Finite Group Databasebeta

← #13153898

Subgroups — #13153898

Order 1728.

1485 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1485 conjugacy classes of subgroups — too many to draw. Only its 261 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×#88740C211yesmaximal, characteristic
2C22×#6229C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
4#11721765C214yesmaximal, complemented
5#11721742C214yesmaximal, complemented
6#11721721C212yesmaximal, complemented
7#11721715C212yesmaximal, complemented
8C3×#5526—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235C2212yes—
11C22×#1084C2212yes—
12#88740—24no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2212yes—
15C2×#6229—24no—
16C2×#6229C2212yes—
17C2×#6229C2212yes—
18C2×#6229C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C22⋊C4×H(3,𝔽3)C2212yes—
21C42×H(3,𝔽3)C2212yes—
22#88542C2218yes—
23C4⋊#1084C2218yes—
24C2×C3×#235—34no—
25C22×C3×Dic6—34no—
26C32×#117D314yescomplemented
27C3×#897—316no—
28C3×#886—316no—
29C3×#875—38no—
30C3×#869—38no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272C2312yes—
33C2×C4×H(3,𝔽3)D414yes—
34C2×C4×H(3,𝔽3)C2314yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#6235—24no—
38C2×#1084—24no—
39C2×#1084C2314yes—
40C2×#1084C2312yes—
41C4⋊#272—22no—
42C4×#272—24no—
43#6229—24no—
44#6229—24no—
45#6229—22no—
46C3×#536—91no—
47#5526—34no—
48C22×C32×C4C2×D314yes—
49C2×C3×#67—38no—
50C22×C3×Dic3—38no—
51C3×#235—616no—
52C2×C3×D12—34no—
53C2×C3×C4×D3—38no—
54C2×C3×Dic6—38no—
55C2×C3×Dic6—38no—
56C2×C3×Dic6—34no—
57C2×C3×Dic6—616no—
58C32×C4⋊C4C2×D318yes—
59C32×C22⋊C4C2×D318yes—
60C32×C42C2×D318yes—
61C3×(C2×D3)⋊C4—332no—
62C3×C4⋊Dic3—332no—
63C22×H(3,𝔽3)—22no—
64C22×H(3,𝔽3)C2×D412yes—
65C22×H(3,𝔽3)C2411yescharacteristic
66C2×#272—22no—
67C2×#272—24no—
68C4×H(3,𝔽3)—24no—
69C4×H(3,𝔽3)C2×D412yes—
70C4×H(3,𝔽3)C2×D412yes—
71#1084—24no—
72#1084—22no—
73C2×C3×#41—91no—
74C22×C3×Q8—91no—
75C2×#235—34no—
76C22×Dic6—34no—
77C3×C4⋊Q8—92no—
78C3×#124—92no—
79C3×#123—94no—
80C3×C22⋊Q8—94no—
81C3×#117—34no—
82C3×#117C3⋊D311yescharacteristic
83#897—316no—
84#886—316no—
85#875—38no—
86#869—38no—
87C23×C32C22×D314yes—
88C22×C3×D3—38no—
89C2×C32×C4D12116yes—
90C2×C32×C4C22×D3116yes—
91C2×C32×C4C22×D314yes—
92C2×C32×C4C22×D314yes—
93C3×#67—616no—
94C2×C3×Dic3—38no—
95C2×C3×Dic3—316no—
96C2×C3×Dic3—616no—
97C3×D12—68no—
98C3×C4×D3—616no—
99C3×Dic6—616no—
100C3×Dic6—616no—
101C3×Dic6—68no—
102#536—91noSylow
103C2×H(3,𝔽3)—22no—
104C2×H(3,𝔽3)C22×D411yescharacteristic
105C2×H(3,𝔽3)2−1+412yes—
106#272—42no—
107C3×#41—184no—
108C2×C3×Q8—92no—
109C2×C3×Q8—92no—
110C2×C3×Q8—91no—
111C2×C3×Q8—184no—
112C2×C3×D4—92no—
113C2×C3×D4—91no—
114C22×C3×C4—34no—
115C22×C3×C4—92no—
116C22×C3×C4—92no—
117C22×C3×C4C2×C3⋊D311yescharacteristic
118C2×#67—38no—
119C22×Dic3—38no—
120#235—616no—
121C2×D12—34no—
122C2×C4×D3—38no—
123C2×Dic6—34no—
124C2×Dic6—38no—
125C2×Dic6—38no—
126C2×Dic6—616no—
127C3×C4⋊C4—38no—
128C3×C4⋊C4—98no—
129C3×C4⋊C4C2×C3⋊D312yes—
130C3×C22⋊C4—38no—
131C3×C22⋊C4—98no—
132C3×C22⋊C4C2×C3⋊D312yes—
133C3×C42—38no—
134C3×C42C2×C3⋊D312yes—
135(C2×D3)⋊C4—332no—
136C4⋊Dic3—332no—
137C22×C32—28no—
138C22×C32C2×D1218yes—
139C22×C32C23×D314yes—
140C2×C3×D3—68no—
141C2×C3×D3—616no—
142C32×C4—216no—
143C32×C4C2×D1218yes—
144C32×C4C2×D1218yes—
145C3×Dic3—616no—
146C3×Dic3—68no—
147C2×#41—91no—
148C22×Q8—91no—
149C4⋊Q8—92no—
150#124—92no—
151#123—94no—
152C22⋊Q8—94no—
153#117#27211yescharacteristic, complemented
154H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
155C23×C3—34no—
156C23×C3—92no—
157C23×C3C22×C3⋊D311yescharacteristic
158C22×D3—38no—
159C3×Q8—184no—
160C3×Q8—184no—
161C3×Q8—182no—
162C3×D4—184no—
163C3×D4—182no—
164C2×C3×C4—316no—
165C2×C3×C4—34no—
166C2×C3×C4—34no—
167C2×C3×C4—316no—
168C2×C3×C4—94no—
169C2×C3×C4—92no—
170C2×C3×C4—184no—
171C2×C3×C4—184no—
172C2×C3×C4C4⋊(C3⋊D3)14yes—
173C2×C3×C4C22×C3⋊D314yes—
174C2×C3×C4C22×C3⋊D311yescharacteristic
175C2×C3×C4C22×C3⋊D311yescharacteristic
176#67—616no—
177C2×Dic3—38no—
178C2×Dic3—316no—
179C2×Dic3—616no—
180D12—68no—
181C4×D3—616no—
182Dic6—616no—
183Dic6—616no—
184Dic6—68no—
185C2×C32—28no—
186C2×C32C22×D1214yes—
187C2×C32#101018yes—
188C3×D3—128no—
189#41—184no—
190C2×Q8—92no—
191C2×Q8—92no—
192C2×Q8—91no—
193C2×Q8—184no—
194C2×D4—92no—
195C2×D4—91no—
196C22×C4—92no—
197C22×C4—92no—
198C22×C4C2×#27211yescharacteristic
199C4⋊C4—98no—
200C4⋊C4C2×#27212yes—
201C22⋊C4—98no—
202C22⋊C4C2×#27212yes—
203C42C2×#27212yes—
204C22×C3—22no—
205C22×C3—34no—
206C22×C3—38no—
207C22×C3—68no—
208C22×C3—182no—
209C22×C3—184no—
210C22×C3C2×C4⋊(C3⋊D3)12yes—
211C22×C3C23×C3⋊D311yescharacteristic
212C2×D3—68no—
213C2×D3—616no—
214C3×C4—24no—
215C3×C4—38no—
216C3×C4—38no—
217C3×C4—616no—
218C3×C4—182no—
219C3×C4—184no—
220C3×C4C2×C4⋊(C3⋊D3)12yes—
221C3×C4C2×C4⋊(C3⋊D3)12yes—
222Dic3—616no—
223Dic3—68no—
224C32#552614yescomplemented
225C23—92no—
226C23C22×#27211yescharacteristic
227Q8—184no—
228Q8—184no—
229Q8—182no—
230D4—184no—
231D4—182no—
232C2×C4—94no—
233C2×C4—92no—
234C2×C4—184no—
235C2×C4—184no—
236C2×C4C4⋊#27214yes—
237C2×C4C22×#27214yes—
238C2×C4C22×#27211yescharacteristic
239C2×C4C22×#27211yescharacteristic
240C2×C3—22no—
241C2×C3—34no—
242C2×C3—38no—
243C2×C3—68no—
244C2×C3—362no—
245C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
246C2×C3#6436712yes—
247D3—128no—
248C22—22no—
249C22—182no—
250C22—184no—
251C22C2×C4⋊#27212yes—
252C22C23×#27211yescharacteristic
253C4—24no—
254C4—182no—
255C4—184no—
256C4C2×C4⋊#27212yes—
257C4C2×C4⋊#27212yes—
258C3—34no—
259C3#1059599711yescharacteristic
260C2—22no—
261C2—362no—
262C2C22×C4⋊#27211yescharacteristic
263C2#1172267612yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
prime radicalC2×#88740C2
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#536
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C22×#6229C2semidirect
3. H(3,𝔽3)×#117C2semidirect
4. #11721765C2semidirect
5. #11721742C2semidirect
6. #11721721C2semidirect
7. #11721715C2semidirect
26. C32×#117D3semidirect
153. #117#272semidirect
154. H(3,𝔽3)#536semidirect
224. C32#5526semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C22×#62292
H(3,𝔽3)×#1172
#117217652
#117217422
#117217212
#117217152
C3×#55263