Finite Group Databasebeta

← #13147057

Subgroups — #13147057

Order 1728.

356 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 43 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
1Q8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11722380C211yesmaximal, characteristic
3#11720736C211yesmaximal, characteristic
4C3×#5416—33nomaximal
5C3×#5228—31nomaximal
6C4×C22⋊H(3,𝔽3)—21no—
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8#88613—21no—
9#88550—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C3×Q8×A4D313yes—
12C3×A4⋊Q8—33no—
13C2×C3×#216—31no—
14C2×C3×#216—31no—
15C3×#924—31no—
16C3×A4⋊C8—33no—
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20C2×C22⋊H(3,𝔽3)D411yescharacteristic
21Q8×H(3,𝔽3)—41no—
22#6229—41no—
23#6180—41no—
24#5416—33no—
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C22×C32×C4—21no—
31C22×C32×C4C2×D311yescharacteristic
32C2×C3×Dic6—31no—
33C3×C4×A4—23no—
34C3×C4×A4C2×D313yes—
35C3×SL(2,ℤ/4)—63no—
36C3×#217—61no—
37C3×#216—62no—
38C3×#216—61no—
39C3×#216—61no—
40C3×#216—123no—
41C3×C4⋊Dic3—31no—
42C3×Dic3⋊C4—61no—
43C2×C3×C3⋊C8—31no—
44C2×C3×C3⋊C8—31no—
45C22⋊H(3,𝔽3)Q1611yescharacteristic, complemented
46C4×H(3,𝔽3)—41no—
47C4×H(3,𝔽3)—81no—
48#1084—81no—
49C22×C3×Q8—31no—
50C22×C3×Q8C3⋊D311yescharacteristic
51Q8×A4—33no—
52A4⋊Q8—33no—
53C2×C3×Q16—91no—
54C2×C3×Q16—91no—
55C3×C22⋊Q8—91no—
56C2×#216—31no—
57C2×#216—31no—
58#924—31no—
59A4⋊C8—33no—
60C3×#103—91no—
61C3×#103—91no—
62C3×C22⋊C8—91no—
63#835—31no—
64#835—31no—
65#830—31no—
66C23×C32#6711yescharacteristic
67C2×C3×A4#6713yes—
68C32×Q8—32no—
69C32×Q8—31no—
70C32×Q8—43no—
71C32×Q8—61no—
72C32×Q8S411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C3×Dic6—61no—
80C3×Dic6—61no—
81C3×Dic6—123no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—61no—
84C3×C3⋊C8—123no—
85#451—91noSylow
86C2×H(3,𝔽3)—41no—
87C2×C3×Q8—31no—
88C2×C3×Q8—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—31no—
91C2×C3×Q8—61no—
92C2×C3×Q8—61no—
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—62no—
96C2×C3×Q8—91no—
97C22×C3×C4—21no—
98C22×C3×C4—31no—
99C22×C3×C4—61no—
100C22×C3×C4C2×C3⋊D311yescharacteristic
101C2×Dic6—31no—
102C4×A4—33no—
103C4×A4—63no—
104SL(2,ℤ/4)—63no—
105C3×Q16—182no—
106C3×Q16—181no—
107C3×Q16—181no—
108C2×C3×C8—91no—
109C2×C3×C8—91no—
110C3×C4⋊C4—91no—
111C3×C4⋊C4—181no—
112C3×C22⋊C4—181no—
113#217—61no—
114#216—62no—
115#216—61no—
116#216—61no—
117#216—123no—
118C4⋊Dic3—31no—
119Dic3⋊C4—61no—
120C2×C3⋊C8—31no—
121C2×C3⋊C8—31no—
122C22×C32—31no—
123C22×C32—31no—
124C22×C32#21611yescharacteristic, complemented
125C3×A4#21613yescomplemented
126C32×C4—21no—
127C32×C4—31no—
128C32×C4—43no—
129C32×C4—61no—
130C32×C4—83no—
131C32×C4C2×S411yescharacteristic
132C3×Dic3—121no—
133C3×Dic3—121no—
134C3×Dic3—243no—
135C22×Q8#27211yescharacteristic
136C2×Q16—91no—
137C2×Q16—91no—
138C22⋊Q8—91no—
139#103—91no—
140#103—91no—
141C22⋊C8—91no—
142H(3,𝔽3)—41noSylow
143C23×C3—31no—
144C23×C3#63811yescharacteristic
145C2×A4—33no—
146C3×Q8—32no—
147C3×Q8—31no—
148C3×Q8—32no—
149C3×Q8—31no—
150C3×Q8—31no—
151C3×Q8—61no—
152C3×Q8—62no—
153C3×Q8—61no—
154C3×Q8—61no—
155C3×Q8—62no—
156C3×Q8—123no—
157C3×Q8—181no—
158C3×Q8—181no—
159C3×Q8C3⋊S411yescharacteristic
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—61no—
168C2×C3×C4—61no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—61no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176Dic6—61no—
177Dic6—61no—
178Dic6—123no—
179C3×C8—181no—
180C3×C8—181no—
181C3⋊C8—61no—
182C3⋊C8—61no—
183C3⋊C8—123no—
184C2×C32—31no—
185C2×C32—31no—
186C2×C32—43no—
187C2×C32GL(2,ℤ/4)11yescharacteristic
188C2×Q8—31no—
189C2×Q8—31no—
190C2×Q8—61no—
191C2×Q8—91no—
192C22×C4—21no—
193C22×C4C2×#27211yescharacteristic
194Q16—182no—
195Q16—181no—
196Q16—181no—
197C2×C8—91no—
198C2×C8—91no—
199C4⋊C4—91no—
200C4⋊C4—181no—
201C22⋊C4—181no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3#375611yescharacteristic
210A4—33no—
211C3×C4—21no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—61no—
216C3×C4—61no—
217C3×C4—62no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—123no—
221C3×C4—243no—
222C3×C4—361no—
223C3×C4—361no—
224C3×C4C2×C3⋊S411yescharacteristic
225Dic3—121no—
226Dic3—121no—
227Dic3—243no—
228C32—43no—
229C32#541611yescharacteristic, complemented
230C23#623511yescharacteristic
231Q8—32no—
232Q8—31no—
233Q8—61no—
234Q8—181no—
235Q8—181no—
236Q8#625811yescharacteristic
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—181no—
242C2×C4—181no—
243C8—181no—
244C8—181no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3—123no—
253C2×C3#6426911yescharacteristic
254C22—31no—
255C22—31no—
256C22#8855011yescharacteristic, complemented
257C4—21no—
258C4—31no—
259C4—61no—
260C4—361no—
261C4—361no—
262C4C2×#625811yescharacteristic
263C3—31no—
264C3—123no—
265C3#1059365011yescharacteristic
266C2—31no—
267C2—31no—
268C2#1172239811yescharacteristic
subgroupG/H
centerC2×C3#64269
commutator, cosocleC4×C22⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C3#638
prime radicalC2×C22⋊H(3,𝔽3)D4
hypercenterC3×Q8C3⋊S4
hypocenterC22⋊H(3,𝔽3)Q16
LCSG ▷ C4×C22⋊H(3,𝔽3) ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
45. C22⋊H(3,𝔽3)Q16semidirect
124. C22×C32#216semidirect
125. C3×A4#216semidirect
229. C32#5416semidirect
256. C22#88550semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
#117223802
#117207362
C3×#54163
C3×#52283
#885504