Finite Group Databasebeta

← #13144009

Subgroups — #13144009

Order 1728.

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

normal subgroups only

This group has 869 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×Q8×#272C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4806—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10Q8×#272—22no—
11Q8×#272—22no—
12C2×C4⋊#272—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272—21no—
15C2×C4×#272C2211yescharacteristic
16C2×#6229—21no—
17C2×#6229C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88549—22no—
21#88542—21no—
22C2×#6180C2211yescharacteristic
23#88528—22no—
24C2×C3×D3×Q8—34no—
25C2×C3×#215—34no—
26C2×C3×#204—34no—
27C3×#895—34no—
28C32×#103D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#810—34no—
31C22×#272—21no—
32C22×#272D411yescharacteristic
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D412yescomplemented
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084D411yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C4⋊#272—42no—
43C4×#272—24no—
44C4×#272—21no—
45C4×#272—21no—
46#6229—24no—
47#6229—21no—
48#6229D412yescomplemented
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51C3×#448—91no—
52#4806—34no—
53C2×C32×Q8C2×D314yes—
54C3×D3×Q8—68no—
55C3×D3×Q8—68no—
56C2×C3×D12—34no—
57C2×C3×D12—64no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—64no—
60C2×C3×Dic6—34no—
61C2×C3×Dic6—64no—
62C2×C32×C8C2×D314yes—
63C32×C4⋊C4C2×D314yes—
64C3×#215—68no—
65C3×(C2×D3)⋊C4—64no—
66C2×C3×C3⋊C8—34no—
67C3×#204—68no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×#272—21no—
70C2×#272—21no—
71C2×#272—41no—
72C2×#272—41no—
73C2×#272—41no—
74C2×#272SD1612yescomplemented
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)C2×D411yescharacteristic
79#1084—21no—
80#1084—22no—
81C22×C3×Q8—91no—
82C2×D3×Q8—34no—
83C2×C3×SD16—91no—
84C2×C3×SD16—91no—
85C3×#121—91no—
86C2×#215—34no—
87C2×#204—34no—
88#895—34no—
89C3×#103—34no—
90C3×#103—91no—
91C3×#103C3⋊D311yescharacteristic
92C3×C22⋊C8—91no—
93(C2×D3)⋊C8—34no—
94#810—34no—
95C22×C3×D3—34no—
96C22×C3×D3—64no—
97C32×Q8—24no—
98C32×Q8D1218yescomplemented
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4C22×D314yes—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—64no—
104C3×D12—64no—
105C3×D12—64no—
106C3×D12—128no—
107C3×C4×D3—616no—
108C3×C4×D3—64no—
109C3×C4×D3—64no—
110C3×Dic6—38no—
111C3×Dic6—616no—
112C3×Dic6—64no—
113C32×C8—24no—
114C3×C3⋊C8—64no—
115#448—91noSylow
116C2×H(3,𝔽3)#12011yescharacteristic
117C2×H(3,𝔽3)C2×SD1611yescharacteristic
118C2×H(3,𝔽3)#13711yescharacteristic
119#272—22no—
120#272—81no—
121C2×C3×Q8—34no—
122C2×C3×Q8—91no—
123C2×C3×Q8—181no—
124C2×C3×Q8—182no—
125C2×C3×Q8—182no—
126C2×C3×Q8C2×C3⋊D311yescharacteristic
127C2×C3×D4—91no—
128C2×C3×D4—181no—
129C22×C3×C4—91no—
130C22×C3×C4—181no—
131D3×Q8—68no—
132D3×Q8—68no—
133C2×D12—34no—
134C2×D12—64no—
135C2×C4×D3—34no—
136C2×C4×D3—64no—
137C2×Dic6—34no—
138C2×Dic6—64no—
139C3×SD16—182no—
140C3×SD16—182no—
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C2×C3×C8C2×C3⋊D311yescharacteristic
144C3×C4⋊C4—34no—
145C3×C4⋊C4C2×C3⋊D311yescharacteristic
146C3×C22⋊C4—181no—
147#215—68no—
148(C2×D3)⋊C4—64no—
149C2×C3⋊C8—34no—
150#204—68no—
151C22×C32D3×D414yes—
152C2×C3×D3—38no—
153C2×C3×D3—64no—
154C2×C3×D3—64no—
155C2×C3×D3—124no—
156C2×C3×D3—124no—
157C2×C3×D3—124no—
158C32×C4—28no—
159C32×C4—44no—
160C32×C4C2×D1214yes—
161C32×C4D3×D414yes—
162C3×Dic3—68no—
163C3×Dic3—64no—
164C22×Q8—91no—
165C2×SD16—91no—
166C2×SD16—91no—
167#121—91no—
168#103—91no—
169#103#27211yescharacteristic, complemented
170C22⋊C8—91no—
171H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
172C23×C3—91no—
173C23×C3—181no—
174C22×D3—34no—
175C22×D3—64no—
176C3×Q8—21no—
177C3×Q8—38no—
178C3×Q8—64no—
179C3×Q8—92no—
180C3×Q8—184no—
181C3×Q8—181no—
182C3×Q8C4⋊(C3⋊D3)12yes—
183C3×D4—181no—
184C3×D4—181no—
185C3×D4—362no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—34no—
189C2×C3×C4—64no—
190C2×C3×C4—64no—
191C2×C3×C4—91no—
192C2×C3×C4—184no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4—181no—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—34no—
198C2×Dic3—64no—
199D12—64no—
200D12—64no—
201D12—128no—
202C4×D3—64no—
203C4×D3—64no—
204C4×D3—616no—
205Dic6—38no—
206Dic6—64no—
207Dic6—616no—
208C3×C8—21no—
209C3×C8—64no—
210C3×C8—181no—
211C3⋊C8—64no—
212C2×C32#88214yes—
213C2×C32D3×SD1614yes—
214C2×C32#91814yes—
215C3×D3—68no—
216C3×D3—244no—
217C2×Q8—91no—
218C2×Q8—181no—
219C2×Q8—182no—
220C2×Q8—182no—
221C2×Q8C2×#27211yescharacteristic
222C2×D4—91no—
223C2×D4—181no—
224C22×C4—91no—
225C22×C4—181no—
226SD16—182no—
227SD16—182no—
228C2×C8—91no—
229C2×C8C2×#27211yescharacteristic
230C4⋊C4C2×#27211yescharacteristic
231C22⋊C4—181no—
232C22×C3—34no—
233C22×C3—92no—
234C22×C3—181no—
235C22×C3—181no—
236C22×C3—361no—
237C22×C3—361no—
238C22×C3—361no—
239C22×C3D4×C3⋊D311yescharacteristic
240C2×D3—38no—
241C2×D3—64no—
242C2×D3—64no—
243C2×D3—124no—
244C2×D3—124no—
245C2×D3—124no—
246C3×C4—22no—
247C3×C4—34no—
248C3×C4—34no—
249C3×C4—41no—
250C3×C4—68no—
251C3×C4—124no—
252C3×C4—181no—
253C3×C4—182no—
254C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
255C3×C4D4×C3⋊D311yescharacteristic
256Dic3—68no—
257Dic3—64no—
258C32#480614yescomplemented
259C23—91no—
260C23—181no—
261Q8—21no—
262Q8—92no—
263Q8—184no—
264Q8—181no—
265Q8C4⋊#27212yescomplemented
266D4—181no—
267D4—181no—
268D4—362no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—91no—
272C2×C4—184no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—181no—
276C2×C4C22×#27211yescharacteristic
277C8—21no—
278C8—181no—
279C2×C3—34no—
280C2×C3—34no—
281C2×C3—34no—
282C2×C3—182no—
283C2×C3—721no—
284C2×C3#6409111yescharacteristic
285C2×C3SD16×C3⋊D311yescharacteristic
286C2×C3#6412711yescharacteristic
287D3—68no—
288D3—244no—
289C22—92no—
290C22—181no—
291C22—181no—
292C22—361no—
293C22—361no—
294C22—361no—
295C22D4×#27211yescharacteristic
296C4—22no—
297C4—41no—
298C4—181no—
299C4—182no—
300C4C2×C4⋊#27211yescharacteristic
301C4D4×#27211yescharacteristic
302C3—34no—
303C3#1059250511yescharacteristic
304C2—182no—
305C2—721no—
306C2#1172173811yescharacteristic
307C2SD16×#27211yescharacteristic
308C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radical#11721763C2
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#272C2semidirect
2. C2×#88549C2semidirect
3. C2×#88528C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720553C2semidirect
7. #11720532C2semidirect
28. C32×#103D3semidirect
34. Q8×H(3,𝔽3)D4semidirect
48. #6229D4semidirect
74. C2×#272SD16semidirect
98. C32×Q8D12semidirect
169. #103#272semidirect
171. H(3,𝔽3)#448semidirect
258. C32#4806semidirect
265. Q8C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×#885492
C2×#885282
#117217632
H(3,𝔽3)×#1032
#117205532
#117205322
C3×#48063