Finite Group Databasebeta

← #13143941

Subgroups — #13143941

Order 1728.

1045 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 125 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×D4×#272C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3#11721760C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4770—34nomaximal
9C23×#272—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12D4×#272—22no—
13D4×#272C414yescomplemented
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
18#88542—21no—
19C4⋊#1084C2211yescharacteristic, complemented
20C4×#1084—21no—
21C2×#6180C2211yescharacteristic, complemented
22#88527—22no—
23C2×C3×D3×D4—34no—
24C2×C3×ΓL(1,𝔽25)—34no—
25C3×#892—34no—
26C32×#102D314yescomplemented
27C3×#832—34no—
28C3×#821—34no—
29C3×#809—34no—
30C23×H(3,𝔽3)—21no—
31C22×#272—21no—
32C22×#272—24no—
33C22×#272—42no—
34C22×#272—42no—
35C22×#272D411yescharacteristic
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)C2×C412yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40#6235—24no—
41C2×#1084—21no—
42C2×#1084D411yescharacteristic
43C4⋊#272—21no—
44C4⋊#272C2×C412yes—
45C4×#272C2×C412yes—
46C4×#272D412yes—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49C3×#418—91no—
50#4770—34no—
51C23×C3×D3—64no—
52C2×C32×D4C2×D314yes—
53C2×C3×#67—64no—
54C3×D3×D4—316no—
55C3×D3×D4—68no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—34no—
58C2×C32×C8C2×D314yescomplemented
59C32×C4⋊C4C2×D314yescomplemented
60C3×(C2×D3)⋊C4—64no—
61C3×C4⋊Dic3—34no—
62C3×C4×Dic3—64no—
63C2×C3×C3⋊C8—34no—
64C3×ΓL(1,𝔽25)—68no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×#272—21no—
70C2×#272—21no—
71C2×#272—22no—
72C2×#272—48no—
73C2×#272—41no—
74C2×#272—41no—
75C2×#272C22⋊C412yes—
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C22×C411yescharacteristic
78C4×H(3,𝔽3)C2×D411yescharacteristic
79#1084—41no—
80#1084C22⋊C412yes—
81C22×C3×D4—91no—
82C2×D3×D4—34no—
83C2×C3×M16—91no—
84C3×#117—91no—
85C2×ΓL(1,𝔽25)—34no—
86#892—34no—
87C3×#102—34no—
88C3×#102—91no—
89C3×#102—91no—
90C3×#102—91no—
91C3×#102C3⋊D311yescharacteristic
92#832—34no—
93#821—34no—
94#809—34no—
95C23×C32—24no—
96C22×C3×D3—34no—
97C22×C3×D3—616no—
98C22×C3×D3—64no—
99C22×C3×D3—128no—
100C22×C3×D3—128no—
101C32×D4—24no—
102C32×D4C4×D318yes—
103C2×C32×C4—24no—
104C2×C32×C4C22×D314yes—
105C3×#67—616no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—64no—
108C3×D12—38no—
109C3×D12—64no—
110C3×C4×D3—38no—
111C3×C4×D3—38no—
112C32×C8—24no—
113C3×C3⋊C8—64no—
114#418—91noSylow
115C2×H(3,𝔽3)—42no—
116C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
117C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
119#272—22no—
120#272—42no—
121C24×C3—181no—
122C23×D3—64no—
123C2×C3×D4—34no—
124C2×C3×D4—94no—
125C2×C3×D4—91no—
126C2×C3×D4—182no—
127C2×C3×D4—181no—
128C2×C3×D4C2×C3⋊D311yescharacteristic
129C22×C3×C4—91no—
130C2×#67—64no—
131D3×D4—316no—
132D3×D4—68no—
133C2×D12—34no—
134C2×C4×D3—34no—
135C3×M16—182no—
136C2×C3×C8—34no—
137C2×C3×C8—91no—
138C2×C3×C8C2×C3⋊D311yescharacteristic
139C3×C4⋊C4—34no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4C2×C3⋊D311yescharacteristic
142C3×C22⋊C4—181no—
143C3×C42—181no—
144(C2×D3)⋊C4—64no—
145C4⋊Dic3—34no—
146C4×Dic3—64no—
147C2×C3⋊C8—34no—
148ΓL(1,𝔽25)—68no—
149C22×C32—28no—
150C22×C32—44no—
151C22×C32—44no—
152C22×C32D3×D414yes—
153C2×C3×D3—38no—
154C2×C3×D3—64no—
155C2×C3×D3—64no—
156C2×C3×D3—68no—
157C2×C3×D3—1232no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C32×C4—44no—
161C32×C4C2×C4×D314yes—
162C32×C4D3×D414yes—
163C3×Dic3—38no—
164C3×Dic3—124no—
165C22×D4—91no—
166C2×M16—91no—
167#117—91no—
168#102—91no—
169#102—91no—
170#102—91no—
171#102#27211yescharacteristic, complemented
172H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
173C23×C3—21no—
174C23×C3—64no—
175C23×C3—91no—
176C23×C3—184no—
177C23×C3—181no—
178C23×C3—362no—
179C23×C3—362no—
180C22×D3—34no—
181C22×D3—616no—
182C22×D3—64no—
183C22×D3—128no—
184C22×D3—128no—
185C3×D4—21no—
186C3×D4—38no—
187C3×D4—64no—
188C3×D4—92no—
189C3×D4—184no—
190C3×D4—181no—
191C3×D4C4×C3⋊D312yes—
192C2×C3×C4—21no—
193C2×C3×C4—34no—
194C2×C3×C4—64no—
195C2×C3×C4—91no—
196C2×C3×C4—92no—
197C2×C3×C4—92no—
198C2×C3×C4—181no—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200#67—616no—
201C2×Dic3—34no—
202C2×Dic3—64no—
203D12—38no—
204D12—64no—
205C4×D3—38no—
206C4×D3—38no—
207C3×C8—21no—
208C3×C8—64no—
209C3×C8—181no—
210C3⋊C8—64no—
211C2×C32—48no—
212C2×C32D3×C22⋊C414yes—
213C2×C32#91114yes—
214C2×C32#91414yes—
215C3×D3—68no—
216C3×D3—128no—
217C24—181no—
218C2×D4—94no—
219C2×D4—91no—
220C2×D4—182no—
221C2×D4—181no—
222C2×D4C2×#27211yescharacteristic
223C22×C4—91no—
224M16—182no—
225C2×C8—91no—
226C2×C8C2×#27211yescharacteristic, complemented
227C4⋊C4—91no—
228C4⋊C4C2×#27211yescharacteristic, complemented
229C22⋊C4—181no—
230C42—181no—
231C22×C3—22no—
232C22×C3—34no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—68no—
236C22×C3—92no—
237C22×C3—124no—
238C22×C3—124no—
239C22×C3—181no—
240C22×C3—181no—
241C22×C3—182no—
242C22×C3—368no—
243C22×C3—361no—
244C22×C3—361no—
245C22×C3D4×C3⋊D311yescharacteristic
246C2×D3—38no—
247C2×D3—64no—
248C2×D3—64no—
249C2×D3—68no—
250C2×D3—1232no—
251C2×D3—124no—
252C2×D3—124no—
253C3×C4—34no—
254C3×C4—34no—
255C3×C4—41no—
256C3×C4—92no—
257C3×C4—124no—
258C3×C4—361no—
259C3×C4C2×C4×C3⋊D311yescharacteristic
260C3×C4D4×C3⋊D311yescharacteristic
261Dic3—38no—
262Dic3—124no—
263C32#477014yescomplemented
264C23—21no—
265C23—91no—
266C23—184no—
267C23—181no—
268C23—362no—
269C23—362no—
270D4—21no—
271D4—92no—
272D4—184no—
273D4—181no—
274D4C4×#27212yes—
275C2×C4—21no—
276C2×C4—91no—
277C2×C4—92no—
278C2×C4—92no—
279C2×C4—181no—
280C2×C4C22×#27211yescharacteristic
281C8—21no—
282C8—181no—
283C2×C3—34no—
284C2×C3—34no—
285C2×C3—34no—
286C2×C3—42no—
287C2×C3—128no—
288C2×C3—182no—
289C2×C3—362no—
290C2×C3C22⋊C4×C3⋊D311yescharacteristic
291C2×C3#6412011yescharacteristic
292C2×C3#6412311yescharacteristic
293D3—68no—
294D3—128no—
295C22—22no—
296C22—41no—
297C22—41no—
298C22—92no—
299C22—181no—
300C22—181no—
301C22—182no—
302C22—368no—
303C22—361no—
304C22—361no—
305C22D4×#27211yescharacteristic
306C4—41no—
307C4—92no—
308C4—361no—
309C4C2×C4×#27211yescharacteristic
310C4D4×#27211yescharacteristic
311C3—34no—
312C3#1059246911yescharacteristic
313C2—42no—
314C2—182no—
315C2—362no—
316C2C22⋊C4×#27211yescharacteristic
317C2#1172179311yescharacteristic
318C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×D4×#272C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#418
LCSG ▷ 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×#88527C2semidirect
3. #11721760C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #11720578C2semidirect
6. #11720554C2semidirect
7. #11720531C2semidirect
13. D4×#272C4semidirect
16. C2×C8×H(3,𝔽3)C22semidirect
17. C4⋊C4×H(3,𝔽3)C22semidirect
19. C4⋊#1084C22semidirect
21. C2×#6180C22semidirect
26. C32×#102D3semidirect
58. C2×C32×C8C2×D3semidirect
59. C32×C4⋊C4C2×D3semidirect
171. #102#272semidirect
172. H(3,𝔽3)#418semidirect
226. C2×C8C2×#272semidirect
228. C4⋊C4C2×#272semidirect
263. C32#4770semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#885272
#117217602
H(3,𝔽3)×#1022
#117205782
#117205542
#117205312
C3×#47703