Finite Group Databasebeta

← #10589733⋊C3 #13152982

Subgroups — #10589733⋊C3 #13152982

Order 1728.

913 conjugacy classes of subgroups.

The subgroup lattice has 530 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#63506⋊C3C211yesmaximal, characteristic, complemented
5C4×#611⋊C3C211yesmaximal, characteristic, complemented
6#63459⋊C3C211yesmaximal, characteristic, complemented
7#63456⋊C3C212yesmaximal, complemented
8#63455⋊C3C211yesmaximal, characteristic, complemented
9C4⋊C4×AGL(1,ℤ/9)C212yesmaximal, complemented
10#63446⋊C3C212yesmaximal, complemented
11#63445⋊C3C212yesmaximal, complemented
12C3×#5631—31nomaximal
13#10589733C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2212yes—
17C2×#611⋊C3C2211yescharacteristic
18#88724—24no—
19#3697⋊C3—24no—
20C2×C4⋊AGL(1,ℤ/9)C2212yes—
21C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
22C2×C4×AGL(1,ℤ/9)C2212yes—
23C2×C4×AGL(1,ℤ/9)C2212yes—
24C2×C4×AGL(1,ℤ/9)C2212yes—
25C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
26C2×3−1+2⋊Q8C2211yescharacteristic
27C4⋊C4×3−1+2C2212yes—
28C4⋊C4×3−1+2C2211yescharacteristic
29C22⋊C4×3−1+2C2212yes—
30#3676⋊C3C2211yescharacteristic
31(C2×AGL(1,ℤ/9))⋊C4C2212yes—
32(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
33(C2×AGL(1,ℤ/9))⋊C4C2212yes—
34C4⋊(3−1+2⋊C4)C2212yes—
35(3−1+2⋊C4)⋊C4C2211yescharacteristic
36(3−1+2⋊C4)⋊C4C2212yes—
37(3−1+2⋊C4)⋊C4C2212yes—
38C4×3−1+2⋊C4C2211yescharacteristic
39C2×C3×#239—31no—
40C2×C3×#235—31no—
41C32×C22⋊Q8D311yescharacteristic
42C3×#944—31no—
43C3×C4×#67—31no—
44C3×#897—31no—
45C3×#894—32no—
46C3×#893—31no—
47C3×D3×C4⋊C4—32no—
48C3×#884—32no—
49C3×#883—32no—
50C2×#3701C2×C311yescharacteristic, complemented
51C2×#3697C2×C311yescharacteristic, complemented
52C9×C22⋊Q8—21no—
53C9×C22⋊Q8C2×C311yescharacteristic, complemented
54#63506C2×C311yescharacteristic, complemented
55C4×#611C2×C311yescharacteristic, complemented
56#63459C2×C311yescharacteristic, complemented
57#63456C2×C312yescomplemented
58#63455C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C312yescomplemented
60#63446C2×C312yescomplemented
61#63445C2×C312yescomplemented
62C23×3−1+2C2311yescharacteristic
63C22×AGL(1,ℤ/9)C2311yescharacteristic
64C22×AGL(1,ℤ/9)C2312yes—
65Q8×3−1+2—22no—
66C2×C4×3−1+2—22no—
67C2×C4×3−1+2C2311yescharacteristic
68C2×C4×3−1+2C2312yes—
69C2×C4×3−1+2C2312yes—
70C2×C4×3−1+2C2311yescharacteristic
71#611⋊C3—24no—
72#611⋊C3—22no—
73C2×3−1+2⋊C4C2312yes—
74C2×3−1+2⋊C4C2311yescharacteristic
75C2×3−1+2⋊C4C2311yescharacteristic
76C2×3−1+2⋊C4C2311yescharacteristic
77C4⋊AGL(1,ℤ/9)—24no—
78C4⋊AGL(1,ℤ/9)D414yescomplemented
79C4×AGL(1,ℤ/9)—24no—
80C4×AGL(1,ℤ/9)—24no—
81C4×AGL(1,ℤ/9)—22no—
82C4×AGL(1,ℤ/9)—24no—
833−1+2⋊Q8—22no—
84C3×#547—91no—
85#5631—31no—
86C2×C32×Q8C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×C3×#67—32no—
89C2×C3×#67—31no—
90C3×#239—64no—
91C3×#235—64no—
92C2×C3×D12—31no—
93C2×C3×D12—32no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—31no—
99C32×C4⋊C4C2×D312yes—
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C22⋊C4C2×D312yes—
102C3×#217—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—31no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—31no—
110C3×C4×Dic3—31no—
111C2×Q8×C9—21no—
112C2×Q8×C9C22×C311yescharacteristic
113C22×C4×C9—21no—
114C22×C4×C9C22×C311yescharacteristic
115C2×#611C22×C311yescharacteristic
116C2×#611C22×C312yes—
117#3701—24no—
118#3697—24no—
119C2×D36C22×C312yes—
120C2×D36C22×C311yescharacteristic
121C2×C4×D9C22×C311yescharacteristic
122C2×C4×D9C22×C312yes—
123C2×C4×D9C22×C312yes—
124C2×C4×D9C22×C312yes—
125C2×Dic18C22×C311yescharacteristic
126C9×C4⋊C4—21no—
127C9×C4⋊C4—22no—
128C9×C4⋊C4C22×C312yes—
129C9×C4⋊C4C22×C311yescharacteristic
130C9×C22⋊C4—22no—
131C9×C22⋊C4C22×C312yes—
132#3676C22×C311yescharacteristic
133(C2×D9)⋊C4C22×C311yescharacteristic
134(C2×D9)⋊C4C22×C312yes—
135(C2×D9)⋊C4C22×C312yes—
136C4⋊Dic9C22×C312yes—
137Dic9⋊C4C22×C311yescharacteristic
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140C4×Dic9C22×C311yescharacteristic
141C22×3−1+2—21no—
142C22×3−1+2—21no—
143C22×3−1+2—21no—
144C22×3−1+2C2411yescharacteristic
145C2×AGL(1,ℤ/9)—22no—
146C2×AGL(1,ℤ/9)—22no—
147C2×AGL(1,ℤ/9)—21no—
148C2×AGL(1,ℤ/9)—21no—
149C2×AGL(1,ℤ/9)—21no—
150C2×AGL(1,ℤ/9)C2×D414yes—
151C4×3−1+2—21no—
152C4×3−1+2—22no—
153C4×3−1+2—22no—
154C4×3−1+2C2×D412yes—
1553−1+2⋊C4—22no—
1563−1+2⋊C4—21no—
1573−1+2⋊C4—21no—
1583−1+2⋊C4#4112yescomplemented
159C2×C3×#41—91no—
160C2×C3×#41—91no—
161C2×#239—31no—
162C2×#235—31no—
163C3×C4⋊Q8—91no—
164C3×#123—92no—
165C3×#123—92no—
166C3×C22⋊Q8—31no—
167C3×C22⋊Q8—91no—
168C3×C22⋊Q8C3×D311yescharacteristic
169C3×#121—92no—
170C3×C4×D4—91no—
171C3×C4×D4—91no—
172C2×C3×C4⋊C4—92no—
173#944—31no—
174C4×#67—31no—
175#897—31no—
176#894—32no—
177#893—31no—
178D3×C4⋊C4—32no—
179#884—32no—
180#883—32no—
181C23×C32C22×D311yescharacteristic
182C22×C3×D3—32no—
183C22×C3×D3—31no—
184C32×Q8—22no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C3×#67—62no—
191C3×#67—64no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—31no—
196C3×D12—34no—
197C3×D12—64no—
198C3×C4×D3—64no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×C4×D3—62no—
202C3×Dic6—62no—
203C23×C9—21no—
204C23×C9C23×C311yescharacteristic
205C22×D9C23×C311yescharacteristic
206C22×D9C23×C312yes—
207Q8×C9—22no—
208Q8×C9—42no—
209C2×C4×C9—22no—
210C2×C4×C9—21no—
211C2×C4×C9—22no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—42no—
215C2×C4×C9C23×C312yes—
216C2×C4×C9C23×C311yescharacteristic
217C2×C4×C9C23×C312yes—
218C2×C4×C9C23×C311yescharacteristic
219#611—24no—
220#611—22no—
221C2×Dic9C23×C311yescharacteristic
222C2×Dic9C23×C312yes—
223C2×Dic9C23×C311yescharacteristic
224C2×Dic9C23×C311yescharacteristic
225D36—24no—
226D36C3×D414yescomplemented
227C4×D9—24no—
228C4×D9—24no—
229C4×D9—22no—
230C4×D9—24no—
231Dic18—22no—
232#547—91noSylow
233C2×3−1+2—41no—
234C2×3−1+2C22×D411yescharacteristic
235C2×3−1+2C2×#4111yescharacteristic
236C2×3−1+22−1+411yescharacteristic
237AGL(1,ℤ/9)—24no—
238AGL(1,ℤ/9)—41no—
239C3×#41—184no—
240C3×#41—184no—
241C2×C3×Q8—31no—
242C2×C3×Q8—91no—
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×C3×D4—91no—
245C2×C3×D4—92no—
246C2×C3×D4—91no—
247C2×C3×D4—92no—
248C22×C3×C4—31no—
249C22×C3×C4—92no—
250C22×C3×C4—92no—
251C22×C3×C4—92no—
252C22×C3×C4—91no—
253C22×C3×C4C2×C3×D311yescharacteristic
254C2×#67—32no—
255C2×#67—31no—
256#239—64no—
257#235—64no—
258C2×D12—31no—
259C2×D12—32no—
260C2×C4×D3—32no—
261C2×C4×D3—32no—
262C2×C4×D3—32no—
263C2×C4×D3—31no—
264C2×Dic6—31no—
265C3×C4⋊C4—32no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—92no—
271C3×C4⋊C4C2×C3×D312yes—
272C3×C4⋊C4C2×C3×D311yescharacteristic
273C3×C22⋊C4—32no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—91no—
278C3×C22⋊C4C2×C3×D312yes—
279C3×C42—91no—
280#217—31no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—31no—
284C4⋊Dic3—32no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—31no—
288C4×Dic3—31no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—34no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C32×C4—21no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4D3×D412yes—
303C3×Dic3—32no—
304C3×Dic3—62no—
305C3×Dic3—61no—
306C3×Dic3—61no—
307C22×C9—22no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—21no—
311C22×C9—21no—
312C22×C9—41no—
313C22×C9—41no—
314C22×C9C24×C311yescharacteristic
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—21no—
318C2×D9—21no—
319C2×D9—21no—
320C2×D9C2×C3×D414yes—
321C4×C9—21no—
322C4×C9—22no—
323C4×C9—22no—
324C4×C9—22no—
325C4×C9—42no—
326C4×C9—41no—
327C4×C9—42no—
328C4×C9C2×C3×D412yes—
329Dic9—22no—
330Dic9—21no—
331Dic9—21no—
332Dic9C3×#4112yescomplemented
333C2×#41—91no—
334C2×#41—91no—
335C4⋊Q8—91no—
336#123—92no—
337#123—92no—
338C22⋊Q8—91no—
339C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
340#121—92no—
341C4×D4—91no—
342C4×D4—91no—
343C2×C4⋊C4—92no—
3443−1+2#54711yescharacteristic, complemented, Sylow
345C23×C3—31no—
346C23×C3—92no—
347C23×C3—91no—
348C23×C3C22×C3×D311yescharacteristic
349C22×D3—32no—
350C22×D3—31no—
351C3×Q8—22no—
352C3×Q8—62no—
353C3×Q8—182no—
354C3×D4—94no—
355C3×D4—184no—
356C3×D4—182no—
357C3×D4—184no—
358C2×C3×C4—22no—
359C2×C3×C4—31no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—31no—
363C2×C3×C4—62no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4—182no—
369C2×C3×C4—184no—
370C2×C3×C4—184no—
371C2×C3×C4—184no—
372C2×C3×C4C22×C3×D312yes—
373C2×C3×C4C22×C3×D311yescharacteristic
374C2×C3×C4C22×C3×D312yes—
375C2×C3×C4C22×C3×D311yescharacteristic
376#67—62no—
377#67—64no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—32no—
381C2×Dic3—31no—
382D12—34no—
383D12—64no—
384C4×D3—64no—
385C4×D3—64no—
386C4×D3—64no—
387C4×D3—62no—
388Dic6—62no—
389C2×C32—41no—
390C2×C32C2×D3×D411yescharacteristic
391C2×C32#100711yescharacteristic
392C2×C32D3×#4111yescharacteristic
393C3×D3—64no—
394C3×D3—121no—
395C2×C9—21no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9—42no—
399C2×C9—41no—
400C2×C9C22×C3×D411yescharacteristic
401C2×C9C2×C3×#4111yescharacteristic
402C2×C9C3×2−1+411yescharacteristic
403D9—24no—
404D9—41no—
405#41—184no—
406#41—184no—
407C2×Q8—91no—
408C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
409C2×D4—91no—
410C2×D4—92no—
411C2×D4—91no—
412C2×D4—92no—
413C22×C4—92no—
414C22×C4—92no—
415C22×C4—92no—
416C22×C4—91no—
417C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
418C4⋊C4—91no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4C2×AGL(1,ℤ/9)12yes—
423C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
424C22⋊C4—91no—
425C22⋊C4—92no—
426C22⋊C4—91no—
427C22⋊C4—92no—
428C22⋊C4C2×AGL(1,ℤ/9)12yes—
429C42—91no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—31no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—94no—
438C22×C3—182no—
439C22×C3—182no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3C23×C3×D311yescharacteristic
444C2×D3—34no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—32no—
454C3×C4—61no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—92no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4C3×D3×D412yes—
462Dic3—32no—
463Dic3—62no—
464Dic3—61no—
465Dic3—61no—
466C32#563111yescharacteristic
467C9—21no—
468C9C3×#54711yescharacteristic, complemented
469C23—92no—
470C23—91no—
471C23C22×AGL(1,ℤ/9)11yescharacteristic
472Q8—22no—
473Q8—182no—
474D4—94no—
475D4—184no—
476D4—182no—
477D4—184no—
478C2×C4—22no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—184no—
484C2×C4—184no—
485C2×C4—184no—
486C2×C4—182no—
487C2×C4C22×AGL(1,ℤ/9)12yes—
488C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
489C2×C4C22×AGL(1,ℤ/9)12yes—
490C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—41no—
495C2×C3—121no—
496C2×C3—184no—
497C2×C3—361no—
498C2×C3C2×C3×D3×D411yescharacteristic
499C2×C3C3×#100711yescharacteristic
500C2×C3C3×D3×#4111yescharacteristic
501D3—64no—
502D3—121no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—94no—
507C22—182no—
508C22—182no—
509C22—181no—
510C22—181no—
511C22—181no—
512C22C23×AGL(1,ℤ/9)11yescharacteristic
513C4—21no—
514C4—22no—
515C4—22no—
516C4—92no—
517C4—181no—
518C4—182no—
519C4—181no—
520C4D4×AGL(1,ℤ/9)12yes—
521C3—31no—
522C3C3×#563111yescharacteristic
523C2—41no—
524C2—184no—
525C2—361no—
526C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
527C2#1172263011yescharacteristic
528C2#41×AGL(1,ℤ/9)11yescharacteristic
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×#547
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. #63506⋊C3C2semidirect
5. C4×#611⋊C3C2semidirect
6. #63459⋊C3C2semidirect
7. #63456⋊C3C2semidirect
8. #63455⋊C3C2semidirect
9. C4⋊C4×AGL(1,ℤ/9)C2semidirect
10. #63446⋊C3C2semidirect
11. #63445⋊C3C2semidirect
13. #10589733C3semidirect
50. C2×#3701C2×C3semidirect
51. C2×#3697C2×C3semidirect
53. C9×C22⋊Q8C2×C3semidirect
54. #63506C2×C3semidirect
55. C4×#611C2×C3semidirect
56. #63459C2×C3semidirect
57. #63456C2×C3semidirect
58. #63455C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63446C2×C3semidirect
61. #63445C2×C3semidirect
78. C4⋊AGL(1,ℤ/9)D4semidirect
158. 3−1+2⋊C4#41semidirect
226. D36C3×D4semidirect
332. Dic9C3×#41semidirect
339. C22⋊Q8AGL(1,ℤ/9)semidirect
344. 3−1+2#547semidirect
468. C9C3×#547semidirect

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
#63506⋊C32
C4×#611⋊C32
#63459⋊C32
#63456⋊C32
#63455⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63446⋊C32
#63445⋊C32
C3×#56313
#105897333