Finite Group Databasebeta

← #10589926⋊C3 #13153741

Subgroups — #10589926⋊C3 #13153741

Order 1728.

958 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×3−1+2C211yesmaximal, characteristic, complemented
2C2×#3699⋊C3C211yesmaximal, characteristic, complemented
3#63508⋊C3C211yesmaximal, characteristic, complemented
4#63506⋊C3C211yesmaximal, characteristic, complemented
5#63501⋊C3C212yesmaximal, complemented
6#63497⋊C3C212yesmaximal, complemented
7D4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
8#63491⋊C3C212yesmaximal, complemented
9C4×#611⋊C3C211yesmaximal, characteristic, complemented
10#63486⋊C3C211yesmaximal, characteristic, complemented
11C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
12C3×#5824—31nomaximal
13#10589926C311yesmaximal, characteristic, complemented
14#41×3−1+2—24no—
15C2×Q8×3−1+2C2211yescharacteristic
16C2×D4×3−1+2C2212yes—
17C2×D4×3−1+2C2211yescharacteristic
18C22×C4×3−1+2C2212yes—
19C22×C4×3−1+2C2211yescharacteristic
20C2×#611⋊C3C2212yes—
21C2×#611⋊C3C2211yescharacteristic
22C22×3−1+2⋊C4C2212yes—
23C22×3−1+2⋊C4C2212yes—
24#3699⋊C3—24no—
25C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
26C2×3−1+2⋊Q8C2211yescharacteristic
27#3676⋊C3C2212yes—
28#3676⋊C3C2211yescharacteristic
29#3676⋊C3C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31(C2×AGL(1,ℤ/9))⋊C4C2212yes—
32C4⋊(3−1+2⋊C4)C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2211yescharacteristic
34(3−1+2⋊C4)⋊C4C2212yes—
35(3−1+2⋊C4)⋊C4C2212yes—
36(3−1+2⋊C4)⋊C4C2212yes—
37(3−1+2⋊C4)⋊C4C2212yes—
38C4×3−1+2⋊C4C2211yescharacteristic
39C2×C32×#41D311yescharacteristic
40C2×C3×#237—31no—
41C3×#946—31no—
42C3×#944—31no—
43C3×#939—32no—
44C3×#935—32no—
45C3×D4×Dic3—31no—
46C3×#929—32no—
47C3×C4×#67—31no—
48C3×#924—31no—
49C2×C3×Dic3⋊C4—32no—
50C2×C9×#41—21no—
51C2×C9×#41C2×C311yescharacteristic, complemented
52C2×#3699C2×C311yescharacteristic, complemented
53#63508C2×C311yescharacteristic, complemented
54#63506C2×C311yescharacteristic, complemented
55#63501C2×C312yescomplemented
56#63497C2×C312yescomplemented
57D4×Dic9C2×C311yescharacteristic, complemented
58#63491C2×C312yescomplemented
59C4×#611C2×C311yescharacteristic, complemented
60#63486C2×C311yescharacteristic, complemented
61C2×Dic9⋊C4C2×C312yescomplemented
62C23×3−1+2C2311yescharacteristic
63C23×3−1+2C2312yes—
64C22×AGL(1,ℤ/9)C2311yescharacteristic
65Q8×3−1+2—22no—
66D4×3−1+2—24no—
67D4×3−1+2D414yescomplemented
68C2×C4×3−1+2—24no—
69C2×C4×3−1+2—22no—
70C2×C4×3−1+2C2311yescharacteristic
71C2×C4×3−1+2C2312yes—
72C2×C4×3−1+2C2311yescharacteristic
73#611⋊C3—24no—
74#611⋊C3—22no—
75C2×3−1+2⋊C4—24no—
76C2×3−1+2⋊C4—24no—
77C2×3−1+2⋊C4C2312yes—
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)—22no—
833−1+2⋊Q8—22no—
84C3×#547—91no—
85#5824—31no—
86C32×#41—24no—
87C2×C32×Q8C2×D311yescharacteristic
88C2×C32×D4C2×D312yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4C2×D312yes—
91C22×C32×C4C2×D311yescharacteristic
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C3×#237—64no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—31no—
99C3×#217—32no—
100C3×#217—31no—
101C3×#217—32no—
102C3×(C2×D3)⋊C4—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×C4⋊Dic3—31no—
105C3×Dic3⋊C4—31no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—31no—
111C9×#41—28no—
112C9×#41—24no—
113C2×Q8×C9—21no—
114C2×Q8×C9C22×C311yescharacteristic
115C2×D4×C9—22no—
116C2×D4×C9—21no—
117C2×D4×C9C22×C312yes—
118C2×D4×C9C22×C311yescharacteristic
119C22×C4×C9—22no—
120C22×C4×C9—21no—
121C22×C4×C9C22×C311yescharacteristic
122C22×C4×C9C22×C312yes—
123C2×#611C22×C311yescharacteristic
124C2×#611C22×C312yes—
125C22×Dic9C22×C312yes—
126C22×Dic9C22×C312yes—
127#3699—24no—
128C2×C4×D9C22×C311yescharacteristic
129C2×Dic18C22×C311yescharacteristic
130#3676C22×C311yescharacteristic
131#3676C22×C312yes—
132#3676C22×C312yes—
133(C2×D9)⋊C4C22×C311yescharacteristic
134(C2×D9)⋊C4C22×C312yes—
135C4⋊Dic9C22×C311yescharacteristic
136Dic9⋊C4C22×C311yescharacteristic
137Dic9⋊C4C22×C312yes—
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140Dic9⋊C4C22×C312yes—
141C4×Dic9C22×C311yescharacteristic
142C22×3−1+2—22no—
143C22×3−1+2—22no—
144C22×3−1+2—21no—
145C22×3−1+2—21no—
146C22×3−1+2—21no—
147C22×3−1+2C2×D414yes—
148C22×3−1+2C2411yescharacteristic
149C2×AGL(1,ℤ/9)—21no—
150C2×AGL(1,ℤ/9)—21no—
151C2×AGL(1,ℤ/9)—21no—
152C4×3−1+2—21no—
153C4×3−1+2—22no—
154C4×3−1+2C2×D412yes—
1553−1+2⋊C4—22no—
1563−1+2⋊C4—21no—
1573−1+2⋊C4—22no—
1583−1+2⋊C4—21no—
1593−1+2⋊C4#4112yescomplemented
160C2×C3×#41—31no—
161C2×C3×#41—91no—
162C2×C3×#41C3×D311yescharacteristic
163C2×#237—31no—
164C3×C4⋊Q8—91no—
165C3×#123—92no—
166C3×#123—92no—
167C3×C22⋊Q8—91no—
168C3×C22⋊Q8—91no—
169C3×#121—92no—
170C3×C4×D4—91no—
171C3×C4×D4—91no—
172C2×C3×C4⋊C4—92no—
173#946—31no—
174#944—31no—
175#939—32no—
176#935—32no—
177D4×Dic3—31no—
178#929—32no—
179C4×#67—31no—
180#924—31no—
181C2×Dic3⋊C4—32no—
182C23×C32C22×D311yescharacteristic
183C23×C32C22×D312yes—
184C22×C3×D3—31no—
185C32×Q8—22no—
186C32×D4—24no—
187C32×D4#6714yes—
188C2×C32×C4—24no—
189C2×C32×C4—22no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C3×#67—62no—
194C3×#67—64no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—64no—
201C2×C3×Dic3—64no—
202C3×C4×D3—62no—
203C3×Dic6—62no—
204C23×C9—21no—
205C23×C9—22no—
206C23×C9C23×C311yescharacteristic
207C23×C9C23×C312yes—
208C22×D9C23×C311yescharacteristic
209Q8×C9—24no—
210Q8×C9—22no—
211D4×C9—28no—
212D4×C9—24no—
213D4×C9—24no—
214D4×C9C3×D414yescomplemented
215C2×C4×C9—28no—
216C2×C4×C9—24no—
217C2×C4×C9—24no—
218C2×C4×C9—22no—
219C2×C4×C9—21no—
220C2×C4×C9—22no—
221C2×C4×C9—21no—
222C2×C4×C9C23×C311yescharacteristic
223C2×C4×C9C23×C312yes—
224C2×C4×C9C23×C311yescharacteristic
225#611—24no—
226#611—22no—
227C2×Dic9—24no—
228C2×Dic9—24no—
229C2×Dic9C23×C311yescharacteristic
230C2×Dic9C23×C312yes—
231C2×Dic9C23×C311yescharacteristic
232C2×Dic9C23×C312yes—
233C2×Dic9C23×C311yescharacteristic
234C4×D9—22no—
235Dic18—22no—
236#547—91noSylow
237C2×3−1+2—24no—
238C2×3−1+2—41no—
239C2×3−1+2C22×D411yescharacteristic
240C2×3−1+2C2×#4111yescharacteristic
241C2×3−1+22−1+411yescharacteristic
242AGL(1,ℤ/9)—41no—
243C3×#41—24no—
244C3×#41—64no—
245C3×#41—184no—
246C2×C3×Q8—31no—
247C2×C3×Q8—91no—
248C2×C3×Q8C2×C3×D311yescharacteristic
249C2×C3×D4—31no—
250C2×C3×D4—32no—
251C2×C3×D4—91no—
252C2×C3×D4—92no—
253C2×C3×D4C2×C3×D312yes—
254C2×C3×D4C2×C3×D311yescharacteristic
255C22×C3×C4—32no—
256C22×C3×C4—31no—
257C22×C3×C4—91no—
258C22×C3×C4—92no—
259C22×C3×C4—92no—
260C22×C3×C4C2×C3×D312yes—
261C22×C3×C4C2×C3×D311yescharacteristic
262C2×#67—31no—
263C2×#67—32no—
264C22×Dic3—32no—
265C22×Dic3—32no—
266#237—64no—
267C2×C4×D3—31no—
268C2×Dic6—31no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4—92no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—91no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4—91no—
279C3×C22⋊C4—92no—
280C3×C42—91no—
281#217—32no—
282#217—31no—
283#217—32no—
284(C2×D3)⋊C4—31no—
285(C2×D3)⋊C4—32no—
286C4⋊Dic3—31no—
287Dic3⋊C4—31no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292C4×Dic3—31no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32C2×#6714yes—
299C22×C32C23×D311yescharacteristic
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C32×C4—21no—
304C32×C4—22no—
305C32×C4C2×#6712yes—
306C3×Dic3—32no—
307C3×Dic3—62no—
308C3×Dic3—61no—
309C3×Dic3—62no—
310C3×Dic3—61no—
311C22×C9—22no—
312C22×C9—22no—
313C22×C9—22no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—24no—
319C22×C9—42no—
320C22×C9—42no—
321C22×C9—41no—
322C22×C9—41no—
323C22×C9C2×C3×D414yes—
324C22×C9C24×C311yescharacteristic
325C2×D9—21no—
326C2×D9—21no—
327C2×D9—21no—
328C4×C9—22no—
329C4×C9—24no—
330C4×C9—21no—
331C4×C9—22no—
332C4×C9—22no—
333C4×C9C2×C3×D412yes—
334Dic9—22no—
335Dic9—21no—
336Dic9—22no—
337Dic9—21no—
338Dic9C3×#4112yescomplemented
339C2×#41—91no—
340C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
341C4⋊Q8—91no—
342#123—92no—
343#123—92no—
344C22⋊Q8—91no—
345C22⋊Q8—91no—
346#121—92no—
347C4×D4—91no—
348C4×D4—91no—
349C2×C4⋊C4—92no—
3503−1+2#54711yescharacteristic, complemented, Sylow
351C23×C3—32no—
352C23×C3—31no—
353C23×C3—91no—
354C23×C3C22×C3×D311yescharacteristic
355C23×C3C22×C3×D312yes—
356C22×D3—31no—
357C3×Q8—22no—
358C3×Q8—62no—
359C3×Q8—182no—
360C3×D4—24no—
361C3×D4—34no—
362C3×D4—64no—
363C3×D4—182no—
364C3×D4—184no—
365C3×D4C3×#6714yes—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—32no—
371C2×C3×C4—62no—
372C2×C3×C4—64no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—182no—
379C2×C3×C4—184no—
380C2×C3×C4—184no—
381C2×C3×C4C22×C3×D311yescharacteristic
382C2×C3×C4C22×C3×D312yes—
383C2×C3×C4C22×C3×D311yescharacteristic
384#67—62no—
385#67—64no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—32no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—64no—
392C2×Dic3—64no—
393C4×D3—62no—
394Dic6—62no—
395C2×C32—24no—
396C2×C32—41no—
397C2×C32D3×#4111yescharacteristic
398C2×C32#101011yescharacteristic
399C2×C32C22×#6711yescharacteristic
400C3×D3—121no—
401C2×C9—24no—
402C2×C9—21no—
403C2×C9—21no—
404C2×C9—21no—
405C2×C9—44no—
406C2×C9—42no—
407C2×C9—41no—
408C2×C9C22×C3×D411yescharacteristic
409C2×C9C2×C3×#4111yescharacteristic
410C2×C9C3×2−1+411yescharacteristic
411D9—41no—
412#41—24no—
413#41—184no—
414C2×Q8—91no—
415C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
416C2×D4—91no—
417C2×D4—92no—
418C2×D4C2×AGL(1,ℤ/9)12yes—
419C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
420C22×C4—92no—
421C22×C4—92no—
422C22×C4—91no—
423C22×C4C2×AGL(1,ℤ/9)12yes—
424C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
425C4⋊C4—91no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4—92no—
429C4⋊C4—91no—
430C4⋊C4—92no—
431C22⋊C4—92no—
432C22⋊C4—91no—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4—92no—
436C42—91no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—31no—
443C22×C3—34no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—181no—
450C22×C3—181no—
451C22×C3—181no—
452C22×C3C2×C3×#6714yes—
453C22×C3C23×C3×D311yescharacteristic
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C3×C4—21no—
458C3×C4—22no—
459C3×C4—32no—
460C3×C4—61no—
461C3×C4—62no—
462C3×C4—92no—
463C3×C4—181no—
464C3×C4—182no—
465C3×C4—182no—
466C3×C4—181no—
467C3×C4C2×C3×#6712yes—
468Dic3—32no—
469Dic3—62no—
470Dic3—61no—
471Dic3—62no—
472Dic3—61no—
473C32#582411yescharacteristic
474C9—21no—
475C9C3×#54711yescharacteristic, complemented
476C23—91no—
477C23C22×AGL(1,ℤ/9)11yescharacteristic
478C23C22×AGL(1,ℤ/9)12yes—
479Q8—22no—
480Q8—182no—
481D4—24no—
482D4—182no—
483D4—184no—
484D4#611⋊C314yescomplemented
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—92no—
490C2×C4—91no—
491C2×C4—92no—
492C2×C4—184no—
493C2×C4—184no—
494C2×C4—182no—
495C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
496C2×C4C22×AGL(1,ℤ/9)12yes—
497C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
498C2×C3—24no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—41no—
503C2×C3—64no—
504C2×C3—121no—
505C2×C3—361no—
506C2×C3C3×D3×#4111yescharacteristic
507C2×C3C3×#101011yescharacteristic
508C2×C3C22×C3×#6711yescharacteristic
509D3—121no—
510C22—22no—
511C22—22no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—181no—
516C22—181no—
517C22—181no—
518C22C2×#611⋊C314yes—
519C22C23×AGL(1,ℤ/9)11yescharacteristic
520C4—21no—
521C4—22no—
522C4—92no—
523C4—182no—
524C4—181no—
525C4—181no—
526C4—182no—
527C4C2×#611⋊C312yes—
528C3—31no—
529C3C3×#582411yescharacteristic
530C2—24no—
531C2—41no—
532C2—361no—
533C2#41×AGL(1,ℤ/9)11yescharacteristic
534C2#1172264211yescharacteristic
535C2C22×#611⋊C311yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×3−1+2C2
hypercenterC2×#41AGL(1,ℤ/9)
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×3−1+2C2semidirect
2. C2×#3699⋊C3C2semidirect
3. #63508⋊C3C2semidirect
4. #63506⋊C3C2semidirect
5. #63501⋊C3C2semidirect
6. #63497⋊C3C2semidirect
7. D4×3−1+2⋊C4C2semidirect
8. #63491⋊C3C2semidirect
9. C4×#611⋊C3C2semidirect
10. #63486⋊C3C2semidirect
11. C2×(3−1+2⋊C4)⋊C4C2semidirect
13. #10589926C3semidirect
51. C2×C9×#41C2×C3semidirect
52. C2×#3699C2×C3semidirect
53. #63508C2×C3semidirect
54. #63506C2×C3semidirect
55. #63501C2×C3semidirect
56. #63497C2×C3semidirect
57. D4×Dic9C2×C3semidirect
58. #63491C2×C3semidirect
59. C4×#611C2×C3semidirect
60. #63486C2×C3semidirect
61. C2×Dic9⋊C4C2×C3semidirect
67. D4×3−1+2D4semidirect
159. 3−1+2⋊C4#41semidirect
214. D4×C9C3×D4semidirect
338. Dic9C3×#41semidirect
340. C2×#41AGL(1,ℤ/9)semidirect
350. 3−1+2#547semidirect
475. C9C3×#547semidirect
484. D4#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41×3−1+22
C2×#3699⋊C32
#63508⋊C32
#63506⋊C32
#63501⋊C32
#63497⋊C32
D4×3−1+2⋊C42
#63491⋊C32
C4×#611⋊C32
#63486⋊C32
C2×(3−1+2⋊C4)⋊C42
C3×#58243
#105899263