Finite Group Databasebeta

← #13134256

Subgroups — #13134256

Order 1728.

300 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88403C211yesmaximal, characteristic
2M16×D27C212yesmaximal, complemented
3C27×#101C211yesmaximal, characteristic, complemented
4#11720075C211yesmaximal, characteristic, complemented
5#11720064C212yesmaximal, complemented
6#10588431—31nomaximal
7C2×Q8×C27C2211yescharacteristic
8#88403—22no—
9#88403—22no—
10C2×D108C412yes—
11C2×D108C2211yescharacteristic
12C2×C4×D27C412yes—
13C2×C4×D27C2211yescharacteristic
14M16×C27C2212yes—
15#88363C2212yes—
16#88359—22no—
17C8×D27—22no—
18C2×#3701—31no—
19M16×D9—32no—
20C9×#101D311yescharacteristic
21#63394—31no—
22#63383—32no—
23C22×D27C2×C412yes—
24C22×D27C2×C411yescharacteristic
25Q8×C27—22no—
26C2×C4×C27C2×C412yes—
27C2×C4×C27C2311yescharacteristic
28C2×Dic27C2×C411yescharacteristic
29D108—24no—
30D108—22no—
31C4×D27—22no—
32C4×D27—22no—
33C4×D27D412yes—
34C4×D27D412yes—
35C8×C27—22no—
36C27⋊C8—22no—
37#4751—91no—
38C2×Q8×C9C2×D311yescharacteristic
39#3701—62no—
40#3701—62no—
41C2×D36—32no—
42C2×D36—31no—
43C2×C4×D9—31no—
44C2×C4×D9—32no—
45C9×M16C2×D312yes—
46#3667—32no—
47#3663—62no—
48C8×D9—62no—
49C22×C27C22×C411yescharacteristic
50C2×D27—22no—
51C2×D27—41no—
52C2×D27—42no—
53C2×D27C22⋊C412yes—
54C4×C27—22no—
55C4×C27C2×D412yes—
56Dic27C22⋊C411yescharacteristic
57Dic27C22⋊C411yescharacteristic
58C2×#239—91no—
59D3×M16—92no—
60C3×#101D911yescharacteristic
61#836—91no—
62#823—92no—
63C22×D9—31no—
64C22×D9—32no—
65Q8×C9—22no—
66C2×C4×C9C4×D312yes—
67C2×C4×C9C22×D311yescharacteristic
68C2×Dic9—31no—
69D36—64no—
70D36—62no—
71C4×D9—32no—
72C4×D9—32no—
73C4×D9—62no—
74C4×D9—62no—
75C8×C9—22no—
76C9⋊C8—62no—
77#413—271noSylow
78C2×C27—21no—
79C2×C27C2×C22⋊C411yescharacteristic
80D27—22no—
81D27—42no—
82C2×C3×Q8C2×D911yescharacteristic
83#239—182no—
84#239—182no—
85C2×D12—92no—
86C2×D12—91no—
87C2×C4×D3—91no—
88C2×C4×D3—92no—
89C3×M16C2×D912yes—
90#208—92no—
91ΓL(1,𝔽25)—182no—
92D3×C8—182no—
93C22×C9C2×C4×D311yescharacteristic
94C2×D9—32no—
95C2×D9—62no—
96C2×D9—121no—
97C2×D9—122no—
98C4×C9—22no—
99C4×C9D3×D412yes—
100Dic9—31no—
101Dic9—31no—
102C2×#41—271no—
103C2×M16—272no—
104#101—271no—
105#101D2711yescharacteristic, complemented
106#100—272no—
107C27#41311yescharacteristic, complemented, Sylow
108C22×D3—91no—
109C22×D3—92no—
110C3×Q8—22no—
111C2×C3×C4C4×D912yes—
112C2×C3×C4C22×D911yescharacteristic
113C2×Dic3—91no—
114D12—184no—
115D12—182no—
116C4×D3—92no—
117C4×D3—92no—
118C4×D3—182no—
119C4×D3—182no—
120C3×C8—22no—
121C3⋊C8—182no—
122C2×C9—21no—
123C2×C9D3×C22⋊C411yescharacteristic
124D9—62no—
125D9—122no—
126#41—542no—
127#41—542no—
128C2×Q8C2×D2711yescharacteristic
129C2×D4—272no—
130C2×D4—271no—
131C22×C4—271no—
132C22×C4—272no—
133M16—272no—
134M16—542no—
135M16C2×D2712yes—
136C2×C8—542no—
137C22×C3C2×C4×D911yescharacteristic
138C2×D3—92no—
139C2×D3—182no—
140C2×D3—361no—
141C2×D3—362no—
142C3×C4—22no—
143C3×C4D4×D912yes—
144Dic3—91no—
145Dic3—91no—
146C9#475111yescharacteristic
147C23—271no—
148C23—272no—
149Q8—22no—
150D4—544no—
151D4—542no—
152C2×C4—271no—
153C2×C4—272no—
154C2×C4—272no—
155C2×C4—542no—
156C2×C4—542no—
157C2×C4C4×D2712yes—
158C2×C4C22×D2711yescharacteristic
159C8—22no—
160C8—542no—
161C2×C3—21no—
162C2×C3C22⋊C4×D911yescharacteristic
163D3—182no—
164D3—362no—
165C22—272no—
166C22—542no—
167C22—1081no—
168C22—1082no—
169C22C2×C4×D2711yescharacteristic
170C4—22no—
171C4—271no—
172C4—271no—
173C4D4×D2712yes—
174C3#1058843111yescharacteristic
175C2—21no—
176C2—542no—
177C2—1082no—
178C2C22⋊C4×D2711yescharacteristic
subgroupG/H
centerC2C22⋊C4×D27
commutatorC22×C27C22×C4
FrattiniC2×C4×C9C22×D3
FittingC27×#101C2
socleC2×C3C22⋊C4×D9
prime radicalC2×#88403C2
cosocleC2×C4×C27C23
hypercenter#101D27
hypocenterC27#413
LCSG ▷ C22×C27 ▷ C2×C27 ▷ C27
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. M16×D27C2semidirect
3. C27×#101C2semidirect
4. #11720075C2semidirect
5. #11720064C2semidirect
105. #101D27semidirect
107. C27#413semidirect

Maximal subgroups

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

subgroupindex
C2×#884032
M16×D272
C27×#1012
#117200752
#117200642
#105884313