Finite Group Databasebeta

← #10589609⋊C3 #13152459

Subgroups — #10589609⋊C3 #13152459

Order 1728.

922 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3697⋊C3C212yesmaximal, complemented
2C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3#63492⋊C3C211yesmaximal, characteristic, complemented
4#63491⋊C3C212yesmaximal, complemented
5C4×#611⋊C3C211yesmaximal, characteristic, complemented
6#63458⋊C3C212yesmaximal, complemented
7#63457⋊C3C211yesmaximal, characteristic, complemented
8#63456⋊C3C212yesmaximal, complemented
9#63455⋊C3C211yesmaximal, characteristic, complemented
10C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
11#63450⋊C3C211yesmaximal, characteristic, complemented
12C3×#5507—31nomaximal
13#10589609C311yesmaximal, characteristic, complemented
14C22×C4×3−1+2C2212yes—
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2212yes—
17C2×#611⋊C3C2211yescharacteristic
18#3697⋊C3—28no—
19C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
20C2×C4⋊AGL(1,ℤ/9)C2212yes—
21C2×C4×AGL(1,ℤ/9)C2212yes—
22C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
23C2×C4×AGL(1,ℤ/9)C2212yes—
24C2×3−1+2⋊Q8C2212yes—
25C4⋊C4×3−1+2C2211yescharacteristic
26C4⋊C4×3−1+2C2211yescharacteristic
27C4⋊C4×3−1+2C2212yes—
28#3676⋊C3C2211yescharacteristic
29(C2×AGL(1,ℤ/9))⋊C4C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31(C2×AGL(1,ℤ/9))⋊C4C2212yes—
32(C2×AGL(1,ℤ/9))⋊C4C2212yes—
33C4⋊(3−1+2⋊C4)C2211yescharacteristic
34(3−1+2⋊C4)⋊C4C2212yes—
35(3−1+2⋊C4)⋊C4C2212yes—
36(3−1+2⋊C4)⋊C4C2211yescharacteristic
37C4×3−1+2⋊C4C2211yescharacteristic
38C2×C3×#235—32no—
39C2×C32×C4⋊C4D311yescharacteristic
40C3×#930—31no—
41C3×#929—32no—
42C3×C4×#67—31no—
43C3×#896—32no—
44C3×#895—31no—
45C3×#894—32no—
46C3×#893—31no—
47C3×D3×C4⋊C4—31no—
48C3×#888—31no—
49C2×#3697C2×C312yescomplemented
50C2×C9×C4⋊C4—21no—
51C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
52#63492C2×C311yescharacteristic, complemented
53#63491C2×C312yescomplemented
54C4×#611C2×C311yescharacteristic, complemented
55#63458C2×C312yescomplemented
56#63457C2×C311yescharacteristic, complemented
57#63456C2×C312yescomplemented
58#63455C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C311yescharacteristic, complemented
60#63450C2×C311yescharacteristic, complemented
61C23×3−1+2C2311yescharacteristic
62C22×AGL(1,ℤ/9)C2312yes—
63C22×AGL(1,ℤ/9)C2311yescharacteristic
64C2×C4×3−1+2—24no—
65C2×C4×3−1+2—22no—
66C2×C4×3−1+2C2312yes—
67C2×C4×3−1+2C2312yes—
68C2×C4×3−1+2C2311yescharacteristic
69C2×C4×3−1+2C2311yescharacteristic
70#611⋊C3—24no—
71#611⋊C3D414yescomplemented
72C2×3−1+2⋊C4C2312yes—
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2311yescharacteristic
75C2×3−1+2⋊C4C2311yescharacteristic
76C4⋊AGL(1,ℤ/9)—24no—
77C4⋊AGL(1,ℤ/9)—22no—
78C4×AGL(1,ℤ/9)—24no—
79C4×AGL(1,ℤ/9)—22no—
80C4×AGL(1,ℤ/9)—24no—
813−1+2⋊Q8—24no—
82C3×#547—91no—
83#5507—31no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C2×D312yes—
86C2×C3×#67—31no—
87C2×C3×#67—32no—
88C3×#235—68no—
89C2×C3×D12—31no—
90C2×C3×D12—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—32no—
94C2×C3×Dic6—32no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D312yes—
98C3×#217—31no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—31no—
107C3×C4×Dic3—31no—
108C22×C4×C9—21no—
109C22×C4×C9—22no—
110C22×C4×C9C22×C311yescharacteristic
111C22×C4×C9C22×C312yes—
112C2×#611C22×C312yes—
113C2×#611C22×C311yescharacteristic
114#3697—28no—
115C2×D36C22×C311yescharacteristic
116C2×D36C22×C312yes—
117C2×C4×D9C22×C311yescharacteristic
118C2×C4×D9C22×C312yes—
119C2×C4×D9C22×C312yes—
120C2×Dic18C22×C312yes—
121C9×C4⋊C4—21no—
122C9×C4⋊C4—21no—
123C9×C4⋊C4—22no—
124C9×C4⋊C4C22×C311yescharacteristic
125C9×C4⋊C4C22×C311yescharacteristic
126C9×C4⋊C4C22×C312yes—
127#3676C22×C311yescharacteristic
128(C2×D9)⋊C4C22×C311yescharacteristic
129(C2×D9)⋊C4C22×C312yes—
130(C2×D9)⋊C4C22×C312yes—
131(C2×D9)⋊C4C22×C312yes—
132C4⋊Dic9C22×C311yescharacteristic
133Dic9⋊C4C22×C311yescharacteristic
134Dic9⋊C4C22×C312yes—
135Dic9⋊C4C22×C312yes—
136C4×Dic9C22×C311yescharacteristic
137C22×3−1+2—21no—
138C22×3−1+2—21no—
139C22×3−1+2C2×D412yes—
140C22×3−1+2C2411yescharacteristic
141C2×AGL(1,ℤ/9)—21no—
142C2×AGL(1,ℤ/9)—21no—
143C2×AGL(1,ℤ/9)—22no—
144C2×AGL(1,ℤ/9)—22no—
145C2×AGL(1,ℤ/9)—22no—
146C2×AGL(1,ℤ/9)C2×D412yes—
147C4×3−1+2—22no—
148C4×3−1+2—22no—
149C4×3−1+2—21no—
150C4×3−1+2#4112yescomplemented
1513−1+2⋊C4—22no—
1523−1+2⋊C4—21no—
1533−1+2⋊C4—21no—
1543−1+2⋊C4C2×D412yes—
155C2×C3×#41—92no—
156C2×#235—32no—
157C3×C4⋊Q8—91no—
158C3×#123—92no—
159C3×#123—92no—
160C3×C22⋊Q8—92no—
161C3×#121—91no—
162C3×#121—91no—
163C3×C4×D4—91no—
164C3×C4×D4—91no—
165C2×C3×C4⋊C4—31no—
166C2×C3×C4⋊C4—91no—
167C2×C3×C4⋊C4C3×D311yescharacteristic
168#930—31no—
169#929—32no—
170C4×#67—31no—
171#896—32no—
172#895—31no—
173#894—32no—
174#893—31no—
175D3×C4⋊C4—31no—
176#888—31no—
177C23×C32C22×D311yescharacteristic
178C22×C3×D3—31no—
179C22×C3×D3—32no—
180C2×C32×C4—24no—
181C2×C32×C4—22no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C3×#67—34no—
187C3×#67—64no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—32no—
192C3×D12—64no—
193C3×D12—62no—
194C3×C4×D3—64no—
195C3×C4×D3—64no—
196C3×C4×D3—62no—
197C3×Dic6—64no—
198C23×C9—21no—
199C23×C9C23×C311yescharacteristic
200C22×D9C23×C312yes—
201C22×D9C23×C311yescharacteristic
202C2×C4×C9—24no—
203C2×C4×C9—24no—
204C2×C4×C9—22no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—22no—
208C2×C4×C9—22no—
209C2×C4×C9—44no—
210C2×C4×C9C23×C311yescharacteristic
211C2×C4×C9C23×C312yes—
212C2×C4×C9C23×C312yes—
213C2×C4×C9C23×C311yescharacteristic
214#611—24no—
215#611C3×D414yescomplemented
216C2×Dic9C23×C311yescharacteristic
217C2×Dic9C23×C311yescharacteristic
218C2×Dic9C23×C312yes—
219C2×Dic9C23×C311yescharacteristic
220D36—24no—
221D36—22no—
222C4×D9—24no—
223C4×D9—22no—
224C4×D9—24no—
225Dic18—24no—
226#547—91noSylow
227C2×3−1+2—22no—
228C2×3−1+2C22×D411yescharacteristic
229C2×3−1+2C2×#4111yescharacteristic
230C2×3−1+22−1+411yescharacteristic
231AGL(1,ℤ/9)—22no—
232AGL(1,ℤ/9)—42no—
233C3×#41—188no—
234C2×C3×Q8—92no—
235C2×C3×D4—91no—
236C2×C3×D4—92no—
237C2×C3×D4—91no—
238C2×C3×D4—92no—
239C22×C3×C4—31no—
240C22×C3×C4—32no—
241C22×C3×C4—91no—
242C22×C3×C4—92no—
243C22×C3×C4—92no—
244C22×C3×C4C2×C3×D311yescharacteristic
245C22×C3×C4C2×C3×D312yes—
246C2×#67—31no—
247C2×#67—32no—
248#235—68no—
249C2×D12—31no—
250C2×D12—32no—
251C2×C4×D3—32no—
252C2×C4×D3—31no—
253C2×C4×D3—32no—
254C2×Dic6—32no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—32no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4C2×C3×D311yescharacteristic
263C3×C4⋊C4C2×C3×D311yescharacteristic
264C3×C4⋊C4C2×C3×D312yes—
265C3×C22⋊C4—91no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—91no—
270C3×C42—91no—
271#217—31no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—31no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—32no—
276C4⋊Dic3—31no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—31no—
280C4×Dic3—31no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32D3×D412yes—
284C22×C32C23×D311yescharacteristic
285C2×C3×D3—32no—
286C2×C3×D3—61no—
287C2×C3×D3—61no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4#23512yes—
295C3×Dic3—32no—
296C3×Dic3—61no—
297C3×Dic3—62no—
298C3×Dic3—61no—
299C22×C9—22no—
300C22×C9—22no—
301C22×C9—21no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—22no—
305C22×C9C2×C3×D412yes—
306C22×C9C24×C311yescharacteristic
307C2×D9—21no—
308C2×D9—21no—
309C2×D9—22no—
310C2×D9—22no—
311C2×D9—22no—
312C2×D9C2×C3×D412yes—
313C4×C9—22no—
314C4×C9—22no—
315C4×C9—22no—
316C4×C9—21no—
317C4×C9—22no—
318C4×C9—42no—
319C4×C9—42no—
320C4×C9C3×#4112yescomplemented
321Dic9—22no—
322Dic9—21no—
323Dic9—21no—
324Dic9C2×C3×D412yes—
325C2×#41—92no—
326C4⋊Q8—91no—
327#123—92no—
328#123—92no—
329C22⋊Q8—92no—
330#121—91no—
331#121—91no—
332C4×D4—91no—
333C4×D4—91no—
334C2×C4⋊C4—91no—
335C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
3363−1+2#54711yescharacteristic, complemented, Sylow
337C23×C3—31no—
338C23×C3—91no—
339C23×C3—92no—
340C23×C3C22×C3×D311yescharacteristic
341C22×D3—31no—
342C22×D3—32no—
343C3×Q8—184no—
344C3×D4—94no—
345C3×D4—182no—
346C3×D4—184no—
347C3×D4—184no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—62no—
355C2×C3×C4—64no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—182no—
361C2×C3×C4—184no—
362C2×C3×C4—184no—
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D312yes—
365C2×C3×C4C22×C3×D312yes—
366C2×C3×C4C22×C3×D311yescharacteristic
367#67—34no—
368#67—64no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373D12—64no—
374D12—62no—
375C4×D3—64no—
376C4×D3—64no—
377C4×D3—62no—
378Dic6—64no—
379C2×C32—22no—
380C2×C32C2×#23511yescharacteristic
381C2×C32C2×D3×D411yescharacteristic
382C2×C32#100711yescharacteristic
383C3×D3—62no—
384C3×D3—122no—
385C2×C9—24no—
386C2×C9—22no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9C22×C3×D411yescharacteristic
391C2×C9C2×C3×#4111yescharacteristic
392C2×C9C3×2−1+411yescharacteristic
393D9—22no—
394D9—42no—
395#41—188no—
396C2×Q8—92no—
397C2×D4—91no—
398C2×D4—92no—
399C2×D4—91no—
400C2×D4—92no—
401C22×C4—91no—
402C22×C4—92no—
403C22×C4—92no—
404C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
405C22×C4C2×AGL(1,ℤ/9)12yes—
406C4⋊C4—91no—
407C4⋊C4—92no—
408C4⋊C4—91no—
409C4⋊C4—92no—
410C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
411C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
412C4⋊C4C2×AGL(1,ℤ/9)12yes—
413C22⋊C4—91no—
414C22⋊C4—91no—
415C22⋊C4—92no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C42—91no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—31no—
422C22×C3—32no—
423C22×C3—61no—
424C22×C3—61no—
425C22×C3—92no—
426C22×C3—181no—
427C22×C3—181no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3—182no—
431C22×C3C3×D3×D412yes—
432C22×C3C23×C3×D311yescharacteristic
433C2×D3—32no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—32no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—61no—
446C3×C4—92no—
447C3×C4—182no—
448C3×C4—181no—
449C3×C4—181no—
450C3×C4C3×#23512yes—
451Dic3—32no—
452Dic3—61no—
453Dic3—62no—
454Dic3—61no—
455C32#550711yescharacteristic
456C9—21no—
457C9C3×#54711yescharacteristic, complemented
458C23—91no—
459C23—92no—
460C23C22×AGL(1,ℤ/9)11yescharacteristic
461Q8—184no—
462D4—94no—
463D4—184no—
464D4—184no—
465D4—182no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—91no—
469C2×C4—91no—
470C2×C4—91no—
471C2×C4—92no—
472C2×C4—184no—
473C2×C4—184no—
474C2×C4—182no—
475C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
476C2×C4C22×AGL(1,ℤ/9)12yes—
477C2×C4C22×AGL(1,ℤ/9)12yes—
478C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
479C2×C3—22no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—62no—
484C2×C3—182no—
485C2×C3—362no—
486C2×C3C2×C3×#23511yescharacteristic
487C2×C3C2×C3×D3×D411yescharacteristic
488C2×C3C3×#100711yescharacteristic
489D3—62no—
490D3—122no—
491C22—21no—
492C22—21no—
493C22—92no—
494C22—181no—
495C22—181no—
496C22—182no—
497C22—182no—
498C22—182no—
499C22D4×AGL(1,ℤ/9)12yes—
500C22C23×AGL(1,ℤ/9)11yescharacteristic
501C4—21no—
502C4—22no—
503C4—22no—
504C4—92no—
505C4—181no—
506C4—182no—
507C4—181no—
508C4#3697⋊C312yescomplemented
509C3—31no—
510C3C3×#550711yescharacteristic
511C2—22no—
512C2—182no—
513C2—362no—
514C2C2×#3697⋊C311yescharacteristic
515C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
516C2#1172263011yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3697⋊C3C2semidirect
2. C2×C4⋊C4×3−1+2C2semidirect
3. #63492⋊C3C2semidirect
4. #63491⋊C3C2semidirect
5. C4×#611⋊C3C2semidirect
6. #63458⋊C3C2semidirect
7. #63457⋊C3C2semidirect
8. #63456⋊C3C2semidirect
9. #63455⋊C3C2semidirect
10. C4⋊C4×AGL(1,ℤ/9)C2semidirect
11. #63450⋊C3C2semidirect
13. #10589609C3semidirect
49. C2×#3697C2×C3semidirect
51. C2×C9×C4⋊C4C2×C3semidirect
52. #63492C2×C3semidirect
53. #63491C2×C3semidirect
54. C4×#611C2×C3semidirect
55. #63458C2×C3semidirect
56. #63457C2×C3semidirect
57. #63456C2×C3semidirect
58. #63455C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63450C2×C3semidirect
71. #611⋊C3D4semidirect
150. C4×3−1+2#41semidirect
215. #611C3×D4semidirect
320. C4×C9C3×#41semidirect
335. C2×C4⋊C4AGL(1,ℤ/9)semidirect
336. 3−1+2#547semidirect
457. C9C3×#547semidirect
508. C4#3697⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3697⋊C32
C2×C4⋊C4×3−1+22
#63492⋊C32
#63491⋊C32
C4×#611⋊C32
#63458⋊C32
#63457⋊C32
#63456⋊C32
#63455⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63450⋊C32
C3×#55073
#105896093