Finite Group Databasebeta

← #13154271

Subgroups — #13154271

Order 1728.

1705 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 1705 conjugacy classes of subgroups — too many to draw. Only its 241 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×#88744C212yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#11721848C214yesmaximal, complemented
4#11721846C212yesmaximal, complemented
5D4×#1084C214yesmaximal, complemented
6C42×#272C211yesmaximal, characteristic, complemented
7#11721715C211yesmaximal, characteristic, complemented
8C3×#5716—34nomaximal
9C2×D4×H(3,𝔽3)C2214yes—
10C2×D4×H(3,𝔽3)C2212yes—
11C2×#6235C2214yes—
12C22×#1084C2214yes—
13#88744—28no—
14C2×C4×#272C2212yes—
15C2×C4×#272C2211yescharacteristic
16C2×#6229C2212yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18#88552C2218yes—
19C4⋊#1084C2214yes—
20C4×#1084C2212yes—
21C4×#1084C2211yescharacteristic
22C2×C3×#237—38no—
23C32×#127D314yescomplemented
24C3×#938—316no—
25C3×#936—38no—
26C3×D4×Dic3—316no—
27C3×C42×D3—34no—
28C3×#869—34no—
29C23×H(3,𝔽3)C2314yes—
30C22×#272C2311yescharacteristic
31D4×H(3,𝔽3)—24no—
32D4×H(3,𝔽3)—28no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)C2312yes—
35#6235—28no—
36C2×#1084—28no—
37C2×#1084C2314yes—
38C2×#1084C2312yes—
39C2×#1084C2311yescharacteristic
40C4×#272—24no—
41C4×#272D414yescomplemented
42#6229—24no—
43C3×#532—91no—
44#5716—34no—
45C2×C32×D4C2×D3116yes—
46C2×C32×D4C2×D318yes—
47C2×C3×#67—316no—
48C22×C3×Dic3—316no—
49C3×#237—632no—
50C2×C3×C4×D3—34no—
51C2×C3×C4×D3—38no—
52C2×C3×Dic6—38no—
53C32×C42C2×D314yes—
54C3×#217—332no—
55C3×C4⋊Dic3—316no—
56C3×C4×Dic3—38no—
57C3×C4×Dic3—34no—
58C22×H(3,𝔽3)—24no—
59C22×H(3,𝔽3)—28no—
60C22×H(3,𝔽3)C2411yescharacteristic
61C2×#272—22no—
62C2×#272C2×D412yes—
63C4×H(3,𝔽3)C2×D412yes—
64C4×H(3,𝔽3)#4114yescomplemented
65#1084—24no—
66#1084—22no—
67#1084C2×D412yes—
68C2×C3×#41—92no—
69C2×#237—38no—
70C3×C4⋊Q8—91no—
71C3×#127—34no—
72C3×#127C3⋊D311yescharacteristic
73C3×#124—92no—
74C3×#121—94no—
75C3×C4×D4—94no—
76C2×C3×C42—91no—
77#938—316no—
78#936—38no—
79D4×Dic3—316no—
80C42×D3—34no—
81#869—34no—
82C23×C32C22×D3116yes—
83C22×C3×D3—34no—
84C32×D4—216no—
85C32×D4—232no—
86C2×C32×C4C22×D314yes—
87C2×C32×C4C22×D318yes—
88C3×#67—632no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—316no—
92C2×C3×Dic3—632no—
93C3×C4×D3—316no—
94C3×C4×D3—616no—
95C3×Dic6—616no—
96#532—91noSylow
97C2×H(3,𝔽3)—44no—
98C2×H(3,𝔽3)C22×D411yescharacteristic
99C2×H(3,𝔽3)C2×#4112yes—
100#272—22no—
101C3×#41—188no—
102C2×C3×Q8—92no—
103C2×C3×D4—316no—
104C2×C3×D4—38no—
105C2×C3×D4—94no—
106C2×C3×D4C2×C3⋊D314yes—
107C2×C3×D4C2×C3⋊D312yes—
108C22×C3×C4—91no—
109C22×C3×C4—94no—
110C22×C3×C4—92no—
111C2×#67—316no—
112C22×Dic3—316no—
113#237—632no—
114C2×C4×D3—34no—
115C2×C4×D3—38no—
116C2×Dic6—38no—
117C3×C4⋊C4—94no—
118C3×C22⋊C4—98no—
119C3×C42—34no—
120C3×C42—92no—
121C3×C42—91no—
122C3×C42C2×C3⋊D311yescharacteristic
123#217—332no—
124C4⋊Dic3—316no—
125C4×Dic3—38no—
126C4×Dic3—34no—
127C22×C32—216no—
128C22×C32—232no—
129C22×C32C23×D314yes—
130C2×C3×D3—38no—
131C2×C3×D3—68no—
132C32×C4D3×D418yes—
133C32×C4#237116yescomplemented
134C3×Dic3—38no—
135C3×Dic3—616no—
136C3×Dic3—68no—
137C2×#41—92no—
138C4⋊Q8—91no—
139#127#27211yescharacteristic, complemented
140#124—92no—
141#121—94no—
142C4×D4—94no—
143C2×C42—91no—
144H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
145C23×C3—316no—
146C23×C3—91no—
147C23×C3C22×C3⋊D314yes—
148C22×D3—34no—
149C3×Q8—184no—
150C3×D4—24no—
151C3×D4—28no—
152C3×D4—616no—
153C3×D4—632no—
154C3×D4—188no—
155C2×C3×C4—38no—
156C2×C3×C4—34no—
157C2×C3×C4—91no—
158C2×C3×C4—94no—
159C2×C3×C4—92no—
160C2×C3×C4—94no—
161C2×C3×C4—188no—
162C2×C3×C4—184no—
163C2×C3×C4C22×C3⋊D311yescharacteristic
164C2×C3×C4C22×C3⋊D312yes—
165#67—632no—
166C2×Dic3—34no—
167C2×Dic3—38no—
168C2×Dic3—316no—
169C2×Dic3—632no—
170C4×D3—316no—
171C4×D3—616no—
172Dic6—616no—
173C2×C32—416no—
174C2×C32C2×D3×D414yes—
175C2×C32C2×#23718yes—
176C3×D3—68no—
177#41—188no—
178C2×Q8—92no—
179C2×D4—94no—
180C2×D4C2×#27214yes—
181C2×D4C2×#27212yes—
182C22×C4—91no—
183C22×C4—94no—
184C22×C4—92no—
185C4⋊C4—94no—
186C22⋊C4—98no—
187C42—92no—
188C42—91no—
189C42C2×#27211yescharacteristic
190C22×C3—24no—
191C22×C3—28no—
192C22×C3—34no—
193C22×C3—616no—
194C22×C3—632no—
195C22×C3—92no—
196C22×C3—182no—
197C22×C3C23×C3⋊D311yescharacteristic
198C2×D3—38no—
199C2×D3—68no—
200C3×C4—38no—
201C3×C4—316no—
202C3×C4—92no—
203C3×C4—184no—
204C3×C4—182no—
205C3×C4D4×C3⋊D312yes—
206C3×C4#383014yes—
207Dic3—38no—
208Dic3—616no—
209Dic3—68no—
210C32#571614yescomplemented
211C23—91no—
212C23C22×#27214yes—
213Q8—184no—
214D4—24no—
215D4—28no—
216D4—188no—
217C2×C4—91no—
218C2×C4—94no—
219C2×C4—92no—
220C2×C4—94no—
221C2×C4—188no—
222C2×C4—184no—
223C2×C4C22×#27211yescharacteristic
224C2×C4C22×#27212yes—
225C2×C3—34no—
226C2×C3—38no—
227C2×C3—44no—
228C2×C3—1216no—
229C2×C3—182no—
230C2×C3C2×D4×C3⋊D311yescharacteristic
231C2×C3C2×#383012yes—
232D3—68no—
233C22—24no—
234C22—28no—
235C22—92no—
236C22—182no—
237C22C23×#27211yescharacteristic
238C4—92no—
239C4—184no—
240C4—182no—
241C4D4×#27212yes—
242C4#8874414yescomplemented
243C3—34no—
244C3#1059618711yescharacteristic
245C2—44no—
246C2—182no—
247C2C2×D4×#27211yescharacteristic
248C2C2×#8874412yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C2×#88744C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #11721848C2semidirect
4. #11721846C2semidirect
5. D4×#1084C2semidirect
6. C42×#272C2semidirect
7. #11721715C2semidirect
23. C32×#127D3semidirect
41. C4×#272D4semidirect
64. C4×H(3,𝔽3)#41semidirect
133. C32×C4#237semidirect
139. #127#272semidirect
144. H(3,𝔽3)#532semidirect
210. C32#5716semidirect
242. C4#88744semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
H(3,𝔽3)×#1272
#117218482
#117218462
D4×#10842
C42×#2722
#117217152
C3×#57163