Finite Group Databasebeta

← #13144444

Subgroups — #13144444

Order 1728.

616 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 123 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, complemented
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88549C211yesmaximal, characteristic
4#11721721C211yesmaximal, characteristic
5#11720532C211yesmaximal, characteristic, complemented
6#11720531C211yesmaximal, characteristic
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#5028—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#88550—22no—
17#88549—22no—
18#88542—21no—
19#88542—21no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×C4×Q8D314yescomplemented
23C2×C3×#216—34no—
24C2×C3×#215—34no—
25C3×#875—34no—
26C3×#810—34no—
27C3×#809—34no—
28C3×C4⋊(C3⋊C8)—34no—
29C22×#272—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)D412yes—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C4⋊#272—21no—
39C4⋊#272—21no—
40#6229—21no—
41#6229—21no—
42#6180—21no—
43#6180—21no—
44C3×#464—91no—
45#5028—34no—
46C2×C32×Q8C2×D314yes—
47C2×C3×D12—34no—
48C2×C3×Dic6—34no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4C2×D314yes—
51C32×C42—24no—
52C32×C42C2×D314yes—
53C3×#216—68no—
54C3×#215—68no—
55C3×(C2×D3)⋊C4—64no—
56C3×(C2×D3)⋊C4—64no—
57C2×C3×C3⋊C8—34no—
58C2×C3×C3⋊C8—34no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—22no—
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)#4111yescharacteristic
69#1084—41no—
70C2×C3×Q16—91no—
71C2×C3×SD16—91no—
72C3×#124—91no—
73C3×C4×Q8—34no—
74C3×C4×Q8C3⋊D311yescharacteristic
75C2×#216—34no—
76C2×#215—34no—
77#875—34no—
78C3×C4⋊C8—91no—
79C3×#103—91no—
80C3×#102—91no—
81#810—34no—
82#809—34no—
83C4⋊(C3⋊C8)—34no—
84C22×C3×D3—64no—
85C32×Q8—24no—
86C32×Q8D1218yes—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4#6714yes—
90C2×C32×C4#6714yes—
91C2×C32×C4C22×D314yes—
92C2×C3×Dic3—64no—
93C3×D12—64no—
94C3×D12—64no—
95C3×Dic6—64no—
96C3×Dic6—64no—
97C3×C3⋊C8—64no—
98C3×C3⋊C8—64no—
99#464—91noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)#13511yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103#272—81no—
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×D4—91no—
108C2×D12—34no—
109C2×Dic6—34no—
110C3×Q16—182no—
111C3×SD16—182no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—34no—
116C3×C4⋊C4—64no—
117C3×C4⋊C4C2×C3⋊D311yescharacteristic
118C3×C22⋊C4—181no—
119C3×C22⋊C4—181no—
120C3×C42—21no—
121C3×C42—34no—
122C3×C42—64no—
123C3×C42C2×C3⋊D311yescharacteristic
124#216—68no—
125#215—68no—
126(C2×D3)⋊C4—64no—
127(C2×D3)⋊C4—64no—
128C2×C3⋊C8—34no—
129C2×C3⋊C8—34no—
130C22×C32C2×#6714yes—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—124no—
134C32×C4—28no—
135C32×C4—24no—
136C32×C4—24no—
137C32×C4—44no—
138C32×C4C2×D1214yes—
139C32×C4#23514yes—
140C3×Dic3—124no—
141C2×Q16—91no—
142C2×SD16—91no—
143#124—91no—
144C4×Q8#27211yescharacteristic, complemented
145C4⋊C8—91no—
146#103—91no—
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—181no—
150C22×D3—64no—
151C3×Q8—21no—
152C3×Q8—38no—
153C3×Q8—64no—
154C3×Q8—181no—
155C3×Q8—181no—
156C3×Q8C4⋊(C3⋊D3)12yes—
157C3×D4—181no—
158C3×D4—181no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—34no—
164C2×C3×C4—64no—
165C2×C3×C4—64no—
166C2×C3×C4—181no—
167C2×C3×C4#63811yescharacteristic
168C2×C3×C4#63811yescharacteristic
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—64no—
171D12—64no—
172D12—64no—
173Dic6—64no—
174Dic6—64no—
175C3×C8—181no—
176C3×C8—181no—
177C3⋊C8—64no—
178C3⋊C8—64no—
179C2×C32#93014yes—
180C2×C32#94214yes—
181C2×C32#95014yes—
182C3×D3—244no—
183C2×Q8—91no—
184C2×Q8C2×#27211yescharacteristic
185C2×D4—91no—
186Q16—182no—
187SD16—182no—
188C2×C8—91no—
189C2×C8—91no—
190C4⋊C4—21no—
191C4⋊C4C2×#27211yescharacteristic
192C22⋊C4—181no—
193C22⋊C4—181no—
194C42—21no—
195C42C2×#27211yescharacteristic
196C22×C3—34no—
197C22×C3—361no—
198C22×C3—361no—
199C22×C3—361no—
200C22×C3C2×#63811yescharacteristic
201C2×D3—124no—
202C2×D3—124no—
203C2×D3—124no—
204C3×C4—22no—
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—34no—
208C3×C4—34no—
209C3×C4—41no—
210C3×C4—64no—
211C3×C4—64no—
212C3×C4—68no—
213C3×C4—124no—
214C3×C4—361no—
215C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
216C3×C4#382811yescharacteristic
217Dic3—124no—
218C32#502814yescomplemented
219C23—181no—
220Q8—21no—
221Q8—181no—
222Q8—181no—
223Q8C4⋊#27212yes—
224D4—181no—
225D4—181no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—181no—
229C2×C4#623511yescharacteristic
230C2×C4#623511yescharacteristic
231C2×C4C22×#27211yescharacteristic
232C8—181no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—721no—
238C2×C3#6413911yescharacteristic
239C2×C3#6415111yescharacteristic
240C2×C3#6415911yescharacteristic
241D3—244no—
242C22—361no—
243C22—361no—
244C22—361no—
245C22C2×#623511yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—41no—
250C4—361no—
251C4C2×C4⋊#27211yescharacteristic
252C4#8874011yescharacteristic
253C3—34no—
254C3#1059272711yescharacteristic
255C2—721no—
256C2#1172183011yescharacteristic
257C2#1172185911yescharacteristic
258C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4×Q8C3⋊D3
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×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
1. C4×Q8×H(3,𝔽3)C2semidirect
2. C2×#88550C2semidirect
5. #11720532C2semidirect
7. C4⋊#6180C2semidirect
22. C32×C4×Q8D3semidirect
144. C4×Q8#272semidirect
148. H(3,𝔽3)#464semidirect
218. C32#5028semidirect

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
C2×#885492
#117217212
#117205322
#117205312
C4⋊#61802
C3×#50283