Finite Group Databasebeta

← C3×#5621 #10595718

Subgroups — C3×#5621 #10595718

Order 576.

487 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#121C211yesmaximal, characteristic, complemented
2C3×#939C211yesmaximal, characteristic, complemented
3C3×#939C211yesmaximal, characteristic, complemented
4C3×#938C211yesmaximal, characteristic, complemented
5C3×#936C211yesmaximal, characteristic, complemented
6C3×#935C211yesmaximal, characteristic, complemented
7C3×#935C211yesmaximal, characteristic, complemented
8C3×#929C211yesmaximal, characteristic, complemented
9C3×#924C211yesmaximal, characteristic, complemented
10C3×#896C211yesmaximal, characteristic, complemented
11C3×#889C211yesmaximal, characteristic, complemented
12C3×#885C211yesmaximal, characteristic, complemented
13C3×#883C211yesmaximal, characteristic, complemented
14C3×#878C211yesmaximal, characteristic, complemented
15C3×#878C211yesmaximal, characteristic, complemented
16C3×#562—31nomaximal
17#5621C311yesmaximal, characteristic, direct product factor
18C2×C32×D4C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C2×C32×D4C2211yescharacteristic
21C22×C32×C4C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic
25C22×C3×Dic3C2211yescharacteristic
26C22×C3×Dic3C2211yescharacteristic
27C2×C3×C4×D3C2211yescharacteristic
28C2×C3×Dic6C2211yescharacteristic
29C2×C3×Dic6C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×#217C2211yescharacteristic
39C3×#217C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×C4⋊Dic3C2211yescharacteristic
44C3×C4⋊Dic3C2211yescharacteristic
45C3×Dic3⋊C4C2211yescharacteristic
46C3×Dic3⋊C4C2211yescharacteristic
47C3×Dic3⋊C4C2211yescharacteristic
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—31no—
57C3×#123—31no—
58C3×#123—31no—
59C3×#123—31no—
60C3×C22⋊Q8—31no—
61C3×C22⋊Q8—31no—
62C3×C22⋊Q8—31no—
63C3×C22⋊Q8—31no—
64C3×#121—21no—
65C3×#121—31no—
66C3×#121—31no—
67C3×#121—31no—
68C3×#121D311yescharacteristic, complemented
69C3×#121C2×C311yescharacteristic, complemented
70#939C2×C311yescharacteristic, complemented
71#939C2×C311yescharacteristic, complemented
72#938C2×C311yescharacteristic, complemented
73#936C2×C311yescharacteristic, complemented
74#935C2×C311yescharacteristic, complemented
75#935C2×C311yescharacteristic, complemented
76#929C2×C311yescharacteristic, complemented
77#924C2×C311yescharacteristic, complemented
78#896C2×C311yescharacteristic, complemented
79#889C2×C311yescharacteristic, complemented
80#885C2×C311yescharacteristic, complemented
81#883C2×C311yescharacteristic, complemented
82#878C2×C311yescharacteristic, complemented
83#878C2×C311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C23×C32C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C32×D4—41no—
89C32×D4—41no—
90C32×D4—41no—
91C2×C32×C4—41no—
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95C2×C32×C4C2311yescharacteristic
96C3×#67—41no—
97C3×#67—41no—
98C3×#67—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
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—21no—
116C2×C3×D4—21no—
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C2×C3×D4—31no—
120C2×C3×D4C2×D311yescharacteristic
121C2×C3×D4C2×D311yescharacteristic
122C2×C3×D4C2×D311yescharacteristic
123C2×C3×D4C22×C311yescharacteristic
124C2×C3×D4C22×C311yescharacteristic
125C2×C3×D4C22×C311yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4C2×D311yescharacteristic
131C22×C3×C4C22×C311yescharacteristic
132C2×#67C22×C311yescharacteristic
133C2×#67C22×C311yescharacteristic
134C2×#67C22×C311yescharacteristic
135C22×Dic3C22×C311yescharacteristic
136C22×Dic3C22×C311yescharacteristic
137C2×C4×D3C22×C311yescharacteristic
138C2×Dic6C22×C311yescharacteristic
139C2×Dic6C22×C311yescharacteristic
140C3×C4⋊C4—21no—
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⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4C2×D311yescharacteristic
151C3×C4⋊C4C22×C311yescharacteristic
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—31no—
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⋊C4C22×C311yescharacteristic
167C3×C22⋊C4C22×C311yescharacteristic
168C3×C42—31no—
169#217C22×C311yescharacteristic
170#217C22×C311yescharacteristic
171#217C22×C311yescharacteristic
172#217C22×C311yescharacteristic
173#217C22×C311yescharacteristic
174#217C22×C311yescharacteristic
175#217C22×C311yescharacteristic
176(C2×D3)⋊C4C22×C311yescharacteristic
177(C2×D3)⋊C4C22×C311yescharacteristic
178(C2×D3)⋊C4C22×C311yescharacteristic
179C4⋊Dic3C22×C311yescharacteristic
180C4⋊Dic3C22×C311yescharacteristic
181Dic3⋊C4C22×C311yescharacteristic
182Dic3⋊C4C22×C311yescharacteristic
183Dic3⋊C4C22×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×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2411yescharacteristic
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C3×Dic3—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#123—31no—
220C22⋊Q8—31no—
221C22⋊Q8—31no—
222C22⋊Q8—31no—
223C22⋊Q8—31no—
224#121—31no—
225#121—31no—
226#121—31no—
227#121C3×D311yescharacteristic, complemented
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—31no—
232C23×C3C22×D311yescharacteristic
233C23×C3C22×D311yescharacteristic
234C23×C3C22×D311yescharacteristic
235C23×C3C23×C311yescharacteristic
236C23×C3C23×C311yescharacteristic
237C23×C3C23×C311yescharacteristic
238C22×D3C23×C311yescharacteristic
239C3×Q8—121no—
240C3×Q8—121no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—121no—
251C3×D4—121no—
252C3×D4—121no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—121no—
268C2×C3×C4—121no—
269C2×C3×C4—121no—
270C2×C3×C4C22×D311yescharacteristic
271C2×C3×C4C22×D311yescharacteristic
272C2×C3×C4C22×D311yescharacteristic
273C2×C3×C4C22×D311yescharacteristic
274C2×C3×C4C23×C311yescharacteristic
275C2×C3×C4C23×C311yescharacteristic
276C2×C3×C4C23×C311yescharacteristic
277C2×C3×C4C23×C311yescharacteristic
278#67—41no—
279#67—41no—
280#67—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3C23×C311yescharacteristic
284C2×Dic3C23×C311yescharacteristic
285C2×Dic3C23×C311yescharacteristic
286C2×Dic3C23×C311yescharacteristic
287C2×Dic3C23×C311yescharacteristic
288C2×Dic3C23×C311yescharacteristic
289C2×Dic3C23×C311yescharacteristic
290C4×D3—41no—
291Dic6—41no—
292Dic6—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32H(5,𝔽2)11yescharacteristic
297C2×C32H(5,𝔽2)11yescharacteristic
298C2×C322−1+411yescharacteristic
299C3×D3—41no—
300C2×Q8—31no—
301C2×Q8—31no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—31no—
305C2×D4C2×C3×D311yescharacteristic
306C2×D4C2×C3×D311yescharacteristic
307C2×D4C2×C3×D311yescharacteristic
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4C2×C3×D311yescharacteristic
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—31no—
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4C2×C3×D311yescharacteristic
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—31no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—31no—
328C22⋊C4—31no—
329C22⋊C4—31no—
330C22⋊C4—31no—
331C22⋊C4—31no—
332C22⋊C4C2×C3×D311yescharacteristic
333C22⋊C4C2×C3×D311yescharacteristic
334C42—31no—
335C22×C3—22no—
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—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3C23×D311yescharacteristic
367C22×C3C24×C311yescharacteristic
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—21no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—61no—
390Dic3—21no—
391Dic3—21no—
392Dic3—21no—
393Dic3—21no—
394Dic3—21no—
395Dic3—21no—
396Dic3—21no—
397C32#56211yescharacteristic, complemented, Sylow
398C23—31no—
399C23C22×C3×D311yescharacteristic
400C23C22×C3×D311yescharacteristic
401C23C22×C3×D311yescharacteristic
402Q8—121no—
403Q8—121no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—121no—
408D4—121no—
409D4—121no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—41no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4C22×C3×D311yescharacteristic
422C2×C4C22×C3×D311yescharacteristic
423C2×C4C22×C3×D311yescharacteristic
424C2×C4C22×C3×D311yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—42no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—81no—
436C2×C3—81no—
437C2×C3—121no—
438C2×C3#100411yescharacteristic
439C2×C3#100411yescharacteristic
440C2×C3#101011yescharacteristic
441C2×C3C3×H(5,𝔽2)11yescharacteristic
442C2×C3C3×H(5,𝔽2)11yescharacteristic
443C2×C3C3×2−1+411yescharacteristic
444D3—41no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—61no—
455C22—61no—
456C22—61no—
457C22C23×C3×D311yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—61no—
463C4—61no—
464C4—61no—
465C4—61no—
466C4—61no—
467C4—61no—
468C4—61no—
469C3—21no—
470C3#562111yescharacteristic, direct product factor
471C3C3×#56211yescharacteristic, complemented
472C2—41no—
473C2—41no—
474C2—41no—
475C2—121no—
476C2C3×#100411yescharacteristic
477C2C3×#100411yescharacteristic
478C2C3×#101011yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×#121C2
socleC22×C32C24
hypercenterC3×#121D3
hypocenterC3C3×#562
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C32×#121C2semidirect
2. C3×#939C2semidirect
3. C3×#939C2semidirect
4. C3×#938C2semidirect
5. C3×#936C2semidirect
6. C3×#935C2semidirect
7. C3×#935C2semidirect
8. C3×#929C2semidirect
9. C3×#924C2semidirect
10. C3×#896C2semidirect
11. C3×#889C2semidirect
12. C3×#885C2semidirect
13. C3×#883C2semidirect
14. C3×#878C2semidirect
15. C3×#878C2semidirect
17. #5621C3direct (N × H)
68. C3×#121D3semidirect
69. C3×#121C2×C3semidirect
70. #939C2×C3semidirect
71. #939C2×C3semidirect
72. #938C2×C3semidirect
73. #936C2×C3semidirect
74. #935C2×C3semidirect
75. #935C2×C3semidirect
76. #929C2×C3semidirect
77. #924C2×C3semidirect
78. #896C2×C3semidirect
79. #889C2×C3semidirect
80. #885C2×C3semidirect
81. #883C2×C3semidirect
82. #878C2×C3semidirect
83. #878C2×C3semidirect
227. #121C3×D3semidirect
397. C32#562semidirect
470. C3#5621direct (N × H)
471. C3C3×#562semidirect

Maximal subgroups

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

subgroupindex
C32×#1212
C3×#9392
C3×#9382
C3×#9362
C3×#9352
C3×#9292
C3×#9242
C3×#8962
C3×#8892
C3×#8852
C3×#8832
C3×#8782
C3×#5623
#56213