Finite Group Databasebeta

← #13159295

Subgroups — #13159295

Order 1728.

720 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 176 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×#887C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592720—31nomaximal
9#10590557—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4×Q8—31no—
22#64078—31no—
23C3×#887D314yes—
24#63918—34no—
25#63662—31no—
26#63662—31no—
27#63634—31no—
28#63634—34no—
29#63634—64no—
30#63634D311yescharacteristic
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#807—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63566—34no—
35#63565—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C32×Dic6—21no—
44C32×Dic6Q812yescomplemented
45#6293—42no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48#5021—94no—
49#4875—91no—
50C2×C32×Q8—31no—
51C2×#634—61no—
52C2×C3×Dic6C2×D314yes—
53C2×#627—64no—
54C32×C4⋊C4—24no—
55C32×C4⋊C4—61no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C42—31no—
59C32×C42—61no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—61no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—34no—
70C3×Dic3⋊C4—24no—
71C3×C4×Dic3—24no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—41no—
81C32×Dic3—22no—
82C32×Dic3—21no—
83C32×Dic3Q1612yes—
84C3×C4×Q8—34no—
85#888—91no—
86#887C3⋊D311yescharacteristic
87#869—94no—
88C3×#107—91no—
89#835—94no—
90#835—94no—
91#816—91no—
92Dic3⋊C8—91no—
93#810—91no—
94#807—34no—
95#807—94no—
96#807—91no—
97#807—184no—
98C4⋊(C3⋊C8)—94no—
99C32×Q8—32no—
100C32×Q8—61no—
101C2×C32×C4—21no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4—44no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—124no—
114#634—122no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3#6714yes—
118C2×C3×Dic3#6714yes—
119C3×Dic6—24no—
120C3×Dic6Dic618yescomplemented
121#627—128no—
122C3⋊(C3⋊C8)—21no—
123C3⋊(C3⋊C8)—68no—
124C3⋊(C3⋊C8)—64no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8—64no—
127#477—271noSylow
128C2×C33C22⋊Q811yescharacteristic
129C2×C33C2×Q1611yescharacteristic
130C2×C33AGL(1,ℤ/8)11yescharacteristic
131C2×C3×Q8—34no—
132C2×Dic6—184no—
133C2×Dic6—181no—
134C2×Dic6C2×C3⋊D311yescharacteristic
135C2×C3×C8—91no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—64no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4D3214yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C42—34no—
142C3×C42—64no—
143C4⋊Dic3—34no—
144C4⋊Dic3—94no—
145C4⋊Dic3—91no—
146C4⋊Dic3—184no—
147C4⋊Dic3—184no—
148Dic3⋊C4—21no—
149Dic3⋊C4—181no—
150C4×Dic3—21no—
151C4×Dic3C2×C3⋊D311yescharacteristic
152C2×C3⋊C8—34no—
153C2×C3⋊C8—94no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—184no—
156C22×C32—24no—
157C22×C32D3×D411yescharacteristic
158C22×C32C2×#6714yes—
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—32no—
162C32×C4—41no—
163C32×C4—44no—
164C32×C4—62no—
165C32×C4—61no—
166C32×C4—84no—
167C32×C4C2×Dic614yes—
168C32×C4#23514yes—
169C32×C4#23511yescharacteristic
170C32×C4D3×Q811yescharacteristic
171C3⋊Dic3—41no—
172C3⋊Dic3—121no—
173C3⋊Dic3—124no—
174C3⋊Dic3—244no—
175C3×Dic3—28no—
176C3×Dic3—24no—
177C3×Dic3—124no—
178C3×Dic3#21618yes—
179C4⋊Q8—271no—
180C4×Q8—31no—
181#107—91no—
182#107—271no—
183C4⋊C8—271no—
184#103—271no—
185#103—271no—
186C33#47711yescharacteristic, complemented, Sylow
187C3×Q8—38no—
188C3×Q8—64no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—34no—
193C2×C3×C4—34no—
194C2×C3×C4—44no—
195C2×C3×C4—64no—
196C2×C3×C4—181no—
197C2×C3×C4C2×D3214yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—64no—
201C2×Dic3—184no—
202C2×Dic3—181no—
203C2×Dic3—364no—
204C2×Dic3#63811yescharacteristic
205C2×Dic3#63811yescharacteristic
206Dic6—21no—
207Dic6—362no—
208Dic6—368no—
209Dic6#63412yescomplemented
210C3×C8—181no—
211C3⋊C8—64no—
212C3⋊C8—188no—
213C3⋊C8—184no—
214C3⋊C8—181no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32#89611yescharacteristic
219C2×C32#91111yescharacteristic
220C2×C32D3×Q1611yescharacteristic
221C2×C32#92414yes—
222C2×C32C2×#21614yes—
223C2×C32#94914yes—
224C2×Q8—31no—
225C2×Q8—541no—
226C2×C8—91no—
227C2×C8—271no—
228C4⋊C4—61no—
229C4⋊C4—91no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4D3×C3⋊D311yescharacteristic
233C42—31no—
234C42—61no—
235C22×C3—24no—
236C22×C3D3×#6714yes—
237C22×C3C2×#63811yescharacteristic
238C3×C4—24no—
239C3×C4—24no—
240C3×C4—38no—
241C3×C4—41no—
242C3×C4—44no—
243C3×C4—68no—
244C3×C4—64no—
245C3×C4—84no—
246C3×C4—361no—
247C3×C4D3×Dic614yes—
248C3×C4#379814yes—
249C3×C4C2×#63411yescharacteristic
250C3×C4#382811yescharacteristic
251Dic3—22no—
252Dic3—21no—
253Dic3—124no—
254Dic3—361no—
255Dic3—364no—
256Dic3—724no—
257Dic3#375612yes—
258C32—24no—
259C32#487511yescharacteristic, complemented
260C32#502114yescomplemented
261Q8—32no—
262Q8—61no—
263Q8—1082no—
264C2×C4—21no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—61no—
268C2×C4—181no—
269C2×C4—541no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—181no—
272C8—541no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—24no—
276C2×C3#6385114yes—
277C2×C3#6392514yes—
278C2×C3D3×#21614yes—
279C2×C3#6413311yescharacteristic
280C2×C3C2×#375611yescharacteristic
281C2×C3#6415811yescharacteristic
282C22D3×#63811yescharacteristic
283C4—32no—
284C4—41no—
285C4—61no—
286C4—62no—
287C4—361no—
288C4—1081no—
289C4D3×#63411yescharacteristic
290C4#8902011yescharacteristic
291C3—24no—
292C3#1059055714yescomplemented
293C3#1059272011yescharacteristic, complemented
294C2#1172341511yescharacteristic
295C2#1172354411yescharacteristic
296C2D3×#375611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#477
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
44. C32×Dic6Q8semidirect
120. C3×Dic6Dic6semidirect
186. C33#477semidirect
209. Dic6#634semidirect
259. C32#4875semidirect
260. C32#5021semidirect
292. C3#10590557semidirect
293. C3#10592720semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117235342
#117221772
C3×#636342
Dic3⋊(C3⋊(C3⋊C8))2
#117220382
#117220372
#105927203
#105905573