Finite Group Databasebeta

← #13144734

Subgroups — #13144734

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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11721848C211yesmaximal, characteristic
4C2×#88546C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720549C211yesmaximal, characteristic, complemented
8C3×#5170—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12Q8×#272—22no—
13Q8×#272—22no—
14C2×C4×#272—21no—
15C2×C4×#272C2211yescharacteristic
16C2×#6229—21no—
17C2×#6229C2211yescharacteristic
18SD16×H(3,𝔽3)—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20#88552—21no—
21#88546—22no—
22C4⋊#1084C2211yescharacteristic
23C2×#6180C2211yescharacteristic
24C2×C3×D3×Q8—34no—
25C2×C32×SD16D314yescomplemented
26C3×#938—34no—
27C2×C3×#214—34no—
28C3×#835—34no—
29C3×(C2×D3)⋊C8—34no—
30C3×#816—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272D411yescharacteristic
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D412yescomplemented
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#6235—42no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084D411yescharacteristic
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#5170—34no—
53C2×C32×Q8C2×D314yes—
54C2×C32×D4C2×D314yes—
55C2×C3×#67—64no—
56C3×D3×Q8—68no—
57C3×D3×Q8—68no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—64no—
60C2×C3×Dic6—34no—
61C2×C3×Dic6—64no—
62C32×SD16—28no—
63C2×C32×C8C2×D314yes—
64C3×#217—64no—
65C3×#214—68no—
66C3×C4⋊Dic3—34no—
67C2×C3×C3⋊C8—34no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)—41no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×#272—21no—
73C2×#272—21no—
74C2×#272SD1612yescomplemented
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×D411yescharacteristic
78#1084—21no—
79#1084—22no—
80#1084—41no—
81C22×C3×Q8—91no—
82C2×D3×Q8—34no—
83C2×C3×SD16—34no—
84C2×C3×SD16—91no—
85C2×C3×SD16C3⋊D311yescharacteristic
86C3×#121—91no—
87#938—34no—
88C2×#214—34no—
89C3×#103—91no—
90C3×#103—91no—
91C3×C22⋊C8—91no—
92#835—34no—
93(C2×D3)⋊C8—34no—
94#816—34no—
95C23×C32—24no—
96C22×C3×D3—34no—
97C32×Q8—24no—
98C32×Q8#6718yescomplemented
99C32×D4—24no—
100C32×D4—24no—
101C2×C32×C4—24no—
102C2×C32×C4C22×D314yes—
103C3×#67—128no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—64no—
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)—81no—
117C2×H(3,𝔽3)#12011yescharacteristic
118C2×H(3,𝔽3)C2×SD1611yescharacteristic
119C2×H(3,𝔽3)#13711yescharacteristic
120#272—22no—
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—34no—
128C2×C3×D4—181no—
129C2×C3×D4C2×C3⋊D311yescharacteristic
130C22×C3×C4—91no—
131C22×C3×C4—181no—
132C2×#67—64no—
133D3×Q8—68no—
134D3×Q8—68no—
135C2×C4×D3—34no—
136C2×C4×D3—64no—
137C2×Dic6—34no—
138C2×Dic6—64no—
139C3×SD16—22no—
140C3×SD16—68no—
141C3×SD16—182no—
142C2×C3×C8—34no—
143C2×C3×C8—91no—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—91no—
146C3×C22⋊C4—181no—
147#217—64no—
148#214—68no—
149C4⋊Dic3—34no—
150C2×C3⋊C8—34no—
151C22×C32—44no—
152C22×C32—44no—
153C22×C32—44no—
154C22×C32D3×D414yes—
155C2×C3×D3—38no—
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C32×C4—28no—
159C32×C4D3×D414yes—
160C32×C4C2×#6714yes—
161C3×Dic3—68no—
162C3×Dic3—64no—
163C3×Dic3—124no—
164C22×Q8—91no—
165C2×SD16—91no—
166C2×SD16#27211yescharacteristic, complemented
167#121—91no—
168#103—91no—
169#103—91no—
170C22⋊C8—91no—
171H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
172C23×C3—21no—
173C23×C3—64no—
174C23×C3—91no—
175C22×D3—34no—
176C3×Q8—21no—
177C3×Q8—38no—
178C3×Q8—64no—
179C3×Q8—92no—
180C3×Q8—184no—
181C3×Q8—181no—
182C3×Q8#63812yes—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—64no—
186C3×D4—64no—
187C3×D4—362no—
188C2×C3×C4—21no—
189C2×C3×C4—34no—
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×C4—181no—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198#67—128no—
199C2×Dic3—34no—
200C2×Dic3—64no—
201C2×Dic3—64no—
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—84no—
213C2×C32D3×SD1614yes—
214C2×C32#91514yes—
215C2×C32#93714yes—
216C3×D3—68no—
217C2×Q8—91no—
218C2×Q8—181no—
219C2×Q8—182no—
220C2×Q8—182no—
221C2×Q8C2×#27211yescharacteristic
222C2×D4—181no—
223C2×D4C2×#27211yescharacteristic
224C22×C4—91no—
225C22×C4—181no—
226SD16—22no—
227SD16—182no—
228C2×C8—91no—
229C2×C8C2×#27211yescharacteristic
230C4⋊C4—91no—
231C22⋊C4—181no—
232C22×C3—34no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—92no—
237C22×C3—124no—
238C22×C3—124no—
239C22×C3—124no—
240C22×C3—181no—
241C22×C3—181no—
242C22×C3D4×C3⋊D311yescharacteristic
243C2×D3—38no—
244C2×D3—64no—
245C2×D3—64no—
246C3×C4—22no—
247C3×C4—34no—
248C3×C4—34no—
249C3×C4—68no—
250C3×C4—181no—
251C3×C4—182no—
252C3×C4—361no—
253C3×C4D4×C3⋊D311yescharacteristic
254C3×C4C2×#63811yescharacteristic
255Dic3—68no—
256Dic3—64no—
257Dic3—124no—
258C32#517014yescomplemented
259C23—21no—
260C23—91no—
261Q8—21no—
262Q8—92no—
263Q8—184no—
264Q8—181no—
265Q8#623512yescomplemented
266D4—21no—
267D4—21no—
268D4—362no—
269C2×C4—21no—
270C2×C4—91no—
271C2×C4—184no—
272C2×C4—181no—
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—81no—
283C2×C3—182no—
284C2×C3—244no—
285C2×C3SD16×C3⋊D311yescharacteristic
286C2×C3#6412411yescharacteristic
287C2×C3#6414611yescharacteristic
288D3—68no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—92no—
293C22—181no—
294C22—181no—
295C22D4×#27211yescharacteristic
296C4—22no—
297C4—181no—
298C4—182no—
299C4—361no—
300C4D4×#27211yescharacteristic
301C4C2×#623511yescharacteristic
302C3—34no—
303C3#1059286911yescharacteristic
304C2—81no—
305C2—182no—
306C2SD16×#27211yescharacteristic
307C2#1172180111yescharacteristic
308C2#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 radical#11721848C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#448
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
1. C2×Q8×#272C2semidirect
2. C2×SD16×H(3,𝔽3)C2semidirect
4. C2×#88546C2semidirect
5. #11720583C2semidirect
6. #11720553C2semidirect
7. #11720549C2semidirect
25. C2×C32×SD16D3semidirect
34. Q8×H(3,𝔽3)D4semidirect
48. #6229D4semidirect
74. C2×#272SD16semidirect
98. C32×Q8#67semidirect
166. C2×SD16#272semidirect
171. H(3,𝔽3)#448semidirect
258. C32#5170semidirect
265. Q8#6235semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×SD16×H(3,𝔽3)2
#117218482
C2×#885462
#117205832
#117205532
#117205492
C3×#51703