Finite Group Databasebeta

← #13143930

Subgroups — #13143930

Order 1728.

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

normal subgroups only

This group has 649 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#11721850C211yesmaximal, characteristic
2#11721756C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4760—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#6229—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#940—34no—
21C3×#888—34no—
22C32×#102D314yescomplemented
23C3×#832—34no—
24C3×#821—34no—
25C3×C8×Dic3—34no—
26C3×#809—34no—
27C23×H(3,𝔽3)—21no—
28C22×#272—21no—
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#6235—42no—
34#6235—42no—
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272—21no—
40#6229—42no—
41C8×H(3,𝔽3)—21no—
42#6180—21no—
43C3×#486—91no—
44#4760—34no—
45C2×C32×D4C2×D314yes—
46C2×C3×#67—64no—
47C2×C3×#67—64no—
48C2×C3×D12—34no—
49C2×C3×Dic6—64no—
50C2×C32×C8C2×D314yes—
51C32×C4⋊C4C2×D314yes—
52C3×C4⋊Dic3—34no—
53C3×Dic3⋊C4—64no—
54C3×C4×Dic3—34no—
55C2×C3×C3⋊C8—34no—
56C22×H(3,𝔽3)—41no—
57C22×H(3,𝔽3)—41no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—41no—
64C4×H(3,𝔽3)#4111yescharacteristic
65C4×H(3,𝔽3)#4111yescharacteristic
66#1084—41no—
67#1084D812yes—
68#1084SD1612yes—
69C3×C4⋊Q8—91no—
70C3×#127—91no—
71#940—34no—
72#888—34no—
73C3×#102—34no—
74C3×#102—91no—
75C3×#102—91no—
76C3×#102—91no—
77C3×#102C3⋊D311yescharacteristic
78C3×C4×C8—91no—
79#832—34no—
80#821—34no—
81C8×Dic3—34no—
82#809—34no—
83C23×C32—24no—
84C22×C3×D3—64no—
85C32×D4—24no—
86C32×D4—24no—
87C2×C32×C4—24no—
88C2×C32×C4C22×D314yes—
89C3×#67—128no—
90C3×#67—128no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—64no—
94C3×D12—64no—
95C3×D12—64no—
96C3×Dic6—128no—
97C32×C8—24no—
98C3×C3⋊C8—64no—
99#486—91noSylow
100C2×H(3,𝔽3)—81no—
101C2×H(3,𝔽3)#12411yescharacteristic
102C2×H(3,𝔽3)C2×D811yescharacteristic
103C2×H(3,𝔽3)C2×SD1611yescharacteristic
104#272—81no—
105C2×C3×Q8—181no—
106C2×C3×D4—34no—
107C2×C3×D4—91no—
108C2×C3×D4—181no—
109C2×C3×D4—181no—
110C2×C3×D4C2×C3⋊D311yescharacteristic
111C2×#67—64no—
112C2×#67—64no—
113C2×D12—34no—
114C2×Dic6—64no—
115C2×C3×C8—34no—
116C2×C3×C8—91no—
117C2×C3×C8C2×C3⋊D311yescharacteristic
118C3×C4⋊C4—34no—
119C3×C4⋊C4—91no—
120C3×C4⋊C4—181no—
121C3×C4⋊C4C2×C3⋊D311yescharacteristic
122C3×C42—91no—
123C4⋊Dic3—34no—
124Dic3⋊C4—64no—
125C4×Dic3—34no—
126C2×C3⋊C8—34no—
127C22×C32—44no—
128C22×C32—44no—
129C22×C32—44no—
130C22×C32D3×D414yes—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—124no—
134C32×C4—44no—
135C32×C4#23514yes—
136C32×C4#23714yes—
137C3×Dic3—38no—
138C3×Dic3—38no—
139C3×Dic3—124no—
140C4⋊Q8—91no—
141#127—91no—
142#102—91no—
143#102—91no—
144#102—91no—
145#102#27211yescharacteristic, complemented
146C4×C8—91no—
147H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
148C23×C3—21no—
149C23×C3—64no—
150C23×C3—181no—
151C22×D3—64no—
152C3×Q8—362no—
153C3×D4—21no—
154C3×D4—21no—
155C3×D4—64no—
156C3×D4—64no—
157C3×D4—181no—
158C3×D4—181no—
159C3×D4—362no—
160C3×D4—362no—
161C2×C3×C4—21no—
162C2×C3×C4—34no—
163C2×C3×C4—64no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—181no—
167C2×C3×C4C22×C3⋊D311yescharacteristic
168#67—128no—
169#67—128no—
170C2×Dic3—34no—
171C2×Dic3—34no—
172C2×Dic3—64no—
173D12—64no—
174D12—64no—
175Dic6—128no—
176C3×C8—21no—
177C3×C8—64no—
178C3×C8—181no—
179C3⋊C8—64no—
180C2×C32—84no—
181C2×C32#88514yes—
182C2×C32D3×D814yes—
183C2×C32D3×SD1614yes—
184C3×D3—244no—
185C2×Q8—181no—
186C2×D4—91no—
187C2×D4—181no—
188C2×D4—181no—
189C2×D4C2×#27211yescharacteristic
190C2×C8—91no—
191C2×C8C2×#27211yescharacteristic
192C4⋊C4—91no—
193C4⋊C4—181no—
194C4⋊C4C2×#27211yescharacteristic
195C42—91no—
196C22×C3—34no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—124no—
201C22×C3—124no—
202C22×C3—124no—
203C22×C3—361no—
204C22×C3—361no—
205C22×C3—361no—
206C22×C3D4×C3⋊D311yescharacteristic
207C2×D3—124no—
208C2×D3—124no—
209C2×D3—124no—
210C3×C4—34no—
211C3×C4—34no—
212C3×C4—41no—
213C3×C4—92no—
214C3×C4—92no—
215C3×C4—124no—
216C3×C4—361no—
217C3×C4#382811yescharacteristic
218C3×C4#383011yescharacteristic
219Dic3—38no—
220Dic3—38no—
221Dic3—124no—
222C32#476014yescomplemented
223C23—21no—
224C23—181no—
225Q8—362no—
226D4—21no—
227D4—21no—
228D4—181no—
229D4—181no—
230D4—362no—
231D4—362no—
232C2×C4—21no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—181no—
236C2×C4C22×#27211yescharacteristic
237C8—21no—
238C8—181no—
239C2×C3—34no—
240C2×C3—34no—
241C2×C3—34no—
242C2×C3—81no—
243C2×C3—244no—
244C2×C3—721no—
245C2×C3#6409411yescharacteristic
246C2×C3D8×C3⋊D311yescharacteristic
247C2×C3SD16×C3⋊D311yescharacteristic
248D3—244no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—361no—
253C22—361no—
254C22—361no—
255C22D4×#27211yescharacteristic
256C4—41no—
257C4—92no—
258C4—92no—
259C4—361no—
260C4#8874011yescharacteristic
261C4#8874411yescharacteristic
262C3—34no—
263C3#1059245911yescharacteristic
264C2—81no—
265C2—721no—
266C2#1172174111yescharacteristic
267C2D8×#27211yescharacteristic
268C2SD16×#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#11721850C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#486
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. #11721756C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #11720578C2semidirect
5. #11720554C2semidirect
6. C8×#1084C2semidirect
7. #11720531C2semidirect
22. C32×#102D3semidirect
145. #102#272semidirect
147. H(3,𝔽3)#486semidirect
222. C32#4760semidirect

Maximal subgroups

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

subgroupindex
#117218502
#117217562
H(3,𝔽3)×#1022
#117205782
#117205542
C8×#10842
#117205312
C3×#47603