Finite Group Databasebeta

← #13161252

Subgroups — #13161252

Order 1728.

744 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C32×Q8×Dic3C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592720—31nomaximal
9#10591546—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6293—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4×Q8—31no—
22#64078—31no—
23C3×Q8×Dic3D314yes—
24#63918—34no—
25#63662—31no—
26#63662—34no—
27#63662—64no—
28#63662D311yescharacteristic
29#63634—31no—
30#63634—31no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#835—34no—
33#63576—34no—
34#63575—34no—
35Dic3⋊(C3⋊C8)—34no—
36C33×Q8—21no—
37C33×Q8Q812yescomplemented
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45#6293—42no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48#5021—94no—
49#4793—91no—
50C2×C32×Q8—24no—
51C2×C32×Q8C2×D314yes—
52C2×C32×Q8C2×D311yescharacteristic
53C2×#634—61no—
54C2×#627—64no—
55C32×C4⋊C4—31no—
56C32×C4⋊C4—61no—
57C32×C42—31no—
58C32×C42—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—61no—
62C4⋊(C3⋊Dic3)—64no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—64no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4⋊Dic3—24no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3C2×D314yes—
71C3×C4×Dic3—24no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—22no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—41no—
81C32×Dic3—21no—
82C32×Dic3—41no—
83C32×Dic3Q1612yes—
84C3×C4×Q8—34no—
85Q8×Dic3C3⋊D311yescharacteristic
86#888—91no—
87#869—94no—
88C3×#103—91no—
89#835—34no—
90#835—94no—
91#835—91no—
92#835—184no—
93#818—91no—
94Dic3⋊C8—91no—
95#807—94no—
96#807—94no—
97#807—91no—
98C4⋊(C3⋊C8)—94no—
99C32×Q8—21no—
100C32×Q8—28no—
101C32×Q8—24no—
102C32×Q8—44no—
103C32×Q8Dic618yescomplemented
104C32×Q8Dic612yescomplemented
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—44no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—124no—
118#634—122no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3#6714yes—
122C2×C3×Dic3#6714yes—
123#627—128no—
124C3⋊(C3⋊C8)—21no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—124no—
128C3×C3⋊C8—64no—
129#477—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×Q1611yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8D3214yes—
135C2×C3×Q8C2×C3⋊D311yescharacteristic
136C2×Dic6—184no—
137C2×Dic6—181no—
138C2×C3×C8—91no—
139C3×C4⋊C4—34no—
140C3×C4⋊C4—64no—
141C3×C4⋊C4—91no—
142C3×C42—34no—
143C3×C42—64no—
144C4⋊Dic3—21no—
145C4⋊Dic3—34no—
146C4⋊Dic3—94no—
147C4⋊Dic3—91no—
148C4⋊Dic3—184no—
149C4⋊Dic3—184no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151Dic3⋊C4—181no—
152C4×Dic3—21no—
153C4×Dic3C2×C3⋊D311yescharacteristic
154C2×C3⋊C8—34no—
155C2×C3⋊C8—94no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—184no—
158C22×C32—24no—
159C22×C32D3×D411yescharacteristic
160C22×C32C2×#6714yes—
161C32×C4—22no—
162C32×C4—24no—
163C32×C4—24no—
164C32×C4—28no—
165C32×C4—32no—
166C32×C4—48no—
167C32×C4—61no—
168C32×C4—121no—
169C32×C4C2×Dic614yes—
170C32×C4C2×Dic611yescharacteristic
171C32×C4#23514yes—
172C32×C4#23711yescharacteristic
173C3⋊Dic3—41no—
174C3⋊Dic3—121no—
175C3⋊Dic3—124no—
176C3⋊Dic3—244no—
177C3×Dic3—24no—
178C3×Dic3—44no—
179C3×Dic3—124no—
180C3×Dic3#21618yes—
181C4⋊Q8—271no—
182C4×Q8—31no—
183#107—271no—
184#107—271no—
185C4⋊C8—271no—
186#103—91no—
187#103—271no—
188C33#47711yescharacteristic, complemented, Sylow
189C3×Q8—24no—
190C3×Q8—21no—
191C3×Q8—28no—
192C3×Q8—44no—
193C3×Q8#62718yescomplemented
194C3×Q8#63412yescomplemented
195C2×C3×C4—24no—
196C2×C3×C4—21no—
197C2×C3×C4—24no—
198C2×C3×C4—34no—
199C2×C3×C4—34no—
200C2×C3×C4—44no—
201C2×C3×C4—64no—
202C2×C3×C4—181no—
203C2×C3×C4C2×D3214yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—64no—
207C2×Dic3—184no—
208C2×Dic3—181no—
209C2×Dic3—364no—
210C2×Dic3#63811yescharacteristic
211C2×Dic3#63811yescharacteristic
212Dic6—362no—
213Dic6—368no—
214C3×C8—181no—
215C3⋊C8—64no—
216C3⋊C8—184no—
217C3⋊C8—181no—
218C3⋊C8—364no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32#87811yescharacteristic
223C2×C32#91411yescharacteristic
224C2×C32D3×Q1611yescharacteristic
225C2×C32#92414yes—
226C2×C32C2×#21614yes—
227C2×C32#94914yes—
228C2×Q8—541no—
229C2×Q8D3×C3⋊D311yescharacteristic
230C2×C8—91no—
231C2×C8—271no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4—91no—
235C4⋊C4—271no—
236C4⋊C4—541no—
237C42—31no—
238C42—61no—
239C22×C3—24no—
240C22×C3D3×#6714yes—
241C22×C3C2×#63811yescharacteristic
242C3×C4—22no—
243C3×C4—28no—
244C3×C4—24no—
245C3×C4—24no—
246C3×C4—38no—
247C3×C4—48no—
248C3×C4—64no—
249C3×C4—124no—
250C3×C4—361no—
251C3×C4#380414yes—
252C3×C4C2×#62714yes—
253C3×C4C2×#63411yescharacteristic
254C3×C4#382811yescharacteristic
255Dic3—21no—
256Dic3—41no—
257Dic3—124no—
258Dic3—361no—
259Dic3—364no—
260Dic3—724no—
261Dic3#375612yes—
262C32—24no—
263C32#479311yescharacteristic, complemented
264C32#502114yescomplemented
265Q8—21no—
266Q8—1082no—
267Q8#629312yescomplemented
268C2×C4—21no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—61no—
272C2×C4—181no—
273C2×C4—541no—
274C2×C4C2×D3×C3⋊D311yescharacteristic
275C8—181no—
276C8—541no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3—24no—
280C2×C3#6393914yes—
281C2×C3D3×#21614yes—
282C2×C3#6396414yes—
283C2×C3#6413311yescharacteristic
284C2×C3C2×#375611yescharacteristic
285C2×C3#6415811yescharacteristic
286C22D3×#63811yescharacteristic
287C4—22no—
288C4—32no—
289C4—61no—
290C4—121no—
291C4—361no—
292C4—1081no—
293C4#8903111yescharacteristic
294C4C2×#629311yescharacteristic
295C3—24no—
296C3#1059154614yescomplemented
297C3#1059272011yescharacteristic, complemented
298C2#1172356811yescharacteristic
299C2D3×#375611yescharacteristic
300C2#1172364011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#477
LCSG ▷ C2×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
37. C33×Q8Q8semidirect
103. C32×Q8Dic6semidirect
104. C32×Q8Dic6semidirect
188. C33#477semidirect
193. C3×Q8#627semidirect
194. C3×Q8#634semidirect
263. C32#4793semidirect
264. C32#5021semidirect
267. Q8#6293semidirect
296. C3#10591546semidirect
297. C3#10592720semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×Dic32
#117235342
#117222052
C3×#636622
#117220572
#117220562
Dic3⋊(C3⋊(C3⋊C8))2
#105927203
#105915463