Finite Group Databasebeta

← #10588723⋊C3 #13142890

Subgroups — #10588723⋊C3 #13142890

Order 1728.

349 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 102 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
13−1+2×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
3#63395⋊C3C212yesmaximal, complemented
4#63389⋊C3C211yesmaximal, characteristic
5#63367⋊C3C212yesmaximal, complemented
6C3×#5043—31nomaximal
7#10588723C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic
10C22×3−1+2⋊C4—21no—
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2C2211yescharacteristic
13C22⋊C4×3−1+2—21no—
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2212yes—
16C2×3−1+2⋊C8C2212yes—
17C32×C22⋊Q8D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C3×#835—32no—
20C3×#830—31no—
21C3×#808—32no—
22C9×C22⋊Q8—21no—
23C9×C22⋊Q8C2×C311yescharacteristic
24C2×C4⋊Dic9C2×C311yescharacteristic
25#63395C2×C312yescomplemented
26#63389C2×C311yescharacteristic
27#63367C2×C312yescomplemented
28C23×3−1+2D411yescharacteristic
29Q8×3−1+2—22no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—21no—
32C2×C4×3−1+2—21no—
33C2×C4×3−1+2D411yescharacteristic
34C2×C4×3−1+2C2311yescharacteristic
35C2×3−1+2⋊C4—22no—
36C2×3−1+2⋊C4—42no—
373−1+2⋊C8—22no—
38C3×#483—91no—
39#5043—31no—
40C2×C32×Q8C2×D311yescharacteristic
41C22×C32×C4C2×D311yescharacteristic
42C22×C3×Dic3—61no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×D311yescharacteristic
45C32×C22⋊C4—21no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—61no—
48C2×C3×C3⋊C8—32no—
49C2×Q8×C9—21no—
50C2×Q8×C9C22×C311yescharacteristic
51C22×C4×C9—21no—
52C22×C4×C9C22×C311yescharacteristic
53C22×Dic9—21no—
54C9×C4⋊C4—22no—
55C9×C4⋊C4—21no—
56C9×C4⋊C4—21no—
57C9×C4⋊C4C22×C311yescharacteristic
58C9×C22⋊C4—22no—
59C9×C22⋊C4—21no—
60C4⋊Dic9—21no—
61C4⋊Dic9C22×C312yes—
62C2×C9⋊C8C22×C312yes—
63C22×3−1+2—21no—
64C22×3−1+2—21no—
65C22×3−1+2SD1612yes—
66C22×3−1+2C2×D411yescharacteristic
67C4×3−1+2—21no—
68C4×3−1+2—41no—
69C4×3−1+2—41no—
70C4×3−1+2#4112yes—
713−1+2⋊C4—42no—
72C3×C22⋊Q8—31no—
73C3×C22⋊Q8C3×D311yescharacteristic
74C2×C3×C4⋊C4—91no—
75C2×C4⋊Dic3—31no—
76C3×#106—92no—
77C3×#103—92no—
78C3×C22⋊C8—91no—
79#835—32no—
80#830—31no—
81#808—32no—
82C23×C32#6711yescharacteristic
83C32×Q8—22no—
84C2×C32×C4—22no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4#6711yescharacteristic
88C2×C32×C4C22×D311yescharacteristic
89C2×C3×Dic3—62no—
90C2×C3×Dic3—122no—
91C23×C9—21no—
92C23×C9C3×D411yescharacteristic
93C3×C3⋊C8—62no—
94Q8×C9—22no—
95Q8×C9—42no—
96C2×C4×C9—22no—
97C2×C4×C9—22no—
98C2×C4×C9—22no—
99C2×C4×C9—21no—
100C2×C4×C9—21no—
101C2×C4×C9—21no—
102C2×C4×C9—21no—
103C2×C4×C9—42no—
104C2×C4×C9C3×D411yescharacteristic
105C2×C4×C9C23×C311yescharacteristic
106C2×Dic9—22no—
107C2×Dic9—42no—
108C9⋊C8—22no—
109#483—91noSylow
110C2×3−1+2—22no—
111C2×3−1+2#12311yescharacteristic
112C2×3−1+2C2×SD1611yescharacteristic
113C2×3−1+2#13711yescharacteristic
114C2×C3×Q8—31no—
115C2×C3×Q8C2×C3×D311yescharacteristic
116C22×C3×C4—31no—
117C22×C3×C4—181no—
118C22×C3×C4C2×C3×D311yescharacteristic
119C22×Dic3—61no—
120C2×C3×C8—92no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—92no—
125C3×C4⋊C4—181no—
126C3×C4⋊C4C2×C3×D311yescharacteristic
127C3×C22⋊C4—21no—
128C3×C22⋊C4—61no—
129C4⋊Dic3—32no—
130C4⋊Dic3—61no—
131C2×C3⋊C8—32no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32#21512yes—
135C22×C32C2×#6711yescharacteristic
136C32×C4—21no—
137C32×C4—41no—
138C32×C4—41no—
139C32×C4#23712yes—
140C3×Dic3—122no—
141C22×C9—21no—
142C22×C9—21no—
143C22×C9—22no—
144C22×C9—21no—
145C22×C9—41no—
146C22×C9—41no—
147C22×C9C3×SD1612yes—
148C22×C9C2×C3×D411yescharacteristic
149C4×C9—21no—
150C4×C9—22no—
151C4×C9—41no—
152C4×C9—42no—
153C4×C9—42no—
154C4×C9—41no—
155C4×C9—41no—
156C4×C9C3×#4112yes—
157Dic9—42no—
158C22⋊Q8AGL(1,ℤ/9)11yescharacteristic
159C2×C4⋊C4—91no—
160#106—92no—
161#103—92no—
162C22⋊C8—91no—
1633−1+2#48311yescharacteristic, complemented, Sylow
164C23×C3—31no—
165C23×C3C3×#6711yescharacteristic
166C3×Q8—22no—
167C3×Q8—62no—
168C2×C3×C4—22no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—62no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—182no—
177C2×C3×C4—362no—
178C2×C3×C4C3×#6711yescharacteristic
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—122no—
182C3×C8—182no—
183C3⋊C8—62no—
184C2×C32—22no—
185C2×C32#93511yescharacteristic
186C2×C32C2×#21511yescharacteristic
187C2×C32#95111yescharacteristic
188C2×C9—22no—
189C2×C9—21no—
190C2×C9—21no—
191C2×C9—21no—
192C2×C9—42no—
193C2×C9C3×#12311yescharacteristic
194C2×C9C2×C3×SD1611yescharacteristic
195C2×C9C3×#13711yescharacteristic
196C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
197C22×C4—181no—
198C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
199C2×C8—92no—
200C4⋊C4—21no—
201C4⋊C4—92no—
202C4⋊C4—181no—
203C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
204C22⋊C4—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—31no—
208C22×C3—32no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3C3×#21512yes—
212C22×C3C2×C3×#6711yescharacteristic
213C3×C4—21no—
214C3×C4—32no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—61no—
218C3×C4—121no—
219C3×C4—121no—
220C3×C4—362no—
221C3×C4C3×#23712yes—
222Dic3—122no—
223C32#504311yescharacteristic
224C9—21no—
225C9C3×#48311yescharacteristic, complemented
226C23#611⋊C311yescharacteristic
227Q8—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—182no—
232C2×C4—362no—
233C2×C4#611⋊C311yescharacteristic
234C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
235C8—182no—
236C2×C3—22no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—62no—
241C2×C3C3×#93511yescharacteristic
242C2×C3C2×C3×#21511yescharacteristic
243C2×C3C3×#95111yescharacteristic
244C22—21no—
245C22—21no—
246C22#3675⋊C312yes—
247C22C2×#611⋊C311yescharacteristic
248C4—21no—
249C4—41no—
250C4—41no—
251C4—362no—
252C4#3699⋊C312yes—
253C3—31no—
254C3C3×#504311yescharacteristic
255C2—22no—
256C2#63497⋊C311yescharacteristic
257C2C2×#3675⋊C311yescharacteristic
258C2#63511⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#483
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
3. #63395⋊C3C2semidirect
5. #63367⋊C3C2semidirect
7. #10588723C3semidirect
25. #63395C2×C3semidirect
27. #63367C2×C3semidirect
163. 3−1+2#483semidirect
225. C9C3×#483semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
C2×C4⋊(3−1+2⋊C4)2
#63395⋊C32
#63389⋊C32
#63367⋊C32
C3×#50433
#105887233