Finite Group Databasebeta

← #10594530

Subgroups — #10594530

Order 576.

631 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3799C212yesmaximal, complemented
2#64102C211yesmaximal, characteristic, complemented
3C3×#897C212yesmaximal, complemented
4#63912C212yesmaximal, complemented
5C4×#625C211yesmaximal, characteristic, complemented
6#63900C212yesmaximal, complemented
7#63873C211yesmaximal, characteristic, complemented
8#63854C214yesmaximal, complemented
9#5631—32nomaximal
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C4×C3⋊D3C2212yes—
12C2×C3×C4×D3C2212yes—
13C2×C3×Dic6C2212yes—
14C2×#626C2214yes—
15C2×#625C2211yescharacteristic
16C2×#624C2214yes—
17#3799—28no—
18C32×C4⋊C4C2211yescharacteristic
19#3752C2212yes—
20C4×C3⋊Dic3C2211yescharacteristic
21C3×(C2×D3)⋊C4C2214yes—
22C3×C4⋊Dic3C2214yes—
23#3724C2212yes—
24#3724C2212yes—
25#3724C2211yescharacteristic
26(C2×D3)⋊Dic3C2212yes—
27C2×#239—32no—
28C2×#235—32no—
29C3×C22⋊Q8—32no—
30#944—32no—
31C4×#67—32no—
32#897D312yescomplemented
33#894—34no—
34#893—32no—
35#893—61no—
36D3×C4⋊C4—34no—
37#884—34no—
38#883—34no—
39C22×C3⋊D3C2312yes—
40C22×C3×D3C2312yes—
41C2×C32×C4C2312yes—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3C2311yescharacteristic
44C2×C3⋊Dic3C2311yescharacteristic
45C4⋊(C3⋊D3)D414yescomplemented
46C4×C3⋊D3—24no—
47C2×C3×Dic3C2312yes—
48C2×C3×Dic3C2314yes—
49C3×C4×D3—24no—
50C3×Dic6—24no—
51#626—28no—
52#625—22no—
53#624—28no—
54#547—91noSylow
55C2×C3×Q8—32no—
56C22×C3×C4—32no—
57C2×#67—32no—
58C2×#67—34no—
59#239—68no—
60#235—68no—
61C2×D12—32no—
62C2×D12—34no—
63C2×D12—61no—
64C2×C4×D3—34no—
65C2×C4×D3—34no—
66C2×C4×D3—34no—
67C2×C4×D3—62no—
68C2×C4×D3C2×D312yes—
69C2×Dic6C2×D312yes—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—34no—
72C3×C4⋊C4C2×D312yes—
73C3×C22⋊C4—34no—
74#217—32no—
75(C2×D3)⋊C4—32no—
76(C2×D3)⋊C4—34no—
77(C2×D3)⋊C4—62no—
78(C2×D3)⋊C4C2×D314yes—
79C4⋊Dic3C2×D314yes—
80Dic3⋊C4—34no—
81Dic3⋊C4—34no—
82Dic3⋊C4—32no—
83C4×Dic3—32no—
84C4×Dic3—61no—
85C22×C32C2411yescharacteristic
86C2×C3⋊D3—22no—
87C2×C3⋊D3—22no—
88C2×C3⋊D3C2×D414yes—
89C2×C3×D3—22no—
90C2×C3×D3—22no—
91C2×C3×D3—22no—
92C32×C4—22no—
93C32×C4C2×D412yes—
94C3⋊Dic3—21no—
95C3⋊Dic3#4112yescomplemented
96C3×Dic3—22no—
97C3×Dic3—24no—
98C2×#41—92no—
99C4⋊Q8—91no—
100#123—94no—
101C22⋊Q8—32no—
102#121—92no—
103C4×D4—91no—
104C4×D4—91no—
105C2×C4⋊C4—92no—
106C23×C3—32no—
107C22×D3—34no—
108C22×D3—62no—
109C22×D3C22×D312yes—
110C3×Q8—64no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—34no—
114C2×C3×C4—32no—
115C2×C3×C4—64no—
116C2×C3×C4C22×D314yes—
117C2×C3×C4C22×D312yes—
118#67—68no—
119#67—64no—
120C2×Dic3—32no—
121C2×Dic3—32no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124C2×Dic3C22×D314yes—
125C2×Dic3C22×D312yes—
126D12—38no—
127D12—64no—
128D12—68no—
129C4×D3—24no—
130C4×D3—68no—
131C4×D3—68no—
132C4×D3—68no—
133C4×D3—124no—
134Dic6—24no—
135C2×C32C22×D411yescharacteristic
136C2×C32C2×#4111yescharacteristic
137C2×C322−1+411yescharacteristic
138C3⋊D3—24no—
139C3×D3—42no—
140#41—188no—
141C2×Q8—32no—
142C2×D4—94no—
143C2×D4—91no—
144C2×D4—91no—
145C22×C4—32no—
146C22×C4—94no—
147C22×C4—92no—
148C4⋊C4—34no—
149C4⋊C4—91no—
150C4⋊C4—92no—
151C4⋊C4—92no—
152C4⋊C4D3211yescharacteristic
153C22⋊C4—34no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C42—91no—
157C22×C3—21no—
158C22×C3—62no—
159C22×C3—62no—
160C22×C3—62no—
161C22×C3C23×D312yes—
162C2×D3—22no—
163C2×D3—22no—
164C2×D3—22no—
165C2×D3—38no—
166C2×D3—64no—
167C2×D3—64no—
168C2×D3—64no—
169C2×D3—122no—
170C2×D3—122no—
171C3×C4—24no—
172C3×C4—22no—
173C3×C4—42no—
174C3×C4—62no—
175C3×C4—64no—
176C3×C4D3×D414yes—
177Dic3—24no—
178Dic3—22no—
179Dic3—34no—
180Dic3—62no—
181Dic3—62no—
182Dic3—121no—
183C32#54711yescharacteristic, complemented, Sylow
184C23—32no—
185C23—92no—
186Q8—64no—
187D4—94no—
188D4—188no—
189D4—182no—
190C2×C4—34no—
191C2×C4—32no—
192C2×C4—64no—
193C2×C4—91no—
194C2×C4—91no—
195C2×C4—184no—
196C2×C4—188no—
197C2×C4C2×D3212yes—
198C2×C4C2×D3211yescharacteristic
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—122no—
203C2×C3C2×D3×D412yes—
204C2×C3#100712yes—
205C2×C3D3×#4112yes—
206D3—42no—
207D3—68no—
208D3—124no—
209C22—62no—
210C22—62no—
211C22—62no—
212C22—94no—
213C22—182no—
214C22—182no—
215C22C22×D3211yescharacteristic
216C4—22no—
217C4—62no—
218C4—64no—
219C4—92no—
220C4—181no—
221C4#380212yes—
222C3—21no—
223C3#563112yescomplemented
224C2—122no—
225C2—184no—
226C2C2×#380211yescharacteristic
227C2#6431211yescharacteristic
228C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3799C2semidirect
2. #64102C2semidirect
3. C3×#897C2semidirect
4. #63912C2semidirect
5. C4×#625C2semidirect
6. #63900C2semidirect
7. #63873C2semidirect
8. #63854C2semidirect
32. #897D3semidirect
45. C4⋊(C3⋊D3)D4semidirect
95. C3⋊Dic3#41semidirect
183. C32#547semidirect
223. C3#5631semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
#641022
C3×#8972
#639122
C4×#6252
#639002
#638732
#638542
#56313