Finite Group Databasebeta

← #13154833

Subgroups — #13154833

Order 1728.

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

normal subgroups only

This group has 352 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#11723436C211yesmaximal, characteristic
2#11723099C211yesmaximal, characteristic
3C3×#63774C211yesmaximal, characteristic
4#10590217—31nomaximal
5C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
6C2×C32×Dic6C411yescharacteristic, complemented
7#88994—21no—
8C3×C22⋊#157—21no—
9C4×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
10#88803—21no—
11#88803—21no—
12#64156—31no—
13#63863—62no—
14#63774—31no—
15#63774D311yescharacteristic
16C22×C3×C3⋊D3D411yescharacteristic
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3⋊#157—41no—
19C2×C3×#157—41no—
20C2×C33×C4C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—21no—
22C3×C4⋊(C3⋊D3)—41no—
23C2×C32×Dic3—21no—
24C32×Dic6—41no—
25#4473—91nomaximal
26C2×C32×Q8—31no—
27C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
28C2×C3×D12—62no—
29C2×C3×Dic6—22no—
30C22⋊#157—21no—
31C22⋊#157—61no—
32#3752—61no—
33#3752—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×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3—41no—
43C2×C3×C3⋊D3—41no—
44C3⋊#157—81no—
45C3×#157—81no—
46C33×C4—21no—
47C3⋊(C3⋊Dic3)—41no—
48C32×Dic3—41no—
49#947—182no—
50#936—91no—
51C3×#99—91no—
52#834—91no—
53C22×C3⋊D3D1211yescharacteristic
54C22×C3⋊D3#6711yescharacteristic
55C22×C3×D3—62no—
56C22×C3×D3—62no—
57C2×#157—41no—
58C2×#157—121no—
59C32×Q8—121no—
60C2×C32×C4—22no—
61C2×C32×C4—42no—
62C2×C32×C4—61no—
63C2×C32×C4C4×D311yescharacteristic
64C2×C3⋊Dic3—61no—
65C2×C3⋊Dic3—64no—
66C2×C3⋊Dic3—124no—
67C4⋊(C3⋊D3)—41no—
68C2×C3×Dic3—24no—
69C3×D12—124no—
70C3×Dic6—44no—
71#354—271noSylow
72C2×C33—21no—
73C2×C33#9911yescharacteristic
74C3×C3⋊D3—41no—
75C3×C3⋊D3—41no—
76C2×C3×Q8—62no—
77C2×C3×D4—91no—
78C2×D12—62no—
79C2×Dic6#15711yescharacteristic, complemented
80C3×C22⋊C4—181no—
81#217—181no—
82#217—181no—
83#217—181no—
84(C2×D3)⋊C4—184no—
85(C2×D3)⋊C4—184no—
86C4×Dic3—91no—
87C4×Dic3—182no—
88C4×Dic3—362no—
89C22×C32—22no—
90C22×C32—42no—
91C22×C32(C2×D3)⋊C411yescharacteristic
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—41no—
95C2×C3⋊D3—41no—
96C2×C3×D3—122no—
97C2×C3×D3—122no—
98C2×C3×D3—124no—
99C2×C3×D3—124no—
100#157—81no—
101#157—241no—
102C32×C4—21no—
103C32×C4—24no—
104C32×C4—44no—
105C32×C4—121no—
106C3⋊Dic3—121no—
107C3⋊Dic3—124no—
108C3⋊Dic3—128no—
109C3×Dic3—44no—
110#124—271no—
111#99—91no—
112#99—271no—
113C33#35411yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C23×C3—91no—
116C22×D3—62no—
117C22×D3—62no—
118C3×Q8—124no—
119C3×D4—361no—
120C2×C3×C4—22no—
121C2×C3×C4—42no—
122C2×C3×C4—64no—
123C2×C3×C4—361no—
124C2×C3×C4C2×#15711yescharacteristic
125C2×Dic3—21no—
126C2×Dic3—184no—
127C2×Dic3—181no—
128C2×Dic3—364no—
129C2×Dic3—361no—
130D12—124no—
131Dic6—41no—
132C2×C32—21no—
133C2×C32—24no—
134C2×C32—22no—
135C2×C32—44no—
136C2×C32—42no—
137C2×C32#80611yescharacteristic
138C3⋊D3—41no—
139C3⋊D3—41no—
140C3×D3—242no—
141C3×D3—242no—
142C2×Q8—31no—
143C2×D4—91no—
144C22⋊C4—181no—
145C22⋊C4—541no—
146C22⋊C4—541no—
147C22⋊C4—541no—
148C42—271no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3—181no—
152C22×C3—181no—
153C22×C3—361no—
154C22×C3—361no—
155C22×C3C22⋊#15711yescharacteristic
156C2×D3—122no—
157C2×D3—122no—
158C2×D3—124no—
159C2×D3—124no—
160C3×C4—24no—
161C3×C4—21no—
162C3×C4—44no—
163C3×C4—124no—
164C3×C4—721no—
165Dic3—41no—
166Dic3—364no—
167Dic3—361no—
168Dic3—368no—
169Dic3—721no—
170C32—22no—
171C32—42no—
172C32#447311yescharacteristic, complemented
173C23—91no—
174C23—91no—
175Q8—121no—
176D4—361no—
177C2×C4—61no—
178C2×C4—361no—
179C2×C4—541no—
180C2×C4—1081no—
181C2×C4D3×#15711yescharacteristic
182C2×C3—24no—
183C2×C3—21no—
184C2×C3—22no—
185C2×C3—44no—
186C2×C3—42no—
187C2×C3—361no—
188C2×C3—361no—
189C2×C3#6378611yescharacteristic
190D3—242no—
191D3—242no—
192C22—181no—
193C22—181no—
194C22—361no—
195C22—361no—
196C22(C2×D3)⋊#15711yescharacteristic
197C4—21no—
198C4—121no—
199C4—721no—
200C4—1081no—
201C4—2161no—
202C3—22no—
203C3—42no—
204C3#1059021711yescharacteristic, complemented
205C2—21no—
206C2—361no—
207C2—361no—
208C2#1172272711yescharacteristic
subgroupG/H
centerC2#11722727
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC2×C33#99
prime radical, cosocleC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#354
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
6. C2×C32×Dic6C4semidirect
9. C4×C3⋊(C3⋊Dic3)C4semidirect
79. C2×Dic6#157semidirect
113. C33#354semidirect
172. C32#4473semidirect
204. C3#10590217semidirect

Maximal subgroups

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

subgroupindex
#117234362
#117230992
C3×#637742
#105902173
#44739