Finite Group Databasebeta

← #13154836

Subgroups — #13154836

Order 1728.

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

normal subgroups only

This group has 388 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#11723441C211yesmaximal, characteristic
2#11723105C211yesmaximal, characteristic, complemented
3C3×#63780C211yesmaximal, characteristic, complemented
4#10590145—31nomaximal
5C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
6C2×C3×#634C2211yescharacteristic
7#88993—21no—
8C3×#3792—21no—
9C32×C4×Dic3C411yescharacteristic
10#88804—21no—
11#88804—21no—
12#64084—31no—
13#63864—62no—
14#63780—31no—
15#63780D311yescharacteristic, complemented
16C22×C3⋊(C3⋊D3)—21no—
17C2×C33×C4C2×C411yescharacteristic
18C4⋊(C3⋊(C3⋊D3))—41no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×#634—41no—
22C2×C32×Dic3—21no—
23C3⋊#622—41no—
24C3×#622—41no—
25#4477—91nomaximal
26C2×C4⋊(C3⋊D3)—31no—
27C2×C4⋊(C3⋊D3)—62no—
28C2×C4⋊(C3⋊D3)—122no—
29C2×#634C2×D311yescharacteristic
30C2×C3×Dic6—62no—
31#3792—21no—
32#3792—61no—
33C32×C42—31no—
34#3752—61no—
35#3752—61no—
36C3×C4×Dic3—22no—
37#3722—64no—
38#3722—64no—
39C22×C33C22⋊C411yescharacteristic
40C2×C3⋊(C3⋊D3)—42no—
41C2×C3⋊(C3⋊D3)—41no—
42C33×C4—21no—
43C3×C3⋊Dic3—21no—
44C3×C3⋊Dic3—21no—
45C32×Dic3—41no—
46#947—91no—
47#875—182no—
48C3×#101—91no—
49#836—91no—
50C22×C3⋊D3—61no—
51C22×C3⋊D3—64no—
52C22×C3⋊D3—124no—
53C2×C32×C4—22no—
54C2×C32×C4—42no—
55C2×C32×C4—61no—
56C2×C32×C4C4×D311yescharacteristic
57C2×C3⋊Dic3D1211yescharacteristic
58C2×C3⋊Dic3#6711yescharacteristic
59C4⋊(C3⋊D3)—121no—
60C4⋊(C3⋊D3)—124no—
61C4⋊(C3⋊D3)—128no—
62#634—41no—
63C2×C3×Dic3—24no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C3×Dic6—124no—
67#622—41no—
68#622—121no—
69#355—271noSylow
70C2×C33—21no—
71C2×C33#9911yescharacteristic
72C3⋊(C3⋊D3)—81no—
73C2×C3×Q8—91no—
74C2×D12—91no—
75C2×D12—182no—
76C2×D12—362no—
77C2×Dic6—62no—
78C3×M16—181no—
79C3×C42—62no—
80(C2×D3)⋊C4—181no—
81(C2×D3)⋊C4—181no—
82(C2×D3)⋊C4—184no—
83(C2×D3)⋊C4—184no—
84C4×Dic3#15711yescharacteristic
85#208—181no—
86C22×C32—22no—
87C22×C32—42no—
88C22×C32(C2×D3)⋊C411yescharacteristic
89C2×C3⋊D3—124no—
90C2×C3⋊D3—122no—
91C2×C3⋊D3—128no—
92C2×C3⋊D3—121no—
93C2×C3⋊D3—128no—
94C2×C3⋊D3—248no—
95C32×C4—21no—
96C32×C4—24no—
97C32×C4—44no—
98C32×C4—121no—
99C3⋊Dic3—21no—
100C3⋊Dic3—21no—
101C3×Dic3—44no—
102C3×Dic3—122no—
103C3×Dic3—122no—
104#124—271no—
105#101—91no—
106#101—271no—
107C33#35511yescharacteristic, complemented, Sylow
108C22×D3—184no—
109C22×D3—181no—
110C22×D3—364no—
111C3×Q8—361no—
112C2×C3×C4—22no—
113C2×C3×C4—42no—
114C2×C3×C4—64no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4C2×#15711yescharacteristic
118C2×Dic3—21no—
119C2×Dic3—62no—
120C2×Dic3—62no—
121D12—361no—
122D12—364no—
123D12—368no—
124Dic6—124no—
125C3×C8—361no—
126C3⋊C8—361no—
127C2×C32—21no—
128C2×C32—24no—
129C2×C32—22no—
130C2×C32—44no—
131C2×C32—42no—
132C2×C32#80611yescharacteristic
133C3⋊D3—244no—
134C3⋊D3—248no—
135C3⋊D3—241no—
136C2×Q8—91no—
137C2×D4—271no—
138M16—181no—
139M16—541no—
140C22⋊C4—541no—
141C22⋊C4—541no—
142C42—31no—
143C22×C3—22no—
144C22×C3—42no—
145C22×C3C22⋊#15711yescharacteristic
146C2×D3—364no—
147C2×D3—362no—
148C2×D3—368no—
149C2×D3—368no—
150C2×D3—361no—
151C2×D3—728no—
152C3×C4—24no—
153C3×C4—21no—
154C3×C4—44no—
155C3×C4—124no—
156C3×C4—181no—
157C3×C4—181no—
158Dic3—41no—
159Dic3—122no—
160Dic3—122no—
161C32—22no—
162C32—42no—
163C32#447711yescharacteristic, complemented
164C23—541no—
165Q8—361no—
166D4—1081no—
167C2×C4—61no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4D3×#15711yescharacteristic
171C8—361no—
172C8—1081no—
173C2×C3—24no—
174C2×C3—21no—
175C2×C3—22no—
176C2×C3—44no—
177C2×C3—42no—
178C2×C3#6377411yescharacteristic
179D3—724no—
180D3—728no—
181D3—721no—
182C22—1082no—
183C22—1081no—
184C22(C2×D3)⋊#15711yescharacteristic
185C4—21no—
186C4—121no—
187C4—181no—
188C4—181no—
189C3—22no—
190C3—42no—
191C3#1059014511yescharacteristic, complemented
192C2—21no—
193C2—2161no—
194C2#1172272811yescharacteristic
subgroupG/H
centerC2#11722728
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC2×C33#99
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C4
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
107. C33#355semidirect
163. C32#4477semidirect
191. C3#10590145semidirect

Maximal subgroups

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

subgroupindex
#117234412
#117231052
C3×#637802
#105901453
#44779