Finite Group Databasebeta

← #11723346

Subgroups — #11723346

Order 864.

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

normal subgroups only

This group has 440 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
1#89019C211yesmaximal, characteristic, complemented
2#89017C211yesmaximal, characteristic
3#88850C211yesmaximal, characteristic, complemented
4C3×#3744C211yesmaximal, characteristic, complemented
5C32×Dic12C211yesmaximal, characteristic, complemented
6#88798C211yesmaximal, characteristic, complemented
7#88793C211yesmaximal, characteristic, complemented
8#64127—31nomaximal
9#63811—34nomaximal
10C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×#634C2211yescharacteristic
13C32×Dic6C2211yescharacteristic
14C32×Dic6C2211yescharacteristic
15#6293—21no—
16#6292—21no—
17#6289—21no—
18Dic3×C3⋊D3—21no—
19C33×C8C2211yescharacteristic
20C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#3832—31no—
22Q8×C3⋊D3—31no—
23#3797—34no—
24#3796—34no—
25C32×Q16—31no—
26#3756—31no—
27#3755—31no—
28#3744D311yescharacteristic, complemented
29#3743—31no—
30#3743—34no—
31#3743—64no—
32C3×Dic12D314yescomplemented
33C3×#204—34no—
34#3718—34no—
35#3715—34no—
36C2×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C3×C3⋊D3—21no—
38C33×C4C2311yescharacteristic
39C3⋊(C3⋊Dic3)D411yescharacteristic
40C3×C3⋊Dic3—21no—
41C32×Dic3—21no—
42C32×Dic3—21no—
43#918—94no—
44#915—91no—
45C32×Q8—31no—
46C32×Q8—31no—
47C4⋊(C3⋊D3)—61no—
48C4⋊(C3⋊D3)C2×D311yescharacteristic
49C4×C3⋊D3—31no—
50C4×C3⋊D3—34no—
51C4×C3⋊D3—64no—
52C4×C3⋊D3—61no—
53C4×C3⋊D3—61no—
54#634—61no—
55#634C2×D311yescharacteristic
56C3×D12—34no—
57C3×Dic6—34no—
58C3×Dic6C2×D314yes—
59C3×Dic6C2×D314yes—
60#627—64no—
61#626—64no—
62#624—64no—
63D3×Dic3—64no—
64C32×C8—24no—
65C32×C8C2×D314yes—
66C32×C8C2×D311yescharacteristic
67C3⋊(C3⋊C8)—31no—
68C3⋊(C3⋊C8)—34no—
69C3⋊(C3⋊C8)—64no—
70C2×C33C2×D411yescharacteristic
71C3⋊(C3⋊D3)—21no—
72C3×C3⋊D3—41no—
73#239—94no—
74D3×Q8—94no—
75D3×Q8—91no—
76#237—91no—
77C3×Q16—34no—
78C3×SD16—91no—
79#216—94no—
80#216—91no—
81#215—94no—
82#214—91no—
83Dic12C3⋊D311yescharacteristic, complemented
84#204—34no—
85ΓL(1,𝔽25)—94no—
86ΓL(1,𝔽25)—91no—
87ΓL(1,𝔽25)—184no—
88C2×C3⋊D3—21no—
89C2×C3⋊D3—31no—
90C2×C3⋊D3—34no—
91C2×C3⋊D3—64no—
92C2×C3×D3—64no—
93C32×C4—24no—
94C32×C4—61no—
95C32×C4—61no—
96C32×C4C22×D314yes—
97C32×C4C22×D311yescharacteristic
98C3⋊Dic3—21no—
99C3⋊Dic3—31no—
100C3⋊Dic3—34no—
101C3⋊Dic3—64no—
102C3×Dic3—24no—
103C3×Dic3—24no—
104C3×Dic3—64no—
105#137—271noSylow
106C33#13711yescharacteristic, complemented, Sylow
107C3×Q8—34no—
108C3×Q8—34no—
109C3×Q8—91no—
110C3×D4—91no—
111#67—181no—
112C2×Dic3—181no—
113D12—34no—
114D12—184no—
115C4×D3—94no—
116C4×D3—91no—
117C4×D3—184no—
118C4×D3—181no—
119C4×D3—184no—
120C4×D3—184no—
121Dic6—34no—
122Dic6—181no—
123Dic6—184no—
124Dic6C2×C3⋊D311yescharacteristic
125Dic6C2×C3⋊D311yescharacteristic
126C3×C8—24no—
127C3×C8D3214yes—
128C3×C8C2×C3⋊D311yescharacteristic
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C2×C32—24no—
133C2×C32D3×D414yes—
134C2×C32D3×D411yescharacteristic
135C3⋊D3—41no—
136C3⋊D3—61no—
137C3⋊D3—64no—
138C3⋊D3—124no—
139C3×D3—124no—
140#41—271no—
141C2×Q8—271no—
142Q16—31no—
143Q16—271no—
144SD16—91no—
145SD16—271no—
146M16—271no—
147C22×C3—181no—
148C2×D3—64no—
149C2×D3—94no—
150C2×D3—91no—
151C2×D3—184no—
152C3×C4—24no—
153C3×C4—64no—
154C3×C4—64no—
155C3×C4—181no—
156C3×C4C2×D3214yes—
157C3×C4C22×C3⋊D311yescharacteristic
158Dic3—21no—
159Dic3—21no—
160Dic3—64no—
161Dic3—94no—
162Dic3—91no—
163Dic3—184no—
164C32—24no—
165C32#91511yescharacteristic, complemented
166C32#91814yescomplemented
167Q8—31no—
168Q8—31no—
169Q8—91no—
170Q8—541no—
171D4—91no—
172D4—541no—
173C2×C4—271no—
174C2×C4—541no—
175C2×C4—541no—
176C8—271no—
177C8D3×C3⋊D311yescharacteristic
178C2×C3—24no—
179C2×C3—361no—
180C2×C3#380214yes—
181C2×C3D4×C3⋊D311yescharacteristic
182D3—124no—
183D3—184no—
184D3—181no—
185D3—364no—
186C22—181no—
187C22—271no—
188C4—61no—
189C4—61no—
190C4—181no—
191C4—271no—
192C4C2×D3×C3⋊D311yescharacteristic
193C3—24no—
194C3#6381114yescomplemented
195C3#6412711yescharacteristic, complemented
196C2—361no—
197C2—541no—
198C2#8902611yescharacteristic
subgroupG/H
centerC2#89026
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×C8C22
socleC2×C33C2×D4
prime radical#89017C2
hypercenterC8D3×C3⋊D3
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #89019C2semidirect
3. #88850C2semidirect
4. C3×#3744C2semidirect
5. C32×Dic12C2semidirect
6. #88798C2semidirect
7. #88793C2semidirect
28. #3744D3semidirect
32. C3×Dic12D3semidirect
83. Dic12C3⋊D3semidirect
106. C33#137semidirect
165. C32#915semidirect
166. C32#918semidirect
194. C3#63811semidirect
195. C3#64127semidirect

Maximal subgroups

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

subgroupindex
#890192
#890172
#888502
C3×#37442
C32×Dic122
#887982
#887932
#641273
#638113