Finite Group Databasebeta

← #10588556⋊C3 #13142236

Subgroups — #10588556⋊C3 #13142236

Order 1728.

288 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#63452⋊C3C211yesmaximal, characteristic
2#63450⋊C3C211yesmaximal, characteristic
33−1+2×#107C211yesmaximal, characteristic
4#63377⋊C3C211yesmaximal, characteristic
5#63375⋊C3C211yesmaximal, characteristic
6#63367⋊C3C211yesmaximal, characteristic
7#63366⋊C3C211yesmaximal, characteristic
8C3×#4876—31nomaximal
9#10588556C311yesmaximal, characteristic, complemented
10C2×3−1+2⋊Q8—21no—
11C2×C8×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#890—31no—
22C3×#888—31no—
23C32×#107D311yescharacteristic
24C3×#817—31no—
25C3×#815—31no—
26C3×#808—31no—
27C3×#807—31no—
28#63452C2×C311yescharacteristic
29#63450C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic
32#63377C2×C311yescharacteristic
33#63375C2×C311yescharacteristic
34#63367C2×C311yescharacteristic
35#63366C2×C311yescharacteristic
36C2×C4×3−1+2—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2C2311yescharacteristic
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4D411yescharacteristic
42C2×3−1+2⋊C4D411yescharacteristic
433−1+2⋊Q8—42no—
44C8×3−1+2Q812yescomplemented
453−1+2⋊C8Q812yescomplemented
46C3×#501—91no—
47#4876—31no—
48C2×C3×Dic6—61no—
49C2×C32×C8C2×D311yescharacteristic
50C32×C4⋊C4C2×D311yescharacteristic
51C32×C4⋊C4C2×D311yescharacteristic
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3—31no—
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×Dic18—21no—
60C2×C8×C9—21no—
61C2×C8×C9C22×C311yescharacteristic
62C9×C4⋊C4—21no—
63C9×C4⋊C4—21no—
64C9×C4⋊C4C22×C311yescharacteristic
65C9×C4⋊C4C22×C311yescharacteristic
66C4⋊Dic9—21no—
67C4⋊Dic9C22×C311yescharacteristic
68C4⋊Dic9C22×C311yescharacteristic
69Dic9⋊C4—21no—
70Dic9⋊C4—21no—
71C4×Dic9C22×C311yescharacteristic
72C2×C9⋊C8C22×C311yescharacteristic
73C22×3−1+2C2×D411yescharacteristic
74C4×3−1+2—41no—
75C4×3−1+2—41no—
76C4×3−1+2C2×Q811yescharacteristic
77C4×3−1+2C2×Q811yescharacteristic
783−1+2⋊C4—21no—
793−1+2⋊C4—21no—
803−1+2⋊C4—41no—
813−1+2⋊C4—41no—
82C3×C4⋊Q8—91no—
83C3×#125—91no—
84#890—31no—
85#888—31no—
86C3×#107—31no—
87C3×#107—91no—
88C3×#107C3×D311yescharacteristic
89C3×#106—91no—
90C3×#106—91no—
91C3×#97—91no—
92#817—31no—
93#815—31no—
94#808—31no—
95#807—31no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C3×Dic3—31no—
100C2×C3×Dic3—31no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C3×Dic6—122no—
104C32×C8Dic612yes—
105C3×C3⋊C8—32no—
106C2×C4×C9—21no—
107C2×C4×C9—21no—
108C2×C4×C9—21no—
109C2×C4×C9—41no—
110C2×C4×C9—41no—
111C2×C4×C9C23×C311yescharacteristic
112C2×Dic9—21no—
113C2×Dic9—21no—
114C2×Dic9C3×D411yescharacteristic
115C2×Dic9C3×D411yescharacteristic
116Dic18—42no—
117C8×C9—22no—
118C8×C9C3×Q812yescomplemented
119C9⋊C8C3×Q812yescomplemented
120#501—91noSylow
121C2×3−1+2C4⋊Q811yescharacteristic
122C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
123C2×3−1+2#13711yescharacteristic
124C2×C3×Q8—181no—
125C2×Dic6—61no—
126C2×C3×C8—31no—
127C2×C3×C8—91no—
128C2×C3×C8C2×C3×D311yescharacteristic
129C3×C4⋊C4—31no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—181no—
134C3×C4⋊C4—181no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4C2×C3×D311yescharacteristic
137C3×C4⋊C4C2×C3×D311yescharacteristic
138C3×C42—91no—
139C4⋊Dic3—31no—
140C4⋊Dic3—31no—
141C4⋊Dic3—61no—
142Dic3⋊C4—61no—
143Dic3⋊C4—61no—
144C4×Dic3—31no—
145C2×C3⋊C8—31no—
146C22×C32D3×D411yescharacteristic
147C32×C4—41no—
148C32×C4—41no—
149C32×C4C2×Dic611yescharacteristic
150C32×C4D3×Q811yescharacteristic
151C3×Dic3—61no—
152C3×Dic3—61no—
153C3×Dic3—121no—
154C3×Dic3—121no—
155C22×C9—21no—
156C22×C9C2×C3×D411yescharacteristic
157C4×C9—21no—
158C4×C9—21no—
159C4×C9—41no—
160C4×C9—41no—
161C4×C9—81no—
162C4×C9—81no—
163C4×C9C2×C3×Q811yescharacteristic
164C4×C9C2×C3×Q811yescharacteristic
165Dic9—21no—
166Dic9—21no—
167Dic9—41no—
168Dic9—41no—
169C4⋊Q8—91no—
170#125—91no—
171#107—91no—
172#107AGL(1,ℤ/9)11yescharacteristic
173#106—91no—
174#106—91no—
175#97—91no—
1763−1+2#50111yescharacteristic, complemented, Sylow
177C3×Q8—362no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4C22×C3×D311yescharacteristic
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192Dic6—122no—
193C3×C8—32no—
194C3×C8—92no—
195C3×C8C3×Dic612yes—
196C3⋊C8—32no—
197C2×C32#88811yescharacteristic
198C2×C32#91111yescharacteristic
199C2×C32#91811yescharacteristic
200C2×C9—21no—
201C2×C9—21no—
202C2×C9—21no—
203C2×C9C3×C4⋊Q811yescharacteristic
204C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
205C2×C9C3×#13711yescharacteristic
206C2×Q8—181no—
207C2×C8—91no—
208C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
209C4⋊C4—91no—
210C4⋊C4—91no—
211C4⋊C4—181no—
212C4⋊C4—181no—
213C4⋊C4—181no—
214C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
215C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
216C42—91no—
217C22×C3—31no—
218C22×C3C3×D3×D411yescharacteristic
219C3×C4—31no—
220C3×C4—31no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—121no—
224C3×C4—121no—
225C3×C4—181no—
226C3×C4—181no—
227C3×C4—361no—
228C3×C4—361no—
229C3×C4C2×C3×Dic611yescharacteristic
230C3×C4C3×D3×Q811yescharacteristic
231Dic3—61no—
232Dic3—61no—
233Dic3—121no—
234Dic3—121no—
235C32#487611yescharacteristic
236C9—21no—
237C9C3×#50111yescharacteristic, complemented
238Q8—362no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
246C8—92no—
247C83−1+2⋊Q812yescomplemented
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3C3×#88811yescharacteristic
252C2×C3C3×#91111yescharacteristic
253C2×C3C3×#91811yescharacteristic
254C22D4×AGL(1,ℤ/9)11yescharacteristic
255C4—41no—
256C4—41no—
257C4—181no—
258C4—181no—
259C4—361no—
260C4—361no—
261C4C2×3−1+2⋊Q811yescharacteristic
262C4Q8×AGL(1,ℤ/9)11yescharacteristic
263C3—31no—
264C3C3×#487611yescharacteristic
265C2#63450⋊C311yescharacteristic
266C2#63473⋊C311yescharacteristic
267C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#501
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
9. #10588556C3semidirect
44. C8×3−1+2Q8semidirect
45. 3−1+2⋊C8Q8semidirect
118. C8×C9C3×Q8semidirect
119. C9⋊C8C3×Q8semidirect
176. 3−1+2#501semidirect
237. C9C3×#501semidirect
247. C83−1+2⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#63452⋊C32
#63450⋊C32
3−1+2×#1072
#63377⋊C32
#63375⋊C32
#63367⋊C32
#63366⋊C32
C3×#48763
#105885563