Finite Group Databasebeta

← #13143949

Subgroups — #13143949

Order 1728.

682 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#11721848C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic, complemented
3C2×C8×#272C211yesmaximal, characteristic, complemented
4#11721765C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4778—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×C4×#272C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88552—21no—
16#88546—22no—
17#88542—21no—
18C4⋊#1084—21no—
19C4⋊#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C8×#272—22no—
22C3×#938—34no—
23C2×C3×#214—34no—
24C2×C3×D3×C8—34no—
25C3×#897—34no—
26C32×#102D314yescomplemented
27C3×#816—34no—
28C3×#808—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272D411yescharacteristic
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—42no—
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41#6229—21no—
42#6229—21no—
43C8×H(3,𝔽3)—21no—
44#6180D412yescomplemented
45C3×#465—91no—
46#4778—34no—
47C2×C32×D4C2×D314yes—
48C2×C3×#67—64no—
49C2×C3×C4×D3—34no—
50C2×C3×Dic6—34no—
51C2×C32×C8C2×D314yes—
52C32×C4⋊C4C2×D314yes—
53C3×#217—64no—
54C3×#214—68no—
55C3×(C2×D3)⋊C4—64no—
56C3×C4⋊Dic3—34no—
57C3×C4⋊Dic3—64no—
58C2×C3×C3⋊C8—34no—
59C3×D3×C8—68no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272SD1612yescomplemented
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70#1084—21no—
71#1084—41no—
72#1084—41no—
73C2×C3×SD16—91no—
74C22×C3×C8—91no—
75C3×C22⋊Q8—91no—
76C3×#121—91no—
77#938—34no—
78C2×#214—34no—
79C2×D3×C8—34no—
80#897—34no—
81C3×#106—91no—
82C3×#103—91no—
83C3×#102—34no—
84C3×#102C3⋊D311yescharacteristic
85#816—34no—
86#808—34no—
87C23×C32—24no—
88C22×C3×D3—34no—
89C32×D4—24no—
90C32×D4—24no—
91C2×C32×C4—24no—
92C2×C32×C4C22×D314yes—
93C3×#67—128no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—64no—
97C3×C4×D3—64no—
98C3×C4×D3—64no—
99C3×Dic6—64no—
100C3×Dic6—64no—
101C32×C8—24no—
102C3×C3⋊C8—38no—
103#465—91noSylow
104C2×H(3,𝔽3)—81no—
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)C2×SD1611yescharacteristic
107C2×H(3,𝔽3)#13511yescharacteristic
108#272—22no—
109C2×C3×Q8—91no—
110C2×C3×D4—34no—
111C2×C3×D4—181no—
112C2×C3×D4C2×C3⋊D311yescharacteristic
113C22×C3×C4—91no—
114C2×#67—64no—
115C2×C4×D3—34no—
116C2×Dic6—34no—
117C3×SD16—182no—
118C2×C3×C8—34no—
119C2×C3×C8—91no—
120C2×C3×C8—182no—
121C2×C3×C8C2×C3⋊D311yescharacteristic
122C3×C4⋊C4—34no—
123C3×C4⋊C4—91no—
124C3×C4⋊C4—181no—
125C3×C4⋊C4C2×C3⋊D311yescharacteristic
126C3×C22⋊C4—181no—
127C3×C22⋊C4—181no—
128#217—64no—
129#214—68no—
130(C2×D3)⋊C4—64no—
131C4⋊Dic3—34no—
132C4⋊Dic3—64no—
133C2×C3⋊C8—34no—
134D3×C8—68no—
135C22×C32—44no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32D3×D414yes—
139C2×C3×D3—38no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C32×C4—44no—
143C32×C4#23514yes—
144C32×C4D3×D414yes—
145C3×Dic3—64no—
146C3×Dic3—124no—
147C3×Dic3—124no—
148C2×SD16—91no—
149C22×C8—91no—
150C22⋊Q8—91no—
151#121—91no—
152#106—91no—
153#103—91no—
154#102#27211yescharacteristic, complemented
155H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—64no—
158C23×C3—91no—
159C22×D3—34no—
160C3×Q8—181no—
161C3×Q8—181no—
162C3×D4—21no—
163C3×D4—21no—
164C3×D4—64no—
165C3×D4—64no—
166C3×D4—362no—
167C2×C3×C4—21no—
168C2×C3×C4—34no—
169C2×C3×C4—64no—
170C2×C3×C4—91no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4C22×C3⋊D311yescharacteristic
176#67—128no—
177C2×Dic3—34no—
178C2×Dic3—64no—
179C2×Dic3—64no—
180C4×D3—64no—
181C4×D3—64no—
182Dic6—64no—
183Dic6—64no—
184C3×C8—21no—
185C3×C8—64no—
186C3×C8—92no—
187C3⋊C8—38no—
188C2×C32—84no—
189C2×C32#88414yes—
190C2×C32#91214yes—
191C2×C32D3×SD1614yes—
192C3×D3—68no—
193C2×Q8—91no—
194C2×D4—181no—
195C2×D4C2×#27211yescharacteristic
196C22×C4—91no—
197SD16—182no—
198C2×C8—91no—
199C2×C8—182no—
200C2×C8C2×#27211yescharacteristic
201C4⋊C4—91no—
202C4⋊C4—181no—
203C4⋊C4C2×#27211yescharacteristic
204C22⋊C4—181no—
205C22⋊C4—181no—
206C22×C3—34no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—92no—
211C22×C3—124no—
212C22×C3—124no—
213C22×C3—124no—
214C22×C3—181no—
215C22×C3—181no—
216C22×C3D4×C3⋊D311yescharacteristic
217C2×D3—38no—
218C2×D3—64no—
219C2×D3—64no—
220C3×C4—34no—
221C3×C4—34no—
222C3×C4—41no—
223C3×C4—124no—
224C3×C4—181no—
225C3×C4—361no—
226C3×C4—361no—
227C3×C4#382811yescharacteristic
228C3×C4D4×C3⋊D311yescharacteristic
229Dic3—64no—
230Dic3—124no—
231Dic3—124no—
232C32#477814yescomplemented
233C23—21no—
234C23—91no—
235Q8—181no—
236Q8—181no—
237D4—21no—
238D4—21no—
239D4—362no—
240C2×C4—21no—
241C2×C4—91no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4C22×#27211yescharacteristic
247C8—21no—
248C8—92no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—81no—
253C2×C3—182no—
254C2×C3—244no—
255C2×C3#6409311yescharacteristic
256C2×C3#6412111yescharacteristic
257C2×C3SD16×C3⋊D311yescharacteristic
258D3—68no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—92no—
263C22—181no—
264C22—181no—
265C22D4×#27211yescharacteristic
266C4—41no—
267C4—181no—
268C4—361no—
269C4—361no—
270C4#8874011yescharacteristic
271C4D4×#27211yescharacteristic
272C3—34no—
273C3#1059247711yescharacteristic
274C2—81no—
275C2—182no—
276C2#1172174011yescharacteristic
277C2#1172179411yescharacteristic
278C2SD16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radical#11721848C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#465
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. C2×C8×#272C2semidirect
4. #11721765C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720549C2semidirect
7. #11720530C2semidirect
26. C32×#102D3semidirect
44. #6180D4semidirect
66. C2×#272SD16semidirect
154. #102#272semidirect
155. H(3,𝔽3)#465semidirect
232. C32#4778semidirect

Maximal subgroups

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

subgroupindex
#117218482
C2×#885462
C2×C8×#2722
#117217652
H(3,𝔽3)×#1022
#117205492
#117205302
C3×#47783