Finite Group Databasebeta

← C3×#5667 #10595764

Subgroups — C3×#5667 #10595764

Order 576.

479 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#123C211yesmaximal, characteristic, complemented
2C3×#939C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5C3×#924C211yesmaximal, characteristic, complemented
6C3×#897C211yesmaximal, characteristic, complemented
7C3×#896C211yesmaximal, characteristic, complemented
8C3×#894C211yesmaximal, characteristic, complemented
9C3×#890C211yesmaximal, characteristic, complemented
10C3×#886C211yesmaximal, characteristic, complemented
11C3×#885C211yesmaximal, characteristic, complemented
12C3×#885C211yesmaximal, characteristic, complemented
13C3×#884C211yesmaximal, characteristic, complemented
14C3×#883C211yesmaximal, characteristic, complemented
15C3×#878C211yesmaximal, characteristic, complemented
16C3×#562—31nomaximal
17#5667C311yesmaximal, characteristic, direct product factor
18C2×C32×D4C2211yescharacteristic
19C22×C32×C4C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C22×C3×Dic3C2211yescharacteristic
25C2×C3×D12C2211yescharacteristic
26C2×C3×C4×D3C2211yescharacteristic
27C2×C3×C4×D3C2211yescharacteristic
28C2×C3×Dic6C2211yescharacteristic
29C2×C3×Dic6C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C32×C4⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic
33C32×C22⋊C4C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic
44C3×C4⋊Dic3C2211yescharacteristic
45C3×C4⋊Dic3C2211yescharacteristic
46C3×C4⋊Dic3C2211yescharacteristic
47C3×C4⋊Dic3C2211yescharacteristic
48C3×Dic3⋊C4C2211yescharacteristic
49C3×Dic3⋊C4C2211yescharacteristic
50C3×Dic3⋊C4C2211yescharacteristic
51C3×Dic3⋊C4C2211yescharacteristic
52C3×C4×Dic3C2211yescharacteristic
53C3×#125—31no—
54C3×#124—31no—
55C3×#124—31no—
56C3×#123—21no—
57C3×#123—31no—
58C3×#123—31no—
59C3×#123—31no—
60C3×#123D311yescharacteristic, complemented
61C3×#123C2×C311yescharacteristic, complemented
62C3×C22⋊Q8—31no—
63C3×C22⋊Q8—31no—
64C3×C22⋊Q8—31no—
65C3×C22⋊Q8—31no—
66C3×#121—31no—
67C3×#121—31no—
68C3×#121—31no—
69C3×#121—31no—
70#939C2×C311yescharacteristic, complemented
71#938C2×C311yescharacteristic, complemented
72#930C2×C311yescharacteristic, complemented
73#924C2×C311yescharacteristic, complemented
74#897C2×C311yescharacteristic, complemented
75#896C2×C311yescharacteristic, complemented
76#894C2×C311yescharacteristic, complemented
77#890C2×C311yescharacteristic, complemented
78#886C2×C311yescharacteristic, complemented
79#885C2×C311yescharacteristic, complemented
80#885C2×C311yescharacteristic, complemented
81#884C2×C311yescharacteristic, complemented
82#883C2×C311yescharacteristic, complemented
83#878C2×C311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C32×D4—41no—
89C2×C32×C4—41no—
90C2×C32×C4C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C3×#67—41no—
96C3×#67—41no—
97C3×#67—41no—
98C3×#67—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C3×D12—41no—
107C3×C4×D3—41no—
108C3×C4×D3—41no—
109C3×Dic6—41no—
110C3×Dic6—41no—
111#562—31noSylow
112C2×C3×Q8—31no—
113C2×C3×Q8—31no—
114C2×C3×D4—21no—
115C2×C3×D4—31no—
116C2×C3×D4—31no—
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C2×C3×D4—31no—
120C2×C3×D4C2×D311yescharacteristic
121C2×C3×D4C22×C311yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4C2×D311yescharacteristic
127C22×C3×C4C22×C311yescharacteristic
128C2×#67C22×C311yescharacteristic
129C2×#67C22×C311yescharacteristic
130C2×#67C22×C311yescharacteristic
131C2×#67C22×C311yescharacteristic
132C22×Dic3C22×C311yescharacteristic
133C2×D12C22×C311yescharacteristic
134C2×C4×D3C22×C311yescharacteristic
135C2×C4×D3C22×C311yescharacteristic
136C2×Dic6C22×C311yescharacteristic
137C2×Dic6C22×C311yescharacteristic
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4C2×D311yescharacteristic
149C3×C4⋊C4C2×D311yescharacteristic
150C3×C4⋊C4C22×C311yescharacteristic
151C3×C4⋊C4C22×C311yescharacteristic
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—31no—
159C3×C22⋊C4—31no—
160C3×C22⋊C4—31no—
161C3×C22⋊C4—31no—
162C3×C22⋊C4—31no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4C2×D311yescharacteristic
165C3×C22⋊C4C2×D311yescharacteristic
166C3×C22⋊C4C2×D311yescharacteristic
167C3×C22⋊C4C22×C311yescharacteristic
168C3×C22⋊C4C22×C311yescharacteristic
169C3×C22⋊C4C22×C311yescharacteristic
170C3×C42—31no—
171#217C22×C311yescharacteristic
172#217C22×C311yescharacteristic
173#217C22×C311yescharacteristic
174(C2×D3)⋊C4C22×C311yescharacteristic
175(C2×D3)⋊C4C22×C311yescharacteristic
176(C2×D3)⋊C4C22×C311yescharacteristic
177(C2×D3)⋊C4C22×C311yescharacteristic
178(C2×D3)⋊C4C22×C311yescharacteristic
179(C2×D3)⋊C4C22×C311yescharacteristic
180C4⋊Dic3C22×C311yescharacteristic
181C4⋊Dic3C22×C311yescharacteristic
182C4⋊Dic3C22×C311yescharacteristic
183C4⋊Dic3C22×C311yescharacteristic
184Dic3⋊C4C22×C311yescharacteristic
185Dic3⋊C4C22×C311yescharacteristic
186Dic3⋊C4C22×C311yescharacteristic
187Dic3⋊C4C22×C311yescharacteristic
188C4×Dic3C22×C311yescharacteristic
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32C2411yescharacteristic
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C3×Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213#125—31no—
214#124—31no—
215#124—31no—
216#123—31no—
217#123—31no—
218#123—31no—
219#123C3×D311yescharacteristic, complemented
220C22⋊Q8—31no—
221C22⋊Q8—31no—
222C22⋊Q8—31no—
223C22⋊Q8—31no—
224#121—31no—
225#121—31no—
226#121—31no—
227#121—31no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—31no—
231C23×C3—31no—
232C23×C3C22×D311yescharacteristic
233C23×C3C22×D311yescharacteristic
234C23×C3C23×C311yescharacteristic
235C23×C3C23×C311yescharacteristic
236C22×D3C23×C311yescharacteristic
237C22×D3C23×C311yescharacteristic
238C3×Q8—121no—
239C3×Q8—121no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—42no—
243C3×D4—121no—
244C3×D4—121no—
245C3×D4—121no—
246C3×D4—121no—
247C3×D4—121no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—121no—
263C2×C3×C4—121no—
264C2×C3×C4—121no—
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268C2×C3×C4C22×D311yescharacteristic
269C2×C3×C4C22×D311yescharacteristic
270C2×C3×C4C23×C311yescharacteristic
271C2×C3×C4C23×C311yescharacteristic
272C2×C3×C4C23×C311yescharacteristic
273C2×C3×C4C23×C311yescharacteristic
274C2×C3×C4C23×C311yescharacteristic
275#67—41no—
276#67—41no—
277#67—41no—
278#67—41no—
279C2×Dic3—41no—
280C2×Dic3C23×C311yescharacteristic
281C2×Dic3C23×C311yescharacteristic
282C2×Dic3C23×C311yescharacteristic
283C2×Dic3C23×C311yescharacteristic
284C2×Dic3C23×C311yescharacteristic
285C2×Dic3C23×C311yescharacteristic
286D12—41no—
287C4×D3—41no—
288C4×D3—41no—
289Dic6—41no—
290Dic6—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32H(5,𝔽2)11yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C2×C322−1+411yescharacteristic
296C3×D3—41no—
297C3×D3—41no—
298C2×Q8—31no—
299C2×Q8—31no—
300C2×D4—31no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—31no—
305C2×D4C2×C3×D311yescharacteristic
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4C2×C3×D311yescharacteristic
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—31no—
317C4⋊C4—31no—
318C4⋊C4C2×C3×D311yescharacteristic
319C4⋊C4C2×C3×D311yescharacteristic
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—31no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—31no—
328C22⋊C4—31no—
329C22⋊C4C2×C3×D311yescharacteristic
330C22⋊C4C2×C3×D311yescharacteristic
331C22⋊C4C2×C3×D311yescharacteristic
332C42—31no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3C23×D311yescharacteristic
359C22×C3C24×C311yescharacteristic
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—61no—
387Dic3—21no—
388Dic3—21no—
389Dic3—21no—
390Dic3—21no—
391Dic3—21no—
392Dic3—21no—
393C32#56211yescharacteristic, complemented, Sylow
394C23—31no—
395C23—31no—
396C23C22×C3×D311yescharacteristic
397C23C22×C3×D311yescharacteristic
398Q8—121no—
399Q8—121no—
400D4—41no—
401D4—121no—
402D4—121no—
403D4—121no—
404D4—121no—
405D4—121no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—41no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4C22×C3×D311yescharacteristic
417C2×C4C22×C3×D311yescharacteristic
418C2×C4C22×C3×D311yescharacteristic
419C2×C4C22×C3×D311yescharacteristic
420C2×C4C22×C3×D311yescharacteristic
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—42no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—81no—
430C2×C3—121no—
431C2×C3—121no—
432C2×C3#100411yescharacteristic
433C2×C3#100911yescharacteristic
434C2×C3#101011yescharacteristic
435C2×C3C3×H(5,𝔽2)11yescharacteristic
436C2×C3C3×H(5,𝔽2)11yescharacteristic
437C2×C3C3×2−1+411yescharacteristic
438D3—41no—
439D3—41no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22C23×C3×D311yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—61no—
462C4—61no—
463C4—61no—
464C3—21no—
465C3#566711yescharacteristic, direct product factor
466C3C3×#56211yescharacteristic, complemented
467C2—41no—
468C2—41no—
469C2—121no—
470C2—121no—
471C2C3×#100411yescharacteristic
472C2C3×#100911yescharacteristic
473C2C3×#101011yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×#123C2
socleC22×C32C24
hypercenterC3×#123D3
hypocenterC3C3×#562
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C32×#123C2semidirect
2. C3×#939C2semidirect
3. C3×#938C2semidirect
4. C3×#930C2semidirect
5. C3×#924C2semidirect
6. C3×#897C2semidirect
7. C3×#896C2semidirect
8. C3×#894C2semidirect
9. C3×#890C2semidirect
10. C3×#886C2semidirect
11. C3×#885C2semidirect
12. C3×#885C2semidirect
13. C3×#884C2semidirect
14. C3×#883C2semidirect
15. C3×#878C2semidirect
17. #5667C3direct (N × H)
60. C3×#123D3semidirect
61. C3×#123C2×C3semidirect
70. #939C2×C3semidirect
71. #938C2×C3semidirect
72. #930C2×C3semidirect
73. #924C2×C3semidirect
74. #897C2×C3semidirect
75. #896C2×C3semidirect
76. #894C2×C3semidirect
77. #890C2×C3semidirect
78. #886C2×C3semidirect
79. #885C2×C3semidirect
80. #885C2×C3semidirect
81. #884C2×C3semidirect
82. #883C2×C3semidirect
83. #878C2×C3semidirect
219. #123C3×D3semidirect
393. C32#562semidirect
465. C3#5667direct (N × H)
466. C3C3×#562semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
C3×#9392
C3×#9382
C3×#9302
C3×#9242
C3×#8972
C3×#8962
C3×#8942
C3×#8902
C3×#8862
C3×#8852
C3×#8842
C3×#8832
C3×#8782
C3×#5623
#56673