Finite Group Databasebeta

← #13144080

Subgroups — #13144080

Order 1728.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#88529C211yesmaximal, characteristic
2C2×#88528C211yesmaximal, characteristic, complemented
3#11721721C211yesmaximal, characteristic, complemented
4C4×C4⋊#272C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720549C211yesmaximal, characteristic, complemented
8C3×#4844—34nomaximal
9C2×C4⋊#272C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×#6229C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#88542—21no—
17#88542—21no—
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20#88529—22no—
21#88528—22no—
22C2×C3×D24—34no—
23C2×C3×#204—34no—
24C3×#875—34no—
25C3×C4×D12—34no—
26C32×C4⋊C8D314yescomplemented
27C3×#821—34no—
28C3×#816—34no—
29C22×#272—21no—
30C22×#272—21no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36C4⋊#272—21no—
37C4⋊#272—21no—
38C4⋊#272—21no—
39C4⋊#272D412yes—
40C4×#272—42no—
41#6229—21no—
42#6229—21no—
43C8×H(3,𝔽3)—21no—
44C8×H(3,𝔽3)—21no—
45C3×#463—91no—
46#4844—34no—
47C2×C3×D12—34no—
48C2×C3×D12—34no—
49C2×C3×C4×D3—64no—
50C2×C3×Dic6—34no—
51C2×C32×C8C2×D314yes—
52C2×C32×C8C2×D314yes—
53C32×C42C2×D314yes—
54C3×(C2×D3)⋊C4—64no—
55C3×(C2×D3)⋊C4—64no—
56C3×(C2×D3)⋊C4—64no—
57C3×C4⋊Dic3—34no—
58C3×D24—68no—
59C3×#204—68no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—22no—
62C2×#272—41no—
63C2×#272—41no—
64C2×#272—41no—
65C2×#272—41no—
66C2×#272—41no—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—41no—
72#1084—41no—
73C2×C3×SD16—91no—
74C2×C3×D8—91no—
75C3×#124—91no—
76C3×C4×D4—91no—
77C2×D24—34no—
78C2×#204—34no—
79#875—34no—
80C4×D12—34no—
81C3×C4⋊C8—34no—
82C3×C4⋊C8C3⋊D311yescharacteristic
83C3×#103—91no—
84C3×#102—91no—
85#821—34no—
86#816—34no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C2×C32×C4D1214yes—
90C2×C32×C4D1214yes—
91C2×C32×C4C22×D314yes—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—64no—
94C3×D12—38no—
95C3×D12—64no—
96C3×D12—64no—
97C3×D12—64no—
98C3×C4×D3—128no—
99C3×Dic6—64no—
100C3×Dic6—64no—
101C32×C8—24no—
102C32×C8—24no—
103#463—91noSylow
104C2×H(3,𝔽3)#12111yescharacteristic
105C2×H(3,𝔽3)#13511yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
107#272—42no—
108#272—81no—
109C2×C3×Q8—91no—
110C2×C3×D4—91no—
111C2×C3×D4—91no—
112C22×C3×C4—181no—
113C2×D12—34no—
114C2×D12—34no—
115C2×C4×D3—64no—
116C2×Dic6—34no—
117C3×SD16—182no—
118C3×D8—182no—
119C2×C3×C8—34no—
120C2×C3×C8—34no—
121C2×C3×C8C2×C3⋊D311yescharacteristic
122C2×C3×C8C2×C3⋊D311yescharacteristic
123C3×C4⋊C4—91no—
124C3×C22⋊C4—181no—
125C3×C22⋊C4—181no—
126C3×C22⋊C4—181no—
127C3×C42—34no—
128C3×C42C2×C3⋊D311yescharacteristic
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—64no—
132C4⋊Dic3—34no—
133D24—68no—
134#204—68no—
135C22×C32C2×D1214yes—
136C2×C3×D3—68no—
137C2×C3×D3—124no—
138C2×C3×D3—124no—
139C2×C3×D3—124no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C32×C4—24no—
143C32×C4—24no—
144C32×C4D3×D414yes—
145C32×C4#23914yes—
146C3×Dic3—124no—
147C3×Dic3—124no—
148C2×SD16—91no—
149C2×D8—91no—
150#124—91no—
151C4×D4—91no—
152C4⋊C8#27211yescharacteristic, complemented
153#103—91no—
154#102—91no—
155H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
156C23×C3—181no—
157C23×C3—181no—
158C22×D3—64no—
159C22×D3—64no—
160C3×Q8—181no—
161C3×Q8—181no—
162C3×D4—92no—
163C3×D4—181no—
164C3×D4—181no—
165C3×D4—181no—
166C2×C3×C4—34no—
167C2×C3×C4—34no—
168C2×C3×C4—34no—
169C2×C3×C4—181no—
170C2×C3×C4—181no—
171C2×C3×C4—362no—
172C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
173C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
174C2×C3×C4C22×C3⋊D311yescharacteristic
175C2×Dic3—64no—
176C2×Dic3—64no—
177D12—38no—
178D12—64no—
179D12—64no—
180D12—64no—
181C4×D3—128no—
182Dic6—64no—
183Dic6—64no—
184C3×C8—21no—
185C3×C8—21no—
186C3×C8—64no—
187C3×C8—64no—
188C2×C32#89514yes—
189C2×C32#90414yes—
190C2×C32#90814yes—
191C3×D3—128no—
192C3×D3—244no—
193C2×Q8—91no—
194C2×D4—91no—
195C2×D4—91no—
196C22×C4—181no—
197SD16—182no—
198D8—182no—
199C2×C8C2×#27211yescharacteristic
200C2×C8C2×#27211yescharacteristic
201C4⋊C4—91no—
202C22⋊C4—181no—
203C22⋊C4—181no—
204C22⋊C4—181no—
205C42C2×#27211yescharacteristic
206C22×C3—34no—
207C22×C3—182no—
208C22×C3—361no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3—361no—
212C22×C3—361no—
213C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
214C2×D3—68no—
215C2×D3—124no—
216C2×D3—124no—
217C2×D3—124no—
218C2×D3—124no—
219C2×D3—124no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—34no—
223C3×C4—34no—
224C3×C4—64no—
225C3×C4—64no—
226C3×C4—361no—
227C3×C4—361no—
228C3×C4D4×C3⋊D311yescharacteristic
229C3×C4#383211yescharacteristic
230Dic3—124no—
231Dic3—124no—
232C32#484414yescomplemented
233C23—181no—
234C23—181no—
235Q8—181no—
236Q8—181no—
237D4—92no—
238D4—181no—
239D4—181no—
240D4—181no—
241C2×C4—181no—
242C2×C4—181no—
243C2×C4—362no—
244C2×C4C4⋊#27211yescharacteristic
245C2×C4C4⋊#27211yescharacteristic
246C2×C4C22×#27211yescharacteristic
247C8—21no—
248C8—21no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—362no—
253C2×C3—721no—
254C2×C3#6410411yescharacteristic
255C2×C3#6411311yescharacteristic
256C2×C3#6411711yescharacteristic
257D3—128no—
258D3—244no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22—361no—
263C22—361no—
264C22—361no—
265C22C2×C4⋊#27211yescharacteristic
266C4—21no—
267C4—21no—
268C4—361no—
269C4—361no—
270C4D4×#27211yescharacteristic
271C4#8874811yescharacteristic
272C3—34no—
273C3#1059254311yescharacteristic
274C2—362no—
275C2—721no—
276C2#1172176311yescharacteristic
277C2#1172177911yescharacteristic
278C2#1172178711yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C4⋊C8C2
prime radicalC2×#88529C2
hypercenterC3×C4⋊C8C3⋊D3
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#88528C2semidirect
3. #11721721C2semidirect
4. C4×C4⋊#272C2semidirect
5. H(3,𝔽3)×C4⋊C8C2semidirect
6. #11720554C2semidirect
7. #11720549C2semidirect
26. C32×C4⋊C8D3semidirect
152. C4⋊C8#272semidirect
155. H(3,𝔽3)#463semidirect
232. C32#4844semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×#885282
#117217212
C4×C4⋊#2722
H(3,𝔽3)×C4⋊C82
#117205542
#117205492
C3×#48443