Finite Group Databasebeta

← #13154295

Subgroups — #13154295

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×#88748C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11721864C212yesmaximal, complemented
4#11721763C214yesmaximal, complemented
5#11721761C214yesmaximal, complemented
6#11721720C211yesmaximal, characteristic, complemented
7C42×#272C211yesmaximal, characteristic, complemented
8C3×#5725—34nomaximal
9C2×Q8×H(3,𝔽3)C2212yes—
10#88748—28no—
11C2×C4⋊#272C2214yes—
12C2×C4⋊#272C2214yes—
13C2×C4⋊#272C2212yes—
14C2×C4×#272C2214yes—
15C2×C4×#272C2212yes—
16C2×C4×#272C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2214yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#88542C2218yes—
20C4×#1084C2212yes—
21C4×#1084C2211yescharacteristic
22C2×C3×#239—38no—
23C32×C4⋊Q8D314yescomplemented
24C3×#947—38no—
25C3×#895—316no—
26C3×#893—316no—
27C3×#874—34no—
28C3×C42×D3—34no—
29C22×#272C2314yes—
30C22×#272C2311yescharacteristic
31Q8×H(3,𝔽3)—24no—
32C2×C4×H(3,𝔽3)C2314yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084C2312yes—
36C2×#1084C2311yescharacteristic
37C4⋊#272—28no—
38C4⋊#272—24no—
39C4⋊#272—28no—
40C4×#272—24no—
41C4×#272—28no—
42C4×#272D414yescomplemented
43C3×#532—91no—
44#5725—34no—
45C2×C32×Q8C2×D318yes—
46C3×#239—632no—
47C2×C3×D12—316no—
48C2×C3×D12—316no—
49C2×C3×D12—38no—
50C2×C3×C4×D3—34no—
51C2×C3×C4×D3—316no—
52C2×C3×C4×D3—38no—
53C32×C4⋊C4C2×D3116yes—
54C32×C42C2×D314yes—
55C3×(C2×D3)⋊C4—332no—
56C3×C4×Dic3—38no—
57C3×C4×Dic3—34no—
58C22×H(3,𝔽3)C2411yescharacteristic
59C2×#272—22no—
60C2×#272—24no—
61C2×#272—28no—
62C2×#272C2×D412yes—
63C4×H(3,𝔽3)—24no—
64C4×H(3,𝔽3)C2×D412yes—
65C4×H(3,𝔽3)#4114yescomplemented
66#1084—22no—
67#1084C2×D412yes—
68C2×C3×#41—92no—
69C2×#239—38no—
70C3×C4⋊Q8—34no—
71C3×C4⋊Q8C3⋊D311yescharacteristic
72C3×#127—91no—
73C3×#124—92no—
74C3×#121—94no—
75C3×C4×D4—94no—
76C2×C3×C42—91no—
77#947—38no—
78#895—316no—
79#893—316no—
80#874—34no—
81C42×D3—34no—
82C22×C3×D3—34no—
83C22×C3×D3—316no—
84C32×Q8—216no—
85C2×C32×C4C22×D3116yes—
86C2×C32×C4C22×D314yes—
87C2×C32×C4C22×D318yes—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—38no—
90C3×D12—632no—
91C3×D12—616no—
92C3×D12—632no—
93C3×C4×D3—316no—
94C3×C4×D3—632no—
95C3×C4×D3—616no—
96#532—91noSylow
97C2×H(3,𝔽3)C22×D411yescharacteristic
98C2×H(3,𝔽3)C2×#4112yes—
99#272—22no—
100#272—44no—
101C3×#41—188no—
102C2×C3×Q8—38no—
103C2×C3×Q8C2×C3⋊D312yes—
104C2×C3×D4—94no—
105C2×C3×D4—94no—
106C2×C3×D4—92no—
107C22×C3×C4—91no—
108C22×C3×C4—92no—
109C22×C3×C4—94no—
110#239—632no—
111C2×D12—316no—
112C2×D12—38no—
113C2×D12—316no—
114C2×C4×D3—34no—
115C2×C4×D3—38no—
116C2×C4×D3—316no—
117C3×C4⋊C4—316no—
118C3×C4⋊C4C2×C3⋊D314yes—
119C3×C22⋊C4—98no—
120C3×C42—34no—
121C3×C42—92no—
122C3×C42—91no—
123C3×C42C2×C3⋊D311yescharacteristic
124(C2×D3)⋊C4—332no—
125C4×Dic3—38no—
126C4×Dic3—34no—
127C22×C32C23×D314yes—
128C2×C3×D3—38no—
129C2×C3×D3—68no—
130C2×C3×D3—616no—
131C2×C3×D3—632no—
132C32×C4—216no—
133C32×C4D3×D418yes—
134C32×C4#239116yescomplemented
135C3×Dic3—38no—
136C3×Dic3—68no—
137C2×#41—92no—
138C4⋊Q8#27211yescharacteristic, complemented
139#127—91no—
140#124—92no—
141#121—94no—
142C4×D4—94no—
143C2×C42—91no—
144H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C23×C3—94no—
147C22×D3—34no—
148C22×D3—316no—
149C3×Q8—24no—
150C3×Q8—616no—
151C3×D4—188no—
152C3×D4—184no—
153C3×D4—188no—
154C2×C3×C4—38no—
155C2×C3×C4—34no—
156C2×C3×C4—316no—
157C2×C3×C4—91no—
158C2×C3×C4—94no—
159C2×C3×C4—92no—
160C2×C3×C4—188no—
161C2×C3×C4—184no—
162C2×C3×C4C22×C3⋊D314yes—
163C2×C3×C4C22×C3⋊D311yescharacteristic
164C2×C3×C4C22×C3⋊D312yes—
165C2×Dic3—34no—
166C2×Dic3—38no—
167D12—632no—
168D12—616no—
169D12—632no—
170C4×D3—316no—
171C4×D3—632no—
172C4×D3—616no—
173C2×C32C2×D3×D414yes—
174C2×C32C2×#23918yes—
175C3×D3—68no—
176C3×D3—1216no—
177#41—188no—
178C2×Q8C2×#27212yes—
179C2×D4—94no—
180C2×D4—94no—
181C2×D4—92no—
182C22×C4—91no—
183C22×C4—92no—
184C22×C4—94no—
185C4⋊C4C2×#27214yes—
186C22⋊C4—98no—
187C42—92no—
188C42—91no—
189C42C2×#27211yescharacteristic
190C22×C3—34no—
191C22×C3—92no—
192C22×C3—182no—
193C22×C3—184no—
194C22×C3—188no—
195C22×C3C23×C3⋊D311yescharacteristic
196C2×D3—38no—
197C2×D3—68no—
198C2×D3—616no—
199C2×D3—632no—
200C3×C4—24no—
201C3×C4—38no—
202C3×C4—316no—
203C3×C4—616no—
204C3×C4—92no—
205C3×C4—182no—
206C3×C4D4×C3⋊D312yes—
207C3×C4#383214yes—
208Dic3—38no—
209Dic3—68no—
210C32#572514yescomplemented
211C23—91no—
212C23—94no—
213Q8—24no—
214D4—188no—
215D4—184no—
216D4—188no—
217C2×C4—91no—
218C2×C4—94no—
219C2×C4—92no—
220C2×C4—188no—
221C2×C4—184no—
222C2×C4C22×#27214yes—
223C2×C4C22×#27211yescharacteristic
224C2×C4C22×#27212yes—
225C2×C3—34no—
226C2×C3—38no—
227C2×C3—182no—
228C2×C3—364no—
229C2×C3C2×D4×C3⋊D311yescharacteristic
230C2×C3C2×#383212yes—
231D3—68no—
232D3—1216no—
233C22—92no—
234C22—182no—
235C22—184no—
236C22—188no—
237C22C23×#27211yescharacteristic
238C4—24no—
239C4—92no—
240C4—182no—
241C4D4×#27212yes—
242C4#8874814yescomplemented
243C3—34no—
244C3#1059619611yescharacteristic
245C2—182no—
246C2—364no—
247C2C2×D4×#27211yescharacteristic
248C2C2×#8874812yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #11721864C2semidirect
4. #11721763C2semidirect
5. #11721761C2semidirect
6. #11721720C2semidirect
7. C42×#272C2semidirect
23. C32×C4⋊Q8D3semidirect
42. C4×#272D4semidirect
65. C4×H(3,𝔽3)#41semidirect
134. C32×C4#239semidirect
138. C4⋊Q8#272semidirect
144. H(3,𝔽3)#532semidirect
210. C32#5725semidirect
242. C4#88748semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
H(3,𝔽3)×C4⋊Q82
#117218642
#117217632
#117217612
#117217202
C42×#2722
C3×#57253