Finite Group Databasebeta

← #13143953

Subgroups — #13143953

Order 1728.

715 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 99 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, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3C2×#88527C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4782—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×C4⋊#272—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88552—21no—
17#88545—22no—
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88527—22no—
22C3×#938—34no—
23C2×C3×#213—34no—
24C2×C3×ΓL(1,𝔽25)—34no—
25C3×#895—34no—
26C32×#102D314yescomplemented
27C3×#821—34no—
28C3×#808—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31C22×#272D411yescharacteristic
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—42no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4⋊#272—21no—
40C4⋊#272—21no—
41C4⋊#272—42no—
42C4×#272—21no—
43C4×#272—21no—
44C8×H(3,𝔽3)—21no—
45#6180D412yescomplemented
46C3×#469—91no—
47#4782—34no—
48C2×C32×D4C2×D314yes—
49C2×C3×#67—64no—
50C2×C3×D12—34no—
51C2×C3×D12—64no—
52C2×C3×C4×D3—34no—
53C2×C32×C8C2×D314yes—
54C32×C4⋊C4C2×D314yes—
55C3×#217—64no—
56C3×#213—68no—
57C3×(C2×D3)⋊C4—64no—
58C3×C4⋊Dic3—34no—
59C2×C3×C3⋊C8—34no—
60C3×ΓL(1,𝔽25)—68no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—21no—
68C2×#272—41no—
69C2×#272—41no—
70C2×#272—41no—
71C4×H(3,𝔽3)—41no—
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74#1084—21no—
75#1084—41no—
76C2×C3×D8—91no—
77C2×C3×M16—91no—
78C3×#121—91no—
79C3×#121—91no—
80#938—34no—
81C2×#213—34no—
82C2×ΓL(1,𝔽25)—34no—
83#895—34no—
84C3×#106—91no—
85C3×#102—34no—
86C3×#102—91no—
87C3×#102C3⋊D311yescharacteristic
88#821—34no—
89#808—34no—
90C23×C32—24no—
91C22×C3×D3—34no—
92C22×C3×D3—64no—
93C32×D4—24no—
94C32×D4—24no—
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C3×#67—128no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—64no—
100C3×D12—64no—
101C3×D12—64no—
102C3×D12—128no—
103C3×C4×D3—64no—
104C3×C4×D3—64no—
105C32×C8—24no—
106C3×C3⋊C8—38no—
107#469—91noSylow
108C2×H(3,𝔽3)—81no—
109C2×H(3,𝔽3)#12111yescharacteristic
110C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
111C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
112#272—41no—
113#272—81no—
114C2×C3×D4—34no—
115C2×C3×D4—91no—
116C2×C3×D4—181no—
117C2×C3×D4—181no—
118C2×C3×D4C2×C3⋊D311yescharacteristic
119C22×C3×C4—91no—
120C2×#67—64no—
121C2×D12—34no—
122C2×D12—64no—
123C2×C4×D3—34no—
124C3×D8—182no—
125C3×M16—182no—
126C2×C3×C8—34no—
127C2×C3×C8—91no—
128C2×C3×C8C2×C3⋊D311yescharacteristic
129C3×C4⋊C4—34no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4C2×C3⋊D311yescharacteristic
132C3×C22⋊C4—181no—
133C3×C22⋊C4—181no—
134#217—64no—
135#213—68no—
136(C2×D3)⋊C4—64no—
137C4⋊Dic3—34no—
138C2×C3⋊C8—34no—
139ΓL(1,𝔽25)—68no—
140C22×C32—44no—
141C22×C32—44no—
142C22×C32—44no—
143C22×C32D3×D414yes—
144C2×C3×D3—64no—
145C2×C3×D3—64no—
146C2×C3×D3—64no—
147C2×C3×D3—124no—
148C2×C3×D3—124no—
149C2×C3×D3—124no—
150C32×C4—44no—
151C32×C4#23514yes—
152C32×C4D3×D414yes—
153C3×Dic3—64no—
154C3×Dic3—124no—
155C2×D8—91no—
156C2×M16—91no—
157#121—91no—
158#121—91no—
159#106—91no—
160#102—91no—
161#102#27211yescharacteristic, complemented
162H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
163C23×C3—21no—
164C23×C3—64no—
165C23×C3—91no—
166C23×C3—181no—
167C22×D3—34no—
168C22×D3—64no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—64no—
172C3×D4—64no—
173C3×D4—181no—
174C3×D4—181no—
175C3×D4—362no—
176C3×D4—362no—
177C2×C3×C4—21no—
178C2×C3×C4—34no—
179C2×C3×C4—64no—
180C2×C3×C4—91no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4C22×C3⋊D311yescharacteristic
185#67—128no—
186C2×Dic3—34no—
187C2×Dic3—64no—
188D12—64no—
189D12—64no—
190D12—128no—
191C4×D3—64no—
192C4×D3—64no—
193C3×C8—21no—
194C3×C8—64no—
195C3×C8—92no—
196C3⋊C8—38no—
197C2×C32—84no—
198C2×C32#88414yes—
199C2×C32#91114yes—
200C2×C32#91414yes—
201C3×D3—124no—
202C3×D3—244no—
203C2×D4—91no—
204C2×D4—181no—
205C2×D4—181no—
206C2×D4C2×#27211yescharacteristic
207C22×C4—91no—
208D8—182no—
209M16—182no—
210C2×C8—91no—
211C2×C8C2×#27211yescharacteristic
212C4⋊C4—91no—
213C4⋊C4C2×#27211yescharacteristic
214C22⋊C4—181no—
215C22⋊C4—181no—
216C22×C3—34no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—124no—
221C22×C3—124no—
222C22×C3—124no—
223C22×C3—181no—
224C22×C3—181no—
225C22×C3—181no—
226C22×C3—361no—
227C22×C3—361no—
228C22×C3—361no—
229C22×C3D4×C3⋊D311yescharacteristic
230C2×D3—64no—
231C2×D3—64no—
232C2×D3—64no—
233C2×D3—124no—
234C2×D3—124no—
235C2×D3—124no—
236C3×C4—34no—
237C3×C4—34no—
238C3×C4—41no—
239C3×C4—124no—
240C3×C4—181no—
241C3×C4—361no—
242C3×C4#382811yescharacteristic
243C3×C4D4×C3⋊D311yescharacteristic
244Dic3—64no—
245Dic3—124no—
246C32#478214yescomplemented
247C23—21no—
248C23—91no—
249C23—181no—
250D4—21no—
251D4—21no—
252D4—181no—
253D4—181no—
254D4—362no—
255D4—362no—
256C2×C4—21no—
257C2×C4—91no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4C22×#27211yescharacteristic
262C8—21no—
263C8—92no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—81no—
268C2×C3—244no—
269C2×C3—361no—
270C2×C3—721no—
271C2×C3#6409311yescharacteristic
272C2×C3#6412011yescharacteristic
273C2×C3#6412311yescharacteristic
274D3—124no—
275D3—244no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—181no—
280C22—181no—
281C22—181no—
282C22—361no—
283C22—361no—
284C22—361no—
285C22D4×#27211yescharacteristic
286C4—41no—
287C4—181no—
288C4—361no—
289C4#8874011yescharacteristic
290C4D4×#27211yescharacteristic
291C3—34no—
292C3#1059248111yescharacteristic
293C2—81no—
294C2—361no—
295C2—721no—
296C2#1172174011yescharacteristic
297C2#1172179311yescharacteristic
298C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#469
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
1. #11721848C2semidirect
2. C2×#88545C2semidirect
3. C2×#88527C2semidirect
4. #11721763C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720554C2semidirect
7. #11720530C2semidirect
26. C32×#102D3semidirect
45. #6180D4semidirect
161. #102#272semidirect
162. H(3,𝔽3)#469semidirect
246. C32#4782semidirect

Maximal subgroups

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

subgroupindex
#117218482
C2×#885452
C2×#885272
#117217632
H(3,𝔽3)×#1022
#117205542
#117205302
C3×#47823