Finite Group Databasebeta

← #10588479⋊C3 #13141941

Subgroups — #10588479⋊C3 #13141941

Order 1728.

291 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 96 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
1Q8×3−1+2⋊C4C211yesmaximal, characteristic
2#63452⋊C3C211yesmaximal, characteristic
33−1+2×#103C211yesmaximal, characteristic
4#63395⋊C3C211yesmaximal, characteristic
5#63377⋊C3C211yesmaximal, characteristic
6(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic
7#63366⋊C3C211yesmaximal, characteristic
8C3×#4799—31nomaximal
9#10588479C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C8×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2C2211yescharacteristic
13C4⋊(3−1+2⋊C4)—21no—
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×Q8×Dic3—31no—
22C3×#890—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#817—31no—
26C3×Dic3⋊C8—31no—
27C3×#807—31no—
28Q8×Dic9C2×C311yescharacteristic
29#63452C2×C311yescharacteristic
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic
32#63395C2×C311yescharacteristic
33#63377C2×C311yescharacteristic
34Dic9⋊C8C2×C311yescharacteristic
35#63366C2×C311yescharacteristic
36Q8×3−1+2—21no—
37Q8×3−1+2Q812yes—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D411yescharacteristic
44C2×3−1+2⋊C4D411yescharacteristic
45C8×3−1+2—21no—
463−1+2⋊C8—21no—
47C3×#479—91no—
48#4799—31no—
49C2×C32×Q8C2×D311yescharacteristic
50C2×C32×C8C2×D311yescharacteristic
51C32×C4⋊C4C2×D311yescharacteristic
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3—31no—
54C3×C4⋊Dic3—61no—
55C3×C4⋊Dic3—61no—
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—31no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8—31no—
60C2×Q8×C9—21no—
61C2×Q8×C9C22×C311yescharacteristic
62C2×C8×C9—21no—
63C2×C8×C9C22×C311yescharacteristic
64C9×C4⋊C4—21no—
65C9×C4⋊C4C22×C311yescharacteristic
66C4⋊Dic9—21no—
67C4⋊Dic9—21no—
68C4⋊Dic9C22×C311yescharacteristic
69C4⋊Dic9C22×C311yescharacteristic
70Dic9⋊C4—21no—
71C4×Dic9—21no—
72C4×Dic9C22×C311yescharacteristic
73C2×C9⋊C8C22×C311yescharacteristic
74C22×3−1+2C2×D411yescharacteristic
75C4×3−1+2—22no—
76C4×3−1+2—41no—
77C4×3−1+2C2×Q811yescharacteristic
78C4×3−1+2#4111yescharacteristic
793−1+2⋊C4—21no—
803−1+2⋊C4—21no—
813−1+2⋊C4—41no—
823−1+2⋊C4—41no—
83C3×#125—91no—
84C3×C4×Q8—91no—
85Q8×Dic3—31no—
86#890—31no—
87C3×#107—91no—
88C3×#106—91no—
89C3×C4⋊C8—91no—
90C3×#103—31no—
91C3×#103—91no—
92C3×#103C3×D311yescharacteristic
93#835—31no—
94#817—31no—
95Dic3⋊C8—31no—
96#807—31no—
97C32×Q8—21no—
98C32×Q8Dic612yes—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C32×C8—21no—
107C3×C3⋊C8—61no—
108Q8×C9—21no—
109Q8×C9—22no—
110Q8×C9—41no—
111Q8×C9C3×Q812yes—
112C2×C4×C9—21no—
113C2×C4×C9—21no—
114C2×C4×C9—21no—
115C2×C4×C9—41no—
116C2×C4×C9—41no—
117C2×C4×C9C23×C311yescharacteristic
118C2×Dic9—21no—
119C2×Dic9—21no—
120C2×Dic9C3×D411yescharacteristic
121C2×Dic9C3×D411yescharacteristic
122C8×C9—21no—
123C8×C9—41no—
124C9⋊C8—21no—
125#479—91noSylow
126C2×3−1+2C22⋊Q811yescharacteristic
127C2×3−1+2#13511yescharacteristic
128C2×3−1+2#13711yescharacteristic
129C2×C3×Q8—31no—
130C2×C3×Q8C2×C3×D311yescharacteristic
131C2×C3×C8—31no—
132C2×C3×C8—91no—
133C2×C3×C8C2×C3×D311yescharacteristic
134C3×C4⋊C4—31no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4—181no—
140C3×C4⋊C4C2×C3×D311yescharacteristic
141C3×C42—91no—
142C3×C42—181no—
143C4⋊Dic3—31no—
144C4⋊Dic3—31no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147Dic3⋊C4—61no—
148C4×Dic3—31no—
149C4×Dic3—61no—
150C2×C3⋊C8—31no—
151C22×C32D3×D411yescharacteristic
152C32×C4—22no—
153C32×C4—41no—
154C32×C4C2×Dic611yescharacteristic
155C32×C4#23711yescharacteristic
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—121no—
159C3×Dic3—121no—
160C22×C9—21no—
161C22×C9C2×C3×D411yescharacteristic
162C4×C9—22no—
163C4×C9—21no—
164C4×C9—21no—
165C4×C9—42no—
166C4×C9—41no—
167C4×C9—81no—
168C4×C9C2×C3×Q811yescharacteristic
169C4×C9C3×#4111yescharacteristic
170Dic9—21no—
171Dic9—21no—
172Dic9—41no—
173Dic9—41no—
174#125—91no—
175C4×Q8—91no—
176#107—91no—
177#106—91no—
178C4⋊C8—91no—
179#103—91no—
180#103AGL(1,ℤ/9)11yescharacteristic
1813−1+2#47911yescharacteristic, complemented, Sylow
182C3×Q8—21no—
183C3×Q8—32no—
184C3×Q8—61no—
185C3×Q8C3×Dic612yes—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C22×C3×D311yescharacteristic
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C3×C8—21no—
201C3×C8—61no—
202C3×C8—181no—
203C3⋊C8—61no—
204C2×C32#87811yescharacteristic
205C2×C32#91611yescharacteristic
206C2×C32#91811yescharacteristic
207C2×C9—21no—
208C2×C9—21no—
209C2×C9—21no—
210C2×C9C3×C22⋊Q811yescharacteristic
211C2×C9C3×#13511yescharacteristic
212C2×C9C3×#13711yescharacteristic
213C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
214C2×C8—91no—
215C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
216C4⋊C4—91no—
217C4⋊C4—91no—
218C4⋊C4—181no—
219C4⋊C4—181no—
220C4⋊C4—181no—
221C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
222C42—91no—
223C42—181no—
224C22×C3—31no—
225C22×C3C3×D3×D411yescharacteristic
226C3×C4—22no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—41no—
230C3×C4—62no—
231C3×C4—121no—
232C3×C4—181no—
233C3×C4—181no—
234C3×C4—361no—
235C3×C4—361no—
236C3×C4C2×C3×Dic611yescharacteristic
237C3×C4C3×#23711yescharacteristic
238Dic3—61no—
239Dic3—61no—
240Dic3—121no—
241Dic3—121no—
242C32#479911yescharacteristic
243C9—21no—
244C9C3×#47911yescharacteristic, complemented
245Q8—21no—
246Q83−1+2⋊Q812yes—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—91no—
250C2×C4—91no—
251C2×C4—181no—
252C2×C4—181no—
253C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
254C8—21no—
255C8—181no—
256C2×C3—31no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3C3×#87811yescharacteristic
260C2×C3C3×#91611yescharacteristic
261C2×C3C3×#91811yescharacteristic
262C22D4×AGL(1,ℤ/9)11yescharacteristic
263C4—22no—
264C4—41no—
265C4—181no—
266C4—181no—
267C4—361no—
268C4—361no—
269C4C2×3−1+2⋊Q811yescharacteristic
270C4#3699⋊C311yescharacteristic
271C3—31no—
272C3C3×#479911yescharacteristic
273C2#63440⋊C311yescharacteristic
274C2#63478⋊C311yescharacteristic
275C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#479
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
9. #10588479C3semidirect
181. 3−1+2#479semidirect
244. C9C3×#479semidirect

Maximal subgroups

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

subgroupindex
Q8×3−1+2⋊C42
#63452⋊C32
3−1+2×#1032
#63395⋊C32
#63377⋊C32
(3−1+2⋊C4)⋊C82
#63366⋊C32
C3×#47993
#105884793