Finite Group Databasebeta

← #13147022

Subgroups — #13147022

Order 1728.

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

normal subgroups only

This group has 358 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
1C4⋊#6258C211yesmaximal, characteristic
2#11722380C211yesmaximal, characteristic, complemented
3C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5401—33nomaximal
5C3×#5114—31nomaximal
6C2×#6258—21no—
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8#88613—21no—
9#88528—41nomaximal
10C3×C4⋊S4—33no—
11C3×A4⋊Q8—33no—
12C22×C32×C8D311yescharacteristic, complemented
13C3×#930—31no—
14C3×#924—31no—
15C2×C3×#204—31no—
16C3×C8×A4D313yescomplemented
17C3×#821—31no—
18C3×#817—31no—
19C3×#816—31no—
20C2×C22⋊H(3,𝔽3)D411yescharacteristic
21#6258—41no—
22C4⋊#272—41no—
23#6229—41no—
24C8×H(3,𝔽3)—41no—
25#5401—33no—
26C3×#465—91no—
27#5114—31no—
28C2×C3×S4—63no—
29C22×C32×C4C2×D311yescharacteristic
30C2×C3×#67—61no—
31C2×C3×D12—31no—
32C2×C3×Dic6—31no—
33C3×C4×A4C2×D313yes—
34C3×SL(2,ℤ/4)—63no—
35C2×C32×C8—31no—
36C2×C32×C8—31no—
37C3×#217—61no—
38C3×(C2×D3)⋊C4—61no—
39C3×C4⋊Dic3—31no—
40C3×C4⋊Dic3—31no—
41C3×Dic3⋊C4—61no—
42C3×#204—61no—
43C3×#204—61no—
44C3×#204—123no—
45C2×#272—81no—
46C22⋊H(3,𝔽3)SD1611yescharacteristic, complemented
47C4×H(3,𝔽3)—41no—
48#1084—81no—
49C4⋊S4—33no—
50A4⋊Q8—33no—
51C2×C3×SD16—91no—
52C22×C3×C8—31no—
53C22×C3×C8C3⋊D311yescharacteristic
54C3×C22⋊Q8—91no—
55C3×#121—91no—
56#930—31no—
57#924—31no—
58C2×#204—31no—
59C8×A4—33no—
60C3×#106—91no—
61C3×#103—91no—
62C3×#102—91no—
63#821—31no—
64#817—31no—
65#816—31no—
66C23×C32D1211yescharacteristic
67C22×C3×D3—61no—
68C2×C3×A4D1213yes—
69C3×S4—123no—
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C3×#67—121no—
73C3×#67—121no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C3×D12—61no—
78C3×D12—61no—
79C3×D12—123no—
80C3×Dic6—61no—
81C3×Dic6—61no—
82C3×Dic6—123no—
83C32×C8—31no—
84C32×C8—43no—
85C32×C8S411yescharacteristic, complemented
86#465—91noSylow
87C2×H(3,𝔽3)—41no—
88#272—161no—
89C2×S4—63no—
90C2×C3×Q8—91no—
91C2×C3×D4—91no—
92C2×C3×D4—181no—
93C22×C3×C4—31no—
94C22×C3×C4C2×C3⋊D311yescharacteristic
95C2×#67—61no—
96C2×D12—31no—
97C2×Dic6—31no—
98C4×A4—33no—
99SL(2,ℤ/4)—63no—
100C3×SD16—181no—
101C3×SD16—181no—
102C2×C3×C8—31no—
103C2×C3×C8—31no—
104C2×C3×C8—31no—
105C2×C3×C8—31no—
106C2×C3×C8—61no—
107C2×C3×C8—61no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110C3×C4⋊C4—181no—
111C3×C22⋊C4—181no—
112C3×C22⋊C4—181no—
113#217—61no—
114(C2×D3)⋊C4—61no—
115C4⋊Dic3—31no—
116C4⋊Dic3—31no—
117Dic3⋊C4—61no—
118#204—61no—
119#204—61no—
120#204—123no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32#20411yescharacteristic, complemented
124C2×C3×D3—121no—
125C2×C3×D3—121no—
126C2×C3×D3—121no—
127C2×C3×D3—243no—
128C3×A4#20413yescomplemented
129C32×C4—31no—
130C32×C4—43no—
131C32×C4C2×S411yescharacteristic
132C3×Dic3—121no—
133C3×Dic3—121no—
134C3×Dic3—121no—
135C3×Dic3—243no—
136C2×SD16—91no—
137C22×C8#27211yescharacteristic, complemented
138C22⋊Q8—91no—
139#121—91no—
140#106—91no—
141#103—91no—
142#102—91no—
143H(3,𝔽3)—41noSylow
144C23×C3—31no—
145C23×C3—181no—
146C23×C3C4⋊(C3⋊D3)11yescharacteristic
147C22×D3—61no—
148C2×A4—33no—
149S4—123no—
150C3×Q8—181no—
151C3×Q8—181no—
152C3×D4—181no—
153C3×D4—181no—
154C3×D4—361no—
155C3×D4—361no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—181no—
163C2×C3×C4—181no—
164C2×C3×C4—181no—
165#67—121no—
166#67—121no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170D12—61no—
171D12—61no—
172D12—123no—
173Dic6—61no—
174Dic6—61no—
175Dic6—123no—
176C3×C8—31no—
177C3×C8—31no—
178C3×C8—31no—
179C3×C8—61no—
180C3×C8—123no—
181C3×C8C3⋊S411yescharacteristic
182C2×C32—31no—
183C2×C32—31no—
184C2×C32—43no—
185C2×C32C4⋊S411yescharacteristic
186C3×D3—241no—
187C3×D3—483no—
188C2×Q8—91no—
189C2×D4—91no—
190C2×D4—181no—
191C22×C4C2×#27211yescharacteristic
192SD16—181no—
193SD16—181no—
194C2×C8—31no—
195C2×C8—31no—
196C4⋊C4—91no—
197C4⋊C4—91no—
198C4⋊C4—181no—
199C22⋊C4—181no—
200C22⋊C4—181no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—361no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3#374411yescharacteristic
212C2×D3—121no—
213C2×D3—121no—
214C2×D3—121no—
215C2×D3—243no—
216A4—33no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—31no—
220C3×C4—61no—
221C3×C4—123no—
222C3×C4—361no—
223C3×C4—361no—
224C3×C4—361no—
225C3×C4C2×C3⋊S411yescharacteristic
226Dic3—121no—
227Dic3—121no—
228Dic3—121no—
229Dic3—243no—
230C32—43no—
231C32#540111yescharacteristic, complemented
232C23—181no—
233C23C4⋊#27211yescharacteristic
234Q8—181no—
235Q8—181no—
236D4—181no—
237D4—181no—
238D4—361no—
239D4—361no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C4—181no—
245C8—31no—
246C8#625811yescharacteristic, complemented
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—61no—
253C2×C3—61no—
254C2×C3—123no—
255C2×C3—721no—
256C2×C3C4⋊(C3⋊S4)11yescharacteristic
257D3—241no—
258D3—483no—
259C22—31no—
260C22—31no—
261C22—361no—
262C22—361no—
263C22—361no—
264C22#8852811yescharacteristic, complemented
265C4—31no—
266C4—361no—
267C4—361no—
268C4—361no—
269C4C2×#625811yescharacteristic
270C3—31no—
271C3—123no—
272C3#1059363511yescharacteristic
273C2—31no—
274C2—31no—
275C2—721no—
276C2C4⋊#625811yescharacteristic
subgroupG/H
centerC2×C3C4⋊(C3⋊S4)
commutator, cosocleC4×C22⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×C8D3
socleC23×C3C4⋊(C3⋊D3)
prime radicalC4⋊#6258C2
hypercenterC3×C8C3⋊S4
hypocenterC22⋊H(3,𝔽3)SD16
LCSG ▷ C4×C22⋊H(3,𝔽3) ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊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. #11722380C2semidirect
3. C8×C22⋊H(3,𝔽3)C2semidirect
12. C22×C32×C8D3semidirect
16. C3×C8×A4D3semidirect
46. C22⋊H(3,𝔽3)SD16semidirect
85. C32×C8S4semidirect
123. C22×C32#204semidirect
128. C3×A4#204semidirect
137. C22×C8#272semidirect
231. C32#5401semidirect
246. C8#6258semidirect
264. C22#88528semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62582
#117223802
C8×C22⋊H(3,𝔽3)2
C3×#54013
C3×#51143
#885284