Finite Group Databasebeta

← #13136464

Subgroups — #13136464

Order 1728.

620 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 163 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×#88403C212yesmaximal, complemented
2C27×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11720196C212yesmaximal, complemented
4#11720144C214yesmaximal, complemented
5#11720142C214yesmaximal, complemented
6#11720123C211yesmaximal, characteristic, complemented
7C42×D27C211yesmaximal, characteristic, complemented
8#10589827—31nomaximal
9C2×Q8×C27C2212yes—
10#88403—28no—
11C2×D108C2212yes—
12C2×D108C2214yes—
13C2×D108C2214yes—
14C2×C4×D27C2212yes—
15C2×C4×D27C2214yes—
16C2×C4×D27C2211yescharacteristic
17C4⋊C4×C27C2214yes—
18C42×C27C2211yescharacteristic
19(C2×D27)⋊C4C2218yes—
20C4×Dic27C2211yescharacteristic
21C4×Dic27C2212yes—
22C2×#3701—32no—
23C9×C4⋊Q8D311yescharacteristic
24#63509—32no—
25#63457—34no—
26#63455—34no—
27#63436—31no—
28C42×D9—31no—
29C22×D27C2314yes—
30C22×D27C2311yescharacteristic
31Q8×C27—24no—
32C2×C4×C27C2314yes—
33C2×C4×C27C2311yescharacteristic
34C2×C4×C27C2312yes—
35C2×Dic27C2312yes—
36C2×Dic27C2311yescharacteristic
37D108—28no—
38D108—24no—
39D108—28no—
40C4×D27—24no—
41C4×D27—28no—
42C4×D27D414yescomplemented
43#5725—91no—
44C2×Q8×C9C2×D312yes—
45#3701—68no—
46C2×D36—32no—
47C2×D36—34no—
48C2×D36—34no—
49C2×C4×D9—31no—
50C2×C4×D9—32no—
51C2×C4×D9—34no—
52C9×C4⋊C4C2×D314yes—
53C42×C9C2×D311yescharacteristic
54(C2×D9)⋊C4—38no—
55C4×Dic9—31no—
56C4×Dic9—32no—
57C22×C27C2411yescharacteristic
58C2×D27—22no—
59C2×D27—24no—
60C2×D27—28no—
61C2×D27C2×D412yes—
62C4×C27—24no—
63C4×C27C2×D412yes—
64C4×C27#4114yescomplemented
65Dic27—22no—
66Dic27C2×D412yes—
67C2×#239—92no—
68C3×C4⋊Q8D911yescharacteristic
69#947—92no—
70#895—94no—
71#893—94no—
72#874—91no—
73C42×D3—91no—
74C22×D9—31no—
75C22×D9—34no—
76Q8×C9—24no—
77C2×C4×C9C22×D314yes—
78C2×C4×C9C22×D311yescharacteristic
79C2×C4×C9C22×D312yes—
80C2×Dic9—31no—
81C2×Dic9—32no—
82D36—68no—
83D36—64no—
84D36—68no—
85C4×D9—34no—
86C4×D9—68no—
87C4×D9—64no—
88#532—271noSylow
89C2×C27C22×D411yescharacteristic
90C2×C27C2×#4112yes—
91D27—22no—
92D27—44no—
93C2×C3×Q8C2×D912yes—
94#239—188no—
95C2×D12—94no—
96C2×D12—92no—
97C2×D12—94no—
98C2×C4×D3—91no—
99C2×C4×D3—92no—
100C2×C4×D3—94no—
101C3×C4⋊C4C2×D914yes—
102C3×C42C2×D911yescharacteristic
103(C2×D3)⋊C4—98no—
104C4×Dic3—92no—
105C4×Dic3—91no—
106C22×C9C23×D311yescharacteristic
107C2×D9—32no—
108C2×D9—62no—
109C2×D9—64no—
110C2×D9—68no—
111C4×C9—24no—
112C4×C9D3×D412yes—
113C4×C9#23914yes—
114Dic9—32no—
115Dic9—62no—
116C2×#41—272no—
117C4⋊Q8D2711yescharacteristic, complemented
118#127—271no—
119#124—272no—
120#121—274no—
121C4×D4—274no—
122C2×C42—271no—
123C27#53211yescharacteristic, complemented, Sylow
124C22×D3—91no—
125C22×D3—94no—
126C3×Q8—24no—
127C2×C3×C4C22×D914yes—
128C2×C3×C4C22×D911yescharacteristic
129C2×C3×C4C22×D912yes—
130C2×Dic3—91no—
131C2×Dic3—92no—
132D12—188no—
133D12—184no—
134D12—188no—
135C4×D3—94no—
136C4×D3—188no—
137C4×D3—184no—
138C2×C9C2×D3×D411yescharacteristic
139C2×C9C2×#23912yes—
140D9—62no—
141D9—124no—
142#41—548no—
143C2×Q8C2×D2712yes—
144C2×D4—274no—
145C2×D4—274no—
146C2×D4—272no—
147C22×C4—271no—
148C22×C4—272no—
149C22×C4—274no—
150C4⋊C4C2×D2714yes—
151C22⋊C4—278no—
152C42—272no—
153C42—271no—
154C42C2×D2711yescharacteristic
155C22×C3C23×D911yescharacteristic
156C2×D3—92no—
157C2×D3—182no—
158C2×D3—184no—
159C2×D3—188no—
160C3×C4—24no—
161C3×C4D4×D912yes—
162C3×C4#370114yes—
163Dic3—92no—
164Dic3—182no—
165C9#572511yescharacteristic
166C23—271no—
167C23—274no—
168Q8—24no—
169D4—548no—
170D4—544no—
171D4—548no—
172C2×C4—271no—
173C2×C4—274no—
174C2×C4—272no—
175C2×C4—548no—
176C2×C4—544no—
177C2×C4C22×D2714yes—
178C2×C4C22×D2711yescharacteristic
179C2×C4C22×D2712yes—
180C2×C3C2×D4×D911yescharacteristic
181C2×C3C2×#370112yes—
182D3—182no—
183D3—364no—
184C22—272no—
185C22—542no—
186C22—544no—
187C22—548no—
188C22C23×D2711yescharacteristic
189C4—24no—
190C4—272no—
191C4—542no—
192C4D4×D2712yes—
193C4#8840314yescomplemented
194C3#1058982711yescharacteristic
195C2—542no—
196C2—1084no—
197C2C2×D4×D2711yescharacteristic
198C2C2×#8840312yes—
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×C4⋊Q8C2
socleC22×C3C23×D9
hypercenterC4⋊Q8D27
hypocenterC27#532
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88403C2semidirect
2. C27×C4⋊Q8C2semidirect
3. #11720196C2semidirect
4. #11720144C2semidirect
5. #11720142C2semidirect
6. #11720123C2semidirect
7. C42×D27C2semidirect
42. C4×D27D4semidirect
64. C4×C27#41semidirect
117. C4⋊Q8D27semidirect
123. C27#532semidirect
193. C4#88403semidirect

Maximal subgroups

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

subgroupindex
C2×#884032
C27×C4⋊Q82
#117201962
#117201442
#117201422
#117201232
C42×D272
#105898273