Finite Group Databasebeta

← #10588674⋊C3 #13142654

Subgroups — #10588674⋊C3 #13142654

Order 1728.

604 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
5C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4994—31nomaximal
9#10588674C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C4⋊C4×3−1+2—22no—
18(C2×AGL(1,ℤ/9))⋊C4—22no—
19(C2×AGL(1,ℤ/9))⋊C4—22no—
20(C2×AGL(1,ℤ/9))⋊C4—22no—
21C4⋊(3−1+2⋊C4)—22no—
22(3−1+2⋊C4)⋊C4—22no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×C4⋊Dic3—31no—
28C2×C3×Dic3⋊C4—31no—
29C3×C3⋊#95—31no—
30C2×C9×C4⋊C4—21no—
31C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
32C2×(C2×D9)⋊C4C2×C311yescharacteristic
33C2×(C2×D9)⋊C4C2×C311yescharacteristic
34C2×(C2×D9)⋊C4C2×C311yescharacteristic
35C2×C4⋊Dic9C2×C311yescharacteristic, complemented
36C2×Dic9⋊C4C2×C311yescharacteristic, complemented
37C9⋊#95C2×C311yescharacteristic, complemented
38C23×3−1+2C2311yescharacteristic
39C22×AGL(1,ℤ/9)—21no—
40C22×AGL(1,ℤ/9)—21no—
41C22×AGL(1,ℤ/9)—21no—
42C22×AGL(1,ℤ/9)—21no—
43C22×AGL(1,ℤ/9)—21no—
44C22×AGL(1,ℤ/9)—21no—
45C22×AGL(1,ℤ/9)Q812yes—
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2—21no—
50C2×C4×3−1+2—21no—
51C2×C4×3−1+2—21no—
52C2×C4×3−1+2—21no—
53C2×C4×3−1+2D412yes—
54C2×C4×3−1+2D412yes—
55C2×3−1+2⋊C4—21no—
56C2×3−1+2⋊C4—21no—
57C2×3−1+2⋊C4—21no—
58C2×3−1+2⋊C4—21no—
59C2×3−1+2⋊C4—21no—
60C2×3−1+2⋊C4—21no—
61C2×3−1+2⋊C4—21no—
62C2×3−1+2⋊C4—21no—
63C2×3−1+2⋊C4D412yes—
64C3×#396—91no—
65#4994—31no—
66C23×C3×D3—31no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C23×D9C22×C311yescharacteristic
74C32×C4⋊C4—22no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C3×(C2×D3)⋊C4—62no—
78C3×C4⋊Dic3—62no—
79C3×Dic3⋊C4—62no—
80C22×C4×C9—21no—
81C22×C4×C9—21no—
82C22×C4×C9—21no—
83C22×C4×C9C22×C311yescharacteristic
84C22×C4×C9C22×C311yescharacteristic
85C22×C4×C9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C22×Dic9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C9×C4⋊C4—24no—
90C9×C4⋊C4—22no—
91(C2×D9)⋊C4—22no—
92(C2×D9)⋊C4—22no—
93(C2×D9)⋊C4—22no—
94C4⋊Dic9—22no—
95Dic9⋊C4—22no—
96C22×3−1+2C2×D411yescharacteristic
97C22×3−1+2C2×D411yescharacteristic
98C22×3−1+2C2×D411yescharacteristic
99C22×3−1+2C2×Q811yescharacteristic
100C22×3−1+2#4111yescharacteristic
101C22×3−1+2#4111yescharacteristic
102C22×3−1+2#4111yescharacteristic
103C2×AGL(1,ℤ/9)—22no—
104C2×AGL(1,ℤ/9)—22no—
105C2×AGL(1,ℤ/9)—22no—
106C2×AGL(1,ℤ/9)—41no—
107C2×AGL(1,ℤ/9)—41no—
108C2×AGL(1,ℤ/9)—41no—
109C2×AGL(1,ℤ/9)—41no—
110C4×3−1+2—22no—
111C4×3−1+2—22no—
112C4×3−1+2—41no—
1133−1+2⋊C4—22no—
1143−1+2⋊C4—41no—
1153−1+2⋊C4—41no—
116C2×C3×C4⋊C4—31no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C4⋊C4C3×D311yescharacteristic
120C2×C3×C22⋊C4—91no—
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4—91no—
123C2×(C2×D3)⋊C4—31no—
124C2×(C2×D3)⋊C4—31no—
125C2×(C2×D3)⋊C4—31no—
126C2×C4⋊Dic3—31no—
127C2×Dic3⋊C4—31no—
128C3×#95—91no—
129C3⋊#95—31no—
130C23×C32C22×D311yescharacteristic
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4D1212yes—
146C2×C32×C4#6712yes—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C23×C9—21no—
157C23×C9C23×C311yescharacteristic
158C22×D9—21no—
159C22×D9—21no—
160C22×D9—21no—
161C22×D9—21no—
162C22×D9—21no—
163C22×D9—21no—
164C22×D9C3×Q812yes—
165C2×C4×C9—22no—
166C2×C4×C9—22no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—22no—
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—22no—
176C2×C4×C9—22no—
177C2×C4×C9—41no—
178C2×C4×C9—41no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9C3×D412yes—
182C2×C4×C9C3×D412yes—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9C3×D412yes—
192#396—91noSylow
193C2×3−1+2#12111yescharacteristic
194C2×3−1+2#12111yescharacteristic
195C2×3−1+2#12111yescharacteristic
196C2×3−1+2C22⋊Q811yescharacteristic
197C2×3−1+2C22⋊Q811yescharacteristic
198C2×3−1+2C22⋊Q811yescharacteristic
199C2×3−1+2#12611yescharacteristic
200AGL(1,ℤ/9)—42no—
201C24×C3—91no—
202C23×D3—31no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4C2×C3×D311yescharacteristic
210C22×C3×C4C2×C3×D311yescharacteristic
211C22×C3×C4C2×C3×D311yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—62no—
217C3×C4⋊C4—182no—
218C3×C4⋊C4—182no—
219C3×C22⋊C4—182no—
220C3×C22⋊C4—182no—
221C3×C22⋊C4—182no—
222(C2×D3)⋊C4—62no—
223(C2×D3)⋊C4—62no—
224(C2×D3)⋊C4—62no—
225C4⋊Dic3—62no—
226Dic3⋊C4—62no—
227C22×C32C2×D1211yescharacteristic
228C22×C32#23511yescharacteristic
229C22×C32D3×D411yescharacteristic
230C22×C32#23711yescharacteristic
231C22×C32D3×Q811yescharacteristic
232C22×C32#23911yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—41no—
244C3×Dic3—62no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9C2×C3×D411yescharacteristic
255C22×C9C2×C3×D411yescharacteristic
256C22×C9C2×C3×D411yescharacteristic
257C22×C9C2×C3×Q811yescharacteristic
258C22×C9C3×#4111yescharacteristic
259C22×C9C3×#4111yescharacteristic
260C22×C9C3×#4111yescharacteristic
261C2×D9—22no—
262C2×D9—22no—
263C2×D9—22no—
264C2×D9—41no—
265C2×D9—41no—
266C2×D9—41no—
267C2×D9—41no—
268C4×C9—24no—
269C4×C9—22no—
270C4×C9—22no—
271C4×C9—42no—
272C4×C9—42no—
273C4×C9—41no—
274Dic9—22no—
275Dic9—41no—
276Dic9—41no—
277C2×C4⋊C4—91no—
278C2×C4⋊C4—91no—
279C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
280C2×C22⋊C4—91no—
281C2×C22⋊C4—91no—
282C2×C22⋊C4—91no—
283#95—91no—
2843−1+2#39611yescharacteristic, complemented, Sylow
285C23×C3—31no—
286C23×C3—92no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3—181no—
292C23×C3—181no—
293C23×C3C22×C3×D311yescharacteristic
294C22×D3—32no—
295C22×D3—61no—
296C22×D3—61no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—61no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4C3×D1212yes—
327C2×C3×C4C3×#6712yes—
328C2×Dic3—32no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×C32#89511yescharacteristic
338C2×C32#89611yescharacteristic
339C2×C32#89711yescharacteristic
340C2×C32#89811yescharacteristic
341C2×C32#93011yescharacteristic
342C2×C32#93911yescharacteristic
343C2×C32#94611yescharacteristic
344C3×D3—122no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9C3×#12111yescharacteristic
353C2×C9C3×#12111yescharacteristic
354C2×C9C3×#12111yescharacteristic
355C2×C9C3×C22⋊Q811yescharacteristic
356C2×C9C3×C22⋊Q811yescharacteristic
357C2×C9C3×C22⋊Q811yescharacteristic
358C2×C9C3×#12611yescharacteristic
359D9—42no—
360C24—91no—
361C22×C4—91no—
362C22×C4—91no—
363C22×C4—91no—
364C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
365C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
366C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
367C4⋊C4—22no—
368C4⋊C4—182no—
369C4⋊C4—182no—
370C22⋊C4—182no—
371C22⋊C4—182no—
372C22⋊C4—182no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—182no—
381C22×C3—182no—
382C22×C3—182no—
383C22×C3—361no—
384C22×C3—361no—
385C22×C3—361no—
386C22×C3—361no—
387C22×C3C2×C3×D1211yescharacteristic
388C22×C3C3×#23511yescharacteristic
389C22×C3C3×D3×D411yescharacteristic
390C22×C3C3×#23711yescharacteristic
391C22×C3C3×D3×Q811yescharacteristic
392C22×C3C3×#23911yescharacteristic
393C22×C3C2×C3×#6711yescharacteristic
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—121no—
398C2×D3—121no—
399C2×D3—121no—
400C2×D3—121no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—62no—
405C3×C4—62no—
406C3×C4—121no—
407C3×C4—182no—
408C3×C4—361no—
409C3×C4—361no—
410Dic3—62no—
411Dic3—121no—
412Dic3—121no—
413C32#499411yescharacteristic
414C9—21no—
415C9C3×#39611yescharacteristic, complemented
416C23—92no—
417C23—181no—
418C23—181no—
419C23—181no—
420C23—181no—
421C23—181no—
422C23—181no—
423C23C22×AGL(1,ℤ/9)11yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—92no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4—181no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4C4⋊AGL(1,ℤ/9)12yes—
441C2×C4#611⋊C312yes—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—362no—
450C2×C3C3×#89511yescharacteristic
451C2×C3C3×#89611yescharacteristic
452C2×C3C3×#89711yescharacteristic
453C2×C3C3×#89811yescharacteristic
454C2×C3C3×#93011yescharacteristic
455C2×C3C3×#93911yescharacteristic
456C2×C3C3×#94611yescharacteristic
457D3—122no—
458C22—182no—
459C22—182no—
460C22—182no—
461C22—361no—
462C22—361no—
463C22—361no—
464C22—361no—
465C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
466C22#3697⋊C311yescharacteristic
467C22D4×AGL(1,ℤ/9)11yescharacteristic
468C22#3699⋊C311yescharacteristic
469C22Q8×AGL(1,ℤ/9)11yescharacteristic
470C22#8872411yescharacteristic
471C22C2×#611⋊C311yescharacteristic
472C4—22no—
473C4—22no—
474C4—41no—
475C4—182no—
476C4—361no—
477C4—361no—
478C3—31no—
479C3C3×#499411yescharacteristic
480C2—362no—
481C2#63457⋊C311yescharacteristic
482C2#63458⋊C311yescharacteristic
483C2#63459⋊C311yescharacteristic
484C2#63460⋊C311yescharacteristic
485C2#63492⋊C311yescharacteristic
486C2#63501⋊C311yescharacteristic
487C2#63508⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#396
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
5. C2×C4⋊(3−1+2⋊C4)C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588674C3semidirect
31. C2×C9×C4⋊C4C2×C3semidirect
35. C2×C4⋊Dic9C2×C3semidirect
36. C2×Dic9⋊C4C2×C3semidirect
37. C9⋊#95C2×C3semidirect
279. C2×C4⋊C4AGL(1,ℤ/9)semidirect
284. 3−1+2#396semidirect
415. C9C3×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49943
#105886743