Finite Group Databasebeta

← #13158823

Subgroups — #13158823

Order 1728.

872 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723910C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic
3C32×#803C211yesmaximal, characteristic, complemented
4#11722040C212yesmaximal, complemented
5#11722034C212yesmaximal
6#10592413—31nomaximal
7#10590276—34nomaximal
8C2×#6308—21no—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2212yes—
14C3×#3752—21no—
15C3×#3752—21no—
16C2×C32×C3⋊C8C2212yes—
17#64084D311yescharacteristic
18#64078—31no—
19#64078—34no—
20#64078—64no—
21C3×#875—34no—
22C32×#97—31no—
23#63648—32no—
24#63643—32no—
25C3×#803D314yescomplemented
26#63566—38no—
27#63563—38no—
28C22×C3×C3⋊D3—21no—
29C2×C33×C4D411yescharacteristic
30C2×C33×C4D411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—22no—
33#6308—42no—
34C2×C3×C3⋊Dic3—21no—
35C3×C4⋊(C3⋊D3)—22no—
36C3×#634—22no—
37C32×C3⋊C8—22no—
38#5055—91no—
39#4714—94no—
40C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
41C2×#634—68no—
42C2×#634—64no—
43C2×#634—61no—
44C2×#634C2×D311yescharacteristic
45C2×C3×D12—34no—
46C2×C3×Dic6—34no—
47C2×C32×C8—32no—
48C32×C42—24no—
49C32×C42C2×D314yes—
50C32×C42C2×D311yescharacteristic
51#3752—21no—
52#3752—21no—
53C4⋊(C3⋊Dic3)—32no—
54C4⋊(C3⋊Dic3)—38no—
55C4⋊(C3⋊Dic3)—68no—
56C4⋊(C3⋊Dic3)—68no—
57C4⋊(C3⋊Dic3)—64no—
58C4⋊(C3⋊Dic3)—61no—
59C3×(C2×D3)⋊C4—64no—
60C3×(C2×D3)⋊C4—64no—
61C2×C3×C3⋊C8C2×D318yes—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3—41no—
66C33×C4—21no—
67C33×C4—21no—
68C33×C4#4112yes—
69C3⋊(C3⋊Dic3)—42no—
70C3×C3⋊Dic3—41no—
71C3×#124—91no—
72#875—34no—
73#869—94no—
74#869—91no—
75#869—184no—
76C3×#97—34no—
77#835—92no—
78#832—92no—
79#821—98no—
80#816—98no—
81#803C3⋊D311yescharacteristic, complemented
82C22×C3⋊D3—21no—
83C22×C3×D3—64no—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4D1214yes—
88C2×C32×C4D1214yes—
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4#6711yescharacteristic
91C2×C32×C4C22×D314yes—
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—616no—
95C2×C3⋊Dic3—68no—
96C2×C3⋊Dic3—62no—
97C4⋊(C3⋊D3)—22no—
98#634—22no—
99#634—1216no—
100#634—128no—
101#634—122no—
102C2×C3×Dic3—64no—
103C3×D12—68no—
104C3×Dic6—68no—
105C32×C8—62no—
106C3×C3⋊C8—28no—
107#489—271noSylow
108C2×C33#12411yescharacteristic
109C2×C33AGL(1,ℤ/8)11yescharacteristic
110C2×C33#13711yescharacteristic
111C3×C3⋊D3—81no—
112C2×C3×Q8—91no—
113C2×C3×D4—91no—
114C2×D12—34no—
115C2×Dic6—34no—
116C2×Dic6—188no—
117C2×Dic6—184no—
118C2×Dic6—181no—
119C2×C3×C8—38no—
120C3×C22⋊C4—181no—
121C3×C22⋊C4—181no—
122C3×C42—24no—
123C3×C42D3214yes—
124C3×C42C2×C3⋊D311yescharacteristic
125(C2×D3)⋊C4—64no—
126(C2×D3)⋊C4—64no—
127C4⋊Dic3—98no—
128C4⋊Dic3—92no—
129C4⋊Dic3—188no—
130C4⋊Dic3—188no—
131C4⋊Dic3—184no—
132C4⋊Dic3—181no—
133C2×C3⋊C8C2×C3⋊D312yes—
134C22×C32—24no—
135C22×C32C2×D1214yes—
136C22×C32C2×#6711yescharacteristic
137C2×C3⋊D3—41no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—41no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C2×C3×D3—124no—
143C32×C4—28no—
144C32×C4—28no—
145C32×C4—24no—
146C32×C4—24no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—28no—
150C32×C4#23518yes—
151C32×C4#23712yes—
152C3⋊Dic3—41no—
153C3⋊Dic3—1216no—
154C3⋊Dic3—128no—
155C3⋊Dic3—122no—
156C3×Dic3—124no—
157C4⋊Q8—271no—
158#124—91no—
159#103—272no—
160#102—272no—
161#97—31no—
162C33#48911yescharacteristic, complemented, Sylow
163C23×C3—181no—
164C22×D3—64no—
165C3×Q8—182no—
166C3×D4—182no—
167C2×C3×C4—24no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—181no—
171C2×C3×C4#62614yes—
172C2×C3×C4#62614yes—
173C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
174C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
175C2×C3×C4C2×D3214yes—
176C2×C3×C4C22×C3⋊D311yescharacteristic
177C2×Dic3—64no—
178C2×Dic3—1816no—
179C2×Dic3—188no—
180C2×Dic3—182no—
181D12—68no—
182Dic6—68no—
183Dic6—3616no—
184Dic6—368no—
185Dic6—362no—
186C3×C8—68no—
187C3⋊C8—22no—
188C2×C32—24no—
189C2×C32—24no—
190C2×C32—24no—
191C2×C32#87514yes—
192C2×C32#90814yes—
193C2×C32#90914yes—
194C2×C32#93611yescharacteristic
195C2×C32#94911yescharacteristic
196C2×C32#95111yescharacteristic
197C3⋊D3—81no—
198C3×D3—244no—
199C2×Q8—91no—
200C2×Q8—541no—
201C2×D4—91no—
202C2×C8—32no—
203C4⋊C4—272no—
204C4⋊C4—541no—
205C22⋊C4—181no—
206C22⋊C4—181no—
207C42D3×C3⋊D311yescharacteristic
208C22×C3—24no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3—361no—
212C22×C3C2×#62614yes—
213C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
214C2×D3—124no—
215C2×D3—124no—
216C2×D3—124no—
217C3×C4—28no—
218C3×C4—28no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—28no—
224C3×C4—361no—
225C3×C4#379518yes—
226C3×C4#382812yes—
227Dic3—124no—
228Dic3—3616no—
229Dic3—368no—
230Dic3—362no—
231C32—24no—
232C32#471414yescomplemented
233C32#505511yescharacteristic, complemented
234C23—181no—
235Q8—182no—
236Q8—1082no—
237D4—182no—
238C2×C4—181no—
239C2×C4—542no—
240C2×C4#629211yescharacteristic
241C2×C4#629211yescharacteristic
242C2×C4C2×D3×C3⋊D311yescharacteristic
243C8—62no—
244C2×C3—24no—
245C2×C3—24no—
246C2×C3—24no—
247C2×C3—721no—
248C2×C3#6382514yes—
249C2×C3#6383114yes—
250C2×C3#6386014yes—
251C2×C3#6408411yescharacteristic
252C2×C3#6411711yescharacteristic
253C2×C3#6411811yescharacteristic
254D3—244no—
255C22—361no—
256C22—361no—
257C22—361no—
258C22C2×#629211yescharacteristic
259C4—21no—
260C4—21no—
261C4—361no—
262C4—1082no—
263C4#8902112yes—
264C3—24no—
265C3#1059027614yescomplemented
266C3#1059241311yescharacteristic, complemented
267C2—721no—
268C2#1172337111yescharacteristic
269C2#1172338811yescharacteristic
270C2#1172343911yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42D3×C3⋊D3
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723910C2semidirect
3. C32×#803C2semidirect
4. #11722040C2semidirect
25. C3×#803D3semidirect
81. #803C3⋊D3semidirect
162. C33#489semidirect
232. C32#4714semidirect
233. C32#5055semidirect
265. C3#10590276semidirect
266. C3#10592413semidirect

Maximal subgroups

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

subgroupindex
#117239102
C3×#640842
C32×#8032
#117220402
#117220342
#105924133
#105902763