Finite Group Databasebeta

← #13161166

Subgroups — #13161166

Order 1728.

900 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 180 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×#88863C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
3#11723406C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic
6#11722042C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
8#10592489—31nomaximal
9#10591537—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12#88863C414yescomplemented
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21Dic3×C3⋊Dic3—21no—
22Q8×C3⋊Dic3D311yescharacteristic
23C2×#3756—31no—
24C2×#3756—34no—
25C2×#3756—64no—
26#64096—31no—
27C3×Q8×Dic3—34no—
28#63844—34no—
29C32×#103—31no—
30#63643—31no—
31C8×C3⋊Dic3—31no—
32#63634—31no—
33C3×#835D314yes—
34#63576—34no—
35#63567—34no—
36Dic3×C3⋊C8—34no—
37C33×Q8—21no—
38C33×Q8C2×C412yes—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308C2×C412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C32×Dic3—21no—
48C3⋊(C3⋊(C3⋊C8))C2×C412yes—
49C32×C3⋊C8—21no—
50C4×#216—91no—
51#4790—94no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C32×C8—31no—
59C32×C4⋊C4—31no—
60#3756—34no—
61#3756—316no—
62#3756—616no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—61no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—64no—
74C3×C4⋊Dic3C2×D314yes—
75C3×C4×Dic3—34no—
76C3×C4×Dic3—64no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79Dic32—64no—
80C22×C33C2×D411yescharacteristic
81C33×C4—22no—
82C33×C4C22×C411yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—22no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3Q1612yescomplemented
88C32×Dic3—41no—
89C3×C4×Q8—91no—
90Q8×Dic3—34no—
91C2×#216—94no—
92C2×#216—91no—
93C2×#216—184no—
94#887—94no—
95C4×Dic6—91no—
96C3×#103—34no—
97#835C3⋊D311yescharacteristic
98#816—94no—
99C8×Dic3—94no—
100#810—91no—
101#807—94no—
102#807—91no—
103C4×C3⋊C8—91no—
104C32×Q8—21no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—44no—
108C32×Q8C4×D318yes—
109C32×Q8C4×D312yes—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—124no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—64no—
133C32×C8—61no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—38no—
136C3⋊(C3⋊C8)—68no—
137C3×C3⋊C8—24no—
138C4×Q16—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C3×Q8—24no—
143C2×C3×Q8D3214yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×Dic6—94no—
146C2×Dic6—91no—
147C2×Dic6—184no—
148C2×C3×C8—34no—
149C3×C4⋊C4—34no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C42—91no—
153C3×C42—181no—
154#216—916no—
155#216—94no—
156#216—1816no—
157C4⋊Dic3—34no—
158C4⋊Dic3—64no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—184no—
162C4×Dic3—34no—
163C4×Dic3—64no—
164C4×Dic3—181no—
165C4×Dic3—184no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—24no—
171C22×C32D3×D414yes—
172C22×C32C2×#6711yescharacteristic
173C32×C4—22no—
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—28no—
177C32×C4—48no—
178C32×C4—121no—
179C32×C4C2×C4×D314yes—
180C32×C4C2×C4×D311yescharacteristic
181C32×C4#23511yescharacteristic
182C32×C4#23714yes—
183C3⋊Dic3—21no—
184C3⋊Dic3—41no—
185C3⋊Dic3—62no—
186C3⋊Dic3—68no—
187C3⋊Dic3—128no—
188C3⋊Dic3#21612yescomplemented
189C3×Dic3—38no—
190C3×Dic3—44no—
191C3×Dic3—64no—
192C3×Dic3—124no—
193C2×Q16—271no—
194C4×Q8—91no—
195C4×Q8—271no—
196#107—271no—
197#103—31no—
198#103—271no—
199C4×C8—271no—
200C33C4×Q1611yescharacteristic, complemented, Sylow
201C3×Q8—24no—
202C3×Q8—21no—
203C3×Q8—28no—
204C3×Q8—44no—
205C3×Q8#62418yes—
206C3×Q8C4×C3⋊D312yes—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—44no—
211C2×C3×C4—64no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—34no—
219C2×Dic3—34no—
220C2×Dic3—64no—
221C2×Dic3—184no—
222C2×Dic3—181no—
223C2×Dic3—364no—
224Dic6—98no—
225Dic6—92no—
226Dic6—188no—
227Dic6—184no—
228Dic6—181no—
229Dic6—364no—
230C3×C8—64no—
231C3⋊C8—21no—
232C3⋊C8—98no—
233C3⋊C8—92no—
234C3⋊C8—188no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32#67⋊C414yes—
239C2×C32#91614yes—
240C2×C32D3×Q1614yes—
241C2×C32C4×#6711yescharacteristic
242C2×C32C2×#21611yescharacteristic
243C2×C32#95011yescharacteristic
244C2×Q8—271no—
245C2×Q8D3×C3⋊D311yescharacteristic
246Q16—274no—
247C2×C8—31no—
248C2×C8—271no—
249C4⋊C4—31no—
250C4⋊C4—91no—
251C4⋊C4—181no—
252C4⋊C4—541no—
253C42—91no—
254C42—181no—
255C42—541no—
256C22×C3—24no—
257C22×C3D3×#6714yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C3×C4—22no—
260C3×C4—28no—
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—48no—
264C3×C4—92no—
265C3×C4—124no—
266C3×C4—181no—
267C3×C4—361no—
268C3×C4#380414yes—
269C3×C4C2×#62414yes—
270C3×C4C2×C4×C3⋊D311yescharacteristic
271C3×C4#383011yescharacteristic
272Dic3—38no—
273Dic3—41no—
274Dic3—64no—
275Dic3—124no—
276Dic3—182no—
277Dic3—188no—
278Dic3—368no—
279C32—24no—
280C32#479014yescomplemented
281C32C4×#21611yescharacteristic, complemented
282Q8—21no—
283Q8—272no—
284Q8—541no—
285Q8#628912yes—
286C2×C4—21no—
287C2×C4—61no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—541no—
292C2×C4C2×D3×C3⋊D311yescharacteristic
293C8—61no—
294C8—272no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3D3×#21614yes—
299C2×C3#6394714yes—
300C2×C3#6395214yes—
301C2×C3#6409011yescharacteristic
302C2×C3#6412511yescharacteristic
303C2×C3Q16×C3⋊D311yescharacteristic
304C22C3⋊D3×#6711yescharacteristic
305C4—22no—
306C4—92no—
307C4—121no—
308C4—181no—
309C4—361no—
310C4—542no—
311C4#8902811yescharacteristic
312C4C2×#628911yescharacteristic
313C3—24no—
314C3#1059153714yescomplemented
315C3#1059248911yescharacteristic, complemented
316C2C3⋊D3×#21611yescharacteristic
317C2#1172358311yescharacteristic
318C2#1172362211yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
12. #88863C4semidirect
87. C3×C3⋊Dic3Q16semidirect
188. C3⋊Dic3#216semidirect
200. C33C4×Q16semidirect
280. C32#4790semidirect
281. C32C4×#216semidirect
314. C3#10591537semidirect
315. C3#10592489semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×Q8×C3⋊Dic32
#117234062
C32×#8352
#117220562
#117220422
C3⋊C8×C3⋊Dic32
#105924893
#105915373