Finite Group Databasebeta

← #10589835

Subgroups — #10589835

Order 576.

324 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C9×C4⋊Q8C211yesmaximal, characteristic, complemented
2#63509C212yesmaximal, complemented
3#63508C212yesmaximal, complemented
4#63460C212yesmaximal, complemented
5#63459C211yesmaximal, characteristic, complemented
6#63457C211yesmaximal, characteristic
7#63456C212yesmaximal, complemented
8#63455C211yesmaximal, characteristic, complemented
9#63449C211yesmaximal, characteristic, complemented
10#63437C211yesmaximal, characteristic, complemented
11#63434C211yesmaximal, characteristic, complemented
12#5733—31nomaximal
13C2×Q8×C9C2212yes—
14C2×D36C2211yescharacteristic
15C2×D36C2212yes—
16C2×C4×D9C2211yescharacteristic
17C2×C4×D9C2212yes—
18C2×Dic18C2211yescharacteristic
19C9×C4⋊C4C2211yescharacteristic
20C9×C4⋊C4C2211yescharacteristic
21C9×C4⋊C4C2212yes—
22C42×C9C2211yescharacteristic
23(C2×D9)⋊C4C2212yes—
24(C2×D9)⋊C4C2212yes—
25(C2×D9)⋊C4C2211yescharacteristic
26(C2×D9)⋊C4C2212yes—
27(C2×D9)⋊C4C2212yes—
28(C2×D9)⋊C4C2211yescharacteristic
29(C2×D9)⋊C4C2212yes—
30C4⋊Dic9C2212yes—
31Dic9⋊C4C2211yescharacteristic
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2211yescharacteristic
34C4×Dic9C2212yes—
35C4×Dic9C2211yescharacteristic
36C3×C4⋊Q8D311yescharacteristic
37#947—32no—
38#946—32no—
39#898—32no—
40#897—31no—
41#895—31no—
42#894—32no—
43#893—31no—
44#887—31no—
45#875—31no—
46#872—31no—
47C22×D9C2311yescharacteristic
48C22×D9C2312yes—
49Q8×C9—42no—
50C2×C4×C9C2312yes—
51C2×C4×C9C2311yescharacteristic
52C2×C4×C9C2311yescharacteristic
53C2×C4×C9C2312yes—
54C2×C4×C9C2311yescharacteristic
55C2×Dic9C2311yescharacteristic
56C2×Dic9C2311yescharacteristic
57C2×Dic9C2312yes—
58C2×Dic9C2311yescharacteristic
59D36—22no—
60D36—42no—
61C4×D9—22no—
62C4×D9—42no—
63Dic18—22no—
64#542—91noSylow
65C2×C3×Q8C2×D312yes—
66C2×D12—31no—
67C2×D12—32no—
68C2×C4×D3—31no—
69C2×C4×D3—32no—
70C2×Dic6—31no—
71C3×C4⋊C4C2×D311yescharacteristic
72C3×C4⋊C4C2×D311yescharacteristic
73C3×C4⋊C4C2×D312yes—
74C3×C42C2×D311yescharacteristic
75(C2×D3)⋊C4—31no—
76(C2×D3)⋊C4—31no—
77(C2×D3)⋊C4—32no—
78(C2×D3)⋊C4—32no—
79(C2×D3)⋊C4—32no—
80(C2×D3)⋊C4—32no—
81(C2×D3)⋊C4—32no—
82C4⋊Dic3—32no—
83Dic3⋊C4—31no—
84Dic3⋊C4—31no—
85Dic3⋊C4—32no—
86C4×Dic3—31no—
87C4×Dic3—32no—
88C22×C9C2411yescharacteristic
89C2×D9—21no—
90C2×D9—22no—
91C2×D9—22no—
92C2×D9—21no—
93C2×D9—21no—
94C2×D9—22no—
95C4×C9—22no—
96C4×C9—22no—
97C4×C9—21no—
98C4×C9—21no—
99C4×C9#4112yes—
100Dic9—22no—
101Dic9—21no—
102Dic9—21no—
103Dic9—21no—
104C4⋊Q8D911yescharacteristic, complemented
105#126—92no—
106#124—92no—
107#124—91no—
108#123—92no—
109C22⋊Q8—92no—
110C22⋊Q8—91no—
111#121—91no—
112C4×Q8—91no—
113C4×D4—91no—
114#117—91no—
115C22×D3—32no—
116C22×D3—31no—
117C3×Q8—42no—
118C2×C3×C4C22×D312yes—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D312yes—
122C2×C3×C4C22×D311yescharacteristic
123C2×Dic3—31no—
124C2×Dic3—31no—
125C2×Dic3—32no—
126C2×Dic3—31no—
127D12—62no—
128D12—122no—
129C4×D3—62no—
130C4×D3—122no—
131Dic6—62no—
132C2×C9C2×#4111yescharacteristic
133C2×C9H(5,𝔽2)11yescharacteristic
134C2×C92−1+411yescharacteristic
135D9—41no—
136D9—42no—
137C2×Q8—91no—
138C2×Q8C2×D912yes—
139C2×D4—91no—
140C2×D4—92no—
141C22×C4—91no—
142C22×C4—92no—
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—92no—
146C4⋊C4—92no—
147C4⋊C4C2×D911yescharacteristic
148C4⋊C4C2×D911yescharacteristic
149C4⋊C4C2×D912yes—
150C22⋊C4—91no—
151C22⋊C4—91no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C22⋊C4—92no—
157C42—91no—
158C42—92no—
159C42C2×D911yescharacteristic
160C22×C3C23×D311yescharacteristic
161C2×D3—62no—
162C2×D3—61no—
163C2×D3—61no—
164C2×D3—62no—
165C2×D3—62no—
166C2×D3—61no—
167C3×C4—22no—
168C3×C4—22no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4#23912yes—
172Dic3—61no—
173Dic3—62no—
174Dic3—61no—
175Dic3—61no—
176C9#54211yescharacteristic, complemented, Sylow
177C23—92no—
178C23—91no—
179Q8—42no—
180Q8—182no—
181D4—182no—
182D4—362no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—182no—
188C2×C4—362no—
189C2×C4C22×D912yes—
190C2×C4C22×D911yescharacteristic
191C2×C4C22×D911yescharacteristic
192C2×C4C22×D912yes—
193C2×C4C22×D911yescharacteristic
194C2×C3#100411yescharacteristic
195C2×C3C2×#23911yescharacteristic
196C2×C3#100711yescharacteristic
197D3—122no—
198D3—121no—
199C22—182no—
200C22—181no—
201C22—181no—
202C22—182no—
203C22—182no—
204C22—181no—
205C22C23×D911yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—21no—
210C4—181no—
211C4—181no—
212C4—182no—
213C4—181no—
214C4#370112yes—
215C3#573311yescharacteristic
216C2—362no—
217C2—361no—
218C2#6371011yescharacteristic
219C2C2×#370111yescharacteristic
220C2#6371311yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C4⋊Q8C2
prime radical#63457C2
hypercenterC4⋊Q8D9
hypocenterC9#542
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C9×C4⋊Q8C2semidirect
2. #63509C2semidirect
3. #63508C2semidirect
4. #63460C2semidirect
5. #63459C2semidirect
7. #63456C2semidirect
8. #63455C2semidirect
9. #63449C2semidirect
10. #63437C2semidirect
11. #63434C2semidirect
104. C4⋊Q8D9semidirect
176. C9#542semidirect

Maximal subgroups

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

subgroupindex
C9×C4⋊Q82
#635092
#635082
#634602
#634592
#634572
#634562
#634552
#634492
#634372
#634342
#57333