Finite Group Databasebeta

← #13159253

Subgroups — #13159253

Order 1728.

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

normal subgroups only

This group has 756 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×#893C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C3×#63635C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic
7#11722031C211yesmaximal, characteristic
8#10592708—31nomaximal
9#10590545—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×C4×D4—31no—
22#64079—31no—
23C3×#893D314yes—
24#63869—34no—
25#63659—31no—
26#63659—31no—
27#63635D311yescharacteristic, complemented
28#63634—31no—
29#63634—34no—
30#63634—64no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#808—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63563—34no—
35#63562—34no—
36C22×C32×D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C32×D12—21no—
44C32×D12Q812yes—
45C32×C4×D3—42no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48#5009—94no—
49#4871—91no—
50C2×C32×D4—31no—
51C22×C32×C4—61no—
52C2×C3×D12C2×D314yes—
53C2×C3×C4×D3—24no—
54C32×C4⋊C4—24no—
55C32×C4⋊C4C2×D314yes—
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C22⋊C4—61no—
58C32×C42—31no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64#3750—61no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—34no—
71C3×C4×Dic3C2×D314yes—
72C2×C3×C3⋊C8—34no—
73#3724—64no—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C2×C32×D3—22no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C33×C4—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—21no—
85C32×Dic3—21no—
86C3×C4×D4—34no—
87#893C3⋊D311yescharacteristic
88#890—91no—
89#870—94no—
90C3×#106—91no—
91#832—94no—
92#832—94no—
93#821—91no—
94Dic3⋊C8—91no—
95#809—91no—
96#808—34no—
97#807—94no—
98#807—91no—
99#807—184no—
100C4⋊(C3⋊C8)—94no—
101C23×C32—61no—
102C22×C3×D3—24no—
103C32×D4—32no—
104C32×D4—61no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—44no—
111C2×C32×C4—122no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—124no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3#6714yes—
120C2×C3×Dic3#6714yes—
121C3×D12—24no—
122C3×D12Dic618yes—
123C3×C4×D3—48no—
124C3⋊(C3⋊C8)—21no—
125C3⋊(C3⋊C8)—68no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—61no—
128C3×C3⋊C8—64no—
129#478—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C32×D3—42no—
134C2×C3×D4—34no—
135C22×C3×C4—64no—
136C2×D12C2×C3⋊D311yescharacteristic
137C2×C4×D3—21no—
138C2×C3×C8—91no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4D3214yes—
142C3×C4⋊C4C2×C3⋊D311yescharacteristic
143C3×C22⋊C4—64no—
144C3×C42—34no—
145(C2×D3)⋊C4—21no—
146C4⋊Dic3—34no—
147C4⋊Dic3—94no—
148C4⋊Dic3—91no—
149C4⋊Dic3—184no—
150C4⋊Dic3—181no—
151Dic3⋊C4—184no—
152Dic3⋊C4—184no—
153Dic3⋊C4—181no—
154C4×Dic3C2×C3⋊D311yescharacteristic
155C2×C3⋊C8—34no—
156C2×C3⋊C8—94no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—184no—
159C22×C32—24no—
160C22×C32—62no—
161C22×C32—121no—
162C22×C32—121no—
163C22×C32D3×D411yescharacteristic
164C22×C32C2×#6714yes—
165C2×C3×D3—28no—
166C2×C3×D3—44no—
167C2×C3×D3—44no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—41no—
171C32×C4—44no—
172C32×C4—61no—
173C32×C4—61no—
174C32×C4—84no—
175C32×C4C2×Dic614yes—
176C32×C4#23514yes—
177C32×C4#23511yescharacteristic
178C32×C4D3×Q811yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3—121no—
181C3⋊Dic3—124no—
182C3⋊Dic3—244no—
183C3×Dic3—24no—
184C3×Dic3—24no—
185C3×Dic3—124no—
186#125—271no—
187C4×D4—31no—
188#107—271no—
189#106—91no—
190C4⋊C8—271no—
191#102—271no—
192#102—271no—
193C33#47811yescharacteristic, complemented, Sylow
194C23×C3—64no—
195C22×D3—21no—
196C3×D4—38no—
197C3×D4—64no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—34no—
202C2×C3×C4—34no—
203C2×C3×C4—44no—
204C2×C3×C4—128no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—64no—
209C2×Dic3—184no—
210C2×Dic3—181no—
211C2×Dic3—364no—
212C2×Dic3#63811yescharacteristic
213C2×Dic3#63811yescharacteristic
214D12—21no—
215D12#63412yes—
216C4×D3—42no—
217C3×C8—181no—
218C3⋊C8—64no—
219C3⋊C8—188no—
220C3⋊C8—184no—
221C3⋊C8—181no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—122no—
226C2×C32#89611yescharacteristic
227C2×C32#91411yescharacteristic
228C2×C32#91611yescharacteristic
229C2×C32#92414yes—
230C2×C32#93214yes—
231C2×C32#95014yes—
232C3×D3—48no—
233C2×D4—31no—
234C22×C4—61no—
235C2×C8—91no—
236C2×C8—271no—
237C4⋊C4—91no—
238C4⋊C4—271no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C4⋊C4D3×C3⋊D311yescharacteristic
242C22⋊C4—61no—
243C42—31no—
244C22×C3—24no—
245C22×C3—68no—
246C22×C3—124no—
247C22×C3—124no—
248C22×C3D3×#6714yes—
249C22×C3C2×#63811yescharacteristic
250C2×D3—22no—
251C2×D3—41no—
252C2×D3—41no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—41no—
256C3×C4—44no—
257C3×C4—64no—
258C3×C4—64no—
259C3×C4—84no—
260C3×C4—361no—
261C3×C4D3×Dic614yes—
262C3×C4#379814yes—
263C3×C4C2×#63411yescharacteristic
264C3×C4#382811yescharacteristic
265Dic3—21no—
266Dic3—21no—
267Dic3—124no—
268Dic3—361no—
269Dic3—364no—
270Dic3—724no—
271C32—24no—
272C32#487111yescharacteristic, complemented
273C32#500914yescomplemented
274C23—61no—
275D4—32no—
276D4—61no—
277C2×C4—21no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—122no—
281C2×C4—181no—
282C2×C4—541no—
283C2×C4C2×D3×C3⋊D311yescharacteristic
284C8—181no—
285C8—541no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—128no—
290C2×C3#6385114yes—
291C2×C3#6393414yes—
292C2×C3#6394714yes—
293C2×C3#6413311yescharacteristic
294C2×C3#6414111yescharacteristic
295C2×C3#6415911yescharacteristic
296D3—42no—
297C22—62no—
298C22—121no—
299C22—121no—
300C22D3×#63811yescharacteristic
301C4—41no—
302C4—61no—
303C4—61no—
304C4—361no—
305C4—1081no—
306C4D3×#63411yescharacteristic
307C4#8902011yescharacteristic
308C3—24no—
309C3#1059054514yescomplemented
310C3#1059270811yescharacteristic, complemented
311C2—122no—
312C2#1172341511yescharacteristic
313C2#1172355711yescharacteristic
314C2#1172358011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#478
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
2. #11723454C2semidirect
3. #11722177C2semidirect
4. C3×#63635C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
27. #63635D3semidirect
193. C33#478semidirect
272. C32#4871semidirect
273. C32#5009semidirect
309. C3#10590545semidirect
310. C3#10592708semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117234542
#117221772
C3×#636352
Dic3⋊(C3⋊(C3⋊C8))2
#117220322
#117220312
#105927083
#105905453