Finite Group Databasebeta

← #10589758⋊C3 #13153055

Subgroups — #10589758⋊C3 #13153055

Order 1728.

1029 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3699⋊C3C211yesmaximal, characteristic, complemented
2C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
33−1+2×#123C211yesmaximal, characteristic, complemented
4#63499⋊C3C211yesmaximal, characteristic, complemented
5#63491⋊C3C211yesmaximal, characteristic, complemented
6#63458⋊C3C212yesmaximal, complemented
7#63455⋊C3C212yesmaximal, complemented
8#63446⋊C3C212yesmaximal, complemented
9#63444⋊C3C211yesmaximal, characteristic, complemented
10(#611⋊C3)⋊C4C212yesmaximal, complemented
11#63439⋊C3C211yesmaximal, characteristic, complemented
12C3×#5656—31nomaximal
13#10589758C311yesmaximal, characteristic, complemented
14C23×AGL(1,ℤ/9)C2211yescharacteristic
15C2×D4×3−1+2C2211yescharacteristic
16C22×C4×3−1+2C2211yescharacteristic
17C2×#611⋊C3C2211yescharacteristic
18C2×#611⋊C3C2212yes—
19C2×#611⋊C3C2211yescharacteristic, complemented
20C22×3−1+2⋊C4C2211yescharacteristic
21C22×3−1+2⋊C4C2211yescharacteristic, complemented
22#3699⋊C3—24no—
23C2×C4⋊AGL(1,ℤ/9)C2212yes—
24C2×C4×AGL(1,ℤ/9)—24no—
25C2×C4×AGL(1,ℤ/9)C2212yes—
26C2×C4×AGL(1,ℤ/9)C2212yes—
27C2×C4×AGL(1,ℤ/9)C2211yescharacteristic, complemented
28C2×3−1+2⋊Q8C2211yescharacteristic, complemented
29C4⋊C4×3−1+2C2212yescomplemented
30C22⋊C4×3−1+2C2211yescharacteristic
31C22⋊C4×3−1+2C2212yescomplemented
32#3676⋊C3C2211yescharacteristic
33(C2×AGL(1,ℤ/9))⋊C4C2212yes—
34(C2×AGL(1,ℤ/9))⋊C4C2212yes—
35(C2×AGL(1,ℤ/9))⋊C4C2212yescomplemented
36(3−1+2⋊C4)⋊C4C2212yescomplemented
37(3−1+2⋊C4)⋊C4C2212yescomplemented
38C4×3−1+2⋊C4C2212yescomplemented
39C2×C3×#237—31no—
40C22×C3×C4×D3—31no—
41C32×#123D311yescharacteristic
42C3×#937—31no—
43C3×#929—31no—
44C3×#896—32no—
45C3×#893—32no—
46C3×#884—32no—
47C3×#882—31no—
48C3×#67⋊C4—32no—
49C3×#877—31no—
50C2×#3699C2×C311yescharacteristic, complemented
51C22×C4×D9C2×C311yescharacteristic, complemented
52C9×#123—21no—
53C9×#123C2×C311yescharacteristic, complemented
54#63499C2×C311yescharacteristic, complemented
55#63491C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63455C2×C312yescomplemented
58#63446C2×C312yescomplemented
59#63444C2×C311yescharacteristic, complemented
60#611⋊C4C2×C312yescomplemented
61#63439C2×C311yescharacteristic, complemented
62C23×3−1+2C2311yescharacteristic
63C23×3−1+2C2311yescharacteristic
64C22×AGL(1,ℤ/9)—22no—
65C22×AGL(1,ℤ/9)—24no—
66C22×AGL(1,ℤ/9)C2311yescharacteristic
67C22×AGL(1,ℤ/9)C2312yes—
68D4×3−1+2—22no—
69C2×C4×3−1+2—22no—
70C2×C4×3−1+2C2311yescharacteristic
71C2×C4×3−1+2C2312yes—
72C2×C4×3−1+2C2312yes—
73#611⋊C3—22no—
74#611⋊C3—24no—
75#611⋊C3—22no—
76C2×3−1+2⋊C4—22no—
77C2×3−1+2⋊C4D414yescomplemented
78C2×3−1+2⋊C4C2311yescharacteristic
79C2×3−1+2⋊C4C2312yes—
80C2×3−1+2⋊C4C2311yescharacteristic
81C2×3−1+2⋊C4C2311yescharacteristic
82C4⋊AGL(1,ℤ/9)—24no—
83C4×AGL(1,ℤ/9)—24no—
84C4×AGL(1,ℤ/9)—24no—
85C4×AGL(1,ℤ/9)—22no—
863−1+2⋊Q8—22no—
87C3×#525—91no—
88#5656—31no—
89C23×C3×D3—31no—
90C2×C32×D4C2×D311yescharacteristic
91C22×C32×C4C2×D311yescharacteristic
92C2×C3×#67—32no—
93C2×C3×#67—31no—
94C2×C3×#67—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C3×#237—64no—
98C2×C3×D12—32no—
99C2×C3×C4×D3—32no—
100C2×C3×C4×D3—32no—
101C2×C3×C4×D3—31no—
102C2×C3×C4×D3—64no—
103C2×C3×Dic6—31no—
104C23×D9C22×C311yescharacteristic
105C32×C4⋊C4C2×D312yes—
106C32×C22⋊C4C2×D311yescharacteristic
107C32×C22⋊C4C2×D312yes—
108C3×#217—31no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—32no—
115C2×D4×C9—21no—
116C2×D4×C9C22×C311yescharacteristic
117C22×C4×C9—21no—
118C22×C4×C9C22×C311yescharacteristic
119C2×#611C22×C311yescharacteristic
120C2×#611C22×C312yes—
121C2×#611C22×C311yescharacteristic, complemented
122C22×Dic9C22×C311yescharacteristic
123C22×Dic9C22×C311yescharacteristic, complemented
124#3699—24no—
125C2×D36C22×C312yes—
126C2×C4×D9—24no—
127C2×C4×D9C22×C312yes—
128C2×C4×D9C22×C312yes—
129C2×C4×D9C22×C311yescharacteristic, complemented
130C2×Dic18C22×C311yescharacteristic, complemented
131C9×C4⋊C4—22no—
132C9×C4⋊C4C22×C312yescomplemented
133C9×C22⋊C4—21no—
134C9×C22⋊C4—22no—
135C9×C22⋊C4C22×C311yescharacteristic
136C9×C22⋊C4C22×C312yescomplemented
137#3676C22×C311yescharacteristic
138(C2×D9)⋊C4C22×C312yes—
139(C2×D9)⋊C4C22×C312yes—
140(C2×D9)⋊C4C22×C312yescomplemented
141Dic9⋊C4C22×C312yescomplemented
142Dic9⋊C4C22×C312yescomplemented
143C4×Dic9C22×C312yescomplemented
144C22×3−1+2—22no—
145C22×3−1+2—21no—
146C22×3−1+2—22no—
147C22×3−1+2C2×D412yes—
148C22×3−1+2C2411yescharacteristic
149C2×AGL(1,ℤ/9)—28no—
150C2×AGL(1,ℤ/9)—22no—
151C2×AGL(1,ℤ/9)—22no—
152C2×AGL(1,ℤ/9)—21no—
153C2×AGL(1,ℤ/9)—22no—
154C2×AGL(1,ℤ/9)#4114yescomplemented
155C4×3−1+2—21no—
156C4×3−1+2—22no—
157C4×3−1+2—22no—
1583−1+2⋊C4—21no—
1593−1+2⋊C4—22no—
1603−1+2⋊C4C2×D412yes—
1613−1+2⋊C4C2×D412yes—
162C2×C3×#41—91no—
163C23×C3×C4—91no—
164C2×#237—31no—
165C22×C4×D3—31no—
166C3×#123—31no—
167C3×#123—91no—
168C3×#123C3×D311yescharacteristic
169C3×C22⋊Q8—92no—
170C3×#121—92no—
171C3×#120—91no—
172C3×#120—91no—
173C3×C4×D4—92no—
174C3×C4×D4—92no—
175C3×#117—91no—
176#937—31no—
177#929—31no—
178#896—32no—
179#893—32no—
180#884—32no—
181#882—31no—
182#67⋊C4—32no—
183#877—31no—
184C23×C32C22×D311yescharacteristic
185C23×C32C22×D311yescharacteristic
186C22×C3×D3—32no—
187C22×C3×D3—31no—
188C22×C3×D3—64no—
189C22×C3×D3—62no—
190C32×D4—22no—
191C2×C32×C4—22no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C3×#67—62no—
196C3×#67—64no—
197C3×#67—62no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—62no—
204C3×D12—64no—
205C3×C4×D3—64no—
206C3×C4×D3—64no—
207C3×C4×D3—62no—
208C3×Dic6—62no—
209C23×C9—21no—
210C23×C9—21no—
211C23×C9C23×C311yescharacteristic
212C23×C9C23×C311yescharacteristic
213C22×D9—22no—
214C22×D9—24no—
215C22×D9C23×C311yescharacteristic
216C22×D9C23×C312yes—
217D4×C9—22no—
218D4×C9—42no—
219C2×C4×C9—22no—
220C2×C4×C9—22no—
221C2×C4×C9—22no—
222C2×C4×C9—21no—
223C2×C4×C9—42no—
224C2×C4×C9C23×C312yes—
225C2×C4×C9C23×C312yes—
226C2×C4×C9C23×C311yescharacteristic
227#611—22no—
228#611—24no—
229#611—22no—
230C2×Dic9—22no—
231C2×Dic9C3×D414yescomplemented
232C2×Dic9C23×C312yes—
233C2×Dic9C23×C311yescharacteristic
234C2×Dic9C23×C311yescharacteristic
235C2×Dic9C23×C311yescharacteristic
236D36—24no—
237C4×D9—24no—
238C4×D9—24no—
239C4×D9—22no—
240Dic18—22no—
241#525—91noSylow
242C2×3−1+2—22no—
243C2×3−1+2—41no—
244C2×3−1+2C22×D411yescharacteristic
245C2×3−1+2C2×#4112yes—
246AGL(1,ℤ/9)—24no—
247AGL(1,ℤ/9)—41no—
248C24×C3—91no—
249C23×D3—31no—
250C3×#41—184no—
251C2×C3×Q8—91no—
252C2×C3×D4—31no—
253C2×C3×D4—92no—
254C2×C3×D4—92no—
255C2×C3×D4—91no—
256C2×C3×D4—91no—
257C2×C3×D4C2×C3×D311yescharacteristic
258C22×C3×C4—31no—
259C22×C3×C4—92no—
260C22×C3×C4—91no—
261C22×C3×C4—92no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—184no—
265C22×C3×C4C2×C3×D311yescharacteristic
266C2×#67—32no—
267C2×#67—31no—
268C2×#67—31no—
269C22×Dic3—31no—
270C22×Dic3—31no—
271#237—64no—
272C2×D12—32no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—31no—
276C2×C4×D3—64no—
277C2×Dic6—31no—
278C3×C4⋊C4—32no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4C2×C3×D312yes—
282C3×C22⋊C4—31no—
283C3×C22⋊C4—32no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C22⋊C4—91no—
288C3×C22⋊C4C2×C3×D311yescharacteristic
289C3×C22⋊C4C2×C3×D312yes—
290C3×C42—92no—
291#217—31no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297C4×Dic3—32no—
298C22×C32—22no—
299C22×C32—21no—
300C22×C32—22no—
301C22×C32D3×D412yes—
302C22×C32C23×D311yescharacteristic
303C2×C3×D3—34no—
304C2×C3×D3—68no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—61no—
308C2×C3×D3—62no—
309C32×C4—21no—
310C32×C4—22no—
311C32×C4—22no—
312C3×Dic3—32no—
313C3×Dic3—32no—
314C3×Dic3—61no—
315C3×Dic3—62no—
316C22×C9—22no—
317C22×C9—21no—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—22no—
321C22×C9—42no—
322C22×C9—41no—
323C22×C9—42no—
324C22×C9C2×C3×D412yes—
325C22×C9C24×C311yescharacteristic
326C2×D9—28no—
327C2×D9—22no—
328C2×D9—22no—
329C2×D9—21no—
330C2×D9—22no—
331C2×D9C3×#4114yescomplemented
332C4×C9—21no—
333C4×C9—22no—
334C4×C9—22no—
335C4×C9—42no—
336C4×C9—42no—
337C4×C9—41no—
338Dic9—21no—
339Dic9—22no—
340Dic9C2×C3×D412yes—
341Dic9C2×C3×D412yes—
342C2×#41—91no—
343C23×C4—91no—
344#123—91no—
345#123AGL(1,ℤ/9)11yescharacteristic, complemented
346C22⋊Q8—92no—
347#121—92no—
348#120—91no—
349#120—91no—
350C4×D4—92no—
351C4×D4—92no—
352#117—91no—
3533−1+2#52511yescharacteristic, complemented, Sylow
354C23×C3—31no—
355C23×C3—31no—
356C23×C3—92no—
357C23×C3—91no—
358C23×C3—182no—
359C23×C3—184no—
360C23×C3C22×C3×D311yescharacteristic
361C23×C3C22×C3×D311yescharacteristic
362C22×D3—32no—
363C22×D3—31no—
364C22×D3—64no—
365C22×D3—62no—
366C3×Q8—182no—
367C3×D4—22no—
368C3×D4—62no—
369C3×D4—184no—
370C3×D4—182no—
371C3×D4—182no—
372C3×D4—184no—
373C2×C3×C4—22no—
374C2×C3×C4—32no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—62no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—94no—
381C2×C3×C4—92no—
382C2×C3×C4—91no—
383C2×C3×C4—184no—
384C2×C3×C4—184no—
385C2×C3×C4—182no—
386C2×C3×C4—182no—
387C2×C3×C4C22×C3×D312yes—
388C2×C3×C4C22×C3×D312yes—
389C2×C3×C4C22×C3×D311yescharacteristic
390#67—62no—
391#67—64no—
392#67—62no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—34no—
396C2×Dic3—32no—
397C2×Dic3—31no—
398C2×Dic3—62no—
399D12—64no—
400C4×D3—64no—
401C4×D3—64no—
402C4×D3—62no—
403Dic6—62no—
404C2×C32—22no—
405C2×C32—41no—
406C2×C32C2×D3×D411yescharacteristic
407C2×C32D3×#4112yes—
408C3×D3—64no—
409C3×D3—121no—
410C2×C9—22no—
411C2×C9—21no—
412C2×C9—22no—
413C2×C9—42no—
414C2×C9—41no—
415C2×C9—81no—
416C2×C9C22×C3×D411yescharacteristic
417C2×C9C2×C3×#4112yes—
418D9—24no—
419D9—41no—
420C24—91no—
421#41—184no—
422C2×Q8—91no—
423C2×D4—92no—
424C2×D4—91no—
425C2×D4—91no—
426C2×D4—92no—
427C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—92no—
431C22×C4—91no—
432C22×C4—92no—
433C22×C4—184no—
434C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
435C4⋊C4—92no—
436C4⋊C4—92no—
437C4⋊C4C2×AGL(1,ℤ/9)12yescomplemented
438C22⋊C4—91no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
443C22⋊C4C2×AGL(1,ℤ/9)12yescomplemented
444C42—92no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—31no—
449C22×C3—32no—
450C22×C3—62no—
451C22×C3—61no—
452C22×C3—62no—
453C22×C3—94no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—181no—
457C22×C3—182no—
458C22×C3—188no—
459C22×C3C3×D3×D412yes—
460C22×C3C23×C3×D311yescharacteristic
461C2×D3—34no—
462C2×D3—68no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—61no—
466C2×D3—62no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—62no—
471C3×C4—61no—
472C3×C4—62no—
473C3×C4—92no—
474C3×C4—92no—
475C3×C4—181no—
476C3×C4—182no—
477Dic3—32no—
478Dic3—32no—
479Dic3—61no—
480Dic3—62no—
481C32#565611yescharacteristic
482C9—21no—
483C9C3×#52511yescharacteristic, complemented
484C23—92no—
485C23—91no—
486C23—184no—
487C23—182no—
488C23C22×AGL(1,ℤ/9)11yescharacteristic
489C23C22×AGL(1,ℤ/9)11yescharacteristic
490Q8—182no—
491D4—22no—
492D4—184no—
493D4—182no—
494D4—184no—
495D4—182no—
496C2×C4—22no—
497C2×C4—91no—
498C2×C4—91no—
499C2×C4—94no—
500C2×C4—92no—
501C2×C4—91no—
502C2×C4—184no—
503C2×C4—184no—
504C2×C4—182no—
505C2×C4—182no—
506C2×C4C22×AGL(1,ℤ/9)12yes—
507C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
508C2×C4C22×AGL(1,ℤ/9)12yes—
509C2×C3—22no—
510C2×C3—31no—
511C2×C3—32no—
512C2×C3—41no—
513C2×C3—62no—
514C2×C3—121no—
515C2×C3—184no—
516C2×C3—361no—
517C2×C3C2×C3×D3×D411yescharacteristic
518C2×C3C3×D3×#4112yes—
519D3—64no—
520D3—121no—
521C22—22no—
522C22—21no—
523C22—22no—
524C22—94no—
525C22—188no—
526C22—182no—
527C22—182no—
528C22—181no—
529C22—182no—
530C22D4×AGL(1,ℤ/9)12yes—
531C22C23×AGL(1,ℤ/9)11yescharacteristic
532C4—22no—
533C4—21no—
534C4—22no—
535C4—92no—
536C4—92no—
537C4—181no—
538C4—182no—
539C3—31no—
540C3C3×#565611yescharacteristic
541C2—22no—
542C2—41no—
543C2—184no—
544C2—361no—
545C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
546C2#41×AGL(1,ℤ/9)12yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#123C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#525
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#3699⋊C3C2semidirect
2. C22×C4×AGL(1,ℤ/9)C2semidirect
3. 3−1+2×#123C2semidirect
4. #63499⋊C3C2semidirect
5. #63491⋊C3C2semidirect
6. #63458⋊C3C2semidirect
7. #63455⋊C3C2semidirect
8. #63446⋊C3C2semidirect
9. #63444⋊C3C2semidirect
10. (#611⋊C3)⋊C4C2semidirect
11. #63439⋊C3C2semidirect
13. #10589758C3semidirect
19. C2×#611⋊C3C22semidirect
21. C22×3−1+2⋊C4C22semidirect
27. C2×C4×AGL(1,ℤ/9)C22semidirect
28. C2×3−1+2⋊Q8C22semidirect
29. C4⋊C4×3−1+2C22semidirect
31. C22⋊C4×3−1+2C22semidirect
35. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
36. (3−1+2⋊C4)⋊C4C22semidirect
37. (3−1+2⋊C4)⋊C4C22semidirect
38. C4×3−1+2⋊C4C22semidirect
50. C2×#3699C2×C3semidirect
51. C22×C4×D9C2×C3semidirect
53. C9×#123C2×C3semidirect
54. #63499C2×C3semidirect
55. #63491C2×C3semidirect
56. #63458C2×C3semidirect
57. #63455C2×C3semidirect
58. #63446C2×C3semidirect
59. #63444C2×C3semidirect
60. #611⋊C4C2×C3semidirect
61. #63439C2×C3semidirect
77. C2×3−1+2⋊C4D4semidirect
121. C2×#611C22×C3semidirect
123. C22×Dic9C22×C3semidirect
129. C2×C4×D9C22×C3semidirect
130. C2×Dic18C22×C3semidirect
132. C9×C4⋊C4C22×C3semidirect
136. C9×C22⋊C4C22×C3semidirect
140. (C2×D9)⋊C4C22×C3semidirect
141. Dic9⋊C4C22×C3semidirect
142. Dic9⋊C4C22×C3semidirect
143. C4×Dic9C22×C3semidirect
154. C2×AGL(1,ℤ/9)#41semidirect
231. C2×Dic9C3×D4semidirect
331. C2×D9C3×#41semidirect
345. #123AGL(1,ℤ/9)semidirect
353. 3−1+2#525semidirect
437. C4⋊C4C2×AGL(1,ℤ/9)semidirect
443. C22⋊C4C2×AGL(1,ℤ/9)semidirect
483. C9C3×#525semidirect

Maximal subgroups

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

subgroupindex
C2×#3699⋊C32
C22×C4×AGL(1,ℤ/9)2
3−1+2×#1232
#63499⋊C32
#63491⋊C32
#63458⋊C32
#63455⋊C32
#63446⋊C32
#63444⋊C32
(#611⋊C3)⋊C42
#63439⋊C32
C3×#56563
#105897583