Finite Group Databasebeta

← #13161168

Subgroups — #13161168

Order 1728.

816 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4#11722205C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic
8#10592883—31nomaximal
9#10591576—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
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×#216C414yescomplemented
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22C2×C32×Q16—31no—
23Q8×C3⋊Dic3—31no—
24Q8×C3⋊Dic3D311yescharacteristic
25C3×Q8×Dic3—34no—
26C2×C3×#216D314yes—
27Dic3×Dic6—34no—
28#63662—31no—
29#63662—31no—
30#63662—34no—
31#63662—64no—
32#63644—31no—
33#63642—31no—
34#63573—34no—
35#63565—34no—
36#63554—34no—
37C33×Q8—21no—
38C33×Q8C2×C412yes—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8C2×C412yes—
50#5184—94no—
51#5030—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8—31no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D314yes—
56C2×C3×Dic6C2×D314yes—
57C32×Q16—34no—
58C2×C32×C8—31no—
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—61no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3—21no—
66C4×C3⋊Dic3—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×#216Dic3116yescomplemented
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—64no—
74C3×C4×Dic3—34no—
75C3×C4×Dic3—64no—
76C2×C3×C3⋊C8C2×D314yes—
77Dic3⋊Dic3—64no—
78Dic32—64no—
79C22×C33C2×D411yescharacteristic
80C33×C4—22no—
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—22no—
88C2×C3×Q16—34no—
89C3×C4×Q8—91no—
90Q8×Dic3—34no—
91Q8×Dic3—94no—
92C2×#216C3⋊D311yescharacteristic
93C4×Dic6—91no—
94#835—94no—
95#835—94no—
96#835—91no—
97#835—184no—
98#817—94no—
99#815—94no—
100#810—91no—
101#808—91no—
102#803—91no—
103C32×Q8—21no—
104C32×Q8—28no—
105C32×Q8—24no—
106C32×Q8—32no—
107C32×Q8—44no—
108C32×Q8—61no—
109C32×Q8C4×D312yes—
110C32×Q8C2×Dic318yes—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3⋊Dic3#6711yescharacteristic
124C2×C3×Dic3—24no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—64no—
128C3×Dic6—24no—
129C3×Dic6C2×Dic318yes—
130C32×C8—32no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—124no—
134C3×C3⋊C8C2×Dic318yes—
135#441—271noSylow
136C2×C33C4×D411yescharacteristic
137C2×C33#13711yescharacteristic
138C2×C33#13711yescharacteristic
139C2×C3×Q8—24no—
140C2×C3×Q8—34no—
141C2×C3×Q8D3214yes—
142C2×C3×Q8C2×C3⋊D311yescharacteristic
143C2×Dic6C2×C3⋊D311yescharacteristic
144C3×Q16—316no—
145C2×C3×C8—34no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C42—91no—
149C3×C42—181no—
150#216C3⋊Dic314yescomplemented
151C4⋊Dic3—34no—
152C4⋊Dic3—64no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4⋊Dic3—184no—
157Dic3⋊C4—181no—
158C4×Dic3—34no—
159C4×Dic3—64no—
160C4×Dic3—181no—
161C4×Dic3—184no—
162C2×C3⋊C8—94no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—184no—
165C2×C3⋊C8C2×C3⋊D311yescharacteristic
166C22×C32—24no—
167C22×C32D3×D414yes—
168C22×C32C2×#6711yescharacteristic
169C32×C4—22no—
170C32×C4—24no—
171C32×C4—24no—
172C32×C4—28no—
173C32×C4—48no—
174C32×C4—62no—
175C32×C4C2×C4×D311yescharacteristic
176C32×C4#23511yescharacteristic
177C32×C4#23714yes—
178C32×C4C22×Dic314yes—
179C3⋊Dic3—21no—
180C3⋊Dic3—21no—
181C3⋊Dic3—41no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3×Dic3—28no—
186C3×Dic3—64no—
187C3×Dic3—64no—
188C3×Dic3—124no—
189C2×Q16—31no—
190C4×Q8—91no—
191C4×Q8—271no—
192#106—271no—
193#103—271no—
194#103—271no—
195#97—271no—
196C33#44111yescharacteristic, complemented, Sylow
197C3×Q8—24no—
198C3×Q8—21no—
199C3×Q8—28no—
200C3×Q8—38no—
201C3×Q8—44no—
202C3×Q8—64no—
203C3×Q8D3×Dic318yes—
204C3×Q8C2×C3⋊Dic312yes—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—24no—
208C2×C3×C4—44no—
209C2×C3×C4—64no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—181no—
213C2×C3×C4C2×D3214yes—
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219C2×Dic3—184no—
220C2×Dic3—181no—
221C2×Dic3—364no—
222Dic6—21no—
223Dic6C2×C3⋊Dic312yes—
224C3×C8—38no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C3⋊C8—364no—
228C3⋊C8C2×C3⋊Dic312yes—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#91814yes—
233C2×C32#91814yes—
234C2×C32C4×#6711yescharacteristic
235C2×C32D4×Dic314yes—
236C2×C32#94211yescharacteristic
237C2×C32#95111yescharacteristic
238C2×Q8—31no—
239C2×Q8D3×C3⋊D311yescharacteristic
240Q16—34no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C42—91no—
248C42—181no—
249C42—541no—
250C22×C3—24no—
251C22×C3D3×#6714yes—
252C22×C3D4×C3⋊D311yescharacteristic
253C3×C4—22no—
254C3×C4—28no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—48no—
258C3×C4—68no—
259C3×C4—181no—
260C3×C4—181no—
261C3×C4—361no—
262C3×C4C2×D3×Dic314yes—
263C3×C4#380414yes—
264C3×C4#383011yescharacteristic
265C3×C4C22×C3⋊Dic311yescharacteristic
266Dic3—22no—
267Dic3—64no—
268Dic3—64no—
269Dic3—124no—
270Dic3—361no—
271Dic3—364no—
272Dic3—724no—
273C32—24no—
274C32#503011yescharacteristic, complemented
275C32#518414yescomplemented
276Q8—21no—
277Q8—32no—
278Q8—61no—
279Q8D3×C3⋊Dic312yes—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—32no—
288C8—541no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3#6394314yes—
293C2×C3#6394814yes—
294C2×C3Dic3×#6714yes—
295C2×C3#6412711yescharacteristic
296C2×C3#6412711yescharacteristic
297C2×C3D4×C3⋊Dic311yescharacteristic
298C22C3⋊D3×#6711yescharacteristic
299C4—22no—
300C4—62no—
301C4—181no—
302C4—181no—
303C4—361no—
304C4—1081no—
305C4C2×D3×C3⋊Dic311yescharacteristic
306C4#8902811yescharacteristic
307C3—24no—
308C3#1059157614yescomplemented
309C3#1059288311yescharacteristic, complemented
310C2#1172357611yescharacteristic
311C2#1172358611yescharacteristic
312C2#67×C3⋊Dic311yescharacteristic
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
hypocenterC33#441
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
18. C32×#216C4semidirect
71. C3×#216Dic3semidirect
150. #216C3⋊Dic3semidirect
196. C33#441semidirect
274. C32#5030semidirect
275. C32#5184semidirect
308. C3#10591576semidirect
309. C3#10592883semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C32×#2162
Dic6×C3⋊Dic32
#117222052
#117220522
#117220372
#117220202
#105928833
#105915763