Finite Group Databasebeta

← #13147081

Subgroups — #13147081

Order 1728.

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

normal subgroups only

This group has 350 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11722392C211yesmaximal, characteristic, complemented
3#11720740C211yesmaximal, characteristic, complemented
4C3×#5430—33nomaximal
5C3×#5246—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8#88610—21no—
9#88546—41nomaximal
10C32×2−1+4D311yescharacteristic
11C3×2−1+4⋊C3D313yes—
12C3×#984—33no—
13C3×#951—31no—
14C3×#951—31no—
15C3×#944—31no—
16C3×GU(2,𝔽3)—33no—
17C3×#837—31no—
18C3×#837—31no—
19C3×#836—31no—
20D4×H(3,𝔽3)—41no—
21#6229—41no—
22Q8⋊H(3,𝔽3)D411yescharacteristic
23#6180—41no—
24#5430—33no—
25C3×#456—91no—
26#5246—31no—
27C32×#41—31no—
28C32×#41—61no—
29C32×#41C2×D311yescharacteristic
30C2×C32×Q8—21no—
31C2×C32×Q8—31no—
32C2×C3×Dic6—31no—
33C3×#41⋊C3C2×D313yes—
34C2×C3×SL(2,𝔽3)—23no—
35C3×#226—63no—
36C3×#216—61no—
37C3×#216—61no—
38C3×#214—61no—
39C3×#214—61no—
40C3×#214—123no—
41C3×Dic3⋊C4—62no—
42C3×C4×Dic3—31no—
43C3×#208—31no—
44C3×#208—31no—
45C22×H(3,𝔽3)—81no—
46C4×H(3,𝔽3)—41no—
47#1084—81no—
48C3×2−1+4—31no—
49C3×2−1+4C3⋊D311yescharacteristic
502−1+4⋊C3—33no—
51#984—33no—
52C3×#137—91no—
53C3×#137—91no—
54C3×C4⋊Q8—91no—
55#951—31no—
56#951—31no—
57#944—31no—
58GU(2,𝔽3)—33no—
59C3×#104—91no—
60C3×#104—91no—
61C3×#101—91no—
62#837—31no—
63#837—31no—
64#836—31no—
65C32×Q8—31no—
66C32×Q8—61no—
67C32×Q8#6711yescharacteristic
68C32×D4—31no—
69C32×D4—43no—
70C32×D4—61no—
71C32×D4S411yescharacteristic
72C2×C32×C4—31no—
73C2×C32×C4—32no—
74C2×C32×C4—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C3×Dic6—61no—
78C3×Dic6—61no—
79C3×Dic6—123no—
80C3×SL(2,𝔽3)#6713yes—
81C3×C3⋊C8—61no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—123no—
84#456—91noSylow
85C2×H(3,𝔽3)—41no—
86C2×H(3,𝔽3)—161no—
87C3×#41—31no—
88C3×#41—31no—
89C3×#41—31no—
90C3×#41—61no—
91C3×#41—61no—
92C3×#41—62no—
93C3×#41—61no—
94C3×#41C2×C3⋊D311yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8—31no—
97C2×C3×Q8—31no—
98C2×C3×Q8—61no—
99C2×C3×Q8—61no—
100C2×C3×Q8—91no—
101C2×Dic6—31no—
102#41⋊C3—33no—
103C2×SL(2,𝔽3)—63no—
104#226—63no—
105C3×Q16—181no—
106C3×Q16—181no—
107C3×SD16—181no—
108C3×SD16—181no—
109C3×M16—91no—
110C3×M16—91no—
111C3×C4⋊C4—182no—
112C3×C42—91no—
113#216—61no—
114#216—61no—
115#214—61no—
116#214—61no—
117#214—123no—
118Dic3⋊C4—62no—
119C4×Dic3—31no—
120#208—31no—
121#208—31no—
122C22×C32—21no—
123C22×C32—31no—
124C22×C32—83no—
125C32×C4—31no—
126C32×C4—43no—
127C32×C4—61no—
128C32×C4C2×S411yescharacteristic
129C3×Dic3—62no—
130C3×Dic3—121no—
131C3×Dic3—243no—
1322−1+4#27211yescharacteristic
133#137—91no—
134#137—91no—
135C4⋊Q8—91no—
136#104—91no—
137#104—91no—
138#101—91no—
139H(3,𝔽3)—41noSylow
140C3×Q8—31no—
141C3×Q8—31no—
142C3×Q8—31no—
143C3×Q8—61no—
144C3×Q8—61no—
145C3×Q8—61no—
146C3×Q8—62no—
147C3×Q8—181no—
148C3×Q8—181no—
149C3×Q8#63811yescharacteristic
150C3×D4—31no—
151C3×D4—31no—
152C3×D4—31no—
153C3×D4—61no—
154C3×D4—61no—
155C3×D4—61no—
156C3×D4—62no—
157C3×D4—123no—
158C3×D4C3⋊S411yescharacteristic
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—61no—
164C2×C3×C4—62no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—62no—
168C2×C3×C4—181no—
169C2×C3×C4—181no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172Dic6—61no—
173Dic6—61no—
174Dic6—123no—
175SL(2,𝔽3)—33no—
176C3×C8—181no—
177C3×C8—181no—
178C3⋊C8—61no—
179C3⋊C8—61no—
180C3⋊C8—123no—
181C2×C32—41no—
182C2×C32—43no—
183C2×C32—61no—
184C2×C32—163no—
185C2×C32GL(2,ℤ/4)11yescharacteristic
186#41—31no—
187#41—61no—
188#41C2×#27211yescharacteristic
189C2×Q8—21no—
190C2×Q8—31no—
191C2×Q8—91no—
192Q16—181no—
193Q16—181no—
194SD16—181no—
195SD16—181no—
196M16—91no—
197M16—91no—
198C4⋊C4—182no—
199C42—91no—
200C22×C3—21no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—61no—
204C22×C3—61no—
205C22×C3—243no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—61no—
210C3×C4—62no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—123no—
214C3×C4—182no—
215C3×C4—361no—
216C3×C4C2×C3⋊S411yescharacteristic
217Dic3—62no—
218Dic3—121no—
219Dic3—243no—
220C32—43no—
221C32#543011yescharacteristic, complemented
222Q8—31no—
223Q8—61no—
224Q8—181no—
225Q8—181no—
226Q8#623511yescharacteristic
227D4—31no—
228D4—61no—
229D4#625811yescharacteristic
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—61no—
233C2×C4—181no—
234C2×C4—181no—
235C8—181no—
236C8—181no—
237C2×C3—31no—
238C2×C3—41no—
239C2×C3—61no—
240C2×C3—61no—
241C2×C3—121no—
242C2×C3—121no—
243C2×C3—123no—
244C2×C3—483no—
245C2×C3#6426911yescharacteristic
246C22—21no—
247C22—31no—
248C4—31no—
249C4—61no—
250C4—182no—
251C4—361no—
252C4C2×#625811yescharacteristic
253C3—31no—
254C3—123no—
255C3#1059366411yescharacteristic
256C2—41no—
257C2—61no—
258C2#1172239811yescharacteristic
subgroupG/H
center, socleC2×C3#64269
commutator, cosocle#41⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC32×2−1+4D3
prime radical2−1+4⋊H(3,𝔽3)C2
hypercenterC3×D4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ #41⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
2. #11722392C2semidirect
3. #11720740C2semidirect
221. C32#5430semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117223922
#117207402
C3×#54303
C3×#52463
#885464