Finite Group Databasebeta

← #102×#272 #13143940

Subgroups — #102×#272 #13143940

Order 1728.

1111 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 133 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×C8×#272C211yesmaximal, characteristic, complemented
3C4⋊C4×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×D3×#102—34nomaximal
9C23×#272—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12D4×#272—22no—
13D4×#272C414yescomplemented
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272—21no—
16C2×C4×#272C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
19C4⋊#1084C2211yescharacteristic, complemented
20#88540—21no—
21C2×#6180C2211yescharacteristic, complemented
22C8×#272—22no—
23C2×C3×D3×D4—34no—
24C2×C3×D3×C8—34no—
25C3×D3×C4⋊C4—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×C412yescomplemented
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×C412yescomplemented
45C4×#272—42no—
46C4×#272C2×C412yes—
47C4×#272D412yes—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50C2×C3×#102—91no—
51D3×#102—34no—
52C23×C3×D3—64no—
53C2×C32×D4C2×D314yes—
54C2×C3×#67—64no—
55C3×D3×D4—316no—
56C3×D3×D4—68no—
57C2×C3×D12—34no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—64no—
60C2×C32×C8C2×D314yescomplemented
61C32×C4⋊C4C2×D314yescomplemented
62C3×C4⋊Dic3—34no—
63C3×Dic3⋊C4—64no—
64C2×C3×C3⋊C8—34no—
65C3×D3×C8—68no—
66C22×H(3,𝔽3)—22no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C2×#272—22no—
71C2×#272—41no—
72C2×#272—41no—
73C2×#272—48no—
74C2×#272C22⋊C412yes—
75C2×#272D812yes—
76C2×#272SD1612yes—
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)C22×C411yescharacteristic
79C4×H(3,𝔽3)C2×D411yescharacteristic
80#1084—41no—
81#1084C22⋊C412yes—
82C22×C3×D4—91no—
83C2×D3×D4—34no—
84C22×C3×C8—91no—
85C2×C3×C4⋊C4—91no—
86C2×D3×C8—34no—
87D3×C4⋊C4—34no—
88C3×#102—34no—
89C3×#102—91no—
90C3×#102—91no—
91C3×#102—91no—
92C3×#102C3⋊D311yescharacteristic
93#832—34no—
94#821—34no—
95#809—34no—
96C23×C32—24no—
97C22×C3×D3—34no—
98C22×C3×D3—616no—
99C22×C3×D3—64no—
100C22×C3×D3—128no—
101C22×C3×D3—128no—
102C32×D4—24no—
103C32×D4C4×D318yescomplemented
104C2×C32×C4—24no—
105C2×C32×C4C22×D314yes—
106C3×#67—616no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—64no—
109C3×D12—38no—
110C3×D12—64no—
111C3×C4×D3—38no—
112C3×C4×D3—38no—
113C3×C4×D3—128no—
114C32×C8—24no—
115C3×C3⋊C8—64no—
116C2×#102—91noSylow
117C2×H(3,𝔽3)—42no—
118C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
119C2×H(3,𝔽3)C2×D811yescharacteristic
120C2×H(3,𝔽3)C2×SD1611yescharacteristic
121#272—42no—
122#272#10214yesdirect product factor
123C24×C3—181no—
124C23×D3—64no—
125C2×C3×D4—34no—
126C2×C3×D4—94no—
127C2×C3×D4—91no—
128C2×C3×D4—182no—
129C2×C3×D4—181no—
130C2×C3×D4C2×C3⋊D311yescharacteristic
131C22×C3×C4—91no—
132C22×C3×C4—181no—
133C2×#67—64no—
134D3×D4—316no—
135D3×D4—68no—
136C2×D12—34no—
137C2×C4×D3—34no—
138C2×C4×D3—64no—
139C2×C3×C8—34no—
140C2×C3×C8—91no—
141C2×C3×C8—182no—
142C2×C3×C8C2×C3⋊D311yescharacteristic
143C3×C4⋊C4—34no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C4⋊Dic3—34no—
148Dic3⋊C4—64no—
149C2×C3⋊C8—34no—
150D3×C8—68no—
151C22×C32—28no—
152C22×C32—44no—
153C22×C32—44no—
154C22×C32D3×D414yes—
155C2×C3×D3—38no—
156C2×C3×D3—38no—
157C2×C3×D3—38no—
158C2×C3×D3—68no—
159C2×C3×D3—1232no—
160C2×C3×D3—124no—
161C2×C3×D3—124no—
162C32×C4—44no—
163C32×C4C2×C4×D314yes—
164C32×C4D3×D414yes—
165C3×Dic3—38no—
166C3×Dic3—124no—
167C22×D4—91no—
168C22×C8—91no—
169C2×C4⋊C4—91no—
170#102—91no—
171#102—91no—
172#102—91no—
173#102#27211yescharacteristic, direct product factor
174H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
175C23×C3—21no—
176C23×C3—64no—
177C23×C3—91no—
178C23×C3—184no—
179C23×C3—181no—
180C23×C3—362no—
181C23×C3—362no—
182C22×D3—34no—
183C22×D3—616no—
184C22×D3—64no—
185C22×D3—128no—
186C22×D3—128no—
187C3×D4—21no—
188C3×D4—38no—
189C3×D4—64no—
190C3×D4—92no—
191C3×D4—184no—
192C3×D4—181no—
193C3×D4C4×C3⋊D312yes—
194C2×C3×C4—21no—
195C2×C3×C4—34no—
196C2×C3×C4—64no—
197C2×C3×C4—91no—
198C2×C3×C4—92no—
199C2×C3×C4—92no—
200C2×C3×C4—181no—
201C2×C3×C4—362no—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203#67—616no—
204C2×Dic3—34no—
205C2×Dic3—64no—
206D12—38no—
207D12—64no—
208C4×D3—38no—
209C4×D3—38no—
210C4×D3—128no—
211C3×C8—21no—
212C3×C8—64no—
213C3×C8—181no—
214C3⋊C8—64no—
215C2×C32—48no—
216C2×C32D3×C22⋊C414yes—
217C2×C32D3×D814yes—
218C2×C32D3×SD1614yes—
219C3×D3—316no—
220C3×D3—128no—
221C24—181no—
222C2×D4—94no—
223C2×D4—91no—
224C2×D4—182no—
225C2×D4—181no—
226C2×D4C2×#27211yescharacteristic
227C22×C4—91no—
228C22×C4—181no—
229C2×C8—91no—
230C2×C8—182no—
231C2×C8C2×#27211yescharacteristic, complemented
232C4⋊C4—91no—
233C4⋊C4—181no—
234C4⋊C4C2×#27211yescharacteristic, complemented
235C22×C3—22no—
236C22×C3—34no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—68no—
240C22×C3—92no—
241C22×C3—92no—
242C22×C3—92no—
243C22×C3—124no—
244C22×C3—124no—
245C22×C3—182no—
246C22×C3—368no—
247C22×C3—361no—
248C22×C3—361no—
249C22×C3D4×C3⋊D311yescharacteristic
250C2×D3—38no—
251C2×D3—38no—
252C2×D3—38no—
253C2×D3—68no—
254C2×D3—1232no—
255C2×D3—124no—
256C2×D3—124no—
257C3×C4—34no—
258C3×C4—34no—
259C3×C4—41no—
260C3×C4—92no—
261C3×C4—124no—
262C3×C4—361no—
263C3×C4C2×C4×C3⋊D311yescharacteristic
264C3×C4D4×C3⋊D311yescharacteristic
265Dic3—38no—
266Dic3—124no—
267C32D3×#10214yescomplemented
268C23—21no—
269C23—91no—
270C23—184no—
271C23—181no—
272C23—362no—
273C23—362no—
274D4—21no—
275D4—92no—
276D4—184no—
277D4—181no—
278D4C4×#27212yescomplemented
279C2×C4—21no—
280C2×C4—91no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4—181no—
284C2×C4—362no—
285C2×C4C22×#27211yescharacteristic
286C8—21no—
287C8—181no—
288C2×C3—34no—
289C2×C3—34no—
290C2×C3—34no—
291C2×C3—42no—
292C2×C3—94no—
293C2×C3—128no—
294C2×C3—362no—
295C2×C3C22⋊C4×C3⋊D311yescharacteristic
296C2×C3D8×C3⋊D311yescharacteristic
297C2×C3SD16×C3⋊D311yescharacteristic
298D3—316no—
299D3—128no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—92no—
304C22—92no—
305C22—92no—
306C22—182no—
307C22—368no—
308C22—361no—
309C22—361no—
310C22D4×#27211yescharacteristic
311C4—41no—
312C4—92no—
313C4—361no—
314C4C2×C4×#27211yescharacteristic
315C4D4×#27211yescharacteristic
316C3—34no—
317C3C3⋊D3×#10211yescharacteristic
318C2—42no—
319C2—94no—
320C2—362no—
321C2C22⋊C4×#27211yescharacteristic
322C2D8×#27211yescharacteristic
323C2SD16×#27211yescharacteristic
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)C2×#102
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×C8×#272C2semidirect
3. C4⋊C4×#272C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #11720578C2semidirect
6. #11720554C2semidirect
7. #11720531C2semidirect
13. D4×#272C4semidirect
17. C2×C8×H(3,𝔽3)C22semidirect
18. C4⋊C4×H(3,𝔽3)C22semidirect
19. C4⋊#1084C22semidirect
21. C2×#6180C22semidirect
26. C32×#102D3semidirect
37. D4×H(3,𝔽3)C2×C4semidirect
44. C4⋊#272C2×C4semidirect
60. C2×C32×C8C2×D3semidirect
61. C32×C4⋊C4C2×D3semidirect
103. C32×D4C4×D3semidirect
122. #272#102direct (N × H)
173. #102#272direct (N × H)
174. H(3,𝔽3)C2×#102semidirect
231. C2×C8C2×#272semidirect
234. C4⋊C4C2×#272semidirect
267. C32D3×#102semidirect
278. D4C4×#272semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×C8×#2722
C4⋊C4×#2722
H(3,𝔽3)×#1022
#117205782
#117205542
#117205312
C3×D3×#1023