Finite Group Databasebeta

← #10588669⋊C3 #13142660

Subgroups — #10588669⋊C3 #13142660

Order 1728.

748 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
5C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
63−1+2⋊#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4989—31nomaximal
9#10588669C311yesmaximal, 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
17C2×C4×AGL(1,ℤ/9)—22no—
18C2×C4×AGL(1,ℤ/9)—22no—
19C2×C4×AGL(1,ℤ/9)—22no—
20C4⋊C4×3−1+2—22no—
21(C2×AGL(1,ℤ/9))⋊C4—22no—
22(C2×AGL(1,ℤ/9))⋊C4C414yescomplemented
23(3−1+2⋊C4)⋊C4—22no—
24C22×C3×C4×D3—31no—
25C2×C32×C4⋊C4D311yescharacteristic
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C22×C4×D9C2×C311yescharacteristic
30C3×C3⋊#95—31no—
31C3×C3⋊#95—31no—
32C2×C9×C4⋊C4—21no—
33C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
34C2×(C2×D9)⋊C4C2×C311yescharacteristic
35C2×(C2×D9)⋊C4C2×C311yescharacteristic
36C2×Dic9⋊C4C2×C311yescharacteristic, complemented
37C9⋊#95C2×C311yescharacteristic, complemented
38C9⋊#95C2×C311yescharacteristic, complemented
39C23×3−1+2C2311yescharacteristic
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)C2×C412yes—
45C22×AGL(1,ℤ/9)D412yes—
46C22×AGL(1,ℤ/9)Q812yes—
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+2—21no—
54C2×C4×3−1+2C2×C412yescomplemented
55C2×C4×3−1+2D412yes—
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⋊C4C2×C412yescomplemented
64C2×3−1+2⋊C4D412yes—
65C4×AGL(1,ℤ/9)—28no—
66C3×#385—91no—
67#4989—31no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—62no—
77C2×C3×C4×D3—62no—
78C23×D9C22×C311yescharacteristic
79C32×C4⋊C4—22no—
80C3×(C2×D3)⋊C4—34no—
81C3×(C2×D3)⋊C4—62no—
82C3×Dic3⋊C4—62no—
83C22×C4×C9—21no—
84C22×C4×C9—21no—
85C22×C4×C9—21no—
86C22×C4×C9C22×C311yescharacteristic
87C22×C4×C9C22×C311yescharacteristic
88C22×C4×C9C22×C311yescharacteristic
89C22×Dic9C22×C311yescharacteristic
90C22×Dic9C22×C311yescharacteristic
91C22×Dic9C22×C311yescharacteristic
92C2×C4×D9—22no—
93C2×C4×D9—22no—
94C2×C4×D9—22no—
95C9×C4⋊C4—24no—
96C9×C4⋊C4—22no—
97(C2×D9)⋊C4—22no—
98(C2×D9)⋊C4C3×C414yescomplemented
99Dic9⋊C4—22no—
100C22×3−1+2C22×C411yescharacteristic
101C22×3−1+2C2×D411yescharacteristic
102C22×3−1+2C2×D411yescharacteristic
103C22×3−1+2C2×D411yescharacteristic
104C22×3−1+2C2×Q811yescharacteristic
105C22×3−1+2#4111yescharacteristic
106C22×3−1+2#4111yescharacteristic
107C2×AGL(1,ℤ/9)—22no—
108C2×AGL(1,ℤ/9)—22no—
109C2×AGL(1,ℤ/9)—22no—
110C2×AGL(1,ℤ/9)—22no—
111C2×AGL(1,ℤ/9)—22no—
112C2×AGL(1,ℤ/9)—22no—
113C2×AGL(1,ℤ/9)C4⋊C414yescomplemented
114C4×3−1+2—22no—
115C4×3−1+2—22no—
116C4×3−1+2—41no—
1173−1+2⋊C4—22no—
1183−1+2⋊C4—22no—
1193−1+2⋊C4—41no—
120C23×C3×C4—91no—
121C22×C4×D3—31no—
122C2×C3×C4⋊C4—31no—
123C2×C3×C4⋊C4—91no—
124C2×C3×C4⋊C4C3×D311yescharacteristic
125C2×C3×C22⋊C4—91no—
126C2×C3×C22⋊C4—91no—
127C2×(C2×D3)⋊C4—31no—
128C2×(C2×D3)⋊C4—31no—
129C2×Dic3⋊C4—31no—
130C3×#95—91no—
131C3×#95—91no—
132C3⋊#95—31no—
133C3⋊#95—31no—
134C23×C32C22×D311yescharacteristic
135C22×C3×D3—32no—
136C22×C3×D3—32no—
137C22×C3×D3—32no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4C4×D312yes—
150C2×C32×C4#6712yes—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C3×C4×D3—68no—
161C23×C9—21no—
162C23×C9C23×C311yescharacteristic
163C22×D9—21no—
164C22×D9—21no—
165C22×D9—21no—
166C22×D9—21no—
167C22×D9C2×C3×C412yes—
168C22×D9C3×D412yes—
169C22×D9C3×Q812yes—
170C2×C4×C9—22no—
171C2×C4×C9—22no—
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—22no—
177C2×C4×C9—21no—
178C2×C4×C9—21no—
179C2×C4×C9—21no—
180C2×C4×C9—22no—
181C2×C4×C9—22no—
182C2×C4×C9—41no—
183C2×C4×C9—41no—
184C2×C4×C9—41no—
185C2×C4×C9—41no—
186C2×C4×C9C2×C3×C412yescomplemented
187C2×C4×C9C3×D412yes—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9—21no—
193C2×Dic9—21no—
194C2×Dic9—21no—
195C2×Dic9C2×C3×C412yescomplemented
196C2×Dic9C3×D412yes—
197C4×D9—28no—
198#385—91noSylow
199C2×3−1+2C2×C4⋊C411yescharacteristic
200C2×3−1+2C4×D411yescharacteristic
201C2×3−1+2C4×D411yescharacteristic
202C2×3−1+2#12011yescharacteristic
203C2×3−1+2C22⋊Q811yescharacteristic
204C2×3−1+2C22⋊Q811yescharacteristic
205C2×3−1+2#12311yescharacteristic
206AGL(1,ℤ/9)—24no—
207C24×C3—91no—
208C23×D3—31no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—182no—
216C22×C3×C4—182no—
217C22×C3×C4—182no—
218C22×C3×C4C2×C3×D311yescharacteristic
219C22×C3×C4C2×C3×D311yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C2×C4×D3—62no—
225C2×C4×D3—62no—
226C2×C4×D3—62no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—182no—
230C3×C22⋊C4—94no—
231C3×C22⋊C4—182no—
232(C2×D3)⋊C4—34no—
233(C2×D3)⋊C4—62no—
234Dic3⋊C4—62no—
235C22×C32C2×C4×D311yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32D3×Q811yescharacteristic
240C22×C32#23911yescharacteristic
241C22×C32C2×#6711yescharacteristic
242C2×C3×D3—34no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C32×C4—22no—
250C32×C4—22no—
251C32×C4—41no—
252C3×Dic3—62no—
253C3×Dic3—62no—
254C3×Dic3—121no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9C22×C3×C411yescharacteristic
263C22×C9C2×C3×D411yescharacteristic
264C22×C9C2×C3×D411yescharacteristic
265C22×C9C2×C3×D411yescharacteristic
266C22×C9C2×C3×Q811yescharacteristic
267C22×C9C3×#4111yescharacteristic
268C22×C9C3×#4111yescharacteristic
269C2×D9—22no—
270C2×D9—22no—
271C2×D9—22no—
272C2×D9—22no—
273C2×D9—22no—
274C2×D9—22no—
275C2×D9C3×C4⋊C414yescomplemented
276C4×C9—24no—
277C4×C9—22no—
278C4×C9—22no—
279C4×C9—42no—
280C4×C9—42no—
281C4×C9—41no—
282Dic9—22no—
283Dic9—22no—
284Dic9—41no—
285C23×C4—91no—
286C2×C4⋊C4—91no—
287C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
288C2×C22⋊C4—91no—
289C2×C22⋊C4—91no—
290#95—91no—
291#95—91no—
2923−1+2#38511yescharacteristic, complemented, Sylow
293C23×C3—31no—
294C23×C3—92no—
295C23×C3—92no—
296C23×C3—92no—
297C23×C3—181no—
298C23×C3—181no—
299C23×C3—181no—
300C23×C3—181no—
301C23×C3C22×C3×D311yescharacteristic
302C22×D3—32no—
303C22×D3—32no—
304C22×D3—32no—
305C22×D3—61no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—61no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—32no—
317C2×C3×C4—32no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—188no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4C3×C4×D312yes—
336C2×C3×C4C3×#6712yes—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C4×D3—68no—
347C2×C32D3×C4⋊C411yescharacteristic
348C2×C32#89311yescharacteristic
349C2×C32#89411yescharacteristic
350C2×C32#89611yescharacteristic
351C2×C32C4×#6711yescharacteristic
352C2×C32#93711yescharacteristic
353C2×C32#94611yescharacteristic
354C3×D3—64no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9C2×C3×C4⋊C411yescharacteristic
363C2×C9C3×C4×D411yescharacteristic
364C2×C9C3×C4×D411yescharacteristic
365C2×C9C3×#12011yescharacteristic
366C2×C9C3×C22⋊Q811yescharacteristic
367C2×C9C3×C22⋊Q811yescharacteristic
368C2×C9C3×#12311yescharacteristic
369D9—24no—
370C24—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—182no—
375C22×C4—182no—
376C22×C4—182no—
377C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
378C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
379C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—182no—
382C22⋊C4—94no—
383C22⋊C4—182no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—94no—
392C22×C3—182no—
393C22×C3—182no—
394C22×C3—182no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3C2×C3×C4×D311yescharacteristic
399C22×C3C3×#23511yescharacteristic
400C22×C3C3×D3×D411yescharacteristic
401C22×C3C3×D3×D411yescharacteristic
402C22×C3C3×D3×Q811yescharacteristic
403C22×C3C3×#23911yescharacteristic
404C22×C3C2×C3×#6711yescharacteristic
405C2×D3—34no—
406C2×D3—62no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—41no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—121no—
418C3×C4—182no—
419C3×C4—182no—
420C3×C4—361no—
421Dic3—62no—
422Dic3—62no—
423Dic3—121no—
424C32#498911yescharacteristic
425C9—21no—
426C9C3×#38511yescharacteristic, complemented
427C23—92no—
428C23—92no—
429C23—92no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23C22×AGL(1,ℤ/9)11yescharacteristic
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—188no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4C4×AGL(1,ℤ/9)12yescomplemented
453C2×C4#611⋊C312yes—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—184no—
462C2×C3C3×D3×C4⋊C411yescharacteristic
463C2×C3C3×#89311yescharacteristic
464C2×C3C3×#89411yescharacteristic
465C2×C3C3×#89611yescharacteristic
466C2×C3C3×C4×#6711yescharacteristic
467C2×C3C3×#93711yescharacteristic
468C2×C3C3×#94611yescharacteristic
469D3—64no—
470C22—94no—
471C22—182no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22—182no—
476C22—182no—
477C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
478C22#3697⋊C311yescharacteristic
479C22D4×AGL(1,ℤ/9)11yescharacteristic
480C22D4×AGL(1,ℤ/9)11yescharacteristic
481C22Q8×AGL(1,ℤ/9)11yescharacteristic
482C22#8872411yescharacteristic
483C22C2×#611⋊C311yescharacteristic
484C4—22no—
485C4—22no—
486C4—41no—
487C4—182no—
488C4—182no—
489C4—361no—
490C3—31no—
491C3C3×#498911yescharacteristic
492C2—184no—
493C2C4⋊C4×AGL(1,ℤ/9)11yescharacteristic
494C2#63455⋊C311yescharacteristic
495C2#63456⋊C311yescharacteristic
496C2#63458⋊C311yescharacteristic
497C2C4×#611⋊C311yescharacteristic
498C2#63499⋊C311yescharacteristic
499C2#63508⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×AGL(1,ℤ/9)C22
cosocleC23×C9C23×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#385
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×3−1+2C2semidirect
5. C2×(3−1+2⋊C4)⋊C4C2semidirect
6. 3−1+2⋊#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588669C3semidirect
22. (C2×AGL(1,ℤ/9))⋊C4C4semidirect
33. C2×C9×C4⋊C4C2×C3semidirect
36. C2×Dic9⋊C4C2×C3semidirect
37. C9⋊#95C2×C3semidirect
38. C9⋊#95C2×C3semidirect
54. C2×C4×3−1+2C2×C4semidirect
63. C2×3−1+2⋊C4C2×C4semidirect
98. (C2×D9)⋊C4C3×C4semidirect
113. C2×AGL(1,ℤ/9)C4⋊C4semidirect
186. C2×C4×C9C2×C3×C4semidirect
195. C2×Dic9C2×C3×C4semidirect
275. C2×D9C3×C4⋊C4semidirect
287. C2×C4⋊C4AGL(1,ℤ/9)semidirect
292. 3−1+2#385semidirect
426. C9C3×#385semidirect
452. C2×C4C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C22×C4×AGL(1,ℤ/9)2
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49893
#105886693