Finite Group Databasebeta

← C3×#10592441 #13162780

Subgroups — C3×#10592441 #13162780

Order 1728.

1168 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64361C211yesmaximal, characteristic
2C3×#64144C211yesmaximal, characteristic, complemented
3C3×#64095C212yesmaximal, complemented
4C33×#99C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6#10592441C311yesmaximal, characteristic, direct product factor
7C3×#4742—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C2×C3×#638C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
12C3×#3830—22no—
13C3×D4×C3⋊D3—22no—
14C3×#3828—21no—
15C33×C22⋊C4C2212yescomplemented
16C3×#3757—21no—
17C3×#3757C2211yescharacteristic, complemented
18C3×#3752—22no—
19C3×C4⋊(C3⋊Dic3)—22no—
20C3×#3750—21no—
21#64361C2×C311yescharacteristic
22C3×#1004—34no—
23#64144C2×C311yescharacteristic, complemented
24#64095C2×C312yescomplemented
25C3×#935—34no—
26C3×#886—38no—
27C32×#99—24no—
28C32×#99D314yescomplemented
29C32×#99C2×C311yescharacteristic, complemented
30#63633C2×C312yescomplemented
31C3×#806—38no—
32C23×C33D411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3D412yes—
36C33×D4—22no—
37C2×C33×C4—22no—
38C2×C33×C4D411yescharacteristic
39C3×#638—22no—
40C3×#638—22no—
41C3×#638—22no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3D412yes—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4×C3⋊D3—22no—
49C3×#634—21no—
50C3×#458—91no—
51#4742—34no—
52C2×C32×D4—24no—
53C2×C32×D4C2×D314yes—
54C2×C32×D4C22×C311yescharacteristic
55C2×#638—21no—
56C2×#638C22×C312yes—
57C22×C3⋊Dic3C22×C311yescharacteristic, complemented
58#3830—22no—
59D4×C3⋊D3—22no—
60#3828—21no—
61C2×C3×#67—38no—
62C2×C3×#67—64no—
63C22×C3×Dic3—34no—
64C3×#237—68no—
65C3×D3×D4—68no—
66C3×#235—64no—
67C32×C22⋊C4—28no—
68C32×C22⋊C4C2×D318yescomplemented
69C32×C22⋊C4C22×C312yescomplemented
70#3757—21no—
71#3757C22×C311yescharacteristic, complemented
72#3752—22no—
73C4⋊(C3⋊Dic3)—22no—
74#3750—21no—
75C3×#217—34no—
76C3×#217—64no—
77C3×(C2×D3)⋊C4—68no—
78C3×C4⋊Dic3—68no—
79C3×Dic3⋊C4—64no—
80C22×C33—21no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33C2×D411yescharacteristic
84C22×C33C2×D412yes—
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—42no—
87C2×C3×C3⋊D3—81no—
88C33×C4—21no—
89C33×C4—42no—
90C3×C3⋊Dic3—22no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3—41no—
93C3×H(5,𝔽2)—91no—
94#1004—34no—
95C3×#123—91no—
96C3×#123—92no—
97#935—34no—
98#886—38no—
99C3×#99—24no—
100C3×#99—92no—
101C3×#99C3×D314yescomplemented
102C3×#99C3⋊D311yescharacteristic, complemented
103#806—38no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32D1214yes—
107C23×C32C3×D411yescharacteristic
108C23×C32C22×D314yes—
109C23×C32C23×C311yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3⋊D3C3×D412yes—
112C22×C3×D3—38no—
113C22×C3×D3—64no—
114C32×D4—22no—
115C32×D4—28no—
116C32×D4—48no—
117C2×C32×C4—22no—
118C2×C32×C4—24no—
119C2×C32×C4—28no—
120C2×C32×C4—48no—
121C2×C32×C4D1214yes—
122C2×C32×C4C3×D411yescharacteristic
123#638—22no—
124#638—22no—
125#638—22no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3C3×D412yes—
131C4⋊(C3⋊D3)—21no—
132C4×C3⋊D3—22no—
133#634—21no—
134C3×#67—68no—
135C3×#67—68no—
136C3×#67—68no—
137C2×C3×Dic3—38no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—68no—
142C3×D12—64no—
143C3×C4×D3—68no—
144C3×Dic6—64no—
145#458—91noSylow
146C2×C33—21no—
147C2×C33—22no—
148C2×C33—41no—
149C2×C33#12011yescharacteristic
150C3×C3⋊D3—42no—
151C3×#41—182no—
152C3×#41—181no—
153C2×C3×D4—24no—
154C2×C3×D4—92no—
155C2×C3×D4—182no—
156C2×C3×D4—181no—
157C2×C3×D4C2×C3×D314yes—
158C2×C3×D4C2×C3⋊D311yescharacteristic
159C22×C3×C4—91no—
160C2×#67—38no—
161C2×#67—64no—
162C22×Dic3—34no—
163#237—68no—
164D3×D4—68no—
165#235—64no—
166C3×C4⋊C4—182no—
167C3×C4⋊C4—181no—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—182no—
171C3×C22⋊C4—181no—
172C3×C22⋊C4C2×C3×D318yescomplemented
173C3×C22⋊C4C2×C3⋊D312yescomplemented
174#217—34no—
175#217—64no—
176(C2×D3)⋊C4—68no—
177C4⋊Dic3—68no—
178Dic3⋊C4—64no—
179C22×C32—21no—
180C22×C32—24no—
181C22×C32—28no—
182C22×C32—24no—
183C22×C32—44no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32—84no—
189C22×C32—84no—
190C22×C32C2×D1214yes—
191C22×C32D3×D418yes—
192C22×C32C2×C3×D411yescharacteristic
193C22×C32C2×C3×D412yes—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—42no—
196C2×C3⋊D3—81no—
197C2×C3×D3—68no—
198C2×C3×D3—128no—
199C2×C3×D3—244no—
200C32×C4—21no—
201C32×C4—24no—
202C32×C4—44no—
203C32×C4—42no—
204C32×C4—48no—
205C32×C4—88no—
206C3⋊Dic3—22no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—68no—
210C3×Dic3—124no—
211C3×Dic3—124no—
212H(5,𝔽2)—91no—
213#123—91no—
214#123—92no—
215#99—92no—
216#99C3×C3⋊D311yescharacteristic, complemented
217C33#45811yescharacteristic, complemented, Sylow
218C23×C3—24no—
219C23×C3—24no—
220C23×C3—92no—
221C23×C3—181no—
222C23×C3C3×D1214yes—
223C23×C3C4⋊(C3⋊D3)11yescharacteristic
224C23×C3C22×C3×D314yes—
225C23×C3C22×C3⋊D311yescharacteristic
226C22×D3—38no—
227C22×D3—64no—
228C3×Q8—181no—
229C3×D4—28no—
230C3×D4—22no—
231C3×D4—48no—
232C3×D4—181no—
233C3×D4—182no—
234C3×D4—182no—
235C3×D4—182no—
236C2×C3×C4—28no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—48no—
240C2×C3×C4—92no—
241C2×C3×C4—182no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—182no—
246C2×C3×C4C3×D1214yes—
247C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
248#67—68no—
249#67—68no—
250#67—68no—
251C2×Dic3—38no—
252C2×Dic3—64no—
253C2×Dic3—64no—
254C2×Dic3—64no—
255C2×Dic3—68no—
256D12—64no—
257C4×D3—68no—
258Dic6—64no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—28no—
264C2×C32—48no—
265C2×C32—44no—
266C2×C32—41no—
267C2×C32—44no—
268C2×C32—84no—
269C2×C32#88214yes—
270C2×C32C3×#12011yescharacteristic
271C3⋊D3—42no—
272C3×D3—128no—
273#41—182no—
274#41—181no—
275C2×D4—92no—
276C2×D4—182no—
277C2×D4—181no—
278C2×D4C2×C3×C3⋊D311yescharacteristic
279C22×C4—91no—
280C4⋊C4—182no—
281C4⋊C4—181no—
282C22⋊C4—91no—
283C22⋊C4—182no—
284C22⋊C4—181no—
285C22⋊C4C2×C3×C3⋊D312yescomplemented
286C22×C3—24no—
287C22×C3—21no—
288C22×C3—24no—
289C22×C3—28no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—44no—
294C22×C3—41no—
295C22×C3—84no—
296C22×C3—84no—
297C22×C3—182no—
298C22×C3—362no—
299C22×C3—721no—
300C22×C3C2×C3×D1214yes—
301C22×C3C3×D3×D418yes—
302C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
303C22×C3D4×C3⋊D312yes—
304C2×D3—68no—
305C2×D3—128no—
306C2×D3—244no—
307C3×C4—24no—
308C3×C4—21no—
309C3×C4—44no—
310C3×C4—42no—
311C3×C4—48no—
312C3×C4—88no—
313C3×C4—182no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—68no—
317Dic3—124no—
318Dic3—124no—
319C32—24no—
320C32#474214yescomplemented
321C32C3×#45811yescharacteristic, complemented
322C23—92no—
323C23—181no—
324C23C3×C4⋊(C3⋊D3)11yescharacteristic
325C23C22×C3×C3⋊D311yescharacteristic
326Q8—181no—
327D4—22no—
328D4—181no—
329D4—182no—
330D4—182no—
331D4—182no—
332C2×C4—22no—
333C2×C4—92no—
334C2×C4—182no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—182no—
339C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
340C2×C3—24no—
341C2×C3—28no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—24no—
345C2×C3—48no—
346C2×C3—44no—
347C2×C3—44no—
348C2×C3—41no—
349C2×C3—84no—
350C2×C3—362no—
351C2×C3C3×#88214yes—
352C2×C3#6409111yescharacteristic
353D3—128no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—182no—
358C22—362no—
359C22—721no—
360C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
361C22C3×D4×C3⋊D312yes—
362C4—21no—
363C4—42no—
364C4—182no—
365C4—361no—
366C4—361no—
367C3—24no—
368C3C3×#474214yescomplemented
369C3#1059244111yescharacteristic, direct product factor
370C2—22no—
371C2—21no—
372C2—41no—
373C2—362no—
374C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC3×#99C3⋊D3
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. C3×#64095C2semidirect
4. C33×#99C2semidirect
5. C3×#63633C2semidirect
6. #10592441C3direct (N × H)
11. C22×C3×C3⋊Dic3C22semidirect
15. C33×C22⋊C4C22semidirect
17. C3×#3757C22semidirect
23. #64144C2×C3semidirect
24. #64095C2×C3semidirect
28. C32×#99D3semidirect
29. C32×#99C2×C3semidirect
30. #63633C2×C3semidirect
57. C22×C3⋊Dic3C22×C3semidirect
68. C32×C22⋊C4C2×D3semidirect
69. C32×C22⋊C4C22×C3semidirect
71. #3757C22×C3semidirect
101. C3×#99C3×D3semidirect
102. C3×#99C3⋊D3semidirect
172. C3×C22⋊C4C2×C3×D3semidirect
173. C3×C22⋊C4C2×C3⋊D3semidirect
216. #99C3×C3⋊D3semidirect
217. C33#458semidirect
285. C22⋊C4C2×C3×C3⋊D3semidirect
320. C32#4742semidirect
321. C32C3×#458semidirect
368. C3C3×#4742semidirect
369. C3#10592441direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C3×#641442
C3×#640952
C33×#992
C3×#636332
#105924413
C3×#47423