Finite Group Databasebeta

← #13158846

Subgroups — #13158846

Order 1728.

692 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 168 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×C4×#634C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic
3C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592762—31nomaximal
9#10590286—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×Dic3⋊C4—21no—
21C32×#125—31no—
22C4×#634D311yescharacteristic
23C3×#870D314yes—
24C3×C4×Dic6—34no—
25#63637—31no—
26#63637—31no—
27#63635—31no—
28#63634—31no—
29C4⋊(C3⋊(C3⋊C8))—31no—
30C4⋊(C3⋊(C3⋊C8))—34no—
31C4⋊(C3⋊(C3⋊C8))—64no—
32#63576—34no—
33#63574—34no—
34#63565—34no—
35#63565—34no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×#634—21no—
42C3×#634Q812yes—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47#5063—94no—
48#5022—91no—
49C2×#634C2×D311yescharacteristic
50C2×C3×Dic6—34no—
51C32×C4⋊C4—31no—
52C32×C4⋊C4—31no—
53C32×C4⋊C4—61no—
54C32×C4⋊C4—61no—
55C32×C42—24no—
56C32×C42C2×D314yes—
57C32×C42C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59#3750—21no—
60C4×C3⋊Dic3—21no—
61C2×C3⋊(C3⋊C8)—31no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C2×C3⋊(C3⋊C8)—64no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3C2×D314yes—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—24no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—21no—
76C33×C4—21no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3—41no—
81C32×Dic3—41no—
82C32×Dic3—41no—
83C3×#125—34no—
84C3×C4×Q8—91no—
85#870C3⋊D311yescharacteristic
86C4×Dic6—34no—
87#835—91no—
88#835—91no—
89#810—94no—
90#810—94no—
91#808—94no—
92#808—91no—
93#807—94no—
94#807—91no—
95C4⋊(C3⋊C8)—94no—
96C4⋊(C3⋊C8)—91no—
97C4⋊(C3⋊C8)—184no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—61no—
102C2×C32×C4—61no—
103C2×C32×C4#6714yes—
104C2×C32×C4#6714yes—
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4#6711yescharacteristic
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111#634—21no—
112#634Dic612yes—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—64no—
117C3×Dic6—38no—
118C3×Dic6—64no—
119C3⋊(C3⋊C8)—61no—
120C3⋊(C3⋊C8)—61no—
121C3⋊(C3⋊C8)—64no—
122C3⋊(C3⋊C8)—64no—
123C3⋊(C3⋊C8)—124no—
124C3⋊(C3⋊C8)—124no—
125#479—271noSylow
126C2×C33C22⋊Q811yescharacteristic
127C2×C33#13511yescharacteristic
128C2×C33#13711yescharacteristic
129C2×C3×Q8—91no—
130C2×Dic6—34no—
131C3×C4⋊C4—34no—
132C3×C4⋊C4—34no—
133C3×C4⋊C4—64no—
134C3×C4⋊C4—64no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C42—24no—
138C3×C42—181no—
139C3×C42D3214yes—
140C3×C42C2×C3⋊D311yescharacteristic
141C4⋊Dic3—34no—
142C4⋊Dic3C2×C3⋊D311yescharacteristic
143C4⋊Dic3C2×C3⋊D311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—64no—
147C4×Dic3—64no—
148C2×C3⋊C8—94no—
149C2×C3⋊C8—94no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—184no—
153C2×C3⋊C8—184no—
154C22×C32—24no—
155C22×C32C2×#6714yes—
156C22×C32C2×#6711yescharacteristic
157C32×C4—28no—
158C32×C4—21no—
159C32×C4—24no—
160C32×C4—21no—
161C32×C4—24no—
162C32×C4—24no—
163C32×C4—24no—
164C32×C4—44no—
165C32×C4—121no—
166C32×C4—121no—
167C32×C4C2×Dic611yescharacteristic
168C32×C4#23511yescharacteristic
169C32×C4D3×Q814yes—
170C32×C4#23914yes—
171C3⋊Dic3—22no—
172C3⋊Dic3—41no—
173C3×Dic3—44no—
174C3×Dic3—44no—
175C3×Dic3—68no—
176C3×Dic3—124no—
177#125—31no—
178C4×Q8—91no—
179#107—271no—
180#106—271no—
181C4⋊C8—271no—
182#103—271no—
183#103—271no—
184C33#47911yescharacteristic, complemented, Sylow
185C3×Q8—92no—
186C3×Q8—181no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—64no—
191C2×C3×C4—64no—
192C2×C3×C4—181no—
193C2×C3×C4—181no—
194C2×C3×C4#62514yes—
195C2×C3×C4#62514yes—
196C2×C3×C4#63811yescharacteristic
197C2×C3×C4#63811yescharacteristic
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204Dic6—38no—
205Dic6—64no—
206C3⋊C8—184no—
207C3⋊C8—184no—
208C3⋊C8—181no—
209C3⋊C8—181no—
210C3⋊C8—364no—
211C3⋊C8—364no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—24no—
215C2×C32#92411yescharacteristic
216C2×C32#94211yescharacteristic
217C2×C32#94614yes—
218C2×C32#95014yes—
219C2×C32#95011yescharacteristic
220C2×C32#95114yes—
221C2×Q8—91no—
222C2×C8—271no—
223C2×C8—271no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—91no—
229C4⋊C4—181no—
230C42—181no—
231C42D3×C3⋊D311yescharacteristic
232C22×C3—24no—
233C22×C3C2×#62514yes—
234C22×C3C2×#63811yescharacteristic
235C3×C4—28no—
236C3×C4—24no—
237C3×C4—24no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—24no—
241C3×C4—24no—
242C3×C4—44no—
243C3×C4—124no—
244C3×C4—124no—
245C3×C4—182no—
246C3×C4—361no—
247C3×C4D3×Dic614yes—
248C3×C4#379914yes—
249C3×C4Q8×C3⋊D311yescharacteristic
250C3×C4#383211yescharacteristic
251Dic3—41no—
252Dic3—41no—
253Dic3—68no—
254Dic3—124no—
255C32—24no—
256C32#502211yescharacteristic, complemented
257C32#506314yescomplemented
258Q8—92no—
259Q8—181no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4#629011yescharacteristic
265C2×C4#629011yescharacteristic
266C2×C4C2×D3×C3⋊D311yescharacteristic
267C8—541no—
268C8—541no—
269C2×C3—24no—
270C2×C3—24no—
271C2×C3—24no—
272C2×C3#6382214yes—
273C2×C3#6382714yes—
274C2×C3#6385614yes—
275C2×C3#6415511yescharacteristic
276C2×C3#6415911yescharacteristic
277C2×C3#6416011yescharacteristic
278C22C2×#629011yescharacteristic
279C4—21no—
280C4—21no—
281C4—121no—
282C4—121no—
283C4—182no—
284C4—361no—
285C4C3⋊D3×Dic611yescharacteristic
286C4#8901811yescharacteristic
287C3—24no—
288C3#1059028614yescomplemented
289C3#1059276211yescharacteristic, complemented
290C2#1172336311yescharacteristic
291C2#1172337311yescharacteristic
292C2#1172343511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#479
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
184. C33#479semidirect
256. C32#5022semidirect
257. C32#5063semidirect
288. C3#10590286semidirect
289. C3#10592762semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×#8702
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
#117220562
#117220532
#117220392
#105927623
#105902863