Finite Group Databasebeta

← #10593524

Subgroups — #10593524

Order 576.

271 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 35 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
1C2×#3805C211yesmaximal, characteristic
2C2×#3776C211yesmaximal, characteristic, complemented
3C2×#3775C211yesmaximal, characteristic, complemented
4#63982C211yesmaximal, characteristic, complemented
5C22⋊#622C211yesmaximal, characteristic, complemented
6#63740C211yesmaximal, characteristic, complemented
7#63739C211yesmaximal, characteristic, complemented
8C22×C3⋊Dic3C2211yescharacteristic
9C2×C3×#67—21no—
10C2×#627C2211yescharacteristic
11C2×#627C2211yescharacteristic
12C2×#625C2211yescharacteristic
13#3805—22no—
14#3805—22no—
15C2×D3×Dic3—21no—
16C2×#622C2211yescharacteristic
17C2×#622C2211yescharacteristic
18#3776—22no—
19#3775—22no—
20C3×#217—21no—
21#3724C2211yescharacteristic
22Dic3⋊Dic3—21no—
23C2×#237—61no—
24#878—61no—
25C23×C32D411yescharacteristic
26C22×C3×D3—21no—
27C2×C3⋊Dic3—21no—
28C2×C3⋊Dic3—21no—
29C2×C3⋊Dic3D411yescharacteristic
30C2×C3⋊Dic3C2311yescharacteristic
31C3×#67—24no—
32C2×C3×Dic3—21no—
33C2×C3×Dic3—21no—
34C2×C3×Dic3—21no—
35#627—21no—
36#627—21no—
37#627—21no—
38#627D412yes—
39#625—21no—
40#625D412yes—
41D3×Dic3—24no—
42#622—21no—
43#622—21no—
44#452—91nomaximal, Sylow
45C2×C3×D4—61no—
46C2×#67—21no—
47C2×#67—61no—
48C22×Dic3—61no—
49C22×Dic3—61no—
50C22×Dic3—61no—
51#237—64no—
52#237—64no—
53C2×C4×D3—61no—
54C2×Dic6—61no—
55C2×Dic6—61no—
56C3×C22⋊C4—61no—
57#217—21no—
58C4⋊Dic3—61no—
59Dic3⋊C4—61no—
60Dic3⋊C4—61no—
61C22×C32—21no—
62C22×C32—21no—
63C22×C32—21no—
64C22×C32C2×D411yescharacteristic
65C2×C3×D3—22no—
66C2×C3×D3—41no—
67C2×C3×D3—41no—
68C3⋊Dic3—21no—
69C3⋊Dic3C2×D411yescharacteristic
70C3⋊Dic3C2×D411yescharacteristic
71C3×Dic3—22no—
72C3×Dic3—41no—
73C3×Dic3—41no—
74C2×#41—91no—
75C2×Q16—91no—
76C2×SD16—91no—
77C22⋊Q8—91no—
78#103—91no—
79#102—91no—
80C22⋊C8—91no—
81C23×C3—21no—
82C23×C3—21no—
83C23×C3—61no—
84C22×D3—21no—
85C3×D4—64no—
86C2×C3×C4—61no—
87C2×C3×C4—61no—
88C2×C3×C4—61no—
89#67—24no—
90#67—62no—
91#67—62no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—61no—
96C2×Dic3—61no—
97C2×Dic3—61no—
98C2×Dic3—61no—
99C2×Dic3—64no—
100C2×Dic3—62no—
101C2×Dic3—62no—
102C2×Dic3—121no—
103C2×Dic3—121no—
104C4×D3—64no—
105Dic6—62no—
106Dic6—62no—
107Dic6—121no—
108Dic6—121no—
109C2×C32—41no—
110C2×C32#12011yescharacteristic
111C2×C32#13511yescharacteristic
112C2×C32#13711yescharacteristic
113C3×D3—42no—
114#41—182no—
115#41—182no—
116C2×Q8—91no—
117C2×Q8—91no—
118C2×D4—61no—
119C2×D4—91no—
120C22×C4—91no—
121C22×C4—181no—
122Q16—182no—
123SD16—182no—
124C2×C8—91no—
125C2×C8—91no—
126C4⋊C4—91no—
127C4⋊C4—181no—
128C22⋊C4—61no—
129C22×C3—22no—
130C22×C3—22no—
131C22×C3—21no—
132C22×C3—21no—
133C22×C3—41no—
134C22×C3—41no—
135C22×C3—41no—
136C22×C3—41no—
137C22×C3—62no—
138C22×C3—121no—
139C22×C3—121no—
140C2×D3—22no—
141C2×D3—41no—
142C2×D3—41no—
143C3×C4—62no—
144C3×C4—121no—
145C3×C4—121no—
146Dic3—22no—
147Dic3—41no—
148Dic3—41no—
149Dic3—61no—
150Dic3—61no—
151Dic3—61no—
152Dic3—61no—
153Dic3—62no—
154Dic3—121no—
155C32#45211yescharacteristic, complemented, Sylow
156C23—61no—
157C23GO+(4,𝔽2)11yescharacteristic
158Q8—92no—
159Q8—181no—
160Q8—181no—
161Q8—181no—
162D4—64no—
163D4—92no—
164D4—181no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—184no—
171C2×C4—181no—
172C2×C4—181no—
173C8—181no—
174C8—181no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—42no—
182C2×C3—42no—
183C2×C3—122no—
184D3—42no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—62no—
189C22—121no—
190C22—121no—
191C22C2×GO+(4,𝔽2)11yescharacteristic
192C4—62no—
193C4—91no—
194C4—91no—
195C4—121no—
196C4—121no—
197C4—181no—
198C3—21no—
199C3—21no—
200C2—41no—
201C2—122no—
202C2#6423711yescharacteristic
203C2#6423911yescharacteristic
204C2#6424111yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
FittingC23×C32D4
socleC22×C32C2×D4
prime radicalC2×#3805C2
hypercenterC23GO+(4,𝔽2)
hypocenterC32#452
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#3776C2semidirect
3. C2×#3775C2semidirect
4. #63982C2semidirect
5. C22⋊#622C2semidirect
6. #63740C2semidirect
7. #63739C2semidirect
155. C32#452semidirect

Maximal subgroups

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

subgroupindex
C2×#38052
C2×#37762
C2×#37752
#639822
C22⋊#6222
#637402
#637392
#4529