Finite Group Databasebeta

← #13144446

Subgroups — #13144446

Order 1728.

594 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic
3C4×#6229C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720532C211yesmaximal, characteristic
6#11720530C211yesmaximal, characteristic
7#11720520C211yesmaximal, characteristic
8C3×#5030—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)—21no—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88550C414yescomplemented
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084—21no—
19C2×#6180C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C32×C4×Q8D314yes—
22C2×C3×#216—34no—
23C3×C4×Dic6—34no—
24C3×#835—34no—
25C3×#810—34no—
26C3×#808—34no—
27C3×#803—34no—
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)C2×C412yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084—21no—
37#6229—21no—
38#6229C2×C412yes—
39#6180—21no—
40#6180C2×C412yes—
41C3×#441—91no—
42#5030—34no—
43C2×C32×Q8C2×D314yes—
44C2×C3×Dic6—34no—
45C32×C4⋊C4—24no—
46C32×C4⋊C4C2×D314yes—
47C32×C42—24no—
48C32×C42C2×D314yes—
49C3×#216—316no—
50C3×C4⋊Dic3—34no—
51C3×Dic3⋊C4—64no—
52C3×C4×Dic3—64no—
53C2×C3×C3⋊C8—34no—
54C2×C3×C3⋊C8—34no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)C22×C411yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62#1084—22no—
63#1084—41no—
64C2×C3×Q16—91no—
65C3×C4×Q8—34no—
66C3×C4×Q8—91no—
67C3×C4×Q8C3⋊D311yescharacteristic
68C2×#216—34no—
69C4×Dic6—34no—
70C3×#106—91no—
71C3×#103—91no—
72C3×#103—91no—
73C3×#97—91no—
74#835—34no—
75#810—34no—
76#808—34no—
77#803—34no—
78C32×Q8—24no—
79C32×Q8C4×D318yes—
80C2×C32×C4—24no—
81C2×C32×C4—24no—
82C2×C32×C4#6714yes—
83C2×C32×C4#6714yes—
84C2×C32×C4C22×D314yes—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×Dic6—38no—
88C3×Dic6—64no—
89C3×C3⋊C8—38no—
90C3×C3⋊C8—64no—
91#441—91noSylow
92C2×H(3,𝔽3)C4×D411yescharacteristic
93C2×H(3,𝔽3)#13711yescharacteristic
94C2×H(3,𝔽3)#13711yescharacteristic
95C2×C3×Q8—34no—
96C2×C3×Q8—91no—
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C2×Dic6—34no—
99C3×Q16—94no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—34no—
104C3×C4⋊C4—64no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C42—21no—
109C3×C42—34no—
110C3×C42—64no—
111C3×C42—181no—
112C3×C42C2×C3⋊D311yescharacteristic
113#216—316no—
114C4⋊Dic3—34no—
115Dic3⋊C4—64no—
116C4×Dic3—64no—
117C2×C3⋊C8—34no—
118C2×C3⋊C8—34no—
119C22×C32C2×#6714yes—
120C32×C4—28no—
121C32×C4—24no—
122C32×C4—24no—
123C32×C4—44no—
124C32×C4C2×C4×D314yes—
125C32×C4#23514yes—
126C3×Dic3—68no—
127C3×Dic3—124no—
128C2×Q16—91no—
129C4×Q8—91no—
130C4×Q8#27211yescharacteristic
131#106—91no—
132#103—91no—
133#103—91no—
134#97—91no—
135H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
136C3×Q8—21no—
137C3×Q8—38no—
138C3×Q8—64no—
139C3×Q8—92no—
140C3×Q8—181no—
141C3×Q8C4×C3⋊D312yes—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—34no—
145C2×C3×C4—34no—
146C2×C3×C4—34no—
147C2×C3×C4—64no—
148C2×C3×C4—64no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4#63811yescharacteristic
152C2×C3×C4#63811yescharacteristic
153C2×C3×C4C22×C3⋊D311yescharacteristic
154C2×Dic3—64no—
155C2×Dic3—64no—
156Dic6—38no—
157Dic6—64no—
158C3×C8—92no—
159C3×C8—181no—
160C3⋊C8—38no—
161C3⋊C8—64no—
162C2×C32C4×#6714yes—
163C2×C32#94214yes—
164C2×C32#95114yes—
165C2×Q8—91no—
166C2×Q8C2×#27211yescharacteristic
167Q16—94no—
168C2×C8—91no—
169C2×C8—91no—
170C4⋊C4—21no—
171C4⋊C4—91no—
172C4⋊C4—181no—
173C4⋊C4C2×#27211yescharacteristic
174C42—21no—
175C42—181no—
176C42C2×#27211yescharacteristic
177C22×C3—34no—
178C22×C3C2×#63811yescharacteristic
179C3×C4—22no—
180C3×C4—21no—
181C3×C4—21no—
182C3×C4—34no—
183C3×C4—34no—
184C3×C4—41no—
185C3×C4—64no—
186C3×C4—64no—
187C3×C4—68no—
188C3×C4—124no—
189C3×C4—182no—
190C3×C4—361no—
191C3×C4C2×C4×C3⋊D311yescharacteristic
192C3×C4#382811yescharacteristic
193Dic3—68no—
194Dic3—124no—
195C32#503014yescomplemented
196Q8—21no—
197Q8—92no—
198Q8—181no—
199Q8C4×#27212yes—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C4#623511yescharacteristic
205C2×C4#623511yescharacteristic
206C2×C4C22×#27211yescharacteristic
207C8—92no—
208C8—181no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3C4×#63811yescharacteristic
213C2×C3#6415111yescharacteristic
214C2×C3#6416011yescharacteristic
215C22C2×#623511yescharacteristic
216C4—22no—
217C4—21no—
218C4—21no—
219C4—41no—
220C4—182no—
221C4—361no—
222C4C2×C4×#27211yescharacteristic
223C4#8874011yescharacteristic
224C3—34no—
225C3#1059272911yescharacteristic
226C2C4×#623511yescharacteristic
227C2#1172185911yescharacteristic
228C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4×Q8C3⋊D3
hypocenterH(3,𝔽3)#441
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
15. #88550C4semidirect
135. H(3,𝔽3)#441semidirect
195. C32#5030semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
C2×#885502
C4×#62292
#117205832
#117205322
#117205302
#117205202
C3×#50303