Finite Group Databasebeta

← C3×#10592565 #13162903

Subgroups — C3×#10592565 #13162903

Order 1728.

780 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64104C211yesmaximal, characteristic
2C3×#64101C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592565C311yesmaximal, characteristic, direct product factor
9C3×#4866—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64104C2×C311yescharacteristic
22#64101C2×C311yescharacteristic, complemented
23C3×#895—34no—
24C3×#892—34no—
25C32×#106—24no—
26C32×#106D314yescomplemented
27C32×#106C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#821—34no—
33C3×(C2×D3)⋊C8—34no—
34C3×#809—34no—
35C3×#807—34no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—42no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×#482—91no—
51#4866—34no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×C3×D12—34no—
56C2×C3×D12—64no—
57C2×C3×C4×D3—34no—
58C2×C32×C8—24no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4—24no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67#3752—21no—
68#3752—21no—
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4×C3⋊Dic3—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×(C2×D3)⋊C4—64no—
73C3×(C2×D3)⋊C4—64no—
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3—64no—
76C2×C3×C3⋊C8—34no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—41no—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—41no—
90C3×#121—91no—
91C3×#117—91no—
92#895—34no—
93#892—34no—
94C3×#107—91no—
95C3×#106—24no—
96C3×#106C3×D314yescomplemented
97C3×#106C3⋊D311yescharacteristic, complemented
98C3×#102—91no—
99C3×#102—91no—
100C3×C22⋊C8—91no—
101#821—34no—
102(C2×D3)⋊C8—34no—
103#809—34no—
104#807—34no—
105C22×C3⋊D3—21no—
106C22×C3⋊D3C3×D411yescharacteristic
107C22×C3×D3—34no—
108C22×C3×D3—64no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—44no—
115C2×C32×C4—44no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C23×C311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3C3×D411yescharacteristic
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—42no—
123C4×C3⋊D3—21no—
124C4×C3⋊D3—21no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×D12—64no—
128C3×D12—64no—
129C3×D12—128no—
130C3×C4×D3—64no—
131C3×C4×D3—64no—
132C32×C8—28no—
133C32×C8—24no—
134C32×C8—21no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—64no—
137#482—91noSylow
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—41no—
142C3×C3⋊D3—81no—
143C2×C3×D4—91no—
144C2×C3×D4—181no—
145C22×C3×C4—91no—
146C2×D12—34no—
147C2×D12—64no—
148C2×C4×D3—34no—
149C2×C3×C8—24no—
150C2×C3×C8—91no—
151C2×C3×C8C2×C3×D314yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4C2×C3×D314yes—
157C3×C4⋊C4C2×C3×D314yes—
158C3×C4⋊C4C2×C3⋊D311yescharacteristic
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160C3×C22⋊C4—181no—
161C3×C22⋊C4—181no—
162C3×C42—181no—
163(C2×D3)⋊C4—64no—
164(C2×D3)⋊C4—64no—
165C4⋊Dic3—34no—
166C4×Dic3—64no—
167C2×C3⋊C8—34no—
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32C2×C3×D411yescharacteristic
171C2×C3⋊D3—21no—
172C2×C3⋊D3—21no—
173C2×C3⋊D3—21no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—64no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—124no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—44no—
188C32×C4—44no—
189C32×C4—84no—
190C32×C4—84no—
191C32×C4#23514yes—
192C32×C4#23914yes—
193C32×C4C3×#4111yescharacteristic
194C32×C4C3×#4111yescharacteristic
195C3⋊Dic3—21no—
196C3⋊Dic3—41no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199#121—91no—
200#117—91no—
201#107—91no—
202#106C3×C3⋊D311yescharacteristic, complemented
203#102—91no—
204#102—91no—
205C22⋊C8—91no—
206C33#48211yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C23×C3—181no—
209C22×D3—34no—
210C22×D3—64no—
211C3×D4—181no—
212C3×D4—181no—
213C3×D4—362no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4—91no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C22×C3×D314yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—34no—
228C2×Dic3—64no—
229D12—64no—
230D12—64no—
231D12—128no—
232C4×D3—64no—
233C4×D3—64no—
234C3×C8—28no—
235C3×C8—24no—
236C3×C8—21no—
237C3×C8—181no—
238C3⋊C8—64no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32#89414yes—
243C2×C32#91414yes—
244C2×C32#91614yes—
245C2×C32C3×#12311yescharacteristic
246C2×C32C3×#13511yescharacteristic
247C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
248C3⋊D3—41no—
249C3⋊D3—81no—
250C3×D3—124no—
251C3×D3—244no—
252C2×D4—91no—
253C2×D4—181no—
254C22×C4—91no—
255C2×C8—91no—
256C2×C8C2×C3×C3⋊D311yescharacteristic
257C4⋊C4—91no—
258C4⋊C4C2×C3×C3⋊D311yescharacteristic
259C4⋊C4C2×C3×C3⋊D311yescharacteristic
260C22⋊C4—181no—
261C22⋊C4—181no—
262C42—181no—
263C22×C3—24no—
264C22×C3—181no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3C3×D3×D414yes—
271C22×C3D4×C3⋊D311yescharacteristic
272C2×D3—64no—
273C2×D3—64no—
274C2×D3—64no—
275C2×D3—124no—
276C2×D3—124no—
277C2×D3—124no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—44no—
281C3×C4—44no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—84no—
285C3×C4—84no—
286C3×C4—181no—
287C3×C4—361no—
288C3×C4C3×#23514yes—
289C3×C4C3×#23914yes—
290C3×C4#382811yescharacteristic
291C3×C4#383211yescharacteristic
292Dic3—64no—
293Dic3—124no—
294C32—24no—
295C32#486614yescomplemented
296C32C3×#48211yescharacteristic, complemented
297C23—91no—
298C23—181no—
299D4—181no—
300D4—181no—
301D4—362no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—91no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4C22×C3×C3⋊D311yescharacteristic
309C8—21no—
310C8—181no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—361no—
315C2×C3—721no—
316C2×C3C3×#89414yes—
317C2×C3C3×#91414yes—
318C2×C3C3×#91614yes—
319C2×C3#6410311yescharacteristic
320C2×C3#6412311yescharacteristic
321C2×C3#6412511yescharacteristic
322D3—124no—
323D3—244no—
324C22—181no—
325C22—181no—
326C22—181no—
327C22—361no—
328C22—361no—
329C22—361no—
330C22C3×D4×C3⋊D311yescharacteristic
331C4—41no—
332C4—41no—
333C4—181no—
334C4—361no—
335C4C3×#382811yescharacteristic
336C4C3×#383211yescharacteristic
337C3—24no—
338C3C3×#486614yescomplemented
339C3#1059256511yescharacteristic, direct product factor
340C2—361no—
341C2—721no—
342C2C3×#6410311yescharacteristic
343C2C3×#6412311yescharacteristic
344C2C3×#6412511yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
2. C3×#64101C2semidirect
3. C33×#106C2semidirect
4. C3×#63648C2semidirect
5. C3×#63647C2semidirect
6. C3×#63636C2semidirect
7. C3×#63634C2semidirect
8. #10592565C3direct (N × H)
22. #64101C2×C3semidirect
26. C32×#106D3semidirect
27. C32×#106C2×C3semidirect
28. #63648C2×C3semidirect
29. #63647C2×C3semidirect
30. #63636C2×C3semidirect
31. #63634C2×C3semidirect
96. C3×#106C3×D3semidirect
97. C3×#106C3⋊D3semidirect
202. #106C3×C3⋊D3semidirect
206. C33#482semidirect
295. C32#4866semidirect
296. C32C3×#482semidirect
338. C3C3×#4866semidirect
339. C3#10592565direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641042
C3×#641012
C33×#1062
C3×#636482
C3×#636472
C3×#636362
C3×#636342
#105925653
C3×#48663