Finite Group Databasebeta

← #13158900

Subgroups — #13158900

Order 1728.

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

normal subgroups only

This group has 692 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
1C3×#64079C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic
3C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592721—31nomaximal
9#10590286—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×#3750—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21C32×C4×Q8—31no—
22#64079D311yescharacteristic
23C3×#870—34no—
24C3×C4×Dic6D314yes—
25#63662—31no—
26#63662—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—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C32×Dic6—21no—
44C32×Dic6Q812yes—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47#5063—91no—
48#5022—94no—
49C2×C32×Q8—31no—
50C2×C3×Dic6C2×D314yes—
51C32×C4⋊C4—31no—
52C32×C4⋊C4—61no—
53C32×C42—24no—
54C32×C42—61no—
55C32×C42C2×D314yes—
56C32×C42C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59#3750—21no—
60#3750—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⋊Dic3—34no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—64no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—24no—
74C22×C33C2×D411yescharacteristic
75C33×C4—21no—
76C33×C4—21no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—41no—
81C32×Dic3—22no—
82C32×Dic3—41no—
83C3×#125—91no—
84C3×C4×Q8—34no—
85#870—34no—
86C4×Dic6C3⋊D311yescharacteristic
87#835—94no—
88#835—94no—
89#810—91no—
90#810—91no—
91#808—94no—
92#808—91no—
93#807—94no—
94#807—91no—
95C4⋊(C3⋊C8)—94no—
96C4⋊(C3⋊C8)—91no—
97C4⋊(C3⋊C8)—184no—
98C32×Q8—32no—
99C32×Q8—61no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—61no—
104C2×C32×C4—61no—
105C2×C32×C4#6714yes—
106C2×C32×C4#6714yes—
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4#6711yescharacteristic
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—64no—
117C3×Dic6—24no—
118C3×Dic6Dic618yes—
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—34no—
130C2×Dic6C2×C3⋊D311yescharacteristic
131C3×C4⋊C4—34no—
132C3×C4⋊C4—64no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4—181no—
137C3×C42—24no—
138C3×C42—64no—
139C3×C42D3214yes—
140C3×C42C2×C3⋊D311yescharacteristic
141C4⋊Dic3—34no—
142C4⋊Dic3—34no—
143C4⋊Dic3C2×C3⋊D311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—64no—
146Dic3⋊C4—64no—
147C4×Dic3—21no—
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—24no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—24no—
162C32×C4—24no—
163C32×C4—24no—
164C32×C4—44no—
165C32×C4—62no—
166C32×C4—121no—
167C32×C4C2×Dic614yes—
168C32×C4#23514yes—
169C32×C4D3×Q811yescharacteristic
170C32×C4#23911yescharacteristic
171C3⋊Dic3—41no—
172C3⋊Dic3—41no—
173C3×Dic3—28no—
174C3×Dic3—44no—
175C3×Dic3—124no—
176C3×Dic3—124no—
177#125—91no—
178C4×Q8—31no—
179#107—271no—
180#106—271no—
181C4⋊C8—271no—
182#103—271no—
183#103—271no—
184C33#47911yescharacteristic, complemented, Sylow
185C3×Q8—38no—
186C3×Q8—64no—
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—21no—
205Dic6#63412yes—
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#92414yes—
216C2×C32#94214yes—
217C2×C32#94611yescharacteristic
218C2×C32#95014yes—
219C2×C32#95011yescharacteristic
220C2×C32#95111yescharacteristic
221C2×Q8—31no—
222C2×C8—271no—
223C2×C8—271no—
224C4⋊C4—31no—
225C4⋊C4—61no—
226C4⋊C4—91no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C42—61no—
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—68no—
244C3×C4—124no—
245C3×C4—361no—
246C3×C4—361no—
247C3×C4D3×Dic614yes—
248C3×C4#379914yes—
249C3×C4C2×#63411yescharacteristic
250C3×C4#382811yescharacteristic
251Dic3—22no—
252Dic3—41no—
253Dic3—124no—
254Dic3—124no—
255C32—24no—
256C32#502214yescomplemented
257C32#506311yescharacteristic, complemented
258Q8—32no—
259Q8—61no—
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#6413311yescharacteristic
276C2×C3#6415111yescharacteristic
277C2×C3#6415911yescharacteristic
278C22C2×#629011yescharacteristic
279C4—21no—
280C4—21no—
281C4—62no—
282C4—121no—
283C4—361no—
284C4—361no—
285C4D3×#63411yescharacteristic
286C4#8902211yescharacteristic
287C3—24no—
288C3#1059028614yescomplemented
289C3#1059272111yescharacteristic, complemented
290C2#1172336311yescharacteristic
291C2#1172337511yescharacteristic
292C2#1172342511yescharacteristic
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#10592721semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×C4×Dic62
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
#117220562
#117220532
#117220372
#105927213
#105902863