Finite Group Databasebeta

← #10589731⋊C3 #13152980

Subgroups — #10589731⋊C3 #13152980

Order 1728.

919 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88724C211yesmaximal, characteristic, complemented
2C2×#3697⋊C3C211yesmaximal, characteristic, complemented
33−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63508⋊C3C211yesmaximal, characteristic, complemented
5#63492⋊C3C211yesmaximal, characteristic, complemented
6#63458⋊C3C212yesmaximal, complemented
7#63457⋊C3C211yesmaximal, characteristic, complemented
8#63457⋊C3C212yesmaximal, complemented
9C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
10#63446⋊C3C212yesmaximal, complemented
11#63446⋊C3C212yesmaximal, complemented
12C3×#5629—31nomaximal
13#10589731C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2211yescharacteristic
17C2×#611⋊C3C2211yescharacteristic
18C2×#611⋊C3C2212yes—
19#88724—24no—
20#3697⋊C3—24no—
21C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
22C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
23C2×C4⋊AGL(1,ℤ/9)C2212yes—
24C2×C4⋊AGL(1,ℤ/9)C2212yes—
25C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
26C2×C4×AGL(1,ℤ/9)C2212yes—
27C2×C4×AGL(1,ℤ/9)C2212yes—
28C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
29C2×3−1+2⋊Q8C2211yescharacteristic
30C4⋊C4×3−1+2C2211yescharacteristic
31C4⋊C4×3−1+2C2212yes—
32C22⋊C4×3−1+2C2212yes—
33(C2×AGL(1,ℤ/9))⋊C4C2212yes—
34(C2×AGL(1,ℤ/9))⋊C4C2212yes—
35(C2×AGL(1,ℤ/9))⋊C4C2212yes—
36C4⋊(3−1+2⋊C4)C2211yescharacteristic
37(3−1+2⋊C4)⋊C4C2212yes—
38(3−1+2⋊C4)⋊C4C2212yes—
39C2×C3×#239—31no—
40C2×C3×#235—31no—
41C32×C22⋊Q8D311yescharacteristic
42C3×#946—31no—
43C3×#930—31no—
44C3×#896—32no—
45C3×#895—31no—
46C3×#895—32no—
47C3×D3×C4⋊C4—31no—
48C3×#884—32no—
49C3×#884—32no—
50C2×#3701C2×C311yescharacteristic, complemented
51C2×#3697C2×C311yescharacteristic, complemented
52C9×C22⋊Q8—21no—
53C9×C22⋊Q8C2×C311yescharacteristic, complemented
54#63508C2×C311yescharacteristic, complemented
55#63492C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63457C2×C312yescomplemented
58#63457C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C311yescharacteristic, complemented
60#63446C2×C312yescomplemented
61#63446C2×C312yescomplemented
62C23×3−1+2C2311yescharacteristic
63C22×AGL(1,ℤ/9)C2312yes—
64C22×AGL(1,ℤ/9)C2311yescharacteristic
65C22×AGL(1,ℤ/9)C2311yescharacteristic
66Q8×3−1+2—22no—
67C2×C4×3−1+2—22no—
68C2×C4×3−1+2C2311yescharacteristic
69C2×C4×3−1+2C2312yes—
70C2×C4×3−1+2C2312yes—
71C2×C4×3−1+2C2311yescharacteristic
72#611⋊C3—22no—
73#611⋊C3—22no—
74#611⋊C3—42no—
75C2×3−1+2⋊C4C2311yescharacteristic
76C2×3−1+2⋊C4C2312yes—
77C2×3−1+2⋊C4C2311yescharacteristic
78C4⋊AGL(1,ℤ/9)—22no—
79C4⋊AGL(1,ℤ/9)—24no—
80C4⋊AGL(1,ℤ/9)—22no—
81C4⋊AGL(1,ℤ/9)—42no—
82C4×AGL(1,ℤ/9)—22no—
83C4×AGL(1,ℤ/9)—24no—
84C4×AGL(1,ℤ/9)—42no—
85C4×AGL(1,ℤ/9)D414yescomplemented
863−1+2⋊Q8—22no—
87C3×#537—91no—
88#5629—31no—
89C2×C32×Q8C2×D311yescharacteristic
90C22×C32×C4C2×D311yescharacteristic
91C2×C3×#67—31no—
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C3×#239—64no—
95C3×#235—64no—
96C2×C3×D12—32no—
97C2×C3×D12—31no—
98C2×C3×D12—31no—
99C2×C3×D12—32no—
100C2×C3×C4×D3—31no—
101C2×C3×C4×D3—32no—
102C2×C3×C4×D3—31no—
103C2×C3×C4×D3—32no—
104C2×C3×Dic6—31no—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C4⋊C4C2×D312yes—
107C32×C22⋊C4C2×D312yes—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×C4⋊Dic3—31no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C2×Q8×C9—21no—
115C2×Q8×C9C22×C311yescharacteristic
116C22×C4×C9—21no—
117C22×C4×C9C22×C311yescharacteristic
118C2×#611C22×C311yescharacteristic
119C2×#611C22×C311yescharacteristic
120C2×#611C22×C312yes—
121#3701—24no—
122#3697—24no—
123C2×D36C22×C311yescharacteristic
124C2×D36C22×C312yes—
125C2×D36C22×C311yescharacteristic
126C2×D36C22×C312yes—
127C2×C4×D9C22×C311yescharacteristic
128C2×C4×D9C22×C312yes—
129C2×C4×D9C22×C312yes—
130C2×C4×D9C22×C311yescharacteristic
131C2×Dic18C22×C311yescharacteristic
132C9×C4⋊C4—21no—
133C9×C4⋊C4—22no—
134C9×C4⋊C4C22×C311yescharacteristic
135C9×C4⋊C4C22×C312yes—
136C9×C22⋊C4—22no—
137C9×C22⋊C4C22×C312yes—
138(C2×D9)⋊C4C22×C312yes—
139(C2×D9)⋊C4C22×C312yes—
140(C2×D9)⋊C4C22×C312yes—
141C4⋊Dic9C22×C311yescharacteristic
142Dic9⋊C4C22×C312yes—
143Dic9⋊C4C22×C312yes—
144C22×3−1+2—21no—
145C22×3−1+2—21no—
146C22×3−1+2—21no—
147C22×3−1+2C2411yescharacteristic
148C2×AGL(1,ℤ/9)—21no—
149C2×AGL(1,ℤ/9)—21no—
150C2×AGL(1,ℤ/9)—22no—
151C2×AGL(1,ℤ/9)—21no—
152C2×AGL(1,ℤ/9)—21no—
153C2×AGL(1,ℤ/9)—22no—
154C2×AGL(1,ℤ/9)—22no—
155C2×AGL(1,ℤ/9)—21no—
156C2×AGL(1,ℤ/9)C2×D412yes—
157C4×3−1+2—21no—
158C4×3−1+2—22no—
159C4×3−1+2—22no—
160C4×3−1+2C2×D412yes—
1613−1+2⋊C4—21no—
1623−1+2⋊C4—22no—
1633−1+2⋊C4C2×D412yes—
164C2×C3×#41—91no—
165C2×C3×#41—91no—
166C2×#239—31no—
167C2×#235—31no—
168C3×C22⋊Q8—31no—
169C3×C22⋊Q8—92no—
170C3×C22⋊Q8—91no—
171C3×C22⋊Q8C3×D311yescharacteristic
172C3×#121—92no—
173C3×#121—92no—
174C3×#121—91no—
175C3×#121—92no—
176C3×#121—91no—
177C2×C3×C4⋊C4—91no—
178#946—31no—
179#930—31no—
180#896—32no—
181#895—31no—
182#895—32no—
183D3×C4⋊C4—31no—
184#884—32no—
185#884—32no—
186C23×C32C22×D311yescharacteristic
187C22×C3×D3—31no—
188C22×C3×D3—32no—
189C22×C3×D3—31no—
190C32×Q8—22no—
191C2×C32×C4—22no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C3×#67—62no—
197C3×#67—62no—
198C3×#67—122no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C3×D12—62no—
203C3×D12—64no—
204C3×D12—62no—
205C3×D12—122no—
206C3×C4×D3—34no—
207C3×C4×D3—64no—
208C3×C4×D3—62no—
209C3×C4×D3—122no—
210C3×Dic6—62no—
211C23×C9—21no—
212C23×C9C23×C311yescharacteristic
213C22×D9C23×C312yes—
214C22×D9C23×C311yescharacteristic
215C22×D9C23×C311yescharacteristic
216Q8×C9—22no—
217Q8×C9—42no—
218C2×C4×C9—22no—
219C2×C4×C9—21no—
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
227C2×C4×C9C23×C311yescharacteristic
228#611—22no—
229#611—22no—
230#611—42no—
231C2×Dic9C23×C312yes—
232C2×Dic9C23×C311yescharacteristic
233C2×Dic9C23×C311yescharacteristic
234D36—22no—
235D36—24no—
236D36—22no—
237D36—42no—
238C4×D9—22no—
239C4×D9—24no—
240C4×D9—42no—
241C4×D9C3×D414yescomplemented
242Dic18—22no—
243#537—91noSylow
244C2×3−1+2—41no—
245C2×3−1+2C22×D411yescharacteristic
246C2×3−1+2H(5,𝔽2)11yescharacteristic
247C2×3−1+22−1+411yescharacteristic
248AGL(1,ℤ/9)—22no—
249AGL(1,ℤ/9)—42no—
250AGL(1,ℤ/9)—41no—
251C3×#41—184no—
252C3×#41—184no—
253C2×C3×Q8—31no—
254C2×C3×Q8—91no—
255C2×C3×Q8C2×C3×D311yescharacteristic
256C2×C3×D4—91no—
257C2×C3×D4—92no—
258C2×C3×D4—91no—
259C2×C3×D4—92no—
260C2×C3×D4—91no—
261C2×C3×D4—91no—
262C2×C3×D4—92no—
263C22×C3×C4—31no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×C3×C4—92no—
267C22×C3×C4—92no—
268C22×C3×C4C2×C3×D311yescharacteristic
269C2×#67—31no—
270C2×#67—31no—
271C2×#67—32no—
272#239—64no—
273#235—64no—
274C2×D12—32no—
275C2×D12—31no—
276C2×D12—31no—
277C2×D12—32no—
278C2×C4×D3—31no—
279C2×C4×D3—32no—
280C2×C4×D3—31no—
281C2×C4×D3—32no—
282C2×Dic6—31no—
283C3×C4⋊C4—31no—
284C3×C4⋊C4—32no—
285C3×C4⋊C4—92no—
286C3×C4⋊C4—92no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4C2×C3×D311yescharacteristic
289C3×C4⋊C4C2×C3×D312yes—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4C2×C3×D312yes—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—32no—
298C4⋊Dic3—31no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32C23×D311yescharacteristic
305C2×C3×D3—32no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—62no—
309C2×C3×D3—61no—
310C2×C3×D3—62no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C32×C4—21no—
315C32×C4—22no—
316C32×C4—22no—
317C32×C4D3×D412yes—
318C3×Dic3—32no—
319C3×Dic3—61no—
320C3×Dic3—62no—
321C22×C9—22no—
322C22×C9—21no—
323C22×C9—21no—
324C22×C9—21no—
325C22×C9—21no—
326C22×C9—41no—
327C22×C9—41no—
328C22×C9C24×C311yescharacteristic
329C2×D9—21no—
330C2×D9—21no—
331C2×D9—22no—
332C2×D9—21no—
333C2×D9—21no—
334C2×D9—22no—
335C2×D9—22no—
336C2×D9—21no—
337C2×D9C2×C3×D412yes—
338C4×C9—21no—
339C4×C9—22no—
340C4×C9—22no—
341C4×C9—22no—
342C4×C9—42no—
343C4×C9—41no—
344C4×C9—42no—
345C4×C9C2×C3×D412yes—
346Dic9—21no—
347Dic9—22no—
348Dic9C2×C3×D412yes—
349C2×#41—91no—
350C2×#41—91no—
351C22⋊Q8—91no—
352C22⋊Q8—92no—
353C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
354#121—91no—
355#121—91no—
356#121—92no—
357#121—92no—
358#121—92no—
359C2×C4⋊C4—91no—
3603−1+2#53711yescharacteristic, complemented, Sylow
361C23×C3—31no—
362C23×C3—91no—
363C23×C3—92no—
364C23×C3—91no—
365C23×C3C22×C3×D311yescharacteristic
366C22×D3—31no—
367C22×D3—32no—
368C22×D3—31no—
369C3×Q8—22no—
370C3×Q8—62no—
371C3×Q8—182no—
372C3×D4—182no—
373C3×D4—182no—
374C3×D4—184no—
375C3×D4—182no—
376C3×D4—182no—
377C3×D4—362no—
378C3×D4—362no—
379C2×C3×C4—22no—
380C2×C3×C4—31no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—31no—
384C2×C3×C4—62no—
385C2×C3×C4—94no—
386C2×C3×C4—91no—
387C2×C3×C4—92no—
388C2×C3×C4—91no—
389C2×C3×C4—182no—
390C2×C3×C4—184no—
391C2×C3×C4—362no—
392C2×C3×C4C22×C3×D312yes—
393C2×C3×C4C22×C3×D312yes—
394C2×C3×C4C22×C3×D311yescharacteristic
395C2×C3×C4C22×C3×D311yescharacteristic
396#67—62no—
397#67—62no—
398#67—122no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—31no—
402D12—62no—
403D12—64no—
404D12—62no—
405D12—122no—
406C4×D3—34no—
407C4×D3—64no—
408C4×D3—62no—
409C4×D3—122no—
410Dic6—62no—
411C2×C32—41no—
412C2×C32C2×D3×D411yescharacteristic
413C2×C32#100711yescharacteristic
414C2×C32#100911yescharacteristic
415C3×D3—62no—
416C3×D3—122no—
417C3×D3—121no—
418C2×C9—21no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—42no—
422C2×C9—41no—
423C2×C9C22×C3×D411yescharacteristic
424C2×C9C3×H(5,𝔽2)11yescharacteristic
425C2×C9C3×2−1+411yescharacteristic
426D9—22no—
427D9—42no—
428D9—41no—
429#41—184no—
430#41—184no—
431C2×Q8—91no—
432C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
433C2×D4—91no—
434C2×D4—91no—
435C2×D4—92no—
436C2×D4—91no—
437C2×D4—92no—
438C2×D4—91no—
439C2×D4—92no—
440C22×C4—91no—
441C22×C4—92no—
442C22×C4—91no—
443C22×C4—92no—
444C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
445C4⋊C4—92no—
446C4⋊C4—91no—
447C4⋊C4—92no—
448C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
449C4⋊C4C2×AGL(1,ℤ/9)12yes—
450C22⋊C4—92no—
451C22⋊C4—92no—
452C22⋊C4—92no—
453C22⋊C4C2×AGL(1,ℤ/9)12yes—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—31no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3—92no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—182no—
466C22×C3—181no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—181no—
470C22×C3C23×C3×D311yescharacteristic
471C2×D3—32no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—62no—
475C2×D3—61no—
476C2×D3—62no—
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—62no—
480C3×C4—21no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—32no—
484C3×C4—61no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—92no—
488C3×C4—181no—
489C3×C4—182no—
490C3×C4C3×D3×D412yes—
491Dic3—32no—
492Dic3—61no—
493Dic3—62no—
494C32#562911yescharacteristic
495C9—21no—
496C9C3×#53711yescharacteristic, complemented
497C23—91no—
498C23—91no—
499C23—92no—
500C23C22×AGL(1,ℤ/9)11yescharacteristic
501Q8—22no—
502Q8—182no—
503D4—182no—
504D4—184no—
505D4—182no—
506D4—182no—
507D4—182no—
508D4—362no—
509D4—362no—
510C2×C4—22no—
511C2×C4—91no—
512C2×C4—94no—
513C2×C4—92no—
514C2×C4—91no—
515C2×C4—184no—
516C2×C4—182no—
517C2×C4—362no—
518C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
519C2×C4C22×AGL(1,ℤ/9)12yes—
520C2×C4C22×AGL(1,ℤ/9)12yes—
521C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—41no—
526C2×C3—121no—
527C2×C3—182no—
528C2×C3—361no—
529C2×C3—362no—
530C2×C3C2×C3×D3×D411yescharacteristic
531C2×C3C3×#100711yescharacteristic
532C2×C3C3×#100911yescharacteristic
533D3—62no—
534D3—122no—
535D3—121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—92no—
540C22—181no—
541C22—181no—
542C22—182no—
543C22—181no—
544C22—181no—
545C22—182no—
546C22—181no—
547C22—182no—
548C22C23×AGL(1,ℤ/9)11yescharacteristic
549C4—21no—
550C4—22no—
551C4—22no—
552C4—92no—
553C4—181no—
554C4—182no—
555C4D4×AGL(1,ℤ/9)12yes—
556C3—31no—
557C3C3×#562911yescharacteristic
558C2—41no—
559C2—182no—
560C2—362no—
561C2—361no—
562C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
563C2#1172263011yescharacteristic
564C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#537
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88724C2semidirect
2. C2×#3697⋊C3C2semidirect
3. 3−1+2×C22⋊Q8C2semidirect
4. #63508⋊C3C2semidirect
5. #63492⋊C3C2semidirect
6. #63458⋊C3C2semidirect
7. #63457⋊C3C2semidirect
8. #63457⋊C3C2semidirect
9. C4⋊C4×AGL(1,ℤ/9)C2semidirect
10. #63446⋊C3C2semidirect
11. #63446⋊C3C2semidirect
13. #10589731C3semidirect
50. C2×#3701C2×C3semidirect
51. C2×#3697C2×C3semidirect
53. C9×C22⋊Q8C2×C3semidirect
54. #63508C2×C3semidirect
55. #63492C2×C3semidirect
56. #63458C2×C3semidirect
57. #63457C2×C3semidirect
58. #63457C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63446C2×C3semidirect
61. #63446C2×C3semidirect
85. C4×AGL(1,ℤ/9)D4semidirect
241. C4×D9C3×D4semidirect
353. C22⋊Q8AGL(1,ℤ/9)semidirect
360. 3−1+2#537semidirect
496. C9C3×#537semidirect

Maximal subgroups

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

subgroupindex
C2×#887242
C2×#3697⋊C32
3−1+2×C22⋊Q82
#63508⋊C32
#63492⋊C32
#63458⋊C32
#63457⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63446⋊C32
C3×#56293
#105897313