Finite Group Databasebeta

← #10588405⋊C3 #13141630

Subgroups — #10588405⋊C3 #13141630

Order 1728.

469 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×3−1+2⋊Q8C211yesmaximal, characteristic
2#63486⋊C3C211yesmaximal, characteristic
3C2×Dic36⋊C3C212yesmaximal, complemented
43−1+2×C22⋊C8C211yesmaximal, characteristic
5#63376⋊C3C212yesmaximal, complemented
6C3×#4739—31nomaximal
7#10588405C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C22×3−1+2⋊C4—21no—
10C2×3−1+2⋊Q8—21no—
11C2×3−1+2⋊Q8—24no—
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×3−1+2⋊Q8C2212yes—
14C2×C8×3−1+2C2212yes—
15#3676⋊C3—21no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18Dic36⋊C3—24no—
19C22×C3×Dic6—31no—
20C3×#924—31no—
21C2×C3×Dic12—32no—
22C22×Dic18C2×C311yescharacteristic
23C32×C22⋊C8D311yescharacteristic
24C3×#816—32no—
25#63486C2×C311yescharacteristic
26C2×Dic36C2×C312yescomplemented
27C9×C22⋊C8—21no—
28C9×C22⋊C8C2×C311yescharacteristic
29#63376C2×C312yescomplemented
30C23×3−1+2D411yescharacteristic
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2D411yescharacteristic
33C2×C4×3−1+2C2311yescharacteristic
34C2×3−1+2⋊C4—24no—
35C2×3−1+2⋊C4—22no—
36C2×3−1+2⋊C4—21no—
37C2×3−1+2⋊C4—21no—
383−1+2⋊Q8—24no—
393−1+2⋊Q8—22no—
403−1+2⋊Q8—22no—
413−1+2⋊Q8D414yes—
42C8×3−1+2—22no—
43C3×#451—91no—
44#4739—31no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C3×Dic3—61no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—31no—
49C2×C3×Dic6—64no—
50C2×C3×Dic6—61no—
51C2×C32×C8C2×D312yes—
52C3×#217—61no—
53C3×C4⋊Dic3—31no—
54C3×Dic3⋊C4—61no—
55C3×Dic12—64no—
56C22×C4×C9—21no—
57C22×C4×C9C22×C311yescharacteristic
58C22×Dic9—21no—
59C2×Dic18—21no—
60C2×Dic18—24no—
61C2×Dic18C22×C311yescharacteristic
62C2×Dic18C22×C312yes—
63C2×C8×C9—22no—
64C2×C8×C9C22×C312yes—
65#3676—21no—
66C4⋊Dic9C22×C311yescharacteristic
67Dic9⋊C4—21no—
68Dic36—24no—
69C22×3−1+2—21no—
70C22×3−1+2—21no—
71C22×3−1+2Q1612yescomplemented
72C22×3−1+2C2×D411yescharacteristic
73C4×3−1+2—21no—
74C4×3−1+2C2×D412yes—
753−1+2⋊C4—24no—
763−1+2⋊C4—41no—
773−1+2⋊C4—41no—
78C22×C3×Q8—91no—
79C22×Dic6—31no—
80C2×C3×Q16—92no—
81C3×C22⋊Q8—91no—
82#924—31no—
83C2×Dic12—32no—
84C3×#103—92no—
85C3×C22⋊C8—31no—
86C3×C22⋊C8C3×D311yescharacteristic
87#816—32no—
88C23×C32D1211yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C3×Dic3—64no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C3×Dic6—34no—
97C3×Dic6—64no—
98C3×Dic6—62no—
99C3×Dic6—62no—
100C23×C9—21no—
101C23×C9C3×D411yescharacteristic
102C32×C8—22no—
103C2×C4×C9—22no—
104C2×C4×C9—21no—
105C2×C4×C9—21no—
106C2×C4×C9—42no—
107C2×C4×C9C3×D411yescharacteristic
108C2×C4×C9C23×C311yescharacteristic
109C2×Dic9—24no—
110C2×Dic9—22no—
111C2×Dic9—21no—
112C2×Dic9—21no—
113Dic18—24no—
114Dic18—22no—
115Dic18—22no—
116Dic18C3×D414yes—
117C8×C9—22no—
118C8×C9—42no—
119#451—91noSylow
120C2×3−1+2—22no—
121C2×3−1+2#12011yescharacteristic
122C2×3−1+2C2×Q1611yescharacteristic
123C2×3−1+2#13711yescharacteristic
124C2×C3×Q8—92no—
125C2×C3×Q8—91no—
126C2×C3×Q8—181no—
127C2×C3×Q8—184no—
128C22×C3×C4—31no—
129C22×C3×C4—181no—
130C22×C3×C4C2×C3×D311yescharacteristic
131C22×Dic3—61no—
132C2×Dic6—32no—
133C2×Dic6—31no—
134C2×Dic6—64no—
135C2×Dic6—61no—
136C3×Q16—184no—
137C2×C3×C8—32no—
138C2×C3×C8C2×C3×D312yes—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C22⋊C4—181no—
142#217—61no—
143C4⋊Dic3—31no—
144Dic3⋊C4—61no—
145Dic12—64no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32Dic1212yes—
149C22×C32C2×D1211yescharacteristic
150C32×C4—21no—
151C32×C4D3×D412yes—
152C3×Dic3—64no—
153C3×Dic3—121no—
154C3×Dic3—121no—
155C22×C9—21no—
156C22×C9—21no—
157C22×C9—22no—
158C22×C9—21no—
159C22×C9—41no—
160C22×C9—41no—
161C22×C9C3×Q1612yescomplemented
162C22×C9C2×C3×D411yescharacteristic
163C4×C9—21no—
164C4×C9—22no—
165C4×C9—41no—
166C4×C9C2×C3×D412yes—
167Dic9—24no—
168Dic9—41no—
169Dic9—41no—
170C22×Q8—91no—
171C2×Q16—92no—
172C22⋊Q8—91no—
173#103—92no—
174C22⋊C8AGL(1,ℤ/9)11yescharacteristic
1753−1+2#45111yescharacteristic, complemented, Sylow
176C23×C3—31no—
177C23×C3C3×D1211yescharacteristic
178C3×Q8—94no—
179C3×Q8—182no—
180C3×Q8—182no—
181C3×Q8—184no—
182C2×C3×C4—22no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—62no—
186C2×C3×C4—182no—
187C2×C3×C4—184no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4C3×D1211yescharacteristic
191C2×C3×C4C22×C3×D311yescharacteristic
192C2×Dic3—64no—
193C2×Dic3—62no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196Dic6—34no—
197Dic6—64no—
198Dic6—62no—
199Dic6—62no—
200C3×C8—22no—
201C3×C8—62no—
202C2×C32—22no—
203C2×C32#88211yescharacteristic
204C2×C32C2×Dic1211yescharacteristic
205C2×C32#90911yescharacteristic
206C2×C9—22no—
207C2×C9—21no—
208C2×C9—21no—
209C2×C9—21no—
210C2×C9—42no—
211C2×C9C3×#12011yescharacteristic
212C2×C9C2×C3×Q1611yescharacteristic
213C2×C9C3×#13711yescharacteristic
214C2×Q8—92no—
215C2×Q8—91no—
216C2×Q8—184no—
217C2×Q8—181no—
218C22×C4—181no—
219C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
220Q16—184no—
221C2×C8C2×AGL(1,ℤ/9)12yes—
222C4⋊C4—91no—
223C4⋊C4—181no—
224C22⋊C4—181no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—31no—
228C22×C3—32no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3C3×Dic1212yes—
232C22×C3C2×C3×D1211yescharacteristic
233C3×C4—21no—
234C3×C4—32no—
235C3×C4—61no—
236C3×C4—184no—
237C3×C4—361no—
238C3×C4—361no—
239C3×C4C3×D3×D412yes—
240Dic3—64no—
241Dic3—121no—
242Dic3—121no—
243C32#473911yescharacteristic
244C9—21no—
245C9C3×#45111yescharacteristic, complemented
246C23C4⋊AGL(1,ℤ/9)11yescharacteristic
247Q8—94no—
248Q8—184no—
249Q8—182no—
250Q8—182no—
251C2×C4—22no—
252C2×C4—184no—
253C2×C4—182no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
257C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
258C8—22no—
259C2×C3—22no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—62no—
264C2×C3C3×#88211yescharacteristic
265C2×C3C2×C3×Dic1211yescharacteristic
266C2×C3C3×#90911yescharacteristic
267C22—21no—
268C22—21no—
269C22Dic36⋊C312yescomplemented
270C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
271C4—21no—
272C4—184no—
273C4—361no—
274C4—361no—
275C4D4×AGL(1,ℤ/9)12yes—
276C3—31no—
277C3C3×#473911yescharacteristic
278C2—22no—
279C2#63444⋊C311yescharacteristic
280C2C2×Dic36⋊C311yescharacteristic
281C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC23×C32D12
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#451
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×Dic36⋊C3C2semidirect
5. #63376⋊C3C2semidirect
7. #10588405C3semidirect
26. C2×Dic36C2×C3semidirect
29. #63376C2×C3semidirect
71. C22×3−1+2Q16semidirect
161. C22×C9C3×Q16semidirect
175. 3−1+2#451semidirect
245. C9C3×#451semidirect
269. C22Dic36⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×3−1+2⋊Q82
#63486⋊C32
C2×Dic36⋊C32
3−1+2×C22⋊C82
#63376⋊C32
C3×#47393
#105884053