Finite Group Databasebeta

← #10588670⋊C3 #13142661

Subgroups — #10588670⋊C3 #13142661

Order 1728.

616 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
5C2×C4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
63−1+2⋊#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4990—31nomaximal
9#10588670C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C4⋊C4×3−1+2—22no—
18(C2×AGL(1,ℤ/9))⋊C4—22no—
19(C2×AGL(1,ℤ/9))⋊C4—22no—
20(C2×AGL(1,ℤ/9))⋊C4C414yescomplemented
21C4×3−1+2⋊C4—22no—
22C2×C32×C4⋊C4D311yescharacteristic
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×C4×Dic3—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C2×C9×C4⋊C4—21no—
30C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
31C2×(C2×D9)⋊C4C2×C311yescharacteristic
32C2×(C2×D9)⋊C4C2×C311yescharacteristic
33C2×(C2×D9)⋊C4C2×C311yescharacteristic
34C2×C4×Dic9C2×C311yescharacteristic, complemented
35C9⋊#95C2×C311yescharacteristic, complemented
36C9⋊#95C2×C311yescharacteristic, complemented
37C23×3−1+2C2311yescharacteristic
38C22×AGL(1,ℤ/9)—21no—
39C22×AGL(1,ℤ/9)—21no—
40C22×AGL(1,ℤ/9)—21no—
41C22×AGL(1,ℤ/9)—21no—
42C22×AGL(1,ℤ/9)—21no—
43C22×AGL(1,ℤ/9)—21no—
44C22×AGL(1,ℤ/9)C2×C412yes—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2—21no—
50C2×C4×3−1+2—21no—
51C2×C4×3−1+2—21no—
52C2×C4×3−1+2C2×C412yes—
53C2×C4×3−1+2D412yes—
54C2×3−1+2⋊C4—21no—
55C2×3−1+2⋊C4—21no—
56C2×3−1+2⋊C4—21no—
57C2×3−1+2⋊C4—21no—
58C2×3−1+2⋊C4—21no—
59C2×3−1+2⋊C4—21no—
60C2×3−1+2⋊C4—21no—
61C2×3−1+2⋊C4C2×C412yes—
62C2×3−1+2⋊C4D412yes—
63C3×#388—91no—
64#4990—31no—
65C23×C3×D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C23×D9C22×C311yescharacteristic
73C32×C4⋊C4—22no—
74C3×(C2×D3)⋊C4—34no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C3×C4×Dic3—62no—
78C22×C4×C9—21no—
79C22×C4×C9—21no—
80C22×C4×C9—21no—
81C22×C4×C9C22×C311yescharacteristic
82C22×C4×C9C22×C311yescharacteristic
83C22×C4×C9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C22×Dic9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C9×C4⋊C4—24no—
88C9×C4⋊C4—22no—
89(C2×D9)⋊C4—22no—
90(C2×D9)⋊C4—22no—
91(C2×D9)⋊C4C3×C414yescomplemented
92C4×Dic9—22no—
93C22×3−1+2C22×C411yescharacteristic
94C22×3−1+2C2×D411yescharacteristic
95C22×3−1+2C2×D411yescharacteristic
96C22×3−1+2#4111yescharacteristic
97C22×3−1+2#4111yescharacteristic
98C22×3−1+2#4111yescharacteristic
99C22×3−1+2#4111yescharacteristic
100C2×AGL(1,ℤ/9)—22no—
101C2×AGL(1,ℤ/9)—22no—
102C2×AGL(1,ℤ/9)—22no—
103C2×AGL(1,ℤ/9)—41no—
104C2×AGL(1,ℤ/9)—41no—
105C2×AGL(1,ℤ/9)—41no—
106C2×AGL(1,ℤ/9)—41no—
107C4×3−1+2—22no—
108C4×3−1+2—22no—
109C4×3−1+2—41no—
1103−1+2⋊C4—22no—
1113−1+2⋊C4—22no—
1123−1+2⋊C4—41no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C4⋊C4C3×D311yescharacteristic
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C22⋊C4—91no—
118C2×C3×C42—91no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—31no—
121C2×(C2×D3)⋊C4—31no—
122C2×C4×Dic3—31no—
123C3×#95—91no—
124C3×#95—91no—
125C3⋊#95—31no—
126C3⋊#95—31no—
127C23×C32C22×D311yescharacteristic
128C22×C3×D3—32no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4C4×D312yes—
143C2×C32×C4#6712yes—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C23×C9—21no—
154C23×C9C23×C311yescharacteristic
155C22×D9—21no—
156C22×D9—21no—
157C22×D9—21no—
158C22×D9—21no—
159C22×D9—21no—
160C22×D9—21no—
161C22×D9C2×C3×C412yes—
162C2×C4×C9—22no—
163C2×C4×C9—22no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—22no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9—41no—
178C2×C4×C9C2×C3×C412yes—
179C2×C4×C9C3×D412yes—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9C2×C3×C412yes—
188C2×Dic9C3×D412yes—
189#388—91noSylow
190C2×3−1+2#11711yescharacteristic
191C2×3−1+2C4×D411yescharacteristic
192C2×3−1+2C4×D411yescharacteristic
193C2×3−1+2#12111yescharacteristic
194C2×3−1+2#12311yescharacteristic
195C2×3−1+2#12411yescharacteristic
196C2×3−1+2#12611yescharacteristic
197AGL(1,ℤ/9)—42no—
198C24×C3—91no—
199C23×D3—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—62no—
214C3×C22⋊C4—94no—
215C3×C22⋊C4—182no—
216C3×C22⋊C4—182no—
217C3×C42—182no—
218(C2×D3)⋊C4—34no—
219(C2×D3)⋊C4—62no—
220(C2×D3)⋊C4—62no—
221C4×Dic3—62no—
222C22×C32C2×C4×D311yescharacteristic
223C22×C32#23511yescharacteristic
224C22×C32D3×D411yescharacteristic
225C22×C32#23711yescharacteristic
226C22×C32#23911yescharacteristic
227C22×C32#23911yescharacteristic
228C22×C32C2×#6711yescharacteristic
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—41no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—121no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—21no—
245C22×C9—21no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9C22×C3×C411yescharacteristic
250C22×C9C2×C3×D411yescharacteristic
251C22×C9C2×C3×D411yescharacteristic
252C22×C9C3×#4111yescharacteristic
253C22×C9C3×#4111yescharacteristic
254C22×C9C3×#4111yescharacteristic
255C22×C9C3×#4111yescharacteristic
256C2×D9—22no—
257C2×D9—22no—
258C2×D9—22no—
259C2×D9—41no—
260C2×D9—41no—
261C2×D9—41no—
262C2×D9—41no—
263C4×C9—24no—
264C4×C9—22no—
265C4×C9—22no—
266C4×C9—42no—
267C4×C9—42no—
268C4×C9—41no—
269Dic9—22no—
270Dic9—22no—
271Dic9—41no—
272C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
273C2×C22⋊C4—91no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—91no—
276C2×C42—91no—
277#95—91no—
278#95—91no—
2793−1+2#38811yescharacteristic, complemented, Sylow
280C23×C3—31no—
281C23×C3—92no—
282C23×C3—181no—
283C23×C3—181no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3—181no—
287C23×C3—181no—
288C23×C3C22×C3×D311yescharacteristic
289C22×D3—32no—
290C22×D3—61no—
291C22×D3—61no—
292C22×D3—61no—
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—61no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4C3×C4×D312yes—
322C2×C3×C4C3×#6712yes—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×C32#89211yescharacteristic
333C2×C32#89311yescharacteristic
334C2×C32#89411yescharacteristic
335C2×C32#89811yescharacteristic
336C2×C32C4×#6711yescharacteristic
337C2×C32#93911yescharacteristic
338C2×C32#94711yescharacteristic
339C3×D3—122no—
340C2×C9—21no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9C3×#11711yescharacteristic
348C2×C9C3×C4×D411yescharacteristic
349C2×C9C3×C4×D411yescharacteristic
350C2×C9C3×#12111yescharacteristic
351C2×C9C3×#12311yescharacteristic
352C2×C9C3×#12411yescharacteristic
353C2×C9C3×#12611yescharacteristic
354D9—42no—
355C24—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—91no—
359C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
360C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
361C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
362C4⋊C4—22no—
363C22⋊C4—94no—
364C22⋊C4—182no—
365C22⋊C4—182no—
366C42—182no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3—182no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3C2×C3×C4×D311yescharacteristic
382C22×C3C3×#23511yescharacteristic
383C22×C3C3×D3×D411yescharacteristic
384C22×C3C3×#23711yescharacteristic
385C22×C3C3×#23911yescharacteristic
386C22×C3C3×#23911yescharacteristic
387C22×C3C2×C3×#6711yescharacteristic
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—41no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—121no—
401C3×C4—182no—
402C3×C4—182no—
403C3×C4—361no—
404Dic3—62no—
405Dic3—62no—
406Dic3—121no—
407C32#499011yescharacteristic
408C9—21no—
409C9C3×#38811yescharacteristic, complemented
410C23—92no—
411C23—181no—
412C23—181no—
413C23—181no—
414C23—181no—
415C23—181no—
416C23—181no—
417C23C22×AGL(1,ℤ/9)11yescharacteristic
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4C4×AGL(1,ℤ/9)12yes—
435C2×C4#611⋊C312yes—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—362no—
444C2×C3C3×#89211yescharacteristic
445C2×C3C3×#89311yescharacteristic
446C2×C3C3×#89411yescharacteristic
447C2×C3C3×#89811yescharacteristic
448C2×C3C3×C4×#6711yescharacteristic
449C2×C3C3×#93911yescharacteristic
450C2×C3C3×#94711yescharacteristic
451D3—122no—
452C22—182no—
453C22—182no—
454C22—182no—
455C22—361no—
456C22—361no—
457C22—361no—
458C22—361no—
459C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
460C22#3697⋊C311yescharacteristic
461C22D4×AGL(1,ℤ/9)11yescharacteristic
462C22#3699⋊C311yescharacteristic
463C22#8872411yescharacteristic
464C22#8872411yescharacteristic
465C22C2×#611⋊C311yescharacteristic
466C4—22no—
467C4—22no—
468C4—41no—
469C4—182no—
470C4—182no—
471C4—361no—
472C3—31no—
473C3C3×#499011yescharacteristic
474C2—362no—
475C2#63454⋊C311yescharacteristic
476C2#63455⋊C311yescharacteristic
477C2#63456⋊C311yescharacteristic
478C2#63460⋊C311yescharacteristic
479C2C4×#611⋊C311yescharacteristic
480C2#63501⋊C311yescharacteristic
481C2#63509⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×AGL(1,ℤ/9)C22
cosocleC23×C9C23×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#388
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
5. C2×C4×3−1+2⋊C4C2semidirect
6. 3−1+2⋊#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588670C3semidirect
20. (C2×AGL(1,ℤ/9))⋊C4C4semidirect
30. C2×C9×C4⋊C4C2×C3semidirect
34. C2×C4×Dic9C2×C3semidirect
35. C9⋊#95C2×C3semidirect
36. C9⋊#95C2×C3semidirect
91. (C2×D9)⋊C4C3×C4semidirect
272. C2×C4⋊C4AGL(1,ℤ/9)semidirect
279. 3−1+2#388semidirect
409. C9C3×#388semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4×3−1+2⋊C42
3−1+2⋊#952
C3×#49903
#105886703