Finite Group Databasebeta

← #10588348⋊C3 #13141388

Subgroups — #10588348⋊C3 #13141388

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 500 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×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
63−1+2×#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4668—31nomaximal
9#10588348C311yesmaximal, 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—
20(C2×AGL(1,ℤ/9))⋊C4—22no—
21(C2×AGL(1,ℤ/9))⋊C4C414yescomplemented
22C4⋊(3−1+2⋊C4)—22no—
23(3−1+2⋊C4)⋊C4—22no—
24C22×C3×C4×D3—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×C4⋊Dic3—31no—
28C2×C3×Dic3⋊C4—31no—
29C22×C4×D9C2×C311yescharacteristic
30C32×#95D311yescharacteristic
31C3×C3⋊#95—31no—
32C2×(C2×D9)⋊C4C2×C311yescharacteristic
33C2×(C2×D9)⋊C4C2×C311yescharacteristic
34C2×C4⋊Dic9C2×C311yescharacteristic, complemented
35C2×Dic9⋊C4C2×C311yescharacteristic, complemented
36C9×#95—21no—
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#4668—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
79C3×(C2×D3)⋊C4—34no—
80C3×(C2×D3)⋊C4—62no—
81C3×C4⋊Dic3—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—
95(C2×D9)⋊C4—22no—
96(C2×D9)⋊C4C3×C414yescomplemented
97C4⋊Dic9—22no—
98Dic9⋊C4—22no—
99C22×3−1+2C22×C411yescharacteristic
100C22×3−1+2C2×D411yescharacteristic
101C22×3−1+2C2×D411yescharacteristic
102C22×3−1+2C2×D411yescharacteristic
103C22×3−1+2C2×Q811yescharacteristic
104C22×3−1+2#4111yescharacteristic
105C22×3−1+2#4111yescharacteristic
106C2×AGL(1,ℤ/9)—22no—
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)C4⋊C414yescomplemented
113C4×3−1+2—22no—
114C4×3−1+2—22no—
115C4×3−1+2—41no—
1163−1+2⋊C4—22no—
1173−1+2⋊C4—22no—
1183−1+2⋊C4—41no—
119C23×C3×C4—91no—
120C22×C4×D3—31no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C22⋊C4—91no—
124C2×C3×C22⋊C4—91no—
125C2×(C2×D3)⋊C4—31no—
126C2×(C2×D3)⋊C4—31no—
127C2×C4⋊Dic3—31no—
128C2×Dic3⋊C4—31no—
129C3×#95—31no—
130C3×#95—91no—
131C3×#95C3×D311yescharacteristic
132C3⋊#95—31no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—32no—
135C22×C3×D3—32no—
136C22×C3×D3—32no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C2×C32×C4—21no—
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×C4C4×D312yes—
149C2×C32×C4D1212yes—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C3×C4×D3—68no—
160C23×C9—21no—
161C23×C9C23×C311yescharacteristic
162C22×D9—21no—
163C22×D9—21no—
164C22×D9—21no—
165C22×D9—21no—
166C22×D9C2×C3×C412yes—
167C22×D9C3×D412yes—
168C22×D9C3×Q812yes—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—21no—
173C2×C4×C9—22no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—22no—
178C2×C4×C9—22no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9—41no—
182C2×C4×C9—41no—
183C2×C4×C9—41no—
184C2×C4×C9—41no—
185C2×C4×C9C2×C3×C412yescomplemented
186C2×C4×C9C3×D412yes—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9—21no—
193C2×Dic9—21no—
194C2×Dic9C2×C3×C412yescomplemented
195C2×Dic9C3×D412yes—
196C4×D9—28no—
197#385—91noSylow
198C2×3−1+2C2×C4⋊C411yescharacteristic
199C2×3−1+2C4×D411yescharacteristic
200C2×3−1+2C4×D411yescharacteristic
201C2×3−1+2#12011yescharacteristic
202C2×3−1+2C22⋊Q811yescharacteristic
203C2×3−1+2C22⋊Q811yescharacteristic
204C2×3−1+2#12311yescharacteristic
205AGL(1,ℤ/9)—24no—
206C24×C3—91no—
207C23×D3—31no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—182no—
215C22×C3×C4—182no—
216C22×C3×C4—182no—
217C22×C3×C4C2×C3×D311yescharacteristic
218C22×C3×C4C2×C3×D311yescharacteristic
219C22×C3×C4C2×C3×D311yescharacteristic
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C2×C4×D3—62no—
224C2×C4×D3—62no—
225C2×C4×D3—62no—
226C3×C4⋊C4—182no—
227C3×C4⋊C4—182no—
228C3×C22⋊C4—94no—
229C3×C22⋊C4—182no—
230(C2×D3)⋊C4—34no—
231(C2×D3)⋊C4—62no—
232C4⋊Dic3—62no—
233Dic3⋊C4—62no—
234C22×C32C2×C4×D311yescharacteristic
235C22×C32C2×D1211yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32#23711yescharacteristic
240C22×C32D3×Q811yescharacteristic
241C2×C3×D3—34no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—41no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9C22×C3×C411yescharacteristic
262C22×C9C2×C3×D411yescharacteristic
263C22×C9C2×C3×D411yescharacteristic
264C22×C9C2×C3×D411yescharacteristic
265C22×C9C2×C3×Q811yescharacteristic
266C22×C9C3×#4111yescharacteristic
267C22×C9C3×#4111yescharacteristic
268C2×D9—22no—
269C2×D9—22no—
270C2×D9—22no—
271C2×D9—22no—
272C2×D9—22no—
273C2×D9—22no—
274C2×D9C3×C4⋊C414yescomplemented
275C4×C9—22no—
276C4×C9—22no—
277C4×C9—42no—
278C4×C9—42no—
279C4×C9—42no—
280C4×C9—41no—
281Dic9—22no—
282Dic9—22no—
283Dic9—41no—
284C23×C4—91no—
285C2×C4⋊C4—91no—
286C2×C4⋊C4—91no—
287C2×C22⋊C4—91no—
288C2×C22⋊C4—91no—
289#95—91no—
290#95AGL(1,ℤ/9)11yescharacteristic, complemented
2913−1+2#38511yescharacteristic, complemented, Sylow
292C23×C3—31no—
293C23×C3—92no—
294C23×C3—92no—
295C23×C3—92no—
296C23×C3—181no—
297C23×C3—181no—
298C23×C3—181no—
299C23×C3—181no—
300C23×C3C22×C3×D311yescharacteristic
301C22×D3—32no—
302C22×D3—32no—
303C22×D3—32no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—61no—
307C22×D3—61no—
308C2×C3×C4—21no—
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—32no—
316C2×C3×C4—32no—
317C2×C3×C4—61no—
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—92no—
325C2×C3×C4—92no—
326C2×C3×C4—188no—
327C2×C3×C4—181no—
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×C4C3×C4×D312yes—
335C2×C3×C4C3×D1212yes—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C4×D3—68no—
346C2×C32C4×D1211yescharacteristic
347C2×C32#67⋊C411yescharacteristic
348C2×C32#88211yescharacteristic
349C2×C32#88311yescharacteristic
350C2×C32D3×C4⋊C411yescharacteristic
351C2×C32#89611yescharacteristic
352C2×C32#89711yescharacteristic
353C3×D3—64no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9C2×C3×C4⋊C411yescharacteristic
362C2×C9C3×C4×D411yescharacteristic
363C2×C9C3×C4×D411yescharacteristic
364C2×C9C3×#12011yescharacteristic
365C2×C9C3×C22⋊Q811yescharacteristic
366C2×C9C3×C22⋊Q811yescharacteristic
367C2×C9C3×#12311yescharacteristic
368D9—24no—
369C24—91no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—182no—
374C22×C4—182no—
375C22×C4—182no—
376C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
377C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
378C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
379C4⋊C4—182no—
380C4⋊C4—182no—
381C22⋊C4—94no—
382C22⋊C4—182no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—94no—
391C22×C3—182no—
392C22×C3—182no—
393C22×C3—182no—
394C22×C3—182no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3C2×C3×C4×D311yescharacteristic
398C22×C3C2×C3×D1211yescharacteristic
399C22×C3C3×#23511yescharacteristic
400C22×C3C3×D3×D411yescharacteristic
401C22×C3C3×D3×D411yescharacteristic
402C22×C3C3×#23711yescharacteristic
403C22×C3C3×D3×Q811yescharacteristic
404C2×D3—34no—
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—121no—
417C3×C4—182no—
418C3×C4—182no—
419C3×C4—361no—
420Dic3—62no—
421Dic3—62no—
422Dic3—121no—
423C32#466811yescharacteristic
424C9—21no—
425C9C3×#38511yescharacteristic, complemented
426C23—92no—
427C23—92no—
428C23—92no—
429C23—181no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23C22×AGL(1,ℤ/9)11yescharacteristic
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—92no—
442C2×C4—92no—
443C2×C4—188no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C4C4×AGL(1,ℤ/9)12yescomplemented
452C2×C4C4⋊AGL(1,ℤ/9)12yes—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—184no—
461C2×C3C3×C4×D1211yescharacteristic
462C2×C3C3×#67⋊C411yescharacteristic
463C2×C3C3×#88211yescharacteristic
464C2×C3C3×#88311yescharacteristic
465C2×C3C3×D3×C4⋊C411yescharacteristic
466C2×C3C3×#89611yescharacteristic
467C2×C3C3×#89711yescharacteristic
468D3—64no—
469C22—94no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22—182no—
476C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
477C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
478C22#3697⋊C311yescharacteristic
479C22D4×AGL(1,ℤ/9)11yescharacteristic
480C22D4×AGL(1,ℤ/9)11yescharacteristic
481C22#3699⋊C311yescharacteristic
482C22Q8×AGL(1,ℤ/9)11yescharacteristic
483C4—22no—
484C4—22no—
485C4—41no—
486C4—182no—
487C4—182no—
488C4—361no—
489C3—31no—
490C3C3×#466811yescharacteristic
491C2—184no—
492C2C4×C4⋊AGL(1,ℤ/9)11yescharacteristic
493C2(#611⋊C3)⋊C411yescharacteristic
494C2#63444⋊C311yescharacteristic
495C2#63445⋊C311yescharacteristic
496C2C4⋊C4×AGL(1,ℤ/9)11yescharacteristic
497C2#63458⋊C311yescharacteristic
498C2#63459⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×AGL(1,ℤ/9)C22
cosocleC23×C9C23×C3
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#385
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×(3−1+2⋊C4)⋊C4C2semidirect
6. 3−1+2×#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588348C3semidirect
21. (C2×AGL(1,ℤ/9))⋊C4C4semidirect
34. C2×C4⋊Dic9C2×C3semidirect
35. 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
96. (C2×D9)⋊C4C3×C4semidirect
112. C2×AGL(1,ℤ/9)C4⋊C4semidirect
185. C2×C4×C9C2×C3×C4semidirect
194. C2×Dic9C2×C3×C4semidirect
274. C2×D9C3×C4⋊C4semidirect
290. #95AGL(1,ℤ/9)semidirect
291. 3−1+2#385semidirect
425. C9C3×#385semidirect
451. 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×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
3−1+2×#952
3−1+2⋊#952
C3×#46683
#105883483