Finite Group Databasebeta

← #13159249

Subgroups — #13159249

Order 1728.

756 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64102C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic
7#11722033C211yesmaximal, characteristic
8#10592570—31nomaximal
9#10590545—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64102D311yescharacteristic
22#64099—31no—
23C3×#893—34no—
24#63869—34no—
25C32×#106—31no—
26#63648—31no—
27#63641—31no—
28#63636—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#808D314yescomplemented
33Dic3⋊(C3⋊C8)—34no—
34#63563—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)Q812yes—
44C3×C4×C3⋊D3—42no—
45C2×C32×Dic3—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48#5009—91no—
49#4871—94no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C4×C3⋊D3—21no—
52C2×C3×D12—34no—
53C2×C3×C4×D3—64no—
54C2×C32×C8—31no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—31no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59#3752—21no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—61no—
63C4⋊(C3⋊Dic3)—64no—
64#3750—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×(C2×D3)⋊C4—64no—
70C3×C4⋊Dic3C2×D314yes—
71C3×C4×Dic3—34no—
72C2×C3×C3⋊C8C2×D314yes—
73#3724—64no—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C33×C4—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3—21no—
85C32×Dic3—41no—
86C3×C4×D4—91no—
87#893—34no—
88#890—94no—
89#870—91no—
90C3×#106—34no—
91#832—91no—
92#832—91no—
93#821—94no—
94Dic3⋊C8—94no—
95#809—94no—
96#808C3⋊D311yescharacteristic, complemented
97#807—94no—
98#807—91no—
99#807—184no—
100C4⋊(C3⋊C8)—91no—
101C22×C3⋊D3—21no—
102C22×C3×D3—64no—
103C2×C32×C4—21no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—44no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—124no—
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)Dic612yes—
117C4×C3⋊D3—42no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—34no—
121C3×D12—38no—
122C3×D12—64no—
123C3×C4×D3—128no—
124C32×C8—61no—
125C3⋊(C3⋊C8)—68no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—61no—
128C3×C3⋊C8—24no—
129#478—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C3×C3⋊D3—42no—
134C2×C3×D4—91no—
135C22×C3×C4—181no—
136C2×D12—34no—
137C2×C4×D3—64no—
138C2×C3×C8—34no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4D3214yes—
142C3×C4⋊C4C2×C3⋊D311yescharacteristic
143C3×C22⋊C4—181no—
144C3×C42—91no—
145(C2×D3)⋊C4—64no—
146C4⋊Dic3—94no—
147C4⋊Dic3—91no—
148C4⋊Dic3—184no—
149C4⋊Dic3—184no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—184no—
154C4×Dic3—34no—
155C2×C3⋊C8—94no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—184no—
158C2×C3⋊C8C2×C3⋊D311yescharacteristic
159C22×C32—24no—
160C22×C32D3×D414yes—
161C22×C32C2×#6711yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3×D3—68no—
166C2×C3×D3—124no—
167C2×C3×D3—124no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—41no—
171C32×C4—44no—
172C32×C4—84no—
173C32×C4—121no—
174C32×C4C2×Dic611yescharacteristic
175C32×C4#23514yes—
176C32×C4#23511yescharacteristic
177C32×C4D3×Q814yes—
178C3⋊Dic3—21no—
179C3⋊Dic3—21no—
180C3⋊Dic3—121no—
181C3⋊Dic3—124no—
182C3⋊Dic3—244no—
183C3×Dic3—44no—
184C3×Dic3—64no—
185C3×Dic3—64no—
186#125—271no—
187C4×D4—91no—
188#107—271no—
189#106—31no—
190C4⋊C8—271no—
191#102—271no—
192#102—271no—
193C33#47811yescharacteristic, complemented, Sylow
194C23×C3—181no—
195C22×D3—64no—
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—44no—
202C2×C3×C4—64no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—362no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—34no—
210C2×Dic3—34no—
211C2×Dic3—184no—
212C2×Dic3—181no—
213C2×Dic3—364no—
214D12—38no—
215D12—64no—
216C4×D3—128no—
217C3×C8—64no—
218C3⋊C8—21no—
219C3⋊C8—188no—
220C3⋊C8—184no—
221C3⋊C8—181no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32#89614yes—
226C2×C32#91414yes—
227C2×C32#91614yes—
228C2×C32#92411yescharacteristic
229C2×C32#93211yescharacteristic
230C2×C32#95011yescharacteristic
231C3⋊D3—42no—
232C3×D3—128no—
233C2×D4—91no—
234C22×C4—181no—
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—271no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C4⋊C4D3×C3⋊D311yescharacteristic
242C22⋊C4—181no—
243C42—91no—
244C22×C3—24no—
245C22×C3—182no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3D3×#6714yes—
249C22×C3D4×C3⋊D311yescharacteristic
250C2×D3—68no—
251C2×D3—124no—
252C2×D3—124no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—41no—
256C3×C4—44no—
257C3×C4—84no—
258C3×C4—124no—
259C3×C4—181no—
260C3×C4—181no—
261C3×C4D3×Dic614yes—
262C3×C4#379814yes—
263C3×C4#382811yescharacteristic
264C3×C4Q8×C3⋊D311yescharacteristic
265Dic3—41no—
266Dic3—64no—
267Dic3—64no—
268Dic3—361no—
269Dic3—364no—
270Dic3—724no—
271C32—24no—
272C32#487114yescomplemented
273C32#500911yescharacteristic, complemented
274C23—181no—
275D4—92no—
276D4—181no—
277C2×C4—21no—
278C2×C4—61no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—362no—
282C2×C4—541no—
283C2×C4C2×D3×C3⋊D311yescharacteristic
284C8—61no—
285C8—541no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—362no—
290C2×C3#6385114yes—
291C2×C3#6393414yes—
292C2×C3#6394714yes—
293C2×C3#6410511yescharacteristic
294C2×C3#6412311yescharacteristic
295C2×C3#6412511yescharacteristic
296D3—128no—
297C22—182no—
298C22—361no—
299C22—361no—
300C22C3⋊D3×#6711yescharacteristic
301C4—41no—
302C4—121no—
303C4—181no—
304C4—181no—
305C4—1081no—
306C4C3⋊D3×Dic611yescharacteristic
307C4#8902011yescharacteristic
308C3—24no—
309C3#1059054514yescomplemented
310C3#1059257011yescharacteristic, complemented
311C2—362no—
312C2#1172341811yescharacteristic
313C2#1172356211yescharacteristic
314C2#1172358311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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. #11723447C2semidirect
3. #11722177C2semidirect
4. C32×#808C2semidirect
5. #11722051C2semidirect
32. C3×#808D3semidirect
96. #808C3⋊D3semidirect
193. C33#478semidirect
272. C32#4871semidirect
273. C32#5009semidirect
309. C3#10590545semidirect
310. C3#10592570semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117234472
#117221772
C32×#8082
#117220512
#117220342
#117220332
#105925703
#105905453