—
104C2×Dic9—31no—
105C2×Dic9—34no—
106C2×Dic9C2×A411yescharacteristic
107D36—32no—
108D36—32no—
109C4×D9—32no—
110C4×D9—32no—
111C4×D9—34no—
112C4×D9—32no—
113C4×D9—32no—
114C4×D9C2×A414yes—
115Dic18—32no—
116Dic18—32no—
117Q8⋊C9—21no—
118C22×#41—91noSylow
119C2×3−1+2—42no—
120C2×3−1+2—41no—
121AGL(1,ℤ/9)—44no—
122C23×D3—91no—
123C3×#41—32no—
124C3×#41C2×C3×D312yes—
125C2×C3×Q8C2×C3×D311yescharacteristic
126C2×C3×D4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—361no—
129C2×#67—92no—
130C22×Dic3—91no—
131#239—32no—
132#239—92no—
133D3×Q8—32no—
134D3×Q8—92no—
135#237—94no—
136#237—94no—
137D3×D4—94no—
138D3×D4—94no—
139#235—94no—
140#235—94no—
141C2×D12—91no—
142C2×C4×D3—31no—
143C2×C4×D3—94no—
144C2×C4×D3—94no—
145C2×C4×D3—91no—
146C2×C4×D3—91no—
147C2×C4×D3—91no—
148C2×Dic6—91no—
149#41⋊C3—32no—
150#41⋊C3—92no—
151C2×SL(2,𝔽3)—31no—
152C2×SL(2,𝔽3)—92no—
153C22×C32—41no—
154C2×C3×D3—122no—
155C2×C3×D3—124no—
156C32×C4—42no—
157C3×Dic3—122no—
158C22×C9—32no—
159C22×C9—62no—
160C22×C9—81no—
161C22×C9C22×A411yescharacteristic
162C2×D9—32no—
163C2×D9—62no—
164C2×D9—68no—
165C2×D9#41⋊C314yescomplemented
166C2×D9C22×A412yes—
167C4×C9—31no—
168C4×C9—31no—
169C4×C9—82no—
170C4×C9C22×A412yes—
171Dic9—32no—
172Dic9C22×A412yes—
173C2×#41—94no—
174C2×#41—91no—
175C2×#41—271no—
176C2×#41—271no—
177C2×#41—274no—
178C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
179C22×Q8—91no—
180C22×D4—271no—
181C23×C4—271no—
1823−1+2—41noSylow
183C23×C3—31no—
184C23×C3—361no—
185C22×D3—31no—
186C22×D3—94no—
187C22×D3—91no—
188C22×D3—184no—
189C3×Q8—31no—
190C3×Q8C22×C3×D311yescharacteristic
191C3×D4—32no—
192C3×D4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—121no—
197C2×C3×C4—364no—
198C2×C3×C4—361no—
199C2×C3×C4D3×A411yescharacteristic
200#67—98no—
201C2×Dic3—31no—
202C2×Dic3—94no—
203C2×Dic3—91no—
204D12—92no—
205D12—92no—
206C4×D3—34no—
207C4×D3—92no—
208C4×D3—92no—
209C4×D3—92no—
210C4×D3—92no—
211C4×D3—94no—
212Dic6—92no—
213Dic6—92no—
214SL(2,𝔽3)—31no—
215C2×C32—42no—
216C2×C32—41no—
217C3×D3—124no—
218C2×C9—62no—
219C2×C9—82no—
220C2×C9—81no—
221C2×C9C2×#41⋊C312yescomplemented
222C2×C9C23×A411yescharacteristic
223D9—62no—
224D9C2×#41⋊C314yescomplemented
225C24—271no—
226#41—32no—
227#41—92no—
228#41—274no—
229#41—274no—
230#41—274no—
231#41—274no—
232#41—272no—
233#41C2×AGL(1,ℤ/9)12yescomplemented
234C2×Q8—92no—
235C2×Q8—271no—
236C2×Q8—272no—
237C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
238C2×D4—31no—
239C2×D4—271no—
240C2×D4—274no—
241C2×D4—274no—
242C2×D4—272no—
243C22×C4—31no—
244C22×C4—91no—
245C22×C4—271no—
246C22×C4—274no—
247C22×C4—274no—
248C22×C4—271no—
249C22×C4—271no—
250C22×C4—271no—
251C22×C3—32no—
252C22×C3—62no—
253C22×C3—121no—
254C22×C3—362no—
255C22×C3—364no—
256C22×C3C2×D3×A411yescharacteristic
257C2×D3—34no—
258C2×D3—32no—
259C2×D3—92no—
260C2×D3—188no—
261C2×D3—182no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—122no—
265C3×C4—362no—
266C3×C4C2×D3×A412yes—
267Dic3—32no—
268Dic3—92no—
269C32—41no—
270C9—81no—
271C9C22×#41⋊C311yescharacteristic, complemented
272C23—31no—
273C23—91no—
274C23—274no—
275C23—271no—
276C23—544no—
277Q8—31no—
278Q8—272no—
279Q8—272no—
280Q8C22×AGL(1,ℤ/9)11yescharacteristic
281D4—32no—
282D4—32no—
283D4—272no—
284D4—272no—
285D4—278no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—91no—
290C2×C4—94no—
291C2×C4—272no—
292C2×C4—272no—
293C2×C4—272no—
294C2×C4—272no—
295C2×C4—274no—
296C2×C4—271no—
297C2×C4—274no—
298C2×C4D9⋊A411yescharacteristic
299C2×C3—62no—
300C2×C3—121no—
301C2×C3—122no—
302C2×C3—364no—
303C2×C3D3×#41⋊C312yes—
304C2×C3C22×D3×A411yescharacteristic
305D3—34no—
306D3—182no—
307C22—32no—
308C22—62no—
309C22—92no—
310C22—94no—
311C22—272no—
312C22—548no—
313C22—542no—
314C22C2×D9⋊A411yescharacteristic
315C4—31no—
316C4—31no—
317C4—92no—
318C4—272no—
319C4C2×D9⋊A412yes—
320C3—121no—
321C3C2×D3×#41⋊C311yescharacteristic
322C2—62no—
323C2—94no—
324C2—542no—
325C2D9⋊(#41⋊C3)12yesdirect product factor
326C2C22×D9⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D9⋊A4
commutator, cosocle, hypocenterQ8×C9C23×C3
FrattiniC2×C3C22×D3×A4
FittingC2×C9×#41C2×C3
socleC22×C3C2×D3×A4
LCSG ▷ Q8×C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88683C2semidirect
2. D9⋊(#41⋊C3)C2direct (N × H)
3. C2×D9⋊SL(2,𝔽3)C2semidirect
4. C2×#88614C2semidirect
6. C2×#41×D9C3semidirect
8. #88683C22semidirect
9. C2×C9⋊SL(2,𝔽3)C22semidirect
10. D9⋊SL(2,𝔽3)C22semidirect
11. #88614C22semidirect
16. C2×C9×#41C2×C3semidirect
18. #41×D9C2×C3semidirect
19. C2×#3701C2×C3semidirect
30. C9⋊SL(2,𝔽3)C23semidirect
40. C9×#41C22×C3semidirect
47. #3701C22×C3semidirect
165. C2×D9#41⋊C3semidirect
178. C2×#41AGL(1,ℤ/9)semidirect
221. C2×C9C2×#41⋊C3semidirect
224. D9C2×#41⋊C3semidirect
233. #41C2×AGL(1,ℤ/9)semidirect
271. C9C22×#41⋊C3semidirect
325. C2D9⋊(#41⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886832
D9⋊(#41⋊C3)2
C2×D9⋊SL(2,𝔽3)2
C2×#886142
C2×D3×#41⋊C33
C2×#41×D93
C2×C4×AGL(1,ℤ/9)4