Finite Group Databasebeta

← #13159251

Subgroups — #13159251

Order 1728.

684 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723454C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592721—31nomaximal
9#10590544—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×C4×Q8—31no—
22#64079—31no—
23C3×#887D314yes—
24#63869—34no—
25#63662—31no—
26#63662—31no—
27#63635—31no—
28#63635—34no—
29#63635—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×Dic6Q812yes—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3×C3⋊(C3⋊C8)—21no—
47#5022—94no—
48#4877—91no—
49C2×C32×Q8—31no—
50C2×C3×Dic6C2×D314yes—
51C32×C4⋊C4—24no—
52C32×C4⋊C4—61no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C42—31no—
56C32×C42—61no—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—34no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—61no—
62#3750—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—34no—
68C3×Dic3⋊C4—24no—
69C3×C4×Dic3—24no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72#3724—64no—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—22no—
81C32×Dic3—21no—
82C32×Dic3—21no—
83C3×C4×Q8—34no—
84#890—91no—
85#887C3⋊D311yescharacteristic
86#870—94no—
87C3×#107—91no—
88#835—94no—
89#835—94no—
90#816—91no—
91Dic3⋊C8—91no—
92#810—91no—
93#808—94no—
94#808—91no—
95#808—184no—
96#807—34no—
97C4⋊(C3⋊C8)—94no—
98C32×Q8—32no—
99C32×Q8—61no—
100C2×C32×C4—21no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—31no—
104C2×C32×C4—31no—
105C2×C32×C4—44no—
106C2×C32×C4—61no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—124no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3#6714yes—
116C2×C3×Dic3#6714yes—
117C3×Dic6—24no—
118C3×Dic6Dic618yes—
119C3⋊(C3⋊C8)—21no—
120C3⋊(C3⋊C8)—68no—
121C3⋊(C3⋊C8)—64no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—64no—
124#479—271noSylow
125C2×C33C22⋊Q811yescharacteristic
126C2×C33#13511yescharacteristic
127C2×C33#13711yescharacteristic
128C2×C3×Q8—34no—
129C2×Dic6C2×C3⋊D311yescharacteristic
130C2×C3×C8—91no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—64no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4D3214yes—
135C3×C4⋊C4C2×C3⋊D311yescharacteristic
136C3×C42—34no—
137C3×C42—64no—
138C4⋊Dic3—34no—
139C4⋊Dic3—94no—
140C4⋊Dic3—91no—
141C4⋊Dic3—184no—
142C4⋊Dic3—181no—
143Dic3⋊C4—21no—
144Dic3⋊C4—184no—
145Dic3⋊C4—184no—
146Dic3⋊C4—181no—
147C4×Dic3—21no—
148C4×Dic3C2×C3⋊D311yescharacteristic
149C2×C3⋊C8—34no—
150C2×C3⋊C8—94no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—184no—
153C22×C32—24no—
154C22×C32D3×D411yescharacteristic
155C22×C32C2×#6714yes—
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—41no—
159C32×C4—44no—
160C32×C4—62no—
161C32×C4—61no—
162C32×C4—61no—
163C32×C4—84no—
164C32×C4C2×Dic614yes—
165C32×C4#23514yes—
166C32×C4#23511yescharacteristic
167C32×C4D3×Q811yescharacteristic
168C3⋊Dic3—41no—
169C3⋊Dic3—121no—
170C3⋊Dic3—124no—
171C3⋊Dic3—244no—
172C3×Dic3—28no—
173C3×Dic3—24no—
174C3×Dic3—24no—
175C3×Dic3—124no—
176#125—271no—
177C4×Q8—31no—
178#107—91no—
179#106—271no—
180C4⋊C8—271no—
181#103—271no—
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—34no—
190C2×C3×C4—34no—
191C2×C3×C4—44no—
192C2×C3×C4—64no—
193C2×C3×C4—181no—
194C2×C3×C4C2×D3214yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—64no—
198C2×Dic3—184no—
199C2×Dic3—181no—
200C2×Dic3—364no—
201C2×Dic3#63811yescharacteristic
202C2×Dic3#63811yescharacteristic
203Dic6—21no—
204Dic6#63412yes—
205C3×C8—181no—
206C3⋊C8—64no—
207C3⋊C8—188no—
208C3⋊C8—184no—
209C3⋊C8—181no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#89611yescharacteristic
214C2×C32#91211yescharacteristic
215C2×C32#91811yescharacteristic
216C2×C32#92414yes—
217C2×C32#94214yes—
218C2×C32#95014yes—
219C2×Q8—31no—
220C2×C8—91no—
221C2×C8—271no—
222C4⋊C4—61no—
223C4⋊C4—91no—
224C4⋊C4—271no—
225C4⋊C4—541no—
226C4⋊C4—541no—
227C4⋊C4D3×C3⋊D311yescharacteristic
228C42—31no—
229C42—61no—
230C22×C3—24no—
231C22×C3D3×#6714yes—
232C22×C3C2×#63811yescharacteristic
233C3×C4—24no—
234C3×C4—24no—
235C3×C4—41no—
236C3×C4—44no—
237C3×C4—68no—
238C3×C4—64no—
239C3×C4—64no—
240C3×C4—84no—
241C3×C4—361no—
242C3×C4D3×Dic614yes—
243C3×C4#379814yes—
244C3×C4C2×#63411yescharacteristic
245C3×C4#382811yescharacteristic
246Dic3—22no—
247Dic3—21no—
248Dic3—21no—
249Dic3—124no—
250Dic3—361no—
251Dic3—364no—
252Dic3—724no—
253C32—24no—
254C32#487711yescharacteristic, complemented
255C32#502214yescomplemented
256Q8—32no—
257Q8—61no—
258C2×C4—21no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—61no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×C3⋊D311yescharacteristic
265C8—181no—
266C8—541no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3#6385114yes—
271C2×C3#6393314yes—
272C2×C3#6394814yes—
273C2×C3#6413311yescharacteristic
274C2×C3#6415111yescharacteristic
275C2×C3#6415911yescharacteristic
276C22D3×#63811yescharacteristic
277C4—41no—
278C4—61no—
279C4—61no—
280C4—62no—
281C4—361no—
282C4—1081no—
283C4D3×#63411yescharacteristic
284C4#8902011yescharacteristic
285C3—24no—
286C3#1059054414yescomplemented
287C3#1059272111yescharacteristic, complemented
288C2#1172341511yescharacteristic
289C2#1172355611yescharacteristic
290C2#1172358111yescharacteristic
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#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#4877semidirect
255. C32#5022semidirect
286. C3#10590544semidirect
287. C3#10592721semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117234542
#117221782
C3×#636342
Dic3⋊(C3⋊(C3⋊C8))2
#117220382
#117220372
#105927213
#105905443