Finite Group Databasebeta

← #13143935

Subgroups — #13143935

Order 1728.

605 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721850C211yesmaximal, characteristic
2#11721758C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6#11720548C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4764—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084—21no—
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#940—34no—
21C3×#890—34no—
22C32×#102D314yescomplemented
23C3×#832—34no—
24C3×#821—34no—
25C3×#815—34no—
26C3×#809—34no—
27C23×H(3,𝔽3)—21no—
28C22×#272—21no—
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#6235—42no—
34#6235—42no—
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272—21no—
40C8×H(3,𝔽3)—21no—
41#6180—21no—
42C3×#490—91no—
43#4764—34no—
44C2×C32×D4C2×D314yes—
45C2×C3×#67—64no—
46C2×C3×#67—64no—
47C2×C3×D12—34no—
48C2×C32×C8C2×D314yes—
49C32×C4⋊C4C2×D314yes—
50C3×C4⋊Dic3—34no—
51C3×C4⋊Dic3—64no—
52C3×Dic3⋊C4—64no—
53C3×C4×Dic3—34no—
54C2×C3×C3⋊C8—34no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)—41no—
57C22×H(3,𝔽3)—41no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×#272—41no—
60C2×#272—41no—
61C2×#272—41no—
62C4×H(3,𝔽3)—41no—
63C4×H(3,𝔽3)#4111yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65#1084—21no—
66#1084—21no—
67#1084—41no—
68C3×#127—91no—
69C3×#125—91no—
70#940—34no—
71#890—34no—
72C3×#102—34no—
73C3×#102—91no—
74C3×#102—91no—
75C3×#102—91no—
76C3×#102C3⋊D311yescharacteristic
77C3×#97—91no—
78#832—34no—
79#821—34no—
80#815—34no—
81#809—34no—
82C23×C32—24no—
83C22×C3×D3—64no—
84C32×D4—24no—
85C32×D4—24no—
86C2×C32×C4—24no—
87C2×C32×C4C22×D314yes—
88C3×#67—128no—
89C3×#67—128no—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—64no—
93C3×D12—64no—
94C3×D12—64no—
95C32×C8—24no—
96C3×C3⋊C8—64no—
97#490—91noSylow
98C2×H(3,𝔽3)—81no—
99C2×H(3,𝔽3)#12411yescharacteristic
100C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
101C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
102#272—81no—
103C2×C3×D4—34no—
104C2×C3×D4—91no—
105C2×C3×D4—181no—
106C2×C3×D4—181no—
107C2×C3×D4C2×C3⋊D311yescharacteristic
108C2×#67—64no—
109C2×#67—64no—
110C2×D12—34no—
111C2×C3×C8—34no—
112C2×C3×C8—91no—
113C2×C3×C8C2×C3⋊D311yescharacteristic
114C3×C4⋊C4—34no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—181no—
117C3×C4⋊C4—181no—
118C3×C4⋊C4C2×C3⋊D311yescharacteristic
119C3×C42—91no—
120C4⋊Dic3—34no—
121C4⋊Dic3—64no—
122Dic3⋊C4—64no—
123C4×Dic3—34no—
124C2×C3⋊C8—34no—
125C22×C32—44no—
126C22×C32—44no—
127C22×C32—44no—
128C22×C32D3×D414yes—
129C2×C3×D3—124no—
130C2×C3×D3—124no—
131C2×C3×D3—124no—
132C32×C4—44no—
133C32×C4#23514yes—
134C32×C4#23714yes—
135C3×Dic3—64no—
136C3×Dic3—64no—
137C3×Dic3—124no—
138#127—91no—
139#125—91no—
140#102—91no—
141#102—91no—
142#102—91no—
143#102#27211yescharacteristic, complemented
144#97—91no—
145H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
146C23×C3—21no—
147C23×C3—64no—
148C23×C3—181no—
149C22×D3—64no—
150C3×D4—21no—
151C3×D4—21no—
152C3×D4—64no—
153C3×D4—64no—
154C3×D4—181no—
155C3×D4—181no—
156C3×D4—362no—
157C3×D4—362no—
158C2×C3×C4—21no—
159C2×C3×C4—34no—
160C2×C3×C4—64no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—181no—
164C2×C3×C4C22×C3⋊D311yescharacteristic
165#67—128no—
166#67—128no—
167C2×Dic3—34no—
168C2×Dic3—34no—
169C2×Dic3—64no—
170D12—64no—
171D12—64no—
172C3×C8—21no—
173C3×C8—64no—
174C3×C8—181no—
175C3⋊C8—64no—
176C2×C32—84no—
177C2×C32#88514yes—
178C2×C32#91114yes—
179C2×C32#91414yes—
180C3×D3—244no—
181C2×D4—91no—
182C2×D4—181no—
183C2×D4—181no—
184C2×D4C2×#27211yescharacteristic
185C2×C8—91no—
186C2×C8C2×#27211yescharacteristic
187C4⋊C4—91no—
188C4⋊C4—181no—
189C4⋊C4—181no—
190C4⋊C4C2×#27211yescharacteristic
191C42—91no—
192C22×C3—34no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—124no—
197C22×C3—124no—
198C22×C3—124no—
199C22×C3—361no—
200C22×C3—361no—
201C22×C3—361no—
202C22×C3D4×C3⋊D311yescharacteristic
203C2×D3—124no—
204C2×D3—124no—
205C2×D3—124no—
206C3×C4—34no—
207C3×C4—34no—
208C3×C4—41no—
209C3×C4—124no—
210C3×C4—181no—
211C3×C4—181no—
212C3×C4—361no—
213C3×C4#382811yescharacteristic
214C3×C4#383011yescharacteristic
215Dic3—64no—
216Dic3—64no—
217Dic3—124no—
218C32#476414yescomplemented
219C23—21no—
220C23—181no—
221D4—21no—
222D4—21no—
223D4—181no—
224D4—181no—
225D4—362no—
226D4—362no—
227C2×C4—21no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—81no—
238C2×C3—244no—
239C2×C3—721no—
240C2×C3#6409411yescharacteristic
241C2×C3#6412011yescharacteristic
242C2×C3#6412311yescharacteristic
243D3—244no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—361no—
248C22—361no—
249C22—361no—
250C22D4×#27211yescharacteristic
251C4—41no—
252C4—181no—
253C4—181no—
254C4—361no—
255C4#8874011yescharacteristic
256C4#8874411yescharacteristic
257C3—34no—
258C3#1059246311yescharacteristic
259C2—81no—
260C2—721no—
261C2#1172174111yescharacteristic
262C2#1172179311yescharacteristic
263C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radical#11721850C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. #11721758C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #11720578C2semidirect
5. #11720554C2semidirect
6. #11720548C2semidirect
7. #11720531C2semidirect
22. C32×#102D3semidirect
143. #102#272semidirect
145. H(3,𝔽3)#490semidirect
218. C32#4764semidirect

Maximal subgroups

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

subgroupindex
#117218502
#117217582
H(3,𝔽3)×#1022
#117205782
#117205542
#117205482
#117205312
C3×#47643