Finite Group Databasebeta

← #13159129

Subgroups — #13159129

Order 1728.

684 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 156 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×#64099C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592498—31nomaximal
9#10590493—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21Q8×C3⋊Dic3—31no—
22#64099D311yescharacteristic
23C3×#890—34no—
24Dic3×Dic6—34no—
25C32×#103—31no—
26#63662—31no—
27#63644—31no—
28#63641—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#810D314yes—
33#63573—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63565—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C32×Dic3—21no—
43C32×Dic6—21no—
44C32×Dic6Q812yes—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C32×C3⋊C8—21no—
47#5063—91no—
48#4799—94no—
49C2×C32×Q8—31no—
50C2×C3×Dic6C2×D314yes—
51C2×C32×C8—31no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—61no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—21no—
62C4×C3⋊Dic3—61no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—64no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3—64no—
69C3×Dic3⋊C4—64no—
70C3×C4×Dic3—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—64no—
73Dic32—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—22no—
83C3×#125—91no—
84Q8×Dic3—94no—
85#890—34no—
86C4×Dic6—91no—
87C3×#103—34no—
88#835—94no—
89#817—94no—
90Dic3⋊C8—94no—
91#810—91no—
92#810C3⋊D311yescharacteristic
93#808—91no—
94#807—94no—
95#807—91no—
96#807—184no—
97C4⋊(C3⋊C8)—91no—
98C32×Q8—32no—
99C32×Q8—61no—
100C2×C32×C4—21no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—44no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—124no—
111C2×C3⋊Dic3#6711yescharacteristic
112C2×C3⋊Dic3#6711yescharacteristic
113C2×C3×Dic3—24no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—64no—
117C3×Dic6—24no—
118C3×Dic6Dic618yes—
119C32×C8—61no—
120C3⋊(C3⋊C8)—68no—
121C3⋊(C3⋊C8)—64no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—24no—
124#479—271noSylow
125C2×C33C22⋊Q811yescharacteristic
126C2×C33#13511yescharacteristic
127C2×C33#13711yescharacteristic
128C2×C3×Q8—34no—
129C2×Dic6C2×C3⋊D311yescharacteristic
130C2×C3×C8—34no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—181no—
134C3×C4⋊C4—181no—
135C3×C4⋊C4D3214yes—
136C3×C4⋊C4C2×C3⋊D311yescharacteristic
137C3×C42—91no—
138C4⋊Dic3—34no—
139C4⋊Dic3—64no—
140C4⋊Dic3—94no—
141C4⋊Dic3—91no—
142C4⋊Dic3—184no—
143C4⋊Dic3—184no—
144Dic3⋊C4—64no—
145Dic3⋊C4—181no—
146C4×Dic3—34no—
147C4×Dic3—181no—
148C4×Dic3—184no—
149C2×C3⋊C8—94no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—184no—
152C2×C3⋊C8C2×C3⋊D311yescharacteristic
153C22×C32—24no—
154C22×C32D3×D414yes—
155C22×C32C2×#6711yescharacteristic
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—41no—
159C32×C4—44no—
160C32×C4—62no—
161C32×C4—84no—
162C32×C4C2×Dic614yes—
163C32×C4#23714yes—
164C32×C4D3×Q811yescharacteristic
165C32×C4#23911yescharacteristic
166C3⋊Dic3—21no—
167C3⋊Dic3—21no—
168C3⋊Dic3—41no—
169C3⋊Dic3—121no—
170C3⋊Dic3—124no—
171C3⋊Dic3—244no—
172C3×Dic3—28no—
173C3×Dic3—64no—
174C3×Dic3—64no—
175C3×Dic3—124no—
176#125—91no—
177C4×Q8—271no—
178#107—271no—
179#106—271no—
180C4⋊C8—271no—
181#103—31no—
182#103—271no—
183C33#47911yescharacteristic, complemented, Sylow
184C3×Q8—38no—
185C3×Q8—64no—
186C2×C3×C4—24no—
187C2×C3×C4—21no—
188C2×C3×C4—24no—
189C2×C3×C4—44no—
190C2×C3×C4—64no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4C2×D3214yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—64no—
200C2×Dic3—184no—
201C2×Dic3—181no—
202C2×Dic3—364no—
203Dic6—21no—
204Dic6#63412yes—
205C3×C8—64no—
206C3⋊C8—21no—
207C3⋊C8—188no—
208C3⋊C8—184no—
209C3⋊C8—181no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#87814yes—
214C2×C32#91614yes—
215C2×C32#91814yes—
216C2×C32#94611yescharacteristic
217C2×C32#95011yescharacteristic
218C2×C32#95111yescharacteristic
219C2×Q8—31no—
220C2×C8—31no—
221C2×C8—271no—
222C4⋊C4—91no—
223C4⋊C4—181no—
224C4⋊C4—181no—
225C4⋊C4—271no—
226C4⋊C4—541no—
227C4⋊C4D3×C3⋊D311yescharacteristic
228C42—91no—
229C42—541no—
230C22×C3—24no—
231C22×C3D3×#6714yes—
232C22×C3D4×C3⋊D311yescharacteristic
233C3×C4—24no—
234C3×C4—24no—
235C3×C4—41no—
236C3×C4—44no—
237C3×C4—68no—
238C3×C4—84no—
239C3×C4—181no—
240C3×C4—181no—
241C3×C4—361no—
242C3×C4D3×Dic614yes—
243C3×C4#379614yes—
244C3×C4C2×#63411yescharacteristic
245C3×C4#383011yescharacteristic
246Dic3—22no—
247Dic3—64no—
248Dic3—64no—
249Dic3—124no—
250Dic3—361no—
251Dic3—364no—
252Dic3—724no—
253C32—24no—
254C32#479914yescomplemented
255C32#506311yescharacteristic, complemented
256Q8—32no—
257Q8—61no—
258C2×C4—21no—
259C2×C4—61no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×C3⋊D311yescharacteristic
265C8—61no—
266C8—541no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3#6388714yes—
271C2×C3#6393514yes—
272C2×C3#6394314yes—
273C2×C3#6408711yescharacteristic
274C2×C3#6412511yescharacteristic
275C2×C3#6412711yescharacteristic
276C22C3⋊D3×#6711yescharacteristic
277C4—41no—
278C4—62no—
279C4—181no—
280C4—181no—
281C4—361no—
282C4—1081no—
283C4D3×#63411yescharacteristic
284C4#8901511yescharacteristic
285C3—24no—
286C3#1059049314yescomplemented
287C3#1059249811yescharacteristic, complemented
288C2#1172348711yescharacteristic
289C2#1172356111yescharacteristic
290C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#479
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
183. C33#479semidirect
254. C32#4799semidirect
255. C32#5063semidirect
286. C3#10590493semidirect
287. C3#10592498semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
Dic6×C3⋊Dic32
#117221772
C32×#8102
#117220522
#117220512
#117220372
#105924983
#105904933