Finite Group Databasebeta

← #103×#272 #13144004

Subgroups — #103×#272 #13144004

Order 1728.

891 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 133 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×Q8×#272C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic
3C4⋊C4×#272C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×D3×#103—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10Q8×#272—22no—
11Q8×#272C414yescomplemented
12C2×C4×#272—21no—
13C2×C4×#272—21no—
14C2×C4×#272C2211yescharacteristic
15C2×#6229—21no—
16C2×#6229C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21C2×#6180C2211yescharacteristic
22C8×#272—22no—
23C2×C3×D3×Q8—34no—
24C2×C3×D3×C8—34no—
25C3×D3×C4⋊C4—34no—
26C32×#103D314yescomplemented
27C3×#835—34no—
28C3×#816—34no—
29C3×#810—34no—
30C22×#272D411yescharacteristic
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)C2×C412yescomplemented
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4×#272—24no—
40C4×#272—42no—
41C4×#272C2×C412yes—
42C4×#272D412yes—
43#6229—21no—
44#6229—24no—
45#6229C2×C412yescomplemented
46C8×H(3,𝔽3)—21no—
47#6180—21no—
48C2×C3×#103—91no—
49D3×#103—34no—
50C2×C32×Q8C2×D314yes—
51C3×D3×Q8—316no—
52C3×D3×Q8—68no—
53C2×C3×C4×D3—34no—
54C2×C3×C4×D3—64no—
55C2×C3×C4×D3—64no—
56C2×C3×Dic6—34no—
57C2×C3×Dic6—64no—
58C2×C32×C8C2×D314yes—
59C32×C4⋊C4C2×D314yes—
60C3×C4⋊Dic3—34no—
61C3×Dic3⋊C4—64no—
62C2×C3×C3⋊C8—34no—
63C3×D3×C8—68no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272C22⋊C412yes—
66C2×#272SD1612yes—
67C2×#272Q1612yes—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)C22×C411yescharacteristic
71C4×H(3,𝔽3)C2×D411yescharacteristic
72#1084—22no—
73#1084—41no—
74#1084C22⋊C412yes—
75C22×C3×Q8—91no—
76C2×D3×Q8—34no—
77C22×C3×C8—91no—
78C2×C3×C4⋊C4—91no—
79C2×D3×C8—34no—
80D3×C4⋊C4—34no—
81C3×#103—34no—
82C3×#103—91no—
83C3×#103—91no—
84C3×#103—91no—
85C3×#103C3⋊D311yescharacteristic
86#835—34no—
87#816—34no—
88#810—34no—
89C22×C3×D3—34no—
90C32×Q8—24no—
91C32×Q8C4×D318yescomplemented
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4C22×D314yes—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—64no—
98C3×C4×D3—38no—
99C3×C4×D3—38no—
100C3×C4×D3—616no—
101C3×C4×D3—128no—
102C3×Dic6—38no—
103C3×Dic6—616no—
104C3×Dic6—64no—
105C32×C8—24no—
106C3×C3⋊C8—64no—
107C2×#103—91noSylow
108C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
109C2×H(3,𝔽3)C2×SD1611yescharacteristic
110C2×H(3,𝔽3)C2×Q1611yescharacteristic
111#272#10314yesdirect product factor
112C2×C3×Q8—34no—
113C2×C3×Q8—94no—
114C2×C3×Q8—91no—
115C2×C3×Q8—181no—
116C2×C3×Q8—182no—
117C2×C3×Q8C2×C3⋊D311yescharacteristic
118C22×C3×C4—91no—
119C22×C3×C4—181no—
120C22×C3×C4—181no—
121D3×Q8—316no—
122D3×Q8—68no—
123C2×C4×D3—34no—
124C2×C4×D3—64no—
125C2×C4×D3—64no—
126C2×Dic6—34no—
127C2×Dic6—64no—
128C2×C3×C8—34no—
129C2×C3×C8—91no—
130C2×C3×C8—182no—
131C2×C3×C8C2×C3⋊D311yescharacteristic
132C3×C4⋊C4—34no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—181no—
135C3×C4⋊C4C2×C3⋊D311yescharacteristic
136C4⋊Dic3—34no—
137Dic3⋊C4—64no—
138C2×C3⋊C8—34no—
139D3×C8—68no—
140C22×C32D3×D414yes—
141C2×C3×D3—38no—
142C2×C3×D3—38no—
143C2×C3×D3—38no—
144C32×C4—28no—
145C32×C4—44no—
146C32×C4C2×C4×D314yes—
147C32×C4D3×D414yes—
148C3×Dic3—38no—
149C3×Dic3—68no—
150C3×Dic3—124no—
151C22×Q8—91no—
152C22×C8—91no—
153C2×C4⋊C4—91no—
154#103—91no—
155#103—91no—
156#103—91no—
157#103#27211yescharacteristic, direct product factor
158H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
159C23×C3—91no—
160C22×D3—34no—
161C3×Q8—21no—
162C3×Q8—38no—
163C3×Q8—64no—
164C3×Q8—92no—
165C3×Q8—184no—
166C3×Q8—181no—
167C3×Q8C4×C3⋊D312yes—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—34no—
171C2×C3×C4—64no—
172C2×C3×C4—64no—
173C2×C3×C4—91no—
174C2×C3×C4—92no—
175C2×C3×C4—92no—
176C2×C3×C4—184no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—362no—
180C2×C3×C4C22×C3⋊D311yescharacteristic
181C2×Dic3—34no—
182C2×Dic3—64no—
183C2×Dic3—64no—
184C4×D3—38no—
185C4×D3—38no—
186C4×D3—616no—
187C4×D3—128no—
188Dic6—38no—
189Dic6—616no—
190Dic6—64no—
191C3×C8—21no—
192C3×C8—64no—
193C3×C8—181no—
194C3⋊C8—64no—
195C2×C32D3×C22⋊C414yes—
196C2×C32D3×SD1614yes—
197C2×C32D3×Q1614yes—
198C3×D3—316no—
199C2×Q8—94no—
200C2×Q8—91no—
201C2×Q8—181no—
202C2×Q8—182no—
203C2×Q8C2×#27211yescharacteristic
204C22×C4—91no—
205C22×C4—181no—
206C22×C4—181no—
207C2×C8—91no—
208C2×C8—182no—
209C2×C8C2×#27211yescharacteristic
210C4⋊C4—91no—
211C4⋊C4—181no—
212C4⋊C4C2×#27211yescharacteristic
213C22×C3—34no—
214C22×C3—92no—
215C22×C3—92no—
216C22×C3—92no—
217C22×C3D4×C3⋊D311yescharacteristic
218C2×D3—38no—
219C2×D3—38no—
220C2×D3—38no—
221C3×C4—22no—
222C3×C4—34no—
223C3×C4—34no—
224C3×C4—41no—
225C3×C4—68no—
226C3×C4—92no—
227C3×C4—124no—
228C3×C4—182no—
229C3×C4—361no—
230C3×C4C2×C4×C3⋊D311yescharacteristic
231C3×C4D4×C3⋊D311yescharacteristic
232Dic3—38no—
233Dic3—68no—
234Dic3—124no—
235C32D3×#10314yescomplemented
236C23—91no—
237Q8—21no—
238Q8—92no—
239Q8—184no—
240Q8—181no—
241Q8C4×#27212yescomplemented
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—91no—
245C2×C4—92no—
246C2×C4—92no—
247C2×C4—184no—
248C2×C4—181no—
249C2×C4—181no—
250C2×C4—362no—
251C2×C4C22×#27211yescharacteristic
252C8—21no—
253C8—181no—
254C2×C3—34no—
255C2×C3—34no—
256C2×C3—34no—
257C2×C3—94no—
258C2×C3C22⋊C4×C3⋊D311yescharacteristic
259C2×C3SD16×C3⋊D311yescharacteristic
260C2×C3Q16×C3⋊D311yescharacteristic
261D3—316no—
262C22—92no—
263C22—92no—
264C22—92no—
265C22D4×#27211yescharacteristic
266C4—22no—
267C4—41no—
268C4—92no—
269C4—182no—
270C4—361no—
271C4C2×C4×#27211yescharacteristic
272C4D4×#27211yescharacteristic
273C3—34no—
274C3C3⋊D3×#10311yescharacteristic
275C2—94no—
276C2C22⋊C4×#27211yescharacteristic
277C2SD16×#27211yescharacteristic
278C2Q16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)C2×#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
4. H(3,𝔽3)×#103C2semidirect
5. #11720583C2semidirect
6. #11720549C2semidirect
7. #11720532C2semidirect
11. Q8×#272C4semidirect
26. C32×#103D3semidirect
32. Q8×H(3,𝔽3)C2×C4semidirect
45. #6229C2×C4semidirect
91. C32×Q8C4×D3semidirect
111. #272#103direct (N × H)
157. #103#272direct (N × H)
158. H(3,𝔽3)C2×#103semidirect
235. C32D3×#103semidirect
241. Q8C4×#272semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×C8×#2722
C4⋊C4×#2722
H(3,𝔽3)×#1032
#117205832
#117205492
#117205322
C3×D3×#1033