Finite Group Databasebeta

← #10588907⋊C3 #13143607

Subgroups — #10588907⋊C3 #13143607

Order 1728.

522 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 122 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×Q8×3−1+2C211yesmaximal, characteristic
2C2×#3674⋊C3C212yesmaximal, complemented
3#63486⋊C3C211yesmaximal, characteristic
4#63395⋊C3C212yesmaximal, complemented
5#63389⋊C3C211yesmaximal, characteristic
6C3×#5228—31nomaximal
7#10588907C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—24no—
9C2×Q8×3−1+2—21no—
10C2×Q8×3−1+2C2212yes—
11C22×C4×3−1+2—21no—
12C22×C4×3−1+2C2211yescharacteristic
13C2×3−1+2⋊Q8C2211yescharacteristic
14#3676⋊C3—21no—
15#3674⋊C3—24no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18C2×3−1+2⋊C8C2212yes—
19C22×C32×Q8D311yescharacteristic
20C2×C3×#216—32no—
21C3×#924—31no—
22C22×Q8×C9—21no—
23C22×Q8×C9C2×C311yescharacteristic
24C3×#835—32no—
25C3×#830—31no—
26C2×#3674C2×C312yescomplemented
27#63486C2×C311yescharacteristic
28#63395C2×C312yescomplemented
29#63389C2×C311yescharacteristic
30C23×3−1+2D411yescharacteristic
31Q8×3−1+2—24no—
32Q8×3−1+2—22no—
33Q8×3−1+2D414yes—
34C2×C4×3−1+2—24no—
35C2×C4×3−1+2—22no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2D411yescharacteristic
38C2×C4×3−1+2C2311yescharacteristic
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—21no—
413−1+2⋊Q8—22no—
423−1+2⋊C8—22no—
43C3×#451—91no—
44#5228—31no—
45C2×C32×Q8—24no—
46C2×C32×Q8—21no—
47C2×C32×Q8C2×D312yes—
48C22×C32×C4—21no—
49C22×C32×C4C2×D311yescharacteristic
50C2×C3×Dic6—31no—
51C3×#217—61no—
52C3×#216—64no—
53C3×C4⋊Dic3—31no—
54C3×Dic3⋊C4—61no—
55C2×C3×C3⋊C8—32no—
56C2×Q8×C9—28no—
57C2×Q8×C9—22no—
58C2×Q8×C9—22no—
59C2×Q8×C9—24no—
60C2×Q8×C9—21no—
61C2×Q8×C9C22×C312yes—
62C22×C4×C9—22no—
63C22×C4×C9—21no—
64C22×C4×C9—21no—
65C22×C4×C9C22×C311yescharacteristic
66C2×Dic18C22×C311yescharacteristic
67#3676—21no—
68#3674—24no—
69C4⋊Dic9C22×C311yescharacteristic
70Dic9⋊C4—21no—
71C2×C9⋊C8C22×C312yes—
72C22×3−1+2—21no—
73C22×3−1+2—21no—
74C22×3−1+2Q1612yescomplemented
75C22×3−1+2C2×D411yescharacteristic
76C4×3−1+2—24no—
77C4×3−1+2—21no—
78C4×3−1+2C2×D412yes—
793−1+2⋊C4—41no—
803−1+2⋊C4—41no—
81C22×C3×Q8—31no—
82C22×C3×Q8C3×D311yescharacteristic
83C2×C3×Q16—92no—
84C3×C22⋊Q8—91no—
85C2×#216—32no—
86#924—31no—
87C3×#103—92no—
88C3×C22⋊C8—91no—
89#835—32no—
90#830—31no—
91C23×C32#6711yescharacteristic
92C32×Q8—24no—
93C32×Q8—22no—
94C32×Q8#6714yes—
95C2×C32×C4—24no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C3×Dic6—62no—
103C23×C9—21no—
104C23×C9C3×D411yescharacteristic
105C3×C3⋊C8—62no—
106Q8×C9—28no—
107Q8×C9—24no—
108Q8×C9—24no—
109Q8×C9—22no—
110Q8×C9—24no—
111Q8×C9C3×D414yes—
112C2×C4×C9—28no—
113C2×C4×C9—24no—
114C2×C4×C9—24no—
115C2×C4×C9—24no—
116C2×C4×C9—22no—
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9C3×D411yescharacteristic
121C2×C4×C9C23×C311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—21no—
124Dic18—22no—
125C9⋊C8—22no—
126#451—91noSylow
127C2×3−1+2—22no—
128C2×3−1+2#12011yescharacteristic
129C2×3−1+2C2×Q1611yescharacteristic
130C2×3−1+2#13711yescharacteristic
131C2×C3×Q8—24no—
132C2×C3×Q8—21no—
133C2×C3×Q8—32no—
134C2×C3×Q8—61no—
135C2×C3×Q8—64no—
136C2×C3×Q8—91no—
137C2×C3×Q8C2×C3×D312yes—
138C22×C3×C4—21no—
139C22×C3×C4—31no—
140C22×C3×C4—61no—
141C22×C3×C4C2×C3×D311yescharacteristic
142C2×Dic6—31no—
143C3×Q16—184no—
144C2×C3×C8—92no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147C3×C22⋊C4—181no—
148#217—61no—
149#216—64no—
150C4⋊Dic3—31no—
151Dic3⋊C4—61no—
152C2×C3⋊C8—32no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32#21612yes—
156C22×C32C2×#6711yescharacteristic
157C32×C4—24no—
158C32×C4—21no—
159C32×C4C2×#6712yes—
160C3×Dic3—121no—
161C3×Dic3—121no—
162C22×C9—22no—
163C22×C9—22no—
164C22×C9—21no—
165C22×C9—21no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9C3×Q1612yescomplemented
169C22×C9C2×C3×D411yescharacteristic
170C4×C9—28no—
171C4×C9—22no—
172C4×C9—24no—
173C4×C9—21no—
174C4×C9—22no—
175C4×C9C2×C3×D412yes—
176Dic9—41no—
177Dic9—41no—
178C22×Q8AGL(1,ℤ/9)11yescharacteristic
179C2×Q16—92no—
180C22⋊Q8—91no—
181#103—92no—
182C22⋊C8—91no—
1833−1+2#45111yescharacteristic, complemented, Sylow
184C23×C3—31no—
185C23×C3C3×#6711yescharacteristic
186C3×Q8—24no—
187C3×Q8—22no—
188C3×Q8—34no—
189C3×Q8—62no—
190C3×Q8—64no—
191C3×Q8—182no—
192C3×Q8C3×#6714yes—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—64no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C3×#6711yescharacteristic
204C2×C3×C4C22×C3×D311yescharacteristic
205C2×Dic3—61no—
206C2×Dic3—61no—
207Dic6—62no—
208C3×C8—182no—
209C3⋊C8—62no—
210C2×C32—22no—
211C2×C32#94211yescharacteristic
212C2×C32C2×#21611yescharacteristic
213C2×C32#95311yescharacteristic
214C2×C9—24no—
215C2×C9—22no—
216C2×C9—21no—
217C2×C9—21no—
218C2×C9—21no—
219C2×C9C3×#12011yescharacteristic
220C2×C9C2×C3×Q1611yescharacteristic
221C2×C9C3×#13711yescharacteristic
222C2×Q8—24no—
223C2×Q8—21no—
224C2×Q8—91no—
225C2×Q8C2×AGL(1,ℤ/9)12yes—
226C22×C4—21no—
227C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
228Q16—184no—
229C2×C8—92no—
230C4⋊C4—91no—
231C4⋊C4—181no—
232C22⋊C4—181no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—31no—
236C22×C3—32no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3C3×#21612yes—
240C22×C3C2×C3×#6711yescharacteristic
241C3×C4—24no—
242C3×C4—21no—
243C3×C4—32no—
244C3×C4—61no—
245C3×C4—64no—
246C3×C4—361no—
247C3×C4—361no—
248C3×C4C2×C3×#6712yes—
249Dic3—121no—
250Dic3—121no—
251C32#522811yescharacteristic
252C9—21no—
253C9C3×#45111yescharacteristic, complemented
254C23#611⋊C311yescharacteristic
255Q8—24no—
256Q8—22no—
257Q8—182no—
258Q8#611⋊C314yes—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4#611⋊C311yescharacteristic
265C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
266C8—182no—
267C2×C3—22no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—62no—
272C2×C3C3×#94211yescharacteristic
273C2×C3C2×C3×#21611yescharacteristic
274C2×C3C3×#95311yescharacteristic
275C22—21no—
276C22—21no—
277C22#3674⋊C312yescomplemented
278C22C2×#611⋊C311yescharacteristic
279C4—24no—
280C4—21no—
281C4—361no—
282C4—361no—
283C4C2×#611⋊C312yes—
284C3—31no—
285C3C3×#522811yescharacteristic
286C2—22no—
287C2C2×#3674⋊C311yescharacteristic
288C2#63505⋊C311yescharacteristic
289C2#1172170411yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×Q8×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenterC22×Q8AGL(1,ℤ/9)
hypocenterC9C3×#451
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3674⋊C3C2semidirect
4. #63395⋊C3C2semidirect
7. #10588907C3semidirect
26. C2×#3674C2×C3semidirect
28. #63395C2×C3semidirect
74. C22×3−1+2Q16semidirect
168. C22×C9C3×Q16semidirect
183. 3−1+2#451semidirect
253. C9C3×#451semidirect
277. C22#3674⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×3−1+22
C2×#3674⋊C32
#63486⋊C32
#63395⋊C32
#63389⋊C32
C3×#52283
#105889073