Finite Group Databasebeta

← #10588695⋊C3 #13142759

Subgroups — #10588695⋊C3 #13142759

Order 1728.

407 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C4×D4×3−1+2C211yesmaximal, characteristic
2C2×#3672⋊C3C212yesmaximal
3#63431⋊C3C211yesmaximal, characteristic, complemented
4#63368⋊C3C212yesmaximal, complemented
5C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
6C3×#5019—31nomaximal
7#10588695C311yesmaximal, characteristic, complemented
8C2×D4×3−1+2C2211yescharacteristic
9C22×C4×3−1+2—21no—
10C2×3−1+2⋊Q8C2212yes—
11C4⋊C4×3−1+2C2211yescharacteristic
12C22⋊C4×3−1+2—21no—
13C42×3−1+2C2211yescharacteristic
14#3672⋊C3—24no—
15C4⋊(3−1+2⋊C4)—21no—
16C4⋊(3−1+2⋊C4)—21no—
17C2×3−1+2⋊C8C2212yes—
18C32×C4×D4D311yescharacteristic
19C2×C3×#214—32no—
20C3×#869—31no—
21C3×#810—32no—
22C3×C4⋊(C3⋊C8)—31no—
23C4×D4×C9—21no—
24C4×D4×C9C2×C311yescharacteristic
25C2×#3672C2×C312yes—
26#63431C2×C311yescharacteristic, complemented
27#63368C2×C312yescomplemented
28C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
29C23×3−1+2—21no—
30D4×3−1+2—21no—
31D4×3−1+2D412yes—
32C2×C4×3−1+2—21no—
33C2×C4×3−1+2—42no—
34C2×C4×3−1+2D411yescharacteristic
35C2×C4×3−1+2D411yescharacteristic
36C2×C4×3−1+2C2311yescharacteristic
37C2×3−1+2⋊C4—22no—
383−1+2⋊Q8—22no—
393−1+2⋊Q8—22no—
403−1+2⋊C8—22no—
41C3×#461—91no—
42#5019—31no—
43C2×C32×D4C2×D311yescharacteristic
44C22×C32×C4—21no—
45C2×C3×Dic6—32no—
46C32×C4⋊C4C2×D311yescharacteristic
47C32×C22⋊C4—21no—
48C32×C42C2×D311yescharacteristic
49C3×#214—64no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—61no—
52C2×C3×C3⋊C8—32no—
53C2×D4×C9—21no—
54C2×D4×C9C22×C311yescharacteristic
55C22×C4×C9—22no—
56C22×C4×C9—21no—
57C2×Dic18C22×C312yes—
58C9×C4⋊C4—21no—
59C9×C4⋊C4C22×C311yescharacteristic
60C9×C22⋊C4—22no—
61C9×C22⋊C4—21no—
62C42×C9—21no—
63C42×C9C22×C311yescharacteristic
64#3672—24no—
65C4⋊Dic9—21no—
66C4⋊Dic9—21no—
67C2×C9⋊C8C22×C312yes—
68C22×3−1+2—22no—
69C22×3−1+2—41no—
70C22×3−1+2—41no—
71C22×3−1+2C2×D411yescharacteristic
72C4×3−1+2—21no—
73C4×3−1+2—41no—
74C4×3−1+2SD1612yescomplemented
75C4×3−1+2C2×D411yescharacteristic
76C4×3−1+2#4111yescharacteristic
773−1+2⋊C4—42no—
78C2×C3×SD16—92no—
79C3×C4⋊Q8—91no—
80C3×C4×D4—31no—
81C3×C4×D4C3×D311yescharacteristic
82C2×#214—32no—
83#869—31no—
84C3×C4⋊C8—91no—
85C3×#103—92no—
86#810—32no—
87C4⋊(C3⋊C8)—31no—
88C23×C32—21no—
89C32×D4—21no—
90C32×D4D1212yes—
91C2×C32×C4—21no—
92C2×C32×C4—42no—
93C2×C32×C4#6711yescharacteristic
94C2×C32×C4#6711yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C3×Dic3—62no—
97C3×Dic6—62no—
98C3×Dic6—62no—
99C23×C9—22no—
100C23×C9—21no—
101C3×C3⋊C8—62no—
102D4×C9—22no—
103D4×C9—21no—
104D4×C9—22no—
105D4×C9C3×D412yes—
106C2×C4×C9—22no—
107C2×C4×C9—21no—
108C2×C4×C9—21no—
109C2×C4×C9—21no—
110C2×C4×C9—21no—
111C2×C4×C9—44no—
112C2×C4×C9—42no—
113C2×C4×C9C3×D411yescharacteristic
114C2×C4×C9C3×D411yescharacteristic
115C2×C4×C9C23×C311yescharacteristic
116C2×Dic9—22no—
117Dic18—22no—
118Dic18—22no—
119C9⋊C8—22no—
120#461—91noSylow
121C2×3−1+2—42no—
122C2×3−1+2#12111yescharacteristic
123C2×3−1+2C2×SD1611yescharacteristic
124C2×3−1+2#13711yescharacteristic
125C2×C3×Q8—92no—
126C2×C3×D4—31no—
127C2×C3×D4C2×C3×D311yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4—61no—
130C2×Dic6—32no—
131C3×SD16—184no—
132C2×C3×C8—92no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—181no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4C2×C3×D311yescharacteristic
137C3×C22⋊C4—21no—
138C3×C22⋊C4—61no—
139C3×C42—31no—
140C3×C42C2×C3×D311yescharacteristic
141#214—64no—
142C4⋊Dic3—61no—
143C4⋊Dic3—61no—
144C2×C3⋊C8—32no—
145C22×C32—22no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32C2×#6711yescharacteristic
149C32×C4—21no—
150C32×C4—41no—
151C32×C4#21412yes—
152C32×C4C2×D1211yescharacteristic
153C32×C4#23511yescharacteristic
154C3×Dic3—122no—
155C22×C9—24no—
156C22×C9—22no—
157C22×C9—21no—
158C22×C9—42no—
159C22×C9—42no—
160C22×C9—41no—
161C22×C9—41no—
162C22×C9C2×C3×D411yescharacteristic
163C4×C9—21no—
164C4×C9—22no—
165C4×C9—21no—
166C4×C9—21no—
167C4×C9—41no—
168C4×C9—42no—
169C4×C9—41no—
170C4×C9C3×SD1612yescomplemented
171C4×C9C2×C3×D411yescharacteristic
172C4×C9C3×#4111yescharacteristic
173Dic9—42no—
174C2×SD16—92no—
175C4⋊Q8—91no—
176C4×D4AGL(1,ℤ/9)11yescharacteristic
177C4⋊C8—91no—
178#103—92no—
1793−1+2#46111yescharacteristic, complemented, Sylow
180C23×C3—21no—
181C23×C3—61no—
182C3×Q8—182no—
183C3×Q8—182no—
184C3×D4—21no—
185C3×D4—32no—
186C3×D4—61no—
187C3×D4C3×D1212yes—
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—42no—
193C2×C3×C4—61no—
194C2×C3×C4—122no—
195C2×C3×C4—182no—
196C2×C3×C4C3×#6711yescharacteristic
197C2×C3×C4C3×#6711yescharacteristic
198C2×C3×C4C22×C3×D311yescharacteristic
199C2×Dic3—62no—
200Dic6—62no—
201Dic6—62no—
202C3×C8—182no—
203C3⋊C8—62no—
204C2×C32—42no—
205C2×C32#93011yescharacteristic
206C2×C32C2×#21411yescharacteristic
207C2×C32#95111yescharacteristic
208C2×C9—21no—
209C2×C9—21no—
210C2×C9—21no—
211C2×C9—44no—
212C2×C9—42no—
213C2×C9C3×#12111yescharacteristic
214C2×C9C2×C3×SD1611yescharacteristic
215C2×C9C3×#13711yescharacteristic
216C2×Q8—92no—
217C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
218C22×C4—21no—
219SD16—184no—
220C2×C8—92no—
221C4⋊C4—181no—
222C4⋊C4—181no—
223C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
224C22⋊C4—21no—
225C42C2×AGL(1,ℤ/9)11yescharacteristic
226C22×C3—22no—
227C22×C3—31no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—62no—
231C22×C3—121no—
232C22×C3—121no—
233C22×C3C2×C3×#6711yescharacteristic
234C3×C4—21no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—32no—
238C3×C4—41no—
239C3×C4—61no—
240C3×C4—121no—
241C3×C4—362no—
242C3×C4C3×#21412yes—
243C3×C4C2×C3×D1211yescharacteristic
244C3×C4C3×#23511yescharacteristic
245Dic3—122no—
246C32#501911yescharacteristic
247C9—21no—
248C9C3×#46111yescharacteristic, complemented
249C23—21no—
250Q8—182no—
251Q8—182no—
252D4—21no—
253D4C4⋊AGL(1,ℤ/9)12yes—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—182no—
257C2×C4#611⋊C311yescharacteristic
258C2×C4#611⋊C311yescharacteristic
259C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
260C8—182no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—42no—
265C2×C3—122no—
266C2×C3C3×#93011yescharacteristic
267C2×C3C2×C3×#21411yescharacteristic
268C2×C3C3×#95111yescharacteristic
269C22—22no—
270C22—41no—
271C22—41no—
272C22C2×#611⋊C311yescharacteristic
273C4—21no—
274C4—41no—
275C4—362no—
276C4#3672⋊C312yescomplemented
277C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
278C4#3697⋊C311yescharacteristic
279C3—31no—
280C3C3×#501911yescharacteristic
281C2—42no—
282C2#63492⋊C311yescharacteristic
283C2C2×#3672⋊C311yescharacteristic
284C2#63511⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC2×C32×D4C2×D3
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#461
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #63431⋊C3C2semidirect
4. #63368⋊C3C2semidirect
5. C4⋊(3−1+2⋊C8)C2semidirect
7. #10588695C3semidirect
26. #63431C2×C3semidirect
27. #63368C2×C3semidirect
28. C4⋊(C9⋊C8)C2×C3semidirect
74. C4×3−1+2SD16semidirect
170. C4×C9C3×SD16semidirect
179. 3−1+2#461semidirect
248. C9C3×#461semidirect
276. C4#3672⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×D4×3−1+22
C2×#3672⋊C32
#63431⋊C32
#63368⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50193
#105886953