Finite Group Databasebeta

← C9×#5077 #13142979

Subgroups — C9×#5077 #13142979

Order 1728.

612 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C9×#1004C211yesmaximal, characteristic, complemented
2C3×C9×#127C211yesmaximal, characteristic, complemented
3C9×#932C212yesmaximal, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5C9×#805C212yesmaximal, complemented
6C3×#5077C311yesmaximal, characteristic
7C9×#453—31nomaximal
8C2×C3×D4×C9—22no—
9C2×C3×D4×C9C2211yescharacteristic
10C2×C3×D4×C9C2211yescharacteristic, complemented
11C2×C9×#67—22no—
12C9×#237—22no—
13D3×D4×C9—22no—
14C9×#235C2212yes—
15C3×C42×C9C2211yescharacteristic, complemented
16C9×#214—22no—
17C9×#213—22no—
18C9×#208C2212yescomplemented
19C3×#1004C2×C311yescharacteristic
20C32×#127C2×C311yescharacteristic
21C3×#932C2×C312yes—
22C9×H(5,𝔽2)—31no—
23C3×#833C2×C311yescharacteristic
24C3×#805C2×C312yes—
25C9×AGL(1,ℤ/8)—32no—
26C9×#127—21no—
27C9×#127D311yescharacteristic, complemented
28C9×#104—32no—
29C9×#100—31no—
30C23×C3×C9—21no—
31C23×C3×C9D412yes—
32C22×D3×C9—22no—
33C3×D4×C9—22no—
34C3×D4×C9—22no—
35C3×D4×C9—44no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C9×#67—24no—
39C9×#67—22no—
40C2×C9×Dic3—22no—
41C9×D12D412yescomplemented
42C4×D3×C9—22no—
43C9×Dic6D412yescomplemented
44C9×C3⋊C8—22no—
45C3×#453—31no—
46#5077C911yescharacteristic, direct product factor
47C2×C32×D4—22no—
48C2×C32×D4C22×C311yescharacteristic
49C2×C32×D4C22×C311yescharacteristic
50C2×C3×#67—22no—
51C3×#237—22no—
52C3×D3×D4—22no—
53C3×#235C22×C312yes—
54C32×C42C22×C311yescharacteristic
55C3×#214—22no—
56C3×#213—22no—
57C3×#208C22×C312yes—
58C9×#41—32no—
59C9×#41—62no—
60C2×D4×C9—24no—
61C2×D4×C9—21no—
62C2×D4×C9—21no—
63C2×D4×C9—22no—
64C2×D4×C9—62no—
65C2×D4×C9—62no—
66C2×D4×C9C2×D311yescharacteristic
67C2×D4×C9C2×D311yescharacteristic, complemented
68C9×SD16—62no—
69C9×D8—62no—
70C9×M16—32no—
71C42×C9—21no—
72C42×C9C2×D311yescharacteristic, complemented
73C22×C3×C9—22no—
74C22×C3×C9—42no—
75C22×C3×C9—42no—
76C22×C3×C9—41no—
77C22×C3×C9C2×D411yescharacteristic
78C2×D3×C9—22no—
79C2×D3×C9—82no—
80C3×C4×C9—22no—
81C3×C4×C9C2×D412yes—
82C9×Dic3—22no—
83C3×H(5,𝔽2)—31no—
84#1004C2×C911yescharacteristic, complemented
85C3×AGL(1,ℤ/8)—32no—
86C3×#127—21no—
87C3×#127C2×C911yescharacteristic, complemented
88C3×#127C3×D311yescharacteristic
89#932C2×C912yescomplemented
90C3×#104—32no—
91C3×#100—31no—
92#833C2×C911yescharacteristic, complemented
93#805C2×C912yescomplemented
94C23×C32—21no—
95C23×C32C3×D412yes—
96C22×C3×D3—22no—
97C32×D4—22no—
98C32×D4—22no—
99C32×D4—44no—
100C2×C32×C4—21no—
101C2×C32×C4C23×C311yescharacteristic
102C3×#67—24no—
103C3×#67—22no—
104C2×C3×Dic3—22no—
105C3×D12C3×D412yes—
106C3×C4×D3—22no—
107C3×Dic6C3×D412yes—
108C23×C9—22no—
109C23×C9—21no—
110C23×C9—22no—
111C23×C9—62no—
112C23×C9#6712yes—
113C3×C3⋊C8—22no—
114Q8×C9—32no—
115D4×C9—22no—
116D4×C9—22no—
117D4×C9—32no—
118D4×C9—42no—
119D4×C9—42no—
120D4×C9—44no—
121D4×C9—48no—
122D4×C9—64no—
123D4×C9—62no—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—62no—
128C2×C4×C9—62no—
129C2×C4×C9C22×D311yescharacteristic
130C8×C9—62no—
131#453—31noSylow
132C2×C3×C9—21no—
133C2×C3×C9—42no—
134C2×C3×C9—81no—
135C2×C3×C9#12011yescharacteristic
136D3×C9—42no—
137C3×#41—32no—
138C3×#41—62no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—62no—
145C2×C3×D4—62no—
146C2×C3×D4C22×C911yescharacteristic
147C2×C3×D4C22×C911yescharacteristic, complemented
148C2×C3×D4C2×C3×D311yescharacteristic
149C2×C3×D4C2×C3×D311yescharacteristic
150C2×#67—22no—
151#237—22no—
152D3×D4—22no—
153#235C22×C912yes—
154C3×SD16—62no—
155C3×D8—62no—
156C3×M16—32no—
157C3×C42—21no—
158C3×C42C22×C911yescharacteristic, complemented
159C3×C42C2×C3×D311yescharacteristic
160#214—22no—
161#213—22no—
162#208C22×C912yescomplemented
163C22×C32—22no—
164C22×C32—42no—
165C22×C32—42no—
166C22×C32—41no—
167C22×C32C2×C3×D411yescharacteristic
168C2×C3×D3—22no—
169C2×C3×D3—82no—
170C32×C4—22no—
171C32×C4C2×C3×D412yes—
172C3×Dic3—22no—
173C22×C9—22no—
174C22×C9—21no—
175C22×C9—42no—
176C22×C9—42no—
177C22×C9—44no—
178C22×C9—44no—
179C22×C9—42no—
180C22×C9—42no—
181C22×C9—41no—
182C22×C9—62no—
183C22×C9—242no—
184C22×C9C2×#6711yescharacteristic
185C4×C9—24no—
186C4×C9—22no—
187C4×C9—22no—
188C4×C9—62no—
189C4×C9D3×D412yes—
190H(5,𝔽2)—31no—
191AGL(1,ℤ/8)—32no—
192#127D3×C911yescharacteristic, complemented
193#104—32no—
194#100—31no—
195C3×C9#45311yescharacteristic, complemented, Sylow
196C23×C3—22no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—22no—
200C23×C3—62no—
201C23×C3D4×C912yes—
202C23×C3C3×#6712yes—
203C22×D3—22no—
204C3×Q8—32no—
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—22no—
208C3×D4—22no—
209C3×D4—32no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—44no—
213C3×D4—44no—
214C3×D4—48no—
215C3×D4—64no—
216C3×D4—62no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4C23×C911yescharacteristic
224C2×C3×C4C22×C3×D311yescharacteristic
225#67—24no—
226#67—22no—
227C2×Dic3—22no—
228D12D4×C912yescomplemented
229C4×D3—22no—
230Dic6D4×C912yescomplemented
231C3×C8—62no—
232C3⋊C8—22no—
233C2×C32—21no—
234C2×C32—42no—
235C2×C32—81no—
236C2×C32C3×#12011yescharacteristic
237C3×D3—42no—
238C2×C9—22no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—42no—
242C2×C9—82no—
243C2×C9—82no—
244C2×C9—81no—
245C2×C9—122no—
246C2×C9#93711yescharacteristic
247#41—32no—
248#41—62no—
249C2×D4—22no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4C2×D3×C911yescharacteristic
253C2×D4C2×D3×C911yescharacteristic, complemented
254SD16—62no—
255D8—62no—
256M16—32no—
257C42C2×D3×C911yescharacteristic, complemented
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—21no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—62no—
272C22×C3—242no—
273C22×C3C2×D4×C911yescharacteristic
274C22×C3C2×C3×#6711yescharacteristic
275C2×D3—22no—
276C2×D3—82no—
277C3×C4—22no—
278C3×C4—24no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—62no—
282C3×C4C2×D4×C912yes—
283C3×C4C3×D3×D412yes—
284Dic3—22no—
285C32C3×#45311yescharacteristic
286C9—21no—
287C9#507711yescharacteristic, direct product factor
288C23—21no—
289C23—62no—
290C23C9×#6712yes—
291Q8—32no—
292D4—22no—
293D4—22no—
294D4—32no—
295D4—44no—
296D4—64no—
297D4—62no—
298C2×C4—21no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4C22×D3×C911yescharacteristic
302C8—62no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—82no—
310C2×C3—82no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—122no—
314C2×C3C9×#12011yescharacteristic
315C2×C3C3×#93711yescharacteristic
316D3—42no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—62no—
322C22—242no—
323C22C2×C9×#6711yescharacteristic
324C4—22no—
325C4—62no—
326C4D3×D4×C912yes—
327C3—21no—
328C3C9×#45311yescharacteristic, complemented
329C3C3×#507711yescharacteristic
330C2—21no—
331C2—42no—
332C2—81no—
333C2—122no—
334C2C9×#93711yescharacteristic
subgroupG/H
centerC2×C9#937
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#127C2
socleC2×C32C3×#120
prime radicalC3×#5077C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#127D3
hypocenterC3C9×#453
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#127

Complements

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

NH ≅ G/Ntype
1. C9×#1004C2semidirect
2. C3×C9×#127C2semidirect
3. C9×#932C2semidirect
4. C9×#833C2semidirect
5. C9×#805C2semidirect
10. C2×C3×D4×C9C22semidirect
15. C3×C42×C9C22semidirect
18. C9×#208C22semidirect
27. C9×#127D3semidirect
41. C9×D12D4semidirect
43. C9×Dic6D4semidirect
46. #5077C9direct (N × H)
67. C2×D4×C9C2×D3semidirect
72. C42×C9C2×D3semidirect
84. #1004C2×C9semidirect
87. C3×#127C2×C9semidirect
89. #932C2×C9semidirect
92. #833C2×C9semidirect
93. #805C2×C9semidirect
147. C2×C3×D4C22×C9semidirect
158. C3×C42C22×C9semidirect
162. #208C22×C9semidirect
192. #127D3×C9semidirect
195. C3×C9#453semidirect
228. D12D4×C9semidirect
230. Dic6D4×C9semidirect
253. C2×D4C2×D3×C9semidirect
257. C42C2×D3×C9semidirect
287. C9#5077direct (N × H)
328. C3C9×#453semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
C3×C9×#1272
C9×#9322
C9×#8332
C9×#8052
C3×#50773
C9×#4533