Finite Group Databasebeta

← #13144733

Subgroups — #13144733

Order 1728.

1034 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 115 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11721863C211yesmaximal, characteristic, complemented
4C2×#88549C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720553C211yesmaximal, characteristic, complemented
8C3×#5169—34nomaximal
9C23×#272—21no—
10C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13D4×#272—22no—
14D4×#272—22no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17SD16×H(3,𝔽3)—22no—
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19#88549—22no—
20#88542—21no—
21C4⋊#1084C2211yescharacteristic, complemented
22#88540—21no—
23C2×#6180C2211yescharacteristic, complemented
24C2×C3×D3×D4—34no—
25C2×C32×SD16D314yescomplemented
26C3×#946—34no—
27C2×C3×#215—34no—
28C3×#832—34no—
29C3×#821—34no—
30C3×(C2×D3)⋊C8—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272—24no—
33C22×#272—21no—
34C22×#272—42no—
35C22×#272—42no—
36C22×#272D411yescharacteristic
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—24no—
44C2×#1084—21no—
45C2×#1084D411yescharacteristic
46C4⋊#272—21no—
47C4⋊#272D412yes—
48C4×#272—21no—
49C4×#272—21no—
50C8×H(3,𝔽3)—21no—
51#6180—21no—
52C3×#450—91no—
53#5169—34no—
54C23×C3×D3—64no—
55C2×C32×Q8C2×D314yescomplemented
56C2×C32×D4C2×D314yes—
57C2×C3×#67—64no—
58C3×D3×D4—68no—
59C3×D3×D4—68no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—34no—
62C32×SD16—28no—
63C2×C32×C8C2×D314yescomplemented
64C3×#215—68no—
65C3×(C2×D3)⋊C4—64no—
66C3×C4⋊Dic3—34no—
67C3×Dic3⋊C4—64no—
68C2×C3×C3⋊C8—34no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)C2×D411yescharacteristic
73C2×#272—21no—
74C2×#272—21no—
75C2×#272—22no—
76C2×#272—48no—
77C2×#272—41no—
78C2×#272—41no—
79C2×#272SD1612yescomplemented
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)C2×D411yescharacteristic
83#1084—21no—
84#1084—41no—
85C22×C3×D4—91no—
86C2×D3×D4—34no—
87C2×C3×SD16—34no—
88C2×C3×SD16—91no—
89C2×C3×SD16C3⋊D311yescharacteristic
90C3×C22⋊Q8—91no—
91#946—34no—
92C2×#215—34no—
93C3×#102—91no—
94C3×#102—91no—
95C3×C22⋊C8—91no—
96#832—34no—
97#821—34no—
98(C2×D3)⋊C8—34no—
99C23×C32—24no—
100C22×C3×D3—34no—
101C22×C3×D3—616no—
102C22×C3×D3—64no—
103C22×C3×D3—128no—
104C22×C3×D3—128no—
105C32×Q8—24no—
106C32×Q8—24no—
107C32×D4—24no—
108C32×D4#6718yes—
109C2×C32×C4—24no—
110C2×C32×C4C22×D314yes—
111C3×#67—616no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—64no—
114C3×D12—38no—
115C3×D12—64no—
116C3×C4×D3—64no—
117C3×C4×D3—64no—
118C32×C8—24no—
119C3×C3⋊C8—64no—
120#450—91noSylow
121C2×H(3,𝔽3)—42no—
122C2×H(3,𝔽3)#12011yescharacteristic
123C2×H(3,𝔽3)C2×SD1611yescharacteristic
124C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
125#272—22no—
126#272—42no—
127C24×C3—181no—
128C23×D3—64no—
129C2×C3×Q8—34no—
130C2×C3×Q8C2×C3⋊D311yescharacteristic
131C2×C3×D4—34no—
132C2×C3×D4—91no—
133C2×C3×D4—181no—
134C2×C3×D4—182no—
135C2×C3×D4—182no—
136C2×C3×D4C2×C3⋊D311yescharacteristic
137C22×C3×C4—91no—
138C2×#67—64no—
139D3×D4—68no—
140D3×D4—68no—
141C2×D12—34no—
142C2×C4×D3—34no—
143C3×SD16—22no—
144C3×SD16—68no—
145C3×SD16—182no—
146C2×C3×C8—34no—
147C2×C3×C8—91no—
148C2×C3×C8C2×C3⋊D311yescharacteristic
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C22⋊C4—181no—
152#215—68no—
153(C2×D3)⋊C4—64no—
154C4⋊Dic3—34no—
155Dic3⋊C4—64no—
156C2×C3⋊C8—34no—
157C22×C32—28no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32D3×D414yes—
161C2×C3×D3—38no—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—68no—
165C2×C3×D3—1232no—
166C2×C3×D3—124no—
167C2×C3×D3—124no—
168C32×C4—44no—
169C32×C4D3×D414yes—
170C32×C4C2×#6714yes—
171C3×Dic3—64no—
172C3×Dic3—124no—
173C22×D4—91no—
174C2×SD16—91no—
175C2×SD16#27211yescharacteristic, complemented
176C22⋊Q8—91no—
177#102—91no—
178#102—91no—
179C22⋊C8—91no—
180H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3—64no—
183C23×C3—91no—
184C23×C3—184no—
185C23×C3—181no—
186C23×C3—362no—
187C23×C3—362no—
188C22×D3—34no—
189C22×D3—64no—
190C22×D3—616no—
191C22×D3—128no—
192C22×D3—128no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—64no—
196C3×Q8—64no—
197C3×D4—21no—
198C3×D4—38no—
199C3×D4—64no—
200C3×D4—92no—
201C3×D4—184no—
202C3×D4—181no—
203C3×D4#63812yes—
204C2×C3×C4—21no—
205C2×C3×C4—34no—
206C2×C3×C4—64no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212#67—616no—
213C2×Dic3—34no—
214C2×Dic3—64no—
215D12—38no—
216D12—64no—
217C4×D3—64no—
218C4×D3—64no—
219C3×C8—21no—
220C3×C8—64no—
221C3×C8—181no—
222C3⋊C8—64no—
223C2×C32—48no—
224C2×C32D3×SD1614yes—
225C2×C32#91414yes—
226C2×C32#93714yes—
227C3×D3—68no—
228C3×D3—128no—
229C24—181no—
230C2×Q8C2×#27211yescharacteristic, complemented
231C2×D4—91no—
232C2×D4—181no—
233C2×D4—182no—
234C2×D4—182no—
235C2×D4C2×#27211yescharacteristic
236C22×C4—91no—
237SD16—22no—
238SD16—182no—
239C2×C8—91no—
240C2×C8C2×#27211yescharacteristic, complemented
241C4⋊C4—91no—
242C4⋊C4—181no—
243C22⋊C4—181no—
244C22×C3—22no—
245C22×C3—34no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—68no—
249C22×C3—92no—
250C22×C3—124no—
251C22×C3—124no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3—182no—
255C22×C3—368no—
256C22×C3—361no—
257C22×C3—361no—
258C22×C3D4×C3⋊D311yescharacteristic
259C2×D3—38no—
260C2×D3—64no—
261C2×D3—64no—
262C2×D3—68no—
263C2×D3—1232no—
264C2×D3—124no—
265C2×D3—124no—
266C3×C4—34no—
267C3×C4—34no—
268C3×C4—41no—
269C3×C4—124no—
270C3×C4—181no—
271C3×C4—361no—
272C3×C4D4×C3⋊D311yescharacteristic
273C3×C4C2×#63811yescharacteristic
274Dic3—64no—
275Dic3—124no—
276C32#516914yescomplemented
277C23—21no—
278C23—91no—
279C23—184no—
280C23—181no—
281C23—362no—
282C23—362no—
283Q8—21no—
284Q8—21no—
285D4—21no—
286D4—92no—
287D4—184no—
288D4—181no—
289D4#623512yes—
290C2×C4—21no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4C22×#27211yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—34no—
299C2×C3—34no—
300C2×C3—34no—
301C2×C3—42no—
302C2×C3—128no—
303C2×C3—182no—
304C2×C3—362no—
305C2×C3SD16×C3⋊D311yescharacteristic
306C2×C3#6412311yescharacteristic
307C2×C3#6414611yescharacteristic
308D3—68no—
309D3—128no—
310C22—22no—
311C22—41no—
312C22—41no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22—182no—
317C22—368no—
318C22—361no—
319C22—361no—
320C22D4×#27211yescharacteristic
321C4—41no—
322C4—181no—
323C4—361no—
324C4D4×#27211yescharacteristic
325C4C2×#623511yescharacteristic
326C3—34no—
327C3#1059286811yescharacteristic
328C2—42no—
329C2—182no—
330C2—362no—
331C2SD16×#27211yescharacteristic
332C2#1172180011yescharacteristic
333C2#1172184711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radicalC2×D4×#272C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C2×SD16×H(3,𝔽3)C2semidirect
3. #11721863C2semidirect
4. C2×#88549C2semidirect
5. #11720578C2semidirect
6. #11720554C2semidirect
7. #11720553C2semidirect
10. C2×Q8×H(3,𝔽3)C22semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
21. C4⋊#1084C22semidirect
23. C2×#6180C22semidirect
25. C2×C32×SD16D3semidirect
55. C2×C32×Q8C2×D3semidirect
63. C2×C32×C8C2×D3semidirect
79. C2×#272SD16semidirect
175. C2×SD16#272semidirect
180. H(3,𝔽3)#450semidirect
230. C2×Q8C2×#272semidirect
240. C2×C8C2×#272semidirect
276. C32#5169semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×SD16×H(3,𝔽3)2
#117218632
C2×#885492
#117205782
#117205542
#117205532
C3×#51693