Finite Group Databasebeta

← #10588641⋊C3 #13142531

Subgroups — #10588641⋊C3 #13142531

Order 1728.

780 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#611⋊C3C211yesmaximal, characteristic
2C2×C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3C2×#3676⋊C3C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
6C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4961—31nomaximal
9#10588641C311yesmaximal, characteristic, complemented
10C24×3−1+2C2211yescharacteristic
11C23×AGL(1,ℤ/9)C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C2×#611⋊C3—22no—
15C2×#611⋊C3—22no—
16C2×#611⋊C3—22no—
17C22×3−1+2⋊C4C2211yescharacteristic
18C22×3−1+2⋊C4C2211yescharacteristic
19C22×3−1+2⋊C4C2211yescharacteristic
20C22⋊C4×3−1+2—22no—
21#3676⋊C3—22no—
22(C2×AGL(1,ℤ/9))⋊C4—22no—
23(C2×AGL(1,ℤ/9))⋊C4—22no—
24C4⋊(3−1+2⋊C4)—22no—
25C22×C3×#67—31no—
26C2×C32×C22⋊C4D311yescharacteristic
27C2×C3×#217—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×(C2×D3)⋊C4—31no—
30C2×C3×C4⋊Dic3—31no—
31C22×#611C2×C311yescharacteristic
32C3×C3⋊#95—31no—
33C2×C9×C22⋊C4—21no—
34C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
35C2×#3676C2×C311yescharacteristic, complemented
36C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
37C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
38C2×C4⋊Dic9C2×C311yescharacteristic, complemented
39C9⋊#95C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C23×3−1+2—21no—
42C23×3−1+2—21no—
43C23×3−1+2—21no—
44C23×3−1+2—21no—
45C23×3−1+2—21no—
46C23×3−1+2D412yes—
47C23×3−1+2C2311yescharacteristic
48C22×AGL(1,ℤ/9)—21no—
49C22×AGL(1,ℤ/9)—21no—
50C22×AGL(1,ℤ/9)—21no—
51C22×AGL(1,ℤ/9)—21no—
52C22×AGL(1,ℤ/9)—21no—
53C22×AGL(1,ℤ/9)—21no—
54C22×AGL(1,ℤ/9)D412yes—
55C2×C4×3−1+2—21no—
56C2×C4×3−1+2—21no—
57C2×C4×3−1+2—21no—
58C2×C4×3−1+2—21no—
59C2×C4×3−1+2—21no—
60C2×C4×3−1+2D412yescomplemented
61#611⋊C3—28no—
62C2×3−1+2⋊C4—21no—
63C2×3−1+2⋊C4—21no—
64C2×3−1+2⋊C4—21no—
65C2×3−1+2⋊C4—21no—
66C2×3−1+2⋊C4—21no—
67C2×3−1+2⋊C4—21no—
68C2×3−1+2⋊C4—21no—
69C2×3−1+2⋊C4—21no—
70C2×3−1+2⋊C4D412yes—
71C3×#394—91no—
72#4961—31no—
73C24×C32C2×D311yescharacteristic
74C23×C3×D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×#67—62no—
78C2×C3×#67—62no—
79C2×C3×#67—62no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C24×C9—21no—
84C24×C9C22×C311yescharacteristic
85C23×D9C22×C311yescharacteristic
86C32×C22⋊C4—22no—
87C3×#217—62no—
88C3×(C2×D3)⋊C4—62no—
89C3×(C2×D3)⋊C4—62no—
90C3×C4⋊Dic3—62no—
91C22×C4×C9—21no—
92C22×C4×C9—21no—
93C22×C4×C9C22×C311yescharacteristic
94C22×C4×C9C22×C311yescharacteristic
95C2×#611—22no—
96C2×#611—22no—
97C2×#611—22no—
98C22×Dic9C22×C311yescharacteristic
99C22×Dic9C22×C311yescharacteristic
100C22×Dic9C22×C311yescharacteristic
101C9×C22⋊C4—24no—
102C9×C22⋊C4—22no—
103#3676—22no—
104(C2×D9)⋊C4—22no—
105(C2×D9)⋊C4—22no—
106C4⋊Dic9—22no—
107C22×3−1+2—22no—
108C22×3−1+2—22no—
109C22×3−1+2—22no—
110C22×3−1+2—41no—
111C22×3−1+2—41no—
112C22×3−1+2—41no—
113C22×3−1+2—41no—
114C22×3−1+2C2×D411yescharacteristic
115C22×3−1+2C2×D411yescharacteristic
116C22×3−1+2C2×D411yescharacteristic
117C22×3−1+2C2×D411yescharacteristic
118C22×3−1+2#4111yescharacteristic
119C22×3−1+2#4111yescharacteristic
120C22×3−1+2#4111yescharacteristic
121C2×AGL(1,ℤ/9)—22no—
122C2×AGL(1,ℤ/9)—22no—
123C2×AGL(1,ℤ/9)—22no—
124C2×AGL(1,ℤ/9)—41no—
125C2×AGL(1,ℤ/9)—41no—
126C2×AGL(1,ℤ/9)—41no—
127C2×AGL(1,ℤ/9)—41no—
128C4×3−1+2—22no—
129C4×3−1+2—41no—
1303−1+2⋊C4—22no—
1313−1+2⋊C4—41no—
1323−1+2⋊C4—41no—
133C22×C3×D4—91no—
134C22×#67—31no—
135C2×C3×C4⋊C4—91no—
136C2×C3×C22⋊C4—31no—
137C2×C3×C22⋊C4—91no—
138C2×C3×C22⋊C4—91no—
139C2×C3×C22⋊C4—91no—
140C2×C3×C22⋊C4C3×D311yescharacteristic
141C2×#217—31no—
142C2×(C2×D3)⋊C4—31no—
143C2×(C2×D3)⋊C4—31no—
144C2×C4⋊Dic3—31no—
145C3×#95—91no—
146C3⋊#95—31no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32D1212yes—
154C23×C32C22×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4#6712yes—
168C3×#67—68no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C23×C9—22no—
179C23×C9—22no—
180C23×C9—21no—
181C23×C9—21no—
182C23×C9—21no—
183C23×C9—21no—
184C23×C9—21no—
185C23×C9—21no—
186C23×C9—22no—
187C23×C9—21no—
188C23×C9—41no—
189C23×C9—41no—
190C23×C9—41no—
191C23×C9—41no—
192C23×C9C3×D412yes—
193C23×C9C23×C311yescharacteristic
194C22×D9—21no—
195C22×D9—21no—
196C22×D9—21no—
197C22×D9—21no—
198C22×D9—21no—
199C22×D9—21no—
200C22×D9C3×D412yes—
201C2×C4×C9—21no—
202C2×C4×C9—21no—
203C2×C4×C9—22no—
204C2×C4×C9—21no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—22no—
208C2×C4×C9—41no—
209C2×C4×C9—41no—
210C2×C4×C9—41no—
211C2×C4×C9—41no—
212C2×C4×C9C3×D412yescomplemented
213#611—28no—
214C2×Dic9—21no—
215C2×Dic9—21no—
216C2×Dic9—21no—
217C2×Dic9—21no—
218C2×Dic9—21no—
219C2×Dic9—21no—
220C2×Dic9—21no—
221C2×Dic9—21no—
222C2×Dic9C3×D412yes—
223#394—91noSylow
224C2×3−1+2—42no—
225C2×3−1+2#12011yescharacteristic
226C2×3−1+2#12111yescharacteristic
227C2×3−1+2#12111yescharacteristic
228C2×3−1+2#12111yescharacteristic
229C2×3−1+2#12311yescharacteristic
230C2×3−1+2#12311yescharacteristic
231C2×3−1+2#12411yescharacteristic
232AGL(1,ℤ/9)—42no—
233C24×C3—31no—
234C24×C3—91no—
235C24×C3C2×C3×D311yescharacteristic
236C23×D3—31no—
237C2×C3×D4—182no—
238C2×C3×D4—182no—
239C2×C3×D4—182no—
240C22×C3×C4—31no—
241C22×C3×C4—31no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C22×C3×C4—91no—
245C22×C3×C4C2×C3×D311yescharacteristic
246C22×C3×C4C2×C3×D311yescharacteristic
247C2×#67—62no—
248C2×#67—62no—
249C2×#67—62no—
250C22×Dic3—31no—
251C22×Dic3—31no—
252C22×Dic3—31no—
253C3×C4⋊C4—182no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—62no—
256C3×C22⋊C4—182no—
257C3×C22⋊C4—182no—
258C3×C22⋊C4—182no—
259#217—62no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—62no—
262C4⋊Dic3—62no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32C2×D1211yescharacteristic
271C22×C32#23511yescharacteristic
272C22×C32D3×D411yescharacteristic
273C22×C32D3×D411yescharacteristic
274C22×C32#23711yescharacteristic
275C22×C32#23711yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—121no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C32×C4—22no—
285C32×C4—41no—
286C3×Dic3—62no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C22×C9—24no—
290C22×C9—22no—
291C22×C9—22no—
292C22×C9—22no—
293C22×C9—21no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—21no—
300C22×C9—42no—
301C22×C9—42no—
302C22×C9—42no—
303C22×C9—42no—
304C22×C9—42no—
305C22×C9—42no—
306C22×C9—41no—
307C22×C9—41no—
308C22×C9—41no—
309C22×C9—41no—
310C22×C9C2×C3×D411yescharacteristic
311C22×C9C2×C3×D411yescharacteristic
312C22×C9C2×C3×D411yescharacteristic
313C22×C9C2×C3×D411yescharacteristic
314C22×C9C3×#4111yescharacteristic
315C22×C9C3×#4111yescharacteristic
316C22×C9C3×#4111yescharacteristic
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—22no—
320C2×D9—41no—
321C2×D9—41no—
322C2×D9—41no—
323C2×D9—41no—
324C4×C9—22no—
325C4×C9—42no—
326C4×C9—42no—
327C4×C9—41no—
328Dic9—22no—
329Dic9—41no—
330Dic9—41no—
331C22×D4—91no—
332C2×C4⋊C4—91no—
333C2×C22⋊C4—91no—
334C2×C22⋊C4—91no—
335C2×C22⋊C4—91no—
336C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
337#95—91no—
3383−1+2#39411yescharacteristic, complemented, Sylow
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—32no—
346C23×C3—31no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—61no—
350C23×C3—61no—
351C23×C3—61no—
352C23×C3—61no—
353C23×C3—92no—
354C23×C3—181no—
355C23×C3—181no—
356C23×C3—181no—
357C23×C3—181no—
358C23×C3—181no—
359C23×C3—181no—
360C23×C3C3×D1212yes—
361C23×C3C22×C3×D311yescharacteristic
362C22×D3—32no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—61no—
368C22×D3—61no—
369C3×D4—188no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—32no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—92no—
382C2×C3×C4—181no—
383C2×C3×C4—181no—
384C2×C3×C4—181no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4—181no—
388C2×C3×C4—181no—
389C2×C3×C4—181no—
390C2×C3×C4C3×#6712yes—
391#67—68no—
392C2×Dic3—32no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×C32—42no—
402C2×C32#88211yescharacteristic
403C2×C32#88311yescharacteristic
404C2×C32#88511yescharacteristic
405C2×C32#88611yescharacteristic
406C2×C32#93011yescharacteristic
407C2×C32#93811yescharacteristic
408C2×C32#93911yescharacteristic
409C3×D3—122no—
410C2×C9—21no—
411C2×C9—21no—
412C2×C9—21no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—21no—
417C2×C9—44no—
418C2×C9—42no—
419C2×C9C3×#12011yescharacteristic
420C2×C9C3×#12111yescharacteristic
421C2×C9C3×#12111yescharacteristic
422C2×C9C3×#12111yescharacteristic
423C2×C9C3×#12311yescharacteristic
424C2×C9C3×#12311yescharacteristic
425C2×C9C3×#12411yescharacteristic
426D9—42no—
427C24—91no—
428C24C2×AGL(1,ℤ/9)11yescharacteristic
429C2×D4—182no—
430C2×D4—182no—
431C2×D4—182no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
436C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
437C4⋊C4—182no—
438C22⋊C4—22no—
439C22⋊C4—182no—
440C22⋊C4—182no—
441C22⋊C4—182no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—121no—
460C22×C3—121no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3—182no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—361no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3C2×C3×D1211yescharacteristic
471C22×C3C3×#23511yescharacteristic
472C22×C3C3×D3×D411yescharacteristic
473C22×C3C3×D3×D411yescharacteristic
474C22×C3C3×#23711yescharacteristic
475C22×C3C3×#23711yescharacteristic
476C22×C3C2×C3×#6711yescharacteristic
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—121no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3—121no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—62no—
487C3×C4—121no—
488C3×C4—182no—
489C3×C4—361no—
490C3×C4—361no—
491Dic3—62no—
492Dic3—121no—
493Dic3—121no—
494C32#496111yescharacteristic
495C9—21no—
496C9C3×#39411yescharacteristic, complemented
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—92no—
504C23—181no—
505C23—181no—
506C23—181no—
507C23—181no—
508C23—181no—
509C23—181no—
510C23C4⋊AGL(1,ℤ/9)12yes—
511C23C22×AGL(1,ℤ/9)11yescharacteristic
512D4—188no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—92no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4#611⋊C312yescomplemented
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—42no—
536C2×C3—122no—
537C2×C3—362no—
538C2×C3C3×#88211yescharacteristic
539C2×C3C3×#88311yescharacteristic
540C2×C3C3×#88511yescharacteristic
541C2×C3C3×#88611yescharacteristic
542C2×C3C3×#93011yescharacteristic
543C2×C3C3×#93811yescharacteristic
544C2×C3C3×#93911yescharacteristic
545D3—122no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—182no—
554C22—182no—
555C22—182no—
556C22—361no—
557C22—361no—
558C22—361no—
559C22—361no—
560C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
561C22#3697⋊C311yescharacteristic
562C22D4×AGL(1,ℤ/9)11yescharacteristic
563C22D4×AGL(1,ℤ/9)11yescharacteristic
564C22#3699⋊C311yescharacteristic
565C22#3699⋊C311yescharacteristic
566C22C2×#611⋊C311yescharacteristic
567C4—22no—
568C4—41no—
569C4—182no—
570C4—361no—
571C4—361no—
572C3—31no—
573C3C3×#496111yescharacteristic
574C2—42no—
575C2—362no—
576C2#63444⋊C311yescharacteristic
577C2#63445⋊C311yescharacteristic
578C2#63447⋊C311yescharacteristic
579C2#63448⋊C311yescharacteristic
580C2#63492⋊C311yescharacteristic
581C2#63500⋊C311yescharacteristic
582C2#63501⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×3−1+2C2
prime radicalC22×#611⋊C3C2
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#394
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×3−1+2C2semidirect
3. C2×#3676⋊C3C2semidirect
4. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
5. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
6. C2×C4⋊(3−1+2⋊C4)C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588641C3semidirect
34. C2×C9×C22⋊C4C2×C3semidirect
35. C2×#3676C2×C3semidirect
36. C2×(C2×D9)⋊C4C2×C3semidirect
37. C2×(C2×D9)⋊C4C2×C3semidirect
38. C2×C4⋊Dic9C2×C3semidirect
39. C9⋊#95C2×C3semidirect
60. C2×C4×3−1+2D4semidirect
212. C2×C4×C9C3×D4semidirect
336. C2×C22⋊C4AGL(1,ℤ/9)semidirect
338. 3−1+2#394semidirect
496. C9C3×#394semidirect
527. C2×C4#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#611⋊C32
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
3−1+2⋊#952
C3×#49613
#105886413