Finite Group Databasebeta

← #11723590

Subgroups — #11723590

Order 864.

512 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 82 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
1Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2#89017C211yesmaximal, characteristic
3C3×#3755C211yesmaximal, characteristic, complemented
4C32×#216C211yesmaximal, characteristic, complemented
5#88799C211yesmaximal, characteristic, complemented
6#88796C211yesmaximal, characteristic, complemented
7#88788C211yesmaximal, characteristic, complemented
8#64127—31nomaximal
9#63951—34nomaximal
10C33×Q8C2211yescharacteristic
11C4×C3⋊(C3⋊D3)—21no—
12C4×C3⋊(C3⋊D3)C2211yescharacteristic
13#6308—21no—
14#6308C2211yescharacteristic
15C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C32×Dic6C2211yescharacteristic
17#6292—21no—
18Dic3×C3⋊D3—21no—
19C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3832—31no—
22Q8×C3⋊D3—31no—
23Q8×C3⋊D3—34no—
24Q8×C3⋊D3—64no—
25#3796—34no—
26C32×Q16—31no—
27#3756—31no—
28#3755D311yescharacteristic, complemented
29#3744—31no—
30#3743—31no—
31C3×#216D314yescomplemented
32C3×#215—34no—
33#3719—34no—
34#3716—34no—
35#3712—34no—
36C2×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C3×C3⋊D3—21no—
38C33×C4—21no—
39C33×C4C2311yescharacteristic
40C3⋊(C3⋊Dic3)—21no—
41C3⋊(C3⋊Dic3)D411yescharacteristic
42C32×Dic3—21no—
43#918—94no—
44#915—91no—
45C32×Q8—24no—
46C32×Q8—31no—
47C32×Q8C2×D314yes—
48C32×Q8C2×D311yescharacteristic
49C4⋊(C3⋊D3)—61no—
50C4⋊(C3⋊D3)C2×D311yescharacteristic
51C4×C3⋊D3—31no—
52C4×C3⋊D3—34no—
53C4×C3⋊D3—64no—
54C4×C3⋊D3—61no—
55C4×C3⋊D3—61no—
56C4×C3⋊D3—64no—
57C4×C3⋊D3—68no—
58#634—31no—
59#634—34no—
60#634—64no—
61#634—61no—
62#634—64no—
63#634—68no—
64C3×D12—34no—
65C3×Dic6C2×D314yes—
66#626—64no—
67D3×Dic3—64no—
68C32×C8—31no—
69C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C3⋊C8—34no—
71C3×C3⋊C8C2×D314yes—
72C2×C33C2×D411yescharacteristic
73C3⋊(C3⋊D3)—21no—
74C3×C3⋊D3—41no—
75#239—94no—
76D3×Q8—94no—
77D3×Q8—91no—
78D3×Q8—184no—
79#237—91no—
80C3×Q16—34no—
81C3×SD16—91no—
82#216—94no—
83#216C3⋊D311yescharacteristic, complemented
84#215—34no—
85#214—91no—
86Dic12—91no—
87#204—94no—
88ΓL(1,𝔽25)—94no—
89ΓL(1,𝔽25)—91no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—31no—
92C2×C3⋊D3—34no—
93C2×C3⋊D3—64no—
94C2×C3×D3—64no—
95C32×C4—28no—
96C32×C4—24no—
97C32×C4—21no—
98C32×C4—24no—
99C32×C4—61no—
100C32×C4C22×D314yes—
101C32×C4C22×D311yescharacteristic
102C3⋊Dic3—31no—
103C3⋊Dic3—34no—
104C3⋊Dic3—64no—
105C3⋊Dic3—68no—
106C3⋊Dic3—64no—
107C3⋊Dic3—61no—
108C3×Dic3—24no—
109#137—271noSylow
110C33#13711yescharacteristic, complemented, Sylow
111C3×Q8—24no—
112C3×Q8—34no—
113C3×Q8D3214yes—
114C3×Q8C2×C3⋊D311yescharacteristic
115C3×D4—91no—
116#67—181no—
117C2×Dic3—181no—
118D12—34no—
119D12—184no—
120C4×D3—94no—
121C4×D3—91no—
122C4×D3—184no—
123C4×D3—184no—
124C4×D3—184no—
125C4×D3—181no—
126C4×D3—188no—
127Dic6—94no—
128Dic6—91no—
129Dic6—184no—
130Dic6—184no—
131Dic6—181no—
132Dic6—188no—
133Dic6C2×C3⋊D311yescharacteristic
134C3×C8—34no—
135C3×C8—91no—
136C3⋊C8—34no—
137C3⋊C8C2×C3⋊D311yescharacteristic
138C2×C32—24no—
139C2×C32D3×D414yes—
140C2×C32D3×D411yescharacteristic
141C3⋊D3—41no—
142C3⋊D3—68no—
143C3⋊D3—61no—
144C3⋊D3—64no—
145C3×D3—124no—
146#41—271no—
147C2×Q8—271no—
148Q16—31no—
149Q16—271no—
150SD16—91no—
151SD16—271no—
152M16—271no—
153C22×C3—181no—
154C2×D3—64no—
155C2×D3—94no—
156C2×D3—91no—
157C2×D3—184no—
158C3×C4—28no—
159C3×C4—24no—
160C3×C4—21no—
161C3×C4—24no—
162C3×C4—64no—
163C3×C4C2×D3214yes—
164C3×C4C22×C3⋊D311yescharacteristic
165Dic3—21no—
166Dic3—94no—
167Dic3—91no—
168Dic3—184no—
169Dic3—188no—
170Dic3—184no—
171Dic3—181no—
172C32—24no—
173C32#91511yescharacteristic, complemented
174C32#91814yescomplemented
175Q8—31no—
176Q8—271no—
177Q8—541no—
178Q8D3×C3⋊D311yescharacteristic
179D4—91no—
180D4—541no—
181C2×C4—271no—
182C2×C4—541no—
183C2×C4—541no—
184C8—31no—
185C8—91no—
186C2×C3—24no—
187C2×C3—361no—
188C2×C3#381114yes—
189C2×C3D4×C3⋊D311yescharacteristic
190D3—124no—
191D3—188no—
192D3—181no—
193D3—184no—
194C22—181no—
195C22—271no—
196C4—21no—
197C4—61no—
198C4—271no—
199C4—541no—
200C4C2×D3×C3⋊D311yescharacteristic
201C3—24no—
202C3#6395114yescomplemented
203C3#6412711yescharacteristic, complemented
204C2—361no—
205C2—541no—
206C2#8903911yescharacteristic
subgroupG/H
centerC2#89039
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×Q8C22
socleC2×C33C2×D4
prime radical#89017C2
hypercenterQ8D3×C3⋊D3
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊(C3⋊D3)C2semidirect
3. C3×#3755C2semidirect
4. C32×#216C2semidirect
5. #88799C2semidirect
6. #88796C2semidirect
7. #88788C2semidirect
28. #3755D3semidirect
31. C3×#216D3semidirect
83. #216C3⋊D3semidirect
110. C33#137semidirect
173. C32#915semidirect
174. C32#918semidirect
202. C3#63951semidirect
203. C3#64127semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊D3)2
#890172
C3×#37552
C32×#2162
#887992
#887962
#887882
#641273
#639513