Finite Group Databasebeta

← C3×#10592483 #13162821

Subgroups — C3×#10592483 #13162821

Order 1728.

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

normal subgroups only

This group has 836 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×#64147C211yesmaximal, characteristic
2C3×#64101C211yesmaximal, characteristic, complemented
3C33×#102C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63645C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592483C311yesmaximal, characteristic, direct product factor
9C3×#4784—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3757—21no—
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64147C2×C311yescharacteristic
22#64101C2×C311yescharacteristic, complemented
23C3×#938—34no—
24C3×#892—34no—
25C32×#102—24no—
26C32×#102D314yescomplemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63645C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×#832—34no—
33C3×(C2×D3)⋊C8—34no—
34C3×#818—34no—
35C3×#808—34no—
36C23×C33—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—42no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×#482—91no—
51#4784—34no—
52C2×C32×D4—24no—
53C2×C32×D4C2×D314yes—
54C2×C32×D4C22×C311yescharacteristic
55C2×#638—21no—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—64no—
58C2×C3×C4×D3—34no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C22×C311yescharacteristic
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C22×C311yescharacteristic
65#3757—21no—
66#3752—21no—
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C3×#217—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3—64no—
76C2×C3×C3⋊C8—34no—
77C22×C33—41no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—21no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3—41no—
90C3×#121—91no—
91C3×#117—91no—
92#938—34no—
93#892—34no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#102—24no—
97C3×#102—91no—
98C3×#102C3×D314yescomplemented
99C3×#102C3⋊D311yescharacteristic, complemented
100C3×C22⋊C8—91no—
101#832—34no—
102(C2×D3)⋊C8—34no—
103#818—34no—
104#808—34no—
105C23×C32—21no—
106C23×C32—24no—
107C23×C32—44no—
108C22×C3⋊D3C3×D411yescharacteristic
109C22×C3×D3—34no—
110C32×D4—28no—
111C32×D4—24no—
112C32×D4—21no—
113C32×D4—21no—
114C32×D4—24no—
115C32×D4—44no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—44no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C23×C311yescharacteristic
122#638—42no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3C3×D411yescharacteristic
126C4×C3⋊D3—21no—
127C4×C3⋊D3—21no—
128C3×#67—128no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C32×C8—21no—
135C32×C8—24no—
136C32×C8—44no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—64no—
139#482—91noSylow
140C2×C33—81no—
141C2×C33#12311yescharacteristic
142C2×C33#13511yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C3×C3⋊D3—41no—
145C2×C3×D4—24no—
146C2×C3×D4—181no—
147C2×C3×D4C2×C3×D314yes—
148C2×C3×D4C2×C3⋊D311yescharacteristic
149C22×C3×C4—91no—
150C2×#67—64no—
151C2×C4×D3—34no—
152C2×C3×C8—24no—
153C2×C3×C8—91no—
154C2×C3×C8C2×C3×D314yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—24no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4C2×C3×D314yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161C3×C22⋊C4—181no—
162C3×C22⋊C4—181no—
163C3×C42—181no—
164#217—64no—
165(C2×D3)⋊C4—64no—
166C4⋊Dic3—34no—
167C4⋊Dic3—34no—
168C4×Dic3—64no—
169C2×C3⋊C8—34no—
170C22×C32—24no—
171C22×C32—48no—
172C22×C32—44no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—44no—
177C22×C32—44no—
178C22×C32—84no—
179C22×C32—84no—
180C22×C32D3×D414yes—
181C22×C32C2×C3×D411yescharacteristic
182C2×C3⋊D3—21no—
183C2×C3⋊D3—21no—
184C2×C3⋊D3—21no—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—41no—
191C32×C4—44no—
192C32×C4—84no—
193C32×C4#23514yes—
194C32×C4#23714yes—
195C32×C4C3×#4111yescharacteristic
196C32×C4C3×#4111yescharacteristic
197C3⋊Dic3—21no—
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3×Dic3—64no—
201C3×Dic3—124no—
202C3×Dic3—124no—
203#121—91no—
204#117—91no—
205#107—91no—
206#106—91no—
207#102—91no—
208#102C3×C3⋊D311yescharacteristic, complemented
209C22⋊C8—91no—
210C33#48211yescharacteristic, complemented, Sylow
211C23×C3—24no—
212C23×C3—21no—
213C23×C3—44no—
214C23×C3—91no—
215C22×D3—34no—
216C3×D4—28no—
217C3×D4—24no—
218C3×D4—24no—
219C3×D4—21no—
220C3×D4—21no—
221C3×D4—44no—
222C3×D4—362no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—44no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C22×C3×D314yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234#67—128no—
235C2×Dic3—34no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C4×D3—64no—
239C4×D3—64no—
240C3×C8—24no—
241C3×C8—21no—
242C3×C8—44no—
243C3×C8—181no—
244C3⋊C8—64no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—88no—
249C2×C32—84no—
250C2×C32—81no—
251C2×C32#88314yes—
252C2×C32#91214yes—
253C2×C32#91414yes—
254C2×C32C3×#12311yescharacteristic
255C2×C32C3×#13511yescharacteristic
256C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
257C3⋊D3—41no—
258C3×D3—124no—
259C2×D4—181no—
260C2×D4C2×C3×C3⋊D311yescharacteristic
261C22×C4—91no—
262C2×C8—91no—
263C2×C8C2×C3×C3⋊D311yescharacteristic
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4C2×C3×C3⋊D311yescharacteristic
267C22⋊C4—181no—
268C22⋊C4—181no—
269C42—181no—
270C22×C3—24no—
271C22×C3—48no—
272C22×C3—44no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—84no—
279C22×C3—84no—
280C22×C3—181no—
281C22×C3—181no—
282C22×C3—181no—
283C22×C3C3×D3×D414yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C2×D3—64no—
286C2×D3—64no—
287C2×D3—64no—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—44no—
291C3×C4—41no—
292C3×C4—84no—
293C3×C4—181no—
294C3×C4—361no—
295C3×C4—361no—
296C3×C4C3×#23514yes—
297C3×C4C3×#23714yes—
298C3×C4#382811yescharacteristic
299C3×C4#383011yescharacteristic
300Dic3—64no—
301Dic3—124no—
302Dic3—124no—
303C32—24no—
304C32#478414yescomplemented
305C32C3×#48211yescharacteristic, complemented
306C23—21no—
307C23—91no—
308D4—21no—
309D4—21no—
310D4—362no—
311C2×C4—21no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C22×C3×C3⋊D311yescharacteristic
318C8—21no—
319C8—181no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—88no—
324C2×C3—84no—
325C2×C3—81no—
326C2×C3—361no—
327C2×C3C3×#88314yes—
328C2×C3C3×#91214yes—
329C2×C3C3×#91414yes—
330C2×C3#6409211yescharacteristic
331C2×C3#6412111yescharacteristic
332C2×C3#6412311yescharacteristic
333D3—124no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—181no—
338C22—181no—
339C22—181no—
340C22C3×D4×C3⋊D311yescharacteristic
341C4—41no—
342C4—181no—
343C4—361no—
344C4—361no—
345C4C3×#382811yescharacteristic
346C4C3×#383011yescharacteristic
347C3—24no—
348C3C3×#478414yescomplemented
349C3#1059248311yescharacteristic, direct product factor
350C2—81no—
351C2—361no—
352C2C3×#6409211yescharacteristic
353C2C3×#6412111yescharacteristic
354C2C3×#6412311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C3×#64101C2semidirect
3. C33×#102C2semidirect
4. C3×#63659C2semidirect
5. C3×#63647C2semidirect
6. C3×#63645C2semidirect
7. C3×#63635C2semidirect
8. #10592483C3direct (N × H)
22. #64101C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63647C2×C3semidirect
30. #63645C2×C3semidirect
31. #63635C2×C3semidirect
98. C3×#102C3×D3semidirect
99. C3×#102C3⋊D3semidirect
208. #102C3×C3⋊D3semidirect
210. C33#482semidirect
304. C32#4784semidirect
305. C32C3×#482semidirect
348. C3C3×#4784semidirect
349. C3#10592483direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641472
C3×#641012
C33×#1022
C3×#636592
C3×#636472
C3×#636452
C3×#636352
#105924833
C3×#47843