Finite Group Databasebeta

← #10588714⋊C3 #13142851

Subgroups — #10588714⋊C3 #13142851

Order 1728.

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

normal subgroups only

This group has 389 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×#121C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
3#63392⋊C3C212yesmaximal, complemented
4#63389⋊C3C211yesmaximal, characteristic, complemented
5#63367⋊C3C212yesmaximal, complemented
6C3×#5034—31nomaximal
7#10588714C311yesmaximal, characteristic, complemented
8C2×D4×3−1+2—21no—
9C2×D4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4—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×#121D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C3×#832—32no—
20C3×#830—31no—
21C3×#808—32no—
22C9×#121—21no—
23C9×#121C2×C311yescharacteristic
24C2×C4⋊Dic9C2×C311yescharacteristic, complemented
25#63392C2×C312yescomplemented
26#63389C2×C311yescharacteristic, complemented
27#63367C2×C312yescomplemented
28C23×3−1+2—21no—
29C23×3−1+2D411yescharacteristic
30D4×3−1+2—22no—
31D4×3−1+2—42no—
32C2×C4×3−1+2—22no—
33C2×C4×3−1+2—21no—
34C2×C4×3−1+2D411yescharacteristic
35C2×C4×3−1+2C2311yescharacteristic
36C2×3−1+2⋊C4—22no—
37C2×3−1+2⋊C4—42no—
383−1+2⋊C8—22no—
39C3×#481—91no—
40#5034—31no—
41C2×C32×D4—21no—
42C2×C32×D4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3×Dic3—61no—
45C32×C4⋊C4C2×D311yescharacteristic
46C32×C22⋊C4—21no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—61no—
49C2×C3×C3⋊C8—32no—
50C2×D4×C9—22no—
51C2×D4×C9—21no—
52C2×D4×C9—21no—
53C2×D4×C9C22×C311yescharacteristic
54C22×C4×C9—21no—
55C22×C4×C9C22×C311yescharacteristic
56C22×Dic9—21no—
57C9×C4⋊C4—21no—
58C9×C4⋊C4C22×C311yescharacteristic
59C9×C22⋊C4—22no—
60C9×C22⋊C4—21no—
61C4⋊Dic9—21no—
62C4⋊Dic9C22×C312yes—
63C2×C9⋊C8C22×C312yes—
64C22×3−1+2—21no—
65C22×3−1+2—21no—
66C22×3−1+2—41no—
67C22×3−1+2—41no—
68C22×3−1+2—41no—
69C22×3−1+2SD1612yes—
70C22×3−1+2C2×D411yescharacteristic
71C4×3−1+2—21no—
72C4×3−1+2—41no—
73C4×3−1+2#4112yes—
743−1+2⋊C4—42no—
75C3×#121—31no—
76C3×#121C3×D311yescharacteristic
77C2×C3×C4⋊C4—91no—
78C2×C4⋊Dic3—31no—
79C3×#106—92no—
80C3×#102—92no—
81C3×C22⋊C8—91no—
82#832—32no—
83#830—31no—
84#808—32no—
85C23×C32—21no—
86C23×C32#6711yescharacteristic
87C32×D4—22no—
88C32×D4—42no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4#6711yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C3×Dic3—62no—
94C2×C3×Dic3—122no—
95C23×C9—22no—
96C23×C9—21no—
97C23×C9—21no—
98C23×C9C3×D411yescharacteristic
99C3×C3⋊C8—62no—
100D4×C9—22no—
101D4×C9—42no—
102D4×C9—44no—
103D4×C9—42no—
104C2×C4×C9—22no—
105C2×C4×C9—22no—
106C2×C4×C9—21no—
107C2×C4×C9—21no—
108C2×C4×C9—21no—
109C2×C4×C9—42no—
110C2×C4×C9C3×D411yescharacteristic
111C2×C4×C9C23×C311yescharacteristic
112C2×Dic9—22no—
113C2×Dic9—42no—
114C9⋊C8—22no—
115#481—91noSylow
116C2×3−1+2—22no—
117C2×3−1+2—81no—
118C2×3−1+2#12311yescharacteristic
119C2×3−1+2C2×SD1611yescharacteristic
120C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4—31no—
123C2×C3×D4—61no—
124C2×C3×D4C2×C3×D311yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—181no—
127C22×C3×C4C2×C3×D311yescharacteristic
128C22×Dic3—61no—
129C2×C3×C8—92no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—92no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4C2×C3×D311yescharacteristic
134C3×C22⋊C4—21no—
135C3×C22⋊C4—61no—
136C4⋊Dic3—32no—
137C4⋊Dic3—61no—
138C2×C3⋊C8—32no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32#21412yes—
145C22×C32C2×#6711yescharacteristic
146C32×C4—21no—
147C32×C4—41no—
148C32×C4#23712yes—
149C3×Dic3—122no—
150C22×C9—21no—
151C22×C9—21no—
152C22×C9—22no—
153C22×C9—21no—
154C22×C9—41no—
155C22×C9—41no—
156C22×C9—42no—
157C22×C9—42no—
158C22×C9—42no—
159C22×C9—41no—
160C22×C9—41no—
161C22×C9—41no—
162C22×C9C3×SD1612yes—
163C22×C9C2×C3×D411yescharacteristic
164C4×C9—21no—
165C4×C9—22no—
166C4×C9—41no—
167C4×C9—42no—
168C4×C9—41no—
169C4×C9C3×#4112yes—
170Dic9—42no—
171#121AGL(1,ℤ/9)11yescharacteristic
172C2×C4⋊C4—91no—
173#106—92no—
174#102—92no—
175C22⋊C8—91no—
1763−1+2#48111yescharacteristic, complemented, Sylow
177C23×C3—21no—
178C23×C3—31no—
179C23×C3—61no—
180C23×C3C3×#6711yescharacteristic
181C3×D4—22no—
182C3×D4—42no—
183C3×D4—62no—
184C3×D4—122no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—62no—
190C2×C3×C4—61no—
191C2×C3×C4—182no—
192C2×C3×C4—362no—
193C2×C3×C4C3×#6711yescharacteristic
194C2×C3×C4C22×C3×D311yescharacteristic
195C2×Dic3—62no—
196C2×Dic3—122no—
197C3×C8—182no—
198C3⋊C8—62no—
199C2×C32—22no—
200C2×C32—81no—
201C2×C32C2×#21411yescharacteristic
202C2×C32#93511yescharacteristic
203C2×C32#94911yescharacteristic
204C2×C9—22no—
205C2×C9—21no—
206C2×C9—21no—
207C2×C9—21no—
208C2×C9—42no—
209C2×C9—82no—
210C2×C9—81no—
211C2×C9C3×#12311yescharacteristic
212C2×C9C2×C3×SD1611yescharacteristic
213C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
214C2×D4—21no—
215C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
216C22×C4—181no—
217C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
218C2×C8—92no—
219C4⋊C4—92no—
220C4⋊C4—181no—
221C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
222C22⋊C4—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—31no—
226C22×C3—32no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—121no—
233C22×C3—121no—
234C22×C3—121no—
235C22×C3C3×#21412yes—
236C22×C3C2×C3×#6711yescharacteristic
237C3×C4—21no—
238C3×C4—32no—
239C3×C4—41no—
240C3×C4—61no—
241C3×C4—121no—
242C3×C4—362no—
243C3×C4C3×#23712yes—
244Dic3—122no—
245C32#503411yescharacteristic
246C9—21no—
247C9C3×#48111yescharacteristic, complemented
248C23—21no—
249C23#611⋊C311yescharacteristic
250D4—22no—
251D4—42no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—182no—
255C2×C4—362no—
256C2×C4#611⋊C311yescharacteristic
257C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
258C8—182no—
259C2×C3—22no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—62no—
264C2×C3—81no—
265C2×C3—241no—
266C2×C3C2×C3×#21411yescharacteristic
267C2×C3C3×#93511yescharacteristic
268C2×C3C3×#94911yescharacteristic
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22#3672⋊C312yes—
275C22C2×#611⋊C311yescharacteristic
276C4—21no—
277C4—41no—
278C4—362no—
279C4#3699⋊C312yes—
280C3—31no—
281C3C3×#503411yescharacteristic
282C2—22no—
283C2—81no—
284C2C2×#3672⋊C311yescharacteristic
285C2#63497⋊C311yescharacteristic
286C2#63512⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#121C2
socleC22×C3C2×C3×#67
prime radicalC32×#121D3
hypercenter#121AGL(1,ℤ/9)
hypocenterC9C3×#481
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(3−1+2⋊C4)C2semidirect
3. #63392⋊C3C2semidirect
4. #63389⋊C3C2semidirect
5. #63367⋊C3C2semidirect
7. #10588714C3semidirect
24. C2×C4⋊Dic9C2×C3semidirect
25. #63392C2×C3semidirect
26. #63389C2×C3semidirect
27. #63367C2×C3semidirect
176. 3−1+2#481semidirect
247. C9C3×#481semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1212
C2×C4⋊(3−1+2⋊C4)2
#63392⋊C32
#63389⋊C32
#63367⋊C32
C3×#50343
#105887143