Finite Group Databasebeta

← #13147032

Subgroups — #13147032

Order 1728.

310 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 36 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#11722394C211yesmaximal, characteristic
2#11722392C211yesmaximal, characteristic, complemented
3#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5406—33nomaximal
5C3×#5141—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7#88611—21no—
8#88610—21no—
9#88528—41nomaximal
10C3×#986—33no—
11C3×#984—33no—
12C32×#131D311yescharacteristic, complemented
13C3×#951—31no—
14C3×#949—31no—
15C2×C3×#204—31no—
16C3×#131⋊C3D313yescomplemented
17C3×#824—31no—
18C3×#823—31no—
19C3×#819—31no—
20C4⋊#272—41no—
21#6229—41no—
22Q8⋊H(3,𝔽3)D411yescharacteristic
23C8×H(3,𝔽3)—41no—
24#5406—33no—
25C3×#471—91no—
26#5141—31no—
27C32×#41C2×D311yescharacteristic
28C2×C3×D12—31no—
29C2×C3×Dic6—31no—
30C3×#41⋊C3C2×D313yes—
31C3×GL(2,𝔽3)—63no—
32C3×#226—63no—
33C32×M16—31no—
34C2×C32×C8—31no—
35C3×#216—61no—
36C3×#215—61no—
37C3×#214—61no—
38C3×#213—61no—
39C3×#208—31no—
40C3×#208—31no—
41C3×#204—61no—
42C3×#204—61no—
43C3×#204—123no—
44C2×#272—81no—
45C4×H(3,𝔽3)—41no—
46#1084—81no—
47#986—33no—
48#984—33no—
49C3×#137—91no—
50C3×AGL(1,ℤ/8)—91no—
51C2×C3×SD16—91no—
52C3×#131—31no—
53C3×#131C3⋊D311yescharacteristic
54#951—31no—
55#949—31no—
56C2×#204—31no—
57#131⋊C3—33no—
58C3×#108—91no—
59C3×#101—91no—
60C3×#100—91no—
61#824—31no—
62#823—31no—
63#819—31no—
64C22×C3×D3—61no—
65C32×Q8D1211yescharacteristic
66C32×D4—31no—
67C2×C32×C4—31no—
68C2×C3×Dic3—61no—
69C3×D12—61no—
70C3×D12—61no—
71C3×D12—123no—
72C3×Dic6—61no—
73C3×Dic6—61no—
74C3×Dic6—123no—
75C3×SL(2,𝔽3)D1213yes—
76C32×C8—31no—
77C32×C8—43no—
78C32×C8S411yescharacteristic
79C3×C3⋊C8—61no—
80C3×C3⋊C8—61no—
81#471—91noSylow
82C2×H(3,𝔽3)—41no—
83#272—161no—
84C3×#41—31no—
85C3×#41C2×C3⋊D311yescharacteristic
86C2×C3×Q8—91no—
87C2×C3×D4—91no—
88C2×D12—31no—
89C2×Dic6—31no—
90#41⋊C3—33no—
91GL(2,𝔽3)—63no—
92#226—63no—
93C3×Q16—181no—
94C3×SD16—181no—
95C3×SD16—181no—
96C3×SD16—181no—
97C3×SD16—181no—
98C3×D8—181no—
99C3×M16—31no—
100C3×M16—31no—
101C3×M16—61no—
102C3×M16—91no—
103C3×M16—91no—
104C2×C3×C8—31no—
105C2×C3×C8—31no—
106C2×C3×C8—61no—
107#216—61no—
108#215—61no—
109#214—61no—
110#213—61no—
111#208—31no—
112#208—31no—
113#204—61no—
114#204—61no—
115#204—123no—
116C22×C32—31no—
117C2×C3×D3—121no—
118C2×C3×D3—243no—
119C2×C3×D3—241no—
120C32×C4—31no—
121C32×C4—43no—
122C32×C4C2×S411yescharacteristic
123C3×Dic3—121no—
124C3×Dic3—243no—
125#137—91no—
126AGL(1,ℤ/8)—91no—
127C2×SD16—91no—
128#131#27211yescharacteristic, complemented
129#108—91no—
130#101—91no—
131#100—91no—
132H(3,𝔽3)—41noSylow
133C23×C3—181no—
134C22×D3—61no—
135C3×Q8—31no—
136C3×Q8—181no—
137C3×Q8—181no—
138C3×Q8C4⋊(C3⋊D3)11yescharacteristic
139C3×D4—31no—
140C3×D4—31no—
141C3×D4—61no—
142C3×D4—181no—
143C3×D4—181no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—181no—
148C2×Dic3—61no—
149D12—61no—
150D12—61no—
151D12—123no—
152Dic6—61no—
153Dic6—61no—
154Dic6—123no—
155SL(2,𝔽3)—33no—
156C3×C8—31no—
157C3×C8—31no—
158C3×C8—31no—
159C3×C8—61no—
160C3×C8—123no—
161C3×C8—181no—
162C3×C8—181no—
163C3×C8C3⋊S411yescharacteristic
164C3⋊C8—61no—
165C3⋊C8—61no—
166C2×C32—43no—
167C2×C32—61no—
168C2×C32C4⋊S411yescharacteristic
169C3×D3—241no—
170C3×D3—483no—
171#41C2×#27211yescharacteristic
172C2×Q8—91no—
173C2×D4—91no—
174Q16—181no—
175SD16—181no—
176SD16—181no—
177SD16—181no—
178SD16—181no—
179D8—181no—
180M16—31no—
181M16—91no—
182M16—91no—
183C2×C8—31no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—61no—
187C22×C3—361no—
188C22×C3—721no—
189C2×D3—121no—
190C2×D3—243no—
191C2×D3—241no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—61no—
196C3×C4—123no—
197C3×C4—361no—
198C3×C4C2×C3⋊S411yescharacteristic
199Dic3—121no—
200Dic3—243no—
201C32—43no—
202C32#540611yescharacteristic, complemented
203C23—181no—
204Q8—181no—
205Q8—181no—
206Q8C4⋊#27211yescharacteristic
207D4—31no—
208D4—181no—
209D4—181no—
210C2×C4—31no—
211C2×C4—181no—
212C8—31no—
213C8—181no—
214C8—181no—
215C8#625811yescharacteristic
216C2×C3—31no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—121no—
220C2×C3—123no—
221C2×C3—721no—
222C2×C3C4⋊(C3⋊S4)11yescharacteristic
223D3—241no—
224D3—483no—
225C22—31no—
226C22—361no—
227C22—721no—
228C4—31no—
229C4—361no—
230C4C2×#625811yescharacteristic
231C3—31no—
232C3—123no—
233C3#1059364011yescharacteristic
234C2—61no—
235C2—721no—
236C2C4⋊#625811yescharacteristic
subgroupG/H
center, socleC2×C3C4⋊(C3⋊S4)
commutator, cosocle#41⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC32×#131D3
prime radical#11722394C2
hypercenterC3×C8C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ #41⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. #11722392C2semidirect
3. #131⋊H(3,𝔽3)C2semidirect
12. C32×#131D3semidirect
16. C3×#131⋊C3D3semidirect
128. #131#272semidirect
202. C32#5406semidirect

Maximal subgroups

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

subgroupindex
#117223942
#117223922
#131⋊H(3,𝔽3)2
C3×#54063
C3×#51413
#885284