Finite Group Databasebeta

← #13143868

Subgroups — #13143868

Order 1728.

869 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 111 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×#88744C211yesmaximal, characteristic
2#11721733C212yesmaximal, complemented
3H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
4#11720580C211yesmaximal, characteristic, complemented
5#11720524C212yesmaximal, complemented
6C3×#4740—34nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235C411yescharacteristic, complemented
9C2×#6235C2211yescharacteristic
10C22×#1084C411yescharacteristic, complemented
11C22×#1084C2211yescharacteristic
12#88744—22no—
13#88744—22no—
14C2×C4×#272C411yescharacteristic, complemented
15C2×#6229C411yescharacteristic, complemented
16C22⋊C4×H(3,𝔽3)C2212yes—
17#88552C2212yes—
18#88540—22no—
19C4×#1084—22no—
20C2×C3×#237—34no—
21C3×#877—38no—
22C32×#99D314yescomplemented
23C3×#834—34no—
24C3×#806—38no—
25C23×H(3,𝔽3)C2×C411yescharacteristic
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×#272C2×C411yescharacteristic
28D4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31#6235—22no—
32#6235—22no—
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084—22no—
36C2×#1084C2×C411yescharacteristic
37C2×#1084C2×C411yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C2×#1084D412yes—
40C2×#1084D412yes—
41C4×#272—21no—
42C4×#272—21no—
43#6229—22no—
44C3×#410—91no—
45#4740—34no—
46C2×C32×D4C2×D314yes—
47C2×C3×#67—34no—
48C2×C3×#67—34no—
49C22×C3×Dic3—34no—
50C22×C3×Dic3—34no—
51C3×#237—68no—
52C3×#237—68no—
53C2×C3×C4×D3—34no—
54C2×C3×Dic6—34no—
55C32×C22⋊C4C2×D318yes—
56C3×#217—38no—
57C3×Dic3⋊C4—68no—
58C3×C4×Dic3—68no—
59C22×H(3,𝔽3)—21no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)C22×C411yescharacteristic
63C22×H(3,𝔽3)C2×D412yes—
64C2×#272—21no—
65C2×#272—41no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)—42no—
68#1084—21no—
69#1084—42no—
70#1084C22⋊C412yes—
71#1084C22⋊C411yescharacteristic
72#1084C22⋊C411yescharacteristic
73C2×C3×#41—91no—
74C2×#237—34no—
75C3×#117—92no—
76#877—38no—
77C3×#99—34no—
78C3×#99—91no—
79C3×#99—92no—
80C3×#99C3⋊D311yescharacteristic
81#834—34no—
82#806—38no—
83C23×C32C4×D314yes—
84C23×C32C22×D314yes—
85C22×C3×D3—34no—
86C32×D4—28no—
87C2×C32×C4—28no—
88C2×C32×C4C4×D314yes—
89C3×#67—68no—
90C3×#67—68no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—38no—
94C2×C3×Dic3—38no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—68no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101C3×Dic6—68no—
102#410—91noSylow
103C2×H(3,𝔽3)—22no—
104C2×H(3,𝔽3)—21no—
105C2×H(3,𝔽3)—41no—
106C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
107#272—41no—
108C3×#41—182no—
109C3×#41—182no—
110C2×C3×Q8—91no—
111C2×C3×D4—34no—
112C2×C3×D4—91no—
113C2×C3×D4—91no—
114C2×C3×D4C2×C3⋊D311yescharacteristic
115C22×C3×C4—91no—
116C22×C3×C4—91no—
117C22×C3×C4—91no—
118C2×#67—34no—
119C2×#67—34no—
120C22×Dic3—34no—
121C22×Dic3—34no—
122#237—68no—
123#237—68no—
124C2×C4×D3—34no—
125C2×Dic6—34no—
126C3×C4⋊C4—182no—
127C3×C22⋊C4—38no—
128C3×C22⋊C4—92no—
129C3×C22⋊C4C2×C3⋊D312yes—
130C3×C42—182no—
131#217—38no—
132Dic3⋊C4—68no—
133C4×Dic3—68no—
134C22×C32—24no—
135C22×C32—44no—
136C22×C32—44no—
137C22×C32C2×C4×D314yes—
138C22×C32D3×D418yes—
139C2×C3×D3—64no—
140C2×C3×D3—124no—
141C32×C4—24no—
142C32×C4—48no—
143C3×Dic3—38no—
144C3×Dic3—34no—
145C3×Dic3—34no—
146C3×Dic3—64no—
147C3×Dic3—128no—
148C2×#41—91no—
149#117—92no—
150#99—91no—
151#99—92no—
152#99#27211yescharacteristic, complemented
153H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
154C23×C3—34no—
155C23×C3—34no—
156C23×C3—91no—
157C23×C3C4×C3⋊D311yescharacteristic
158C23×C3C22×C3⋊D311yescharacteristic
159C22×D3—34no—
160C3×Q8—182no—
161C3×D4—22no—
162C3×D4—68no—
163C3×D4—182no—
164C3×D4—182no—
165C2×C3×C4—22no—
166C2×C3×C4—34no—
167C2×C3×C4—68no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—91no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4—182no—
178C2×C3×C4C4×C3⋊D311yescharacteristic
179#67—68no—
180#67—68no—
181C2×Dic3—38no—
182C2×Dic3—38no—
183C2×Dic3—34no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C2×Dic3—64no—
187C2×Dic3—64no—
188C2×Dic3—68no—
189C4×D3—64no—
190C4×D3—64no—
191Dic6—68no—
192C2×C32—28no—
193C2×C32—24no—
194C2×C32—44no—
195C2×C32D3×C22⋊C414yes—
196C3×D3—124no—
197#41—182no—
198#41—182no—
199C2×Q8—91no—
200C2×D4—91no—
201C2×D4—91no—
202C2×D4C2×#27211yescharacteristic
203C22×C4—91no—
204C22×C4—91no—
205C22×C4—91no—
206C4⋊C4—182no—
207C22⋊C4—92no—
208C22⋊C4C2×#27212yes—
209C42—182no—
210C22×C3—21no—
211C22×C3—34no—
212C22×C3—38no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—64no—
216C22×C3—124no—
217C22×C3—124no—
218C22×C3—181no—
219C22×C3—361no—
220C22×C3C2×C4×C3⋊D311yescharacteristic
221C22×C3D4×C3⋊D312yes—
222C2×D3—64no—
223C2×D3—124no—
224C3×C4—21no—
225C3×C4—42no—
226C3×C4—64no—
227C3×C4—91no—
228C3×C4—91no—
229C3×C4—92no—
230C3×C4—128no—
231C3×C4—181no—
232C3×C4—362no—
233Dic3—38no—
234Dic3—34no—
235Dic3—34no—
236Dic3—64no—
237Dic3—128no—
238C32#474014yescomplemented
239C23—91no—
240C23C4×#27211yescharacteristic
241C23C22×#27211yescharacteristic
242Q8—182no—
243D4—22no—
244D4—182no—
245D4—182no—
246C2×C4—22no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—92no—
250C2×C4—92no—
251C2×C4—91no—
252C2×C4—181no—
253C2×C4—181no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4—182no—
257C2×C4C4×#27211yescharacteristic
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—34no—
261C2×C3—41no—
262C2×C3—68no—
263C2×C3—64no—
264C2×C3—124no—
265C2×C3—361no—
266C2×C3C22⋊C4×C3⋊D311yescharacteristic
267D3—124no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—181no—
272C22—361no—
273C22C2×C4×#27211yescharacteristic
274C22D4×#27212yes—
275C4—21no—
276C4—42no—
277C4—91no—
278C4—91no—
279C4—92no—
280C4—181no—
281C4—362no—
282C3—34no—
283C3#1059243911yescharacteristic
284C2—22no—
285C2—21no—
286C2—41no—
287C2—361no—
288C2C22⋊C4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3C22⋊C4×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingH(3,𝔽3)×#99C2
prime radicalC2×#88744C2
cosocleC23×H(3,𝔽3)C23
hypercenterC3×#99C3⋊D3
hypocenterH(3,𝔽3)#410
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. #11721733C2semidirect
3. H(3,𝔽3)×#99C2semidirect
4. #11720580C2semidirect
5. #11720524C2semidirect
8. C2×#6235C4semidirect
10. C22×#1084C4semidirect
14. C2×C4×#272C4semidirect
15. C2×#6229C4semidirect
22. C32×#99D3semidirect
152. #99#272semidirect
153. H(3,𝔽3)#410semidirect
238. C32#4740semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
#117217332
H(3,𝔽3)×#992
#117205802
#117205242
C3×#47403