Finite Group Databasebeta

← #10591102

Subgroups — #10591102

Order 576.

217 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#926C211yesmaximal, characteristic
2C3×#924C211yesmaximal, characteristic
3#63657C211yesmaximal, characteristic
4#63576C211yesmaximal, characteristic, complemented
5#63574C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63565C211yesmaximal, characteristic, complemented
8#5045—31nomaximal
9#4996—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×#217—21no—
15C3×#217—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic3C2211yescharacteristic
19C3×Dic3⋊C4—21no—
20C3×C4×Dic3—21no—
21C3×C22⋊Q8—31no—
22C3×#117—31no—
23#926D311yescharacteristic
24#924D311yescharacteristic
25#835—31no—
26#835—31no—
27#830—31no—
28#830—31no—
29#830—61no—
30#810—31no—
31#810—31no—
32#808—31no—
33#808—31no—
34#807—31no—
35#807—31no—
36C23×C32D411yescharacteristic
37C2×C32×C4—21no—
38C2×C32×C4—21no—
39C2×C32×C4D411yescharacteristic
40C2×C32×C4C2311yescharacteristic
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C3×Dic6—21no—
46C3×Dic6—21no—
47C3⋊(C3⋊C8)—21no—
48C3⋊(C3⋊C8)—21no—
49#485—91noSylow
50C2×C3×Q8—31no—
51C22×C3×C4—21no—
52C22×C3×C4C2×D311yescharacteristic
53C22×C3×C4C2×D311yescharacteristic
54C2×Dic6C2×D311yescharacteristic
55C3×C4⋊C4—31no—
56C3×C4⋊C4—31no—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—61no—
59C3×C22⋊C4—61no—
60C3×C22⋊C4—61no—
61C3×C42—61no—
62#217—21no—
63#217—21no—
64C4⋊Dic3C2×D311yescharacteristic
65C4⋊Dic3C2×D311yescharacteristic
66C4⋊Dic3C2×D311yescharacteristic
67Dic3⋊C4—21no—
68C4×Dic3—21no—
69C2×C3⋊C8—31no—
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—61no—
74C2×C3⋊C8—61no—
75C22×C32—21no—
76C22×C32—21no—
77C22×C32—21no—
78C22×C32C2×D411yescharacteristic
79C32×C4—21no—
80C32×C4#4111yescharacteristic
81C32×C4#4111yescharacteristic
82C3×Dic3—41no—
83C3×Dic3—41no—
84C3×Dic3—41no—
85C3×Dic3—41no—
86C22⋊Q8—31no—
87#117—31no—
88#107—91no—
89#106—91no—
90#103—91no—
91#103—91no—
92C22⋊C8—91no—
93C23×C3—21no—
94C23×C3#6711yescharacteristic
95C23×C3#6711yescharacteristic
96C3×Q8—61no—
97C3×Q8—61no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4#6711yescharacteristic
111C2×C3×C4#6711yescharacteristic
112C2×C3×C4C22×D311yescharacteristic
113C2×C3×C4C22×D311yescharacteristic
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—21no—
118Dic6—21no—
119Dic6—21no—
120C3⋊C8—61no—
121C3⋊C8—61no—
122C3⋊C8—61no—
123C3⋊C8—61no—
124C3⋊C8—121no—
125C3⋊C8—121no—
126C2×C32—41no—
127C2×C32#12311yescharacteristic
128C2×C32#13511yescharacteristic
129C2×C32#13711yescharacteristic
130C2×Q8—31no—
131C22×C4D3211yescharacteristic
132C2×C8—91no—
133C2×C8—91no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C4⋊C4—31no—
137C4⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C42—61no—
141C22×C3—22no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3C2×#6711yescharacteristic
152C22×C3C2×#6711yescharacteristic
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—121no—
159C3×C4—121no—
160C3×C4—121no—
161C3×C4—121no—
162C3×C4#23511yescharacteristic
163C3×C4#23511yescharacteristic
164C3×C4#23711yescharacteristic
165C3×C4#23711yescharacteristic
166Dic3—41no—
167Dic3—41no—
168Dic3—41no—
169Dic3—41no—
170C32#48511yescharacteristic, complemented, Sylow
171C23#62511yescharacteristic
172Q8—61no—
173Q8—61no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4#62511yescharacteristic
181C2×C4C2×D3211yescharacteristic
182C8—181no—
183C8—181no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3—21no—
187C2×C3—42no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3#92911yescharacteristic
191C2×C3#93511yescharacteristic
192C2×C3#94211yescharacteristic
193C2×C3#95011yescharacteristic
194C2×C3#95011yescharacteristic
195C2×C3#95111yescharacteristic
196C22—21no—
197C22—21no—
198C22—21no—
199C22C2×#62511yescharacteristic
200C4—21no—
201C4—121no—
202C4—121no—
203C4—121no—
204C4—121no—
205C4#380411yescharacteristic
206C4#380411yescharacteristic
207C3—21no—
208C3#499611yescharacteristic, complemented
209C3#504511yescharacteristic, complemented
210C2—41no—
211C2#6382211yescharacteristic
212C2#6382711yescharacteristic
213C2#6396111yescharacteristic
subgroupG/H
centerC22C2×#625
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC23×C32D4
hypercenterC22×C4D32
hypocenterC32#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #63576C2semidirect
5. #63574C2semidirect
6. #63565C2semidirect
7. #63565C2semidirect
170. C32#485semidirect
208. C3#4996semidirect
209. C3#5045semidirect

Maximal subgroups

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

subgroupindex
C3×#9262
C3×#9242
#636572
#635762
#635742
#635652
#50453
#49963