Finite Group Databasebeta

← #10589727

Subgroups — #10589727

Order 576.

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

normal subgroups only

This group has 288 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×C22⋊Q8C211yesmaximal, characteristic
2Q8×Dic9C211yesmaximal, characteristic, complemented
3#63506C211yesmaximal, characteristic, complemented
4#63488C211yesmaximal, characteristic
5#63486C211yesmaximal, characteristic
6#63452C211yesmaximal, characteristic, complemented
7#63452C212yesmaximal, complemented
8#63451C212yesmaximal, complemented
9#63449C211yesmaximal, characteristic, complemented
10#63441C212yesmaximal
11#63441C212yesmaximal
12#5625—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×Dic18C2211yescharacteristic
16C9×C4⋊C4C2211yescharacteristic
17C9×C4⋊C4C2212yes—
18C9×C22⋊C4C2212yes—
19#3676C2211yescharacteristic
20#3676C2211yescharacteristic
21#3676C2212yes—
22C4⋊Dic9C2211yescharacteristic
23C4⋊Dic9C2211yescharacteristic
24C4⋊Dic9C2212yes—
25C4⋊Dic9C2211yescharacteristic
26C4⋊Dic9C2212yes—
27Dic9⋊C4C2212yes—
28Dic9⋊C4C2212yes—
29Dic9⋊C4C2212yes—
30Dic9⋊C4C2212yes—
31Dic9⋊C4C2212yes—
32C4×Dic9C2211yescharacteristic
33C4×Dic9C2211yescharacteristic
34C4×Dic9C2212yes—
35C4×Dic9C2212yes—
36C3×C22⋊Q8D311yescharacteristic
37Q8×Dic3—31no—
38#944—31no—
39#926—31no—
40#924—31no—
41#890—32no—
42#890—31no—
43#889—32no—
44#887—31no—
45#879—32no—
46#879—32no—
47C23×C9C2311yescharacteristic
48Q8×C9—22no—
49C2×C4×C9—22no—
50C2×C4×C9C2312yes—
51C2×C4×C9C2312yes—
52C2×C4×C9C2311yescharacteristic
53C2×C4×C9C2311yescharacteristic
54C2×Dic9C2311yescharacteristic
55C2×Dic9C2311yescharacteristic
56C2×Dic9C2311yescharacteristic
57C2×Dic9C2312yes—
58C2×Dic9C2311yescharacteristic
59C2×Dic9C2312yes—
60Dic18—22no—
61#541—91noSylow
62C2×C3×Q8C2×D311yescharacteristic
63C22×C3×C4C2×D311yescharacteristic
64C2×Dic6—31no—
65C3×C4⋊C4C2×D311yescharacteristic
66C3×C4⋊C4C2×D312yes—
67C3×C22⋊C4C2×D312yes—
68#217—32no—
69#217—31no—
70#217—31no—
71C4⋊Dic3—32no—
72C4⋊Dic3—32no—
73C4⋊Dic3—31no—
74C4⋊Dic3—31no—
75C4⋊Dic3—31no—
76Dic3⋊C4—32no—
77Dic3⋊C4—32no—
78Dic3⋊C4—32no—
79Dic3⋊C4—32no—
80Dic3⋊C4—32no—
81C4×Dic3—32no—
82C4×Dic3—32no—
83C4×Dic3—31no—
84C4×Dic3—31no—
85C22×C9—21no—
86C22×C9—21no—
87C22×C9—21no—
88C22×C9C2411yescharacteristic
89C4×C9—21no—
90C4×C9—22no—
91C4×C9—22no—
92C4×C9#4112yes—
93Dic9—22no—
94Dic9—21no—
95Dic9—21no—
96Dic9—22no—
97Dic9—21no—
98Dic9—21no—
99C4⋊Q8—91no—
100#126—92no—
101#126—92no—
102#125—92no—
103#125—91no—
104#125—92no—
105C22⋊Q8—91no—
106C22⋊Q8D911yescharacteristic
107C4×Q8—91no—
108C4×Q8—91no—
109#117—91no—
110C23×C3C22×D311yescharacteristic
111C3×Q8—22no—
112C2×C3×C4—22no—
113C2×C3×C4C22×D312yes—
114C2×C3×C4C22×D312yes—
115C2×C3×C4C22×D311yescharacteristic
116C2×C3×C4C22×D311yescharacteristic
117C2×Dic3—31no—
118C2×Dic3—31no—
119C2×Dic3—31no—
120C2×Dic3—32no—
121C2×Dic3—32no—
122C2×Dic3—31no—
123Dic6—62no—
124C2×C9—41no—
125C2×C9C2×#4111yescharacteristic
126C2×C92−1+411yescharacteristic
127C2×C92−1+411yescharacteristic
128C2×Q8—91no—
129C2×Q8C2×D911yescharacteristic
130C22×C4C2×D911yescharacteristic
131C4⋊C4—91no—
132C4⋊C4—91no—
133C4⋊C4—91no—
134C4⋊C4—92no—
135C4⋊C4—92no—
136C4⋊C4—92no—
137C4⋊C4—92no—
138C4⋊C4—92no—
139C4⋊C4—92no—
140C4⋊C4—92no—
141C4⋊C4C2×D911yescharacteristic
142C4⋊C4C2×D912yes—
143C22⋊C4—91no—
144C22⋊C4—91no—
145C22⋊C4—92no—
146C22⋊C4C2×D912yes—
147C42—91no—
148C42—91no—
149C42—92no—
150C42—92no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3C23×D311yescharacteristic
155C3×C4—21no—
156C3×C4—22no—
157C3×C4—22no—
158C3×C4#23712yes—
159Dic3—62no—
160Dic3—61no—
161Dic3—62no—
162Dic3—61no—
163Dic3—61no—
164Dic3—61no—
165C9#54111yescharacteristic, complemented, Sylow
166C23C22×D911yescharacteristic
167Q8—22no—
168Q8—182no—
169C2×C4—22no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—92no—
174C2×C4—91no—
175C2×C4—92no—
176C2×C4C22×D912yes—
177C2×C4C22×D912yes—
178C2×C4C22×D911yescharacteristic
179C2×C4C22×D911yescharacteristic
180C2×C3—41no—
181C2×C3C2×#23711yescharacteristic
182C2×C3#100711yescharacteristic
183C2×C3#101011yescharacteristic
184C22—21no—
185C22—21no—
186C22—21no—
187C22C23×D911yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—182no—
192C4—181no—
193C4—181no—
194C4—182no—
195C4—181no—
196C4—181no—
197C4#369912yes—
198C3#562511yescharacteristic
199C2—41no—
200C2C2×#369911yescharacteristic
201C2#6371311yescharacteristic
202C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
prime radicalC23×C3C22×D3
hypercenterC22⋊Q8D9
hypocenterC9#541
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. Q8×Dic9C2semidirect
3. #63506C2semidirect
6. #63452C2semidirect
7. #63452C2semidirect
8. #63451C2semidirect
9. #63449C2semidirect
165. C9#541semidirect

Maximal subgroups

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

subgroupindex
C9×C22⋊Q82
Q8×Dic92
#635062
#634882
#634862
#634522
#634512
#634492
#634412
#56253