Finite Group Databasebeta

← C3×#5629 #10595726

Subgroups — C3×#5629 #10595726

Order 576.

625 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#239C211yesmaximal, characteristic, complemented
2C2×C3×#235C211yesmaximal, characteristic, complemented
3C32×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#946C211yesmaximal, characteristic, complemented
5C3×#930C211yesmaximal, characteristic, complemented
6C3×#896C212yesmaximal, complemented
7C3×#895C211yesmaximal, characteristic, complemented
8C3×#895C212yesmaximal, complemented
9C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
10C3×#884C212yesmaximal, complemented
11C3×#884C212yesmaximal, complemented
12C3×#537—31nomaximal
13#5629C311yesmaximal, characteristic, direct product factor
14C2×C32×Q8C2211yescharacteristic
15C22×C32×C4C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C2×C3×#67C2212yes—
19C3×#239—24no—
20C3×#235—24no—
21C2×C3×D12C2212yes—
22C2×C3×D12C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic
24C2×C3×D12C2212yes—
25C2×C3×C4×D3C2212yes—
26C2×C3×C4×D3C2211yescharacteristic
27C2×C3×C4×D3C2212yes—
28C2×C3×C4×D3C2211yescharacteristic
29C2×C3×Dic6C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C32×C4⋊C4C2212yes—
32C32×C22⋊C4C2212yes—
33C3×(C2×D3)⋊C4C2212yes—
34C3×(C2×D3)⋊C4C2212yes—
35C3×(C2×D3)⋊C4C2212yes—
36C3×C4⋊Dic3C2211yescharacteristic
37C3×Dic3⋊C4C2212yes—
38C3×Dic3⋊C4C2212yes—
39C2×C3×#41—31no—
40C2×C3×#41—31no—
41C2×#239C2×C311yescharacteristic, complemented
42C2×#235C2×C311yescharacteristic, complemented
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—31no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8D311yescharacteristic, complemented
47C3×C22⋊Q8C2×C311yescharacteristic, complemented
48C3×#121—31no—
49C3×#121—31no—
50C3×#121—32no—
51C3×#121—32no—
52C3×#121—32no—
53C2×C3×C4⋊C4—31no—
54#946C2×C311yescharacteristic, complemented
55#930C2×C311yescharacteristic, complemented
56#896C2×C312yescomplemented
57#895C2×C311yescharacteristic, complemented
58#895C2×C312yescomplemented
59D3×C4⋊C4C2×C311yescharacteristic, complemented
60#884C2×C312yescomplemented
61#884C2×C312yescomplemented
62C23×C32C2311yescharacteristic
63C22×C3×D3C2312yes—
64C22×C3×D3C2311yescharacteristic
65C22×C3×D3C2311yescharacteristic
66C32×Q8—22no—
67C2×C32×C4—22no—
68C2×C32×C4C2312yes—
69C2×C32×C4C2312yes—
70C2×C32×C4C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C3×#67—22no—
73C3×#67—22no—
74C3×#67—42no—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C3×D12—22no—
79C3×D12—24no—
80C3×D12—22no—
81C3×D12—42no—
82C3×C4×D3—22no—
83C3×C4×D3—24no—
84C3×C4×D3—42no—
85C3×C4×D3D414yescomplemented
86C3×Dic6—22no—
87#537—31noSylow
88C3×#41—64no—
89C3×#41—64no—
90C2×C3×Q8—21no—
91C2×C3×Q8—31no—
92C2×C3×Q8C2×D311yescharacteristic
93C2×C3×Q8C22×C311yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—31no—
96C2×C3×D4—32no—
97C2×C3×D4—31no—
98C2×C3×D4—32no—
99C2×C3×D4—31no—
100C2×C3×D4—32no—
101C22×C3×C4—21no—
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×C3×C4—31no—
105C22×C3×C4—32no—
106C22×C3×C4C2×D311yescharacteristic
107C22×C3×C4C22×C311yescharacteristic
108C2×#67C22×C311yescharacteristic
109C2×#67C22×C311yescharacteristic
110C2×#67C22×C312yes—
111#239—24no—
112#235—24no—
113C2×D12C22×C312yes—
114C2×D12C22×C311yescharacteristic
115C2×D12C22×C311yescharacteristic
116C2×D12C22×C312yes—
117C2×C4×D3C22×C312yes—
118C2×C4×D3C22×C311yescharacteristic
119C2×C4×D3C22×C312yes—
120C2×C4×D3C22×C311yescharacteristic
121C2×Dic6C22×C311yescharacteristic
122C3×C4⋊C4—21no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4C2×D311yescharacteristic
128C3×C4⋊C4C2×D312yes—
129C3×C4⋊C4C22×C311yescharacteristic
130C3×C4⋊C4C22×C312yes—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—32no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—32no—
135C3×C22⋊C4C2×D312yes—
136C3×C22⋊C4C22×C312yes—
137(C2×D3)⋊C4C22×C312yes—
138(C2×D3)⋊C4C22×C312yes—
139(C2×D3)⋊C4C22×C312yes—
140C4⋊Dic3C22×C311yescharacteristic
141Dic3⋊C4C22×C312yes—
142Dic3⋊C4C22×C312yes—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32C2411yescharacteristic
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—22no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—22no—
153C2×C3×D3—22no—
154C2×C3×D3—21no—
155C2×C3×D3C2×D412yes—
156C32×C4—21no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4C2×D412yes—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3C2×D412yes—
163C2×#41—31no—
164C2×#41—31no—
165C22⋊Q8—31no—
166C22⋊Q8—32no—
167C22⋊Q8C3×D311yescharacteristic, complemented
168#121—31no—
169#121—31no—
170#121—32no—
171#121—32no—
172#121—32no—
173C2×C4⋊C4—31no—
174C23×C3—21no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—32no—
178C23×C3C22×D311yescharacteristic
179C23×C3C23×C311yescharacteristic
180C22×D3C23×C312yes—
181C22×D3C23×C311yescharacteristic
182C22×D3C23×C311yescharacteristic
183C3×Q8—22no—
184C3×Q8—22no—
185C3×Q8—42no—
186C3×Q8—62no—
187C3×D4—62no—
188C3×D4—64no—
189C3×D4—62no—
190C3×D4—62no—
191C3×D4—62no—
192C3×D4—122no—
193C3×D4—122no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—31no—
201C2×C3×C4—34no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—42no—
205C2×C3×C4—64no—
206C2×C3×C4—62no—
207C2×C3×C4—122no—
208C2×C3×C4C22×D311yescharacteristic
209C2×C3×C4C22×D312yes—
210C2×C3×C4C22×D312yes—
211C2×C3×C4C22×D311yescharacteristic
212C2×C3×C4C23×C312yes—
213C2×C3×C4C23×C312yes—
214C2×C3×C4C23×C311yescharacteristic
215C2×C3×C4C23×C311yescharacteristic
216#67—22no—
217#67—22no—
218#67—42no—
219C2×Dic3C23×C312yes—
220C2×Dic3C23×C311yescharacteristic
221C2×Dic3C23×C311yescharacteristic
222D12—22no—
223D12—24no—
224D12—22no—
225D12—42no—
226C4×D3—22no—
227C4×D3—24no—
228C4×D3—42no—
229C4×D3C3×D414yescomplemented
230Dic6—22no—
231C2×C32—41no—
232C2×C32C22×D411yescharacteristic
233C2×C32H(5,𝔽2)11yescharacteristic
234C2×C322−1+411yescharacteristic
235C3×D3—22no—
236C3×D3—42no—
237C3×D3—41no—
238#41—64no—
239#41—64no—
240C2×Q8—31no—
241C2×Q8C2×C3×D311yescharacteristic
242C2×D4—31no—
243C2×D4—31no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—32no—
247C2×D4—31no—
248C2×D4—32no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—31no—
252C22×C4—32no—
253C22×C4C2×C3×D311yescharacteristic
254C4⋊C4—32no—
255C4⋊C4—31no—
256C4⋊C4—32no—
257C4⋊C4C2×C3×D311yescharacteristic
258C4⋊C4C2×C3×D312yes—
259C22⋊C4—32no—
260C22⋊C4—32no—
261C22⋊C4—32no—
262C22⋊C4C2×C3×D312yes—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—32no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—62no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—62no—
280C22×C3—61no—
281C22×C3—62no—
282C22×C3C23×D311yescharacteristic
283C22×C3C24×C311yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—21no—
292C2×D3C2×C3×D412yes—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—32no—
301C3×C4—41no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—61no—
305C3×C4—62no—
306C3×C4D3×D412yes—
307C3×C4C2×C3×D412yes—
308Dic3—21no—
309Dic3—22no—
310Dic3C2×C3×D412yes—
311C32#53711yescharacteristic, complemented, Sylow
312C23—31no—
313C23—31no—
314C23—32no—
315C23C22×C3×D311yescharacteristic
316Q8—22no—
317Q8—62no—
318D4—62no—
319D4—64no—
320D4—62no—
321D4—62no—
322D4—62no—
323D4—122no—
324D4—122no—
325C2×C4—22no—
326C2×C4—31no—
327C2×C4—34no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—64no—
331C2×C4—62no—
332C2×C4—122no—
333C2×C4C22×C3×D311yescharacteristic
334C2×C4C22×C3×D312yes—
335C2×C4C22×C3×D312yes—
336C2×C4C22×C3×D311yescharacteristic
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—42no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—62no—
344C2×C3—122no—
345C2×C3—121no—
346C2×C3C2×D3×D411yescharacteristic
347C2×C3#100711yescharacteristic
348C2×C3#100911yescharacteristic
349C2×C3C22×C3×D411yescharacteristic
350C2×C3C3×H(5,𝔽2)11yescharacteristic
351C2×C3C3×2−1+411yescharacteristic
352D3—22no—
353D3—42no—
354D3—41no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—32no—
359C22—61no—
360C22—61no—
361C22—62no—
362C22—61no—
363C22—61no—
364C22—62no—
365C22—61no—
366C22—62no—
367C22C23×C3×D311yescharacteristic
368C4—21no—
369C4—22no—
370C4—22no—
371C4—32no—
372C4—61no—
373C4—62no—
374C4C3×D3×D412yes—
375C3—21no—
376C3#562911yescharacteristic, direct product factor
377C3C3×#53711yescharacteristic, complemented
378C2—41no—
379C2—62no—
380C2—122no—
381C2—121no—
382C2C2×C3×D3×D411yescharacteristic
383C2C3×#100711yescharacteristic
384C2C3×#100911yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×C22⋊Q8C2
socleC22×C32C24
hypercenterC3×C22⋊Q8D3
hypocenterC3C3×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#239C2semidirect
2. C2×C3×#235C2semidirect
3. C32×C22⋊Q8C2semidirect
4. C3×#946C2semidirect
5. C3×#930C2semidirect
6. C3×#896C2semidirect
7. C3×#895C2semidirect
8. C3×#895C2semidirect
9. C3×D3×C4⋊C4C2semidirect
10. C3×#884C2semidirect
11. C3×#884C2semidirect
13. #5629C3direct (N × H)
41. C2×#239C2×C3semidirect
42. C2×#235C2×C3semidirect
46. C3×C22⋊Q8D3semidirect
47. C3×C22⋊Q8C2×C3semidirect
54. #946C2×C3semidirect
55. #930C2×C3semidirect
56. #896C2×C3semidirect
57. #895C2×C3semidirect
58. #895C2×C3semidirect
59. D3×C4⋊C4C2×C3semidirect
60. #884C2×C3semidirect
61. #884C2×C3semidirect
85. C3×C4×D3D4semidirect
167. C22⋊Q8C3×D3semidirect
229. C4×D3C3×D4semidirect
311. C32#537semidirect
376. C3#5629direct (N × H)
377. C3C3×#537semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2392
C2×C3×#2352
C32×C22⋊Q82
C3×#9462
C3×#9302
C3×#8962
C3×#8952
C3×D3×C4⋊C42
C3×#8842
C3×#5373
#56293