Finite Group Databasebeta

← #13154291

Subgroups — #13154291

Order 1728.

1100 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 227 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic
2Q8×#1084C212yesmaximal
3#11721861C212yesmaximal
4#11721758C212yesmaximal
5#11721757C214yesmaximal
6#11721755C212yesmaximal
7C4×#6229C211yesmaximal, characteristic
8C4×#6229C211yesmaximal, characteristic
9C3×#5721—34nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11C2×#6229C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2212yes—
14C4⋊C4×H(3,𝔽3)C2212yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2214yes—
17C4⋊#1084C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19#88540C2214yes—
20#88540C2212yes—
21#88540C2212yes—
22#88540C2214yes—
23C4×#1084C2214yes—
24C4×#1084C2212yes—
25C4×#1084C2212yes—
26C32×C4⋊Q8D314yes—
27C3×Q8×Dic3—38no—
28C3×#944—38no—
29C3×#890—38no—
30C3×#889—316no—
31C3×#887—38no—
32C3×C4×Dic6—34no—
33C3×C4×Dic6—34no—
34Q8×H(3,𝔽3)—24no—
35C2×C4×H(3,𝔽3)C2314yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084C2312yes—
40C2×#1084C2312yes—
41C2×#1084C2312yes—
42C2×#1084C2312yes—
43#6229—22no—
44#6229Q814yescomplemented
45C3×#557—91no—
46#5721—34no—
47C2×C32×Q8C2×D318yes—
48C2×C3×Dic6—34no—
49C2×C3×Dic6—34no—
50C32×C4⋊C4C2×D318yes—
51C32×C4⋊C4C2×D318yes—
52C32×C42C2×D314yes—
53C3×C4⋊Dic3—316no—
54C3×C4⋊Dic3—34no—
55C3×C4⋊Dic3—34no—
56C3×Dic3⋊C4—316no—
57C3×Dic3⋊C4—38no—
58C3×Dic3⋊C4—38no—
59C3×Dic3⋊C4—316no—
60C3×C4×Dic3—316no—
61C3×C4×Dic3—38no—
62C3×C4×Dic3—38no—
63C22×H(3,𝔽3)C2411yescharacteristic
64C4×H(3,𝔽3)—24no—
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)C2×Q812yes—
67C4×H(3,𝔽3)#4112yes—
68#1084—22no—
69#1084—22no—
70#1084—22no—
71#1084C2×Q814yes—
72C3×C4⋊Q8—34no—
73C3×C4⋊Q8—92no—
74C3×C4⋊Q8C3⋊D311yescharacteristic
75C3×#125—94no—
76C3×#125—92no—
77C3×C4×Q8—92no—
78C3×C4×Q8—92no—
79C3×C4×Q8—91no—
80C3×C4×Q8—91no—
81Q8×Dic3—38no—
82#944—38no—
83#890—38no—
84#889—316no—
85#887—38no—
86C4×Dic6—34no—
87C4×Dic6—34no—
88C32×Q8—216no—
89C2×C32×C4C22×D3116yes—
90C2×C32×C4C22×D314yes—
91C2×C32×C4C22×D314yes—
92C2×C32×C4C22×D314yes—
93C2×C3×Dic3—38no—
94C2×C3×Dic3—38no—
95C2×C3×Dic3—38no—
96C2×C3×Dic3—38no—
97C3×Dic6—316no—
98C3×Dic6—68no—
99#557—91noSylow
100C2×H(3,𝔽3)C22×Q811yescharacteristic
101C2×H(3,𝔽3)C2×#4111yescharacteristic
102C2×H(3,𝔽3)2−1+411yescharacteristic
103C2×C3×Q8—38no—
104C2×C3×Q8—91no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D312yes—
107C2×Dic6—34no—
108C2×Dic6—34no—
109C3×C4⋊C4—38no—
110C3×C4⋊C4—38no—
111C3×C4⋊C4—92no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—94no—
114C3×C4⋊C4—94no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—94no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4C2×C3⋊D312yes—
119C3×C4⋊C4C2×C3⋊D312yes—
120C3×C42—34no—
121C3×C42—94no—
122C3×C42—92no—
123C3×C42—92no—
124C3×C42C2×C3⋊D311yescharacteristic
125C4⋊Dic3—34no—
126C4⋊Dic3—34no—
127C4⋊Dic3—316no—
128Dic3⋊C4—38no—
129Dic3⋊C4—38no—
130Dic3⋊C4—316no—
131Dic3⋊C4—316no—
132C4×Dic3—38no—
133C4×Dic3—38no—
134C4×Dic3—316no—
135C22×C32C23×D314yes—
136C32×C4—216no—
137C32×C4—24no—
138C32×C4#23718yes—
139C32×C4D3×Q818yes—
140C3×Dic3—316no—
141C3×Dic3—68no—
142C3×Dic3—68no—
143C3×Dic3—68no—
144C4⋊Q8—92no—
145C4⋊Q8#27211yescharacteristic
146#125—94no—
147#125—92no—
148C4×Q8—92no—
149C4×Q8—92no—
150C4×Q8—91no—
151C4×Q8—91no—
152H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
153C3×Q8—24no—
154C3×Q8—616no—
155C3×Q8—94no—
156C3×Q8—182no—
157C2×C3×C4—34no—
158C2×C3×C4—34no—
159C2×C3×C4—316no—
160C2×C3×C4—34no—
161C2×C3×C4—92no—
162C2×C3×C4—92no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4C22×C3⋊D314yes—
166C2×C3×C4C22×C3⋊D311yescharacteristic
167C2×C3×C4C22×C3⋊D311yescharacteristic
168C2×C3×C4C22×C3⋊D311yescharacteristic
169C2×Dic3—38no—
170C2×Dic3—38no—
171C2×Dic3—38no—
172C2×Dic3—38no—
173Dic6—316no—
174Dic6—68no—
175C2×C32C2×#23714yes—
176C2×C32C2×D3×Q814yes—
177C2×C32#100714yes—
178C2×Q8—91no—
179C2×Q8—91no—
180C2×Q8C2×#27212yes—
181C4⋊C4—91no—
182C4⋊C4—92no—
183C4⋊C4—92no—
184C4⋊C4—94no—
185C4⋊C4—91no—
186C4⋊C4—94no—
187C4⋊C4—94no—
188C4⋊C4C2×#27212yes—
189C4⋊C4C2×#27212yes—
190C42—94no—
191C42—92no—
192C42—92no—
193C42C2×#27211yescharacteristic
194C22×C3—34no—
195C22×C3C23×C3⋊D311yescharacteristic
196C3×C4—24no—
197C3×C4—21no—
198C3×C4—38no—
199C3×C4—38no—
200C3×C4—616no—
201C3×C4—64no—
202C3×C4—94no—
203C3×C4—182no—
204C3×C4—182no—
205C3×C4—182no—
206C3×C4#383012yes—
207C3×C4Q8×C3⋊D312yes—
208Dic3—316no—
209Dic3—68no—
210Dic3—68no—
211Dic3—68no—
212C32#572114yescomplemented
213Q8—24no—
214Q8—94no—
215Q8—182no—
216C2×C4—92no—
217C2×C4—92no—
218C2×C4—92no—
219C2×C4—92no—
220C2×C4C22×#27214yes—
221C2×C4C22×#27211yescharacteristic
222C2×C4C22×#27211yescharacteristic
223C2×C4C22×#27211yescharacteristic
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3C2×#383011yescharacteristic
228C2×C3C2×Q8×C3⋊D311yescharacteristic
229C2×C3#6436411yescharacteristic
230C22C23×#27211yescharacteristic
231C4—24no—
232C4—21no—
233C4—94no—
234C4—182no—
235C4—182no—
236C4—182no—
237C4#8874412yes—
238C4Q8×#27212yes—
239C3—34no—
240C3#1059619211yescharacteristic
241C2C2×#8874411yescharacteristic
242C2C2×Q8×#27211yescharacteristic
243C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#557
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
44. #6229Q8semidirect
152. H(3,𝔽3)#557semidirect
212. C32#5721semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
Q8×#10842
#117218612
#117217582
#117217572
#117217552
C4×#62292
C3×#57213