Finite Group Databasebeta

← #13159130

Subgroups — #13159130

Order 1728.

708 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 146 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
2#11723406C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722042C211yesmaximal, characteristic
8#10592576—31nomaximal
9#10590492—34nomaximal
10C2×#6308C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#64099D311yescharacteristic
22#64096—31no—
23C3×#890—34no—
24#63844—34no—
25C32×#107—31no—
26#63643—31no—
27#63641—31no—
28#63637—31no—
29#63637—34no—
30#63637—64no—
31#63635—31no—
32C3×#807D314yes—
33#63574—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63567—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39#6308—21no—
40#6308Q812yes—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C32×Dic3—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C32×C3⋊C8—21no—
47#5063—91no—
48#4877—94no—
49C2×#634—31no—
50C2×#634—34no—
51C2×#634—64no—
52C2×C32×C8—31no—
53C32×C4⋊C4—24no—
54C32×C4⋊C4—31no—
55C32×C4⋊C4C2×D314yes—
56C32×C4⋊C4C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59#3750—21no—
60#3750—61no—
61C4×C3⋊Dic3—61no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—64no—
68C3×C4⋊Dic3C2×D314yes—
69C3×Dic3⋊C4—64no—
70C3×C4×Dic3—34no—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—64no—
73Dic32—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—41no—
83C3×#125—91no—
84#890—34no—
85#887—94no—
86C4×Dic6—91no—
87C3×#107—34no—
88#816—94no—
89Dic3⋊C8—94no—
90#810—94no—
91#810—91no—
92#810—91no—
93#810—184no—
94#808—94no—
95#808—91no—
96#807C3⋊D311yescharacteristic
97C4⋊(C3⋊C8)—91no—
98C2×C32×C4—21no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—44no—
102C2×C32×C4—61no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3—124no—
109C2×C3⋊Dic3#6711yescharacteristic
110C2×C3⋊Dic3#6711yescharacteristic
111#634—32no—
112#634—38no—
113#634—68no—
114#634—61no—
115#634—64no—
116#634—124no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—64no—
121C32×C8—61no—
122C3⋊(C3⋊C8)—61no—
123C3⋊(C3⋊C8)—64no—
124C3⋊(C3⋊C8)—124no—
125C3×C3⋊C8—24no—
126#479—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33#13711yescharacteristic
130C2×Dic6—94no—
131C2×Dic6—91no—
132C2×Dic6—184no—
133C2×C3×C8—34no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—34no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4D3214yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C42—91no—
142C4⋊Dic3—34no—
143C4⋊Dic3—64no—
144C4⋊Dic3C2×C3⋊D311yescharacteristic
145Dic3⋊C4—64no—
146Dic3⋊C4—181no—
147Dic3⋊C4—184no—
148C4×Dic3—34no—
149C4×Dic3—181no—
150C4×Dic3—184no—
151C2×C3⋊C8—94no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—184no—
154C2×C3⋊C8C2×C3⋊D311yescharacteristic
155C22×C32—24no—
156C22×C32D3×D414yes—
157C22×C32C2×#6711yescharacteristic
158C32×C4—24no—
159C32×C4—24no—
160C32×C4—41no—
161C32×C4—44no—
162C32×C4—84no—
163C32×C4—121no—
164C32×C4#23514yes—
165C32×C4D3×Q814yes—
166C32×C4D3×Q811yescharacteristic
167C32×C4#23911yescharacteristic
168C3⋊Dic3—21no—
169C3⋊Dic3—21no—
170C3⋊Dic3—41no—
171C3⋊Dic3—62no—
172C3⋊Dic3—68no—
173C3⋊Dic3—128no—
174C3×Dic3—44no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C3×Dic3—124no—
178#125—91no—
179C4×Q8—271no—
180#107—31no—
181#106—271no—
182C4⋊C8—271no—
183#103—271no—
184#103—271no—
185C33#47911yescharacteristic, complemented, Sylow
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—98no—
204Dic6—92no—
205Dic6—188no—
206Dic6—184no—
207Dic6—181no—
208Dic6—364no—
209C3×C8—64no—
210C3⋊C8—21no—
211C3⋊C8—184no—
212C3⋊C8—181no—
213C3⋊C8—364no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#89614yes—
218C2×C32#91214yes—
219C2×C32#91814yes—
220C2×C32#94611yescharacteristic
221C2×C32#95011yescharacteristic
222C2×C32#95111yescharacteristic
223C2×Q8—271no—
224C2×C8—31no—
225C2×C8—271no—
226C4⋊C4—31no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C4⋊C4—541no—
231C4⋊C4D3×C3⋊D311yescharacteristic
232C42—91no—
233C42—541no—
234C22×C3—24no—
235C22×C3D3×#6714yes—
236C22×C3D4×C3⋊D311yescharacteristic
237C3×C4—24no—
238C3×C4—24no—
239C3×C4—41no—
240C3×C4—44no—
241C3×C4—84no—
242C3×C4—124no—
243C3×C4—181no—
244C3×C4—181no—
245C3×C4—361no—
246C3×C4#379714yes—
247C3×C4#379914yes—
248C3×C4#382811yescharacteristic
249C3×C4Q8×C3⋊D311yescharacteristic
250Dic3—41no—
251Dic3—64no—
252Dic3—64no—
253Dic3—124no—
254Dic3—182no—
255Dic3—188no—
256Dic3—368no—
257C32—24no—
258C32#487714yescomplemented
259C32#506311yescharacteristic, complemented
260Q8—272no—
261Q8—541no—
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—181no—
267C2×C4—541no—
268C2×C4C2×D3×C3⋊D311yescharacteristic
269C8—61no—
270C8—541no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3#6388814yes—
275C2×C3#6393314yes—
276C2×C3#6394314yes—
277C2×C3#6410511yescharacteristic
278C2×C3#6412111yescharacteristic
279C2×C3#6412711yescharacteristic
280C22C3⋊D3×#6711yescharacteristic
281C4—41no—
282C4—121no—
283C4—181no—
284C4—181no—
285C4—361no—
286C4—542no—
287C4#8901911yescharacteristic
288C4#8902211yescharacteristic
289C3—24no—
290C3#1059049214yescomplemented
291C3#1059257611yescharacteristic, complemented
292C2#1172348911yescharacteristic
293C2#1172355911yescharacteristic
294C2#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
185. C33#479semidirect
258. C32#4877semidirect
259. C32#5063semidirect
290. C3#10590492semidirect
291. C3#10592576semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117234062
#117221802
C32×#8072
#117220532
#117220512
#117220422
#105925763
#105904923