Finite Group Databasebeta

← #13154835

Subgroups — #13154835

Order 1728.

340 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 30 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#11723434C211yesmaximal, characteristic
2#11723105C211yesmaximal, characteristic, complemented
3C3×#63780C211yesmaximal, characteristic, complemented
4#10590212—31nomaximal
5C2×C3×#634C2211yescharacteristic
6C2×C32×D12C411yescharacteristic
7#88993—21no—
8C3×#3792—21no—
9C4×C3⋊(C3⋊Dic3)C411yescharacteristic
10(C2×D3)⋊(C3⋊Dic3)—21no—
11(C2×D3)⋊(C3⋊Dic3)—21no—
12#64145—31no—
13#63863—62no—
14#63780—31no—
15#63780D311yescharacteristic, complemented
16C22×C32×D3—21no—
17C2×C33×C4C2×C411yescharacteristic
18C2×C3⋊(C3⋊Dic3)—21no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×#634—41no—
22C32×D12—41no—
23C3⋊#622—41no—
24C3×#622—41no—
25#4477—91nomaximal
26C2×C32×D4—31no—
27C2×#634C2×D311yescharacteristic
28C2×C3×D12—22no—
29C2×C3×Dic6—62no—
30#3792—21no—
31#3792—61no—
32#3757—61no—
33#3757—61no—
34C4×C3⋊Dic3—31no—
35C4×C3⋊Dic3—62no—
36C4×C3⋊Dic3—122no—
37(C2×D3)⋊Dic3—64no—
38(C2×D3)⋊Dic3—64no—
39C22×C33C22⋊C411yescharacteristic
40C2×C32×D3—42no—
41C2×C32×D3—41no—
42C33×C4—21no—
43C3⋊(C3⋊Dic3)—41no—
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—21no—
46#947—91no—
47#936—182no—
48C3×#101—91no—
49#836—91no—
50C23×C32—61no—
51C22×C3×D3—24no—
52C32×D4—121no—
53C2×C32×C4—22no—
54C2×C32×C4—42no—
55C2×C32×C4C4×D311yescharacteristic
56C2×C3⋊Dic3—61no—
57C2×C3⋊Dic3—64no—
58C2×C3⋊Dic3—124no—
59C2×C3⋊Dic3D1211yescharacteristic
60C2×C3⋊Dic3#6711yescharacteristic
61#634—41no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C3×D12—44no—
65C3×Dic6—124no—
66#622—41no—
67#622—121no—
68#355—271noSylow
69C2×C33—21no—
70C2×C33#9911yescharacteristic
71C32×D3—81no—
72C2×C3×Q8—91no—
73C2×C3×D4—62no—
74C2×D12#15711yescharacteristic
75C2×Dic6—62no—
76C3×M16—181no—
77#217—184no—
78#217—184no—
79(C2×D3)⋊C4—181no—
80(C2×D3)⋊C4—181no—
81C4×Dic3—91no—
82C4×Dic3—182no—
83C4×Dic3—362no—
84#208—181no—
85C22×C32—22no—
86C22×C32—42no—
87C22×C32—122no—
88C22×C32—121no—
89C22×C32(C2×D3)⋊C411yescharacteristic
90C2×C3×D3—44no—
91C2×C3×D3—48no—
92C32×C4—21no—
93C32×C4—24no—
94C32×C4—44no—
95C3⋊Dic3—21no—
96C3⋊Dic3—21no—
97C3⋊Dic3—121no—
98C3⋊Dic3—124no—
99C3⋊Dic3—128no—
100C3×Dic3—122no—
101C3×Dic3—122no—
102#124—271no—
103#101—91no—
104#101—271no—
105C33#35511yescharacteristic, complemented, Sylow
106C23×C3—64no—
107C22×D3—21no—
108C3×Q8—361no—
109C3×D4—124no—
110C2×C3×C4—22no—
111C2×C3×C4—42no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4C2×#15711yescharacteristic
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—184no—
118C2×Dic3—181no—
119C2×Dic3—364no—
120D12—41no—
121Dic6—124no—
122C3×C8—361no—
123C3⋊C8—361no—
124C2×C32—21no—
125C2×C32—24no—
126C2×C32—22no—
127C2×C32—44no—
128C2×C32—42no—
129C2×C32—241no—
130C2×C32#80611yescharacteristic
131C3×D3—84no—
132C2×Q8—91no—
133C2×D4—31no—
134M16—181no—
135M16—541no—
136C22⋊C4—541no—
137C22⋊C4—541no—
138C42—271no—
139C22×C3—22no—
140C22×C3—42no—
141C22×C3—124no—
142C22×C3—128no—
143C22×C3C22⋊#15711yescharacteristic
144C2×D3—42no—
145C2×D3—41no—
146C3×C4—24no—
147C3×C4—21no—
148C3×C4—44no—
149C3×C4—181no—
150C3×C4—181no—
151Dic3—122no—
152Dic3—122no—
153Dic3—364no—
154Dic3—361no—
155Dic3—368no—
156C32—22no—
157C32—42no—
158C32#447711yescharacteristic, complemented
159C23—61no—
160Q8—361no—
161D4—121no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—541no—
165C2×C4D3×#15711yescharacteristic
166C8—361no—
167C8—1081no—
168C2×C3—24no—
169C2×C3—21no—
170C2×C3—22no—
171C2×C3—44no—
172C2×C3—42no—
173C2×C3—244no—
174C2×C3#6378611yescharacteristic
175D3—81no—
176C22—122no—
177C22—121no—
178C22(C2×D3)⋊#15711yescharacteristic
179C4—21no—
180C4—181no—
181C4—181no—
182C4—1081no—
183C3—22no—
184C3—42no—
185C3#1059021211yescharacteristic, complemented
186C2—21no—
187C2—241no—
188C2#1172272711yescharacteristic
subgroupG/H
centerC2#11722727
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC2×C33#99
prime radicalC2×C32×D12C4
cosocleC2×C3×#634C22
hypocenterC33#355
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723105C2semidirect
3. C3×#63780C2semidirect
15. #63780D3semidirect
105. C33#355semidirect
158. C32#4477semidirect
185. C3#10590212semidirect

Maximal subgroups

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

subgroupindex
#117234342
#117231052
C3×#637802
#105902123
#44779