Finite Group Databasebeta

← #10588350⋊C3 #13141390

Subgroups — #10588350⋊C3 #13141390

Order 1728.

608 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
2C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×C4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
63−1+2×#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4670—31nomaximal
9#10588350C311yesmaximal, 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
17(C2×AGL(1,ℤ/9))⋊C4—22no—
18(C2×AGL(1,ℤ/9))⋊C4—22no—
19(C2×AGL(1,ℤ/9))⋊C4C414yescomplemented
20C4⋊(3−1+2⋊C4)—22no—
21C4×3−1+2⋊C4—22no—
22C2×C3×(C2×D3)⋊C4—31no—
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4×Dic3—31no—
27C32×#95D311yescharacteristic
28C3×C3⋊#95—31no—
29C2×(C2×D9)⋊C4C2×C311yescharacteristic
30C2×(C2×D9)⋊C4C2×C311yescharacteristic
31C2×(C2×D9)⋊C4C2×C311yescharacteristic
32C2×C4⋊Dic9C2×C311yescharacteristic, complemented
33C2×C4×Dic9C2×C311yescharacteristic, complemented
34C9×#95—21no—
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#4670—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
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—62no—
76C3×C4⋊Dic3—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
87(C2×D9)⋊C4—22no—
88(C2×D9)⋊C4—22no—
89(C2×D9)⋊C4C3×C414yescomplemented
90C4⋊Dic9—22no—
91C4×Dic9—22no—
92C22×3−1+2C22×C411yescharacteristic
93C22×3−1+2C2×D411yescharacteristic
94C22×3−1+2C2×D411yescharacteristic
95C22×3−1+2#4111yescharacteristic
96C22×3−1+2#4111yescharacteristic
97C22×3−1+2#4111yescharacteristic
98C22×3−1+2#4111yescharacteristic
99C2×AGL(1,ℤ/9)—22no—
100C2×AGL(1,ℤ/9)—22no—
101C2×AGL(1,ℤ/9)—22no—
102C2×AGL(1,ℤ/9)—41no—
103C2×AGL(1,ℤ/9)—41no—
104C2×AGL(1,ℤ/9)—41no—
105C2×AGL(1,ℤ/9)—41no—
106C4×3−1+2—22no—
107C4×3−1+2—22no—
108C4×3−1+2—41no—
1093−1+2⋊C4—22no—
1103−1+2⋊C4—22no—
1113−1+2⋊C4—41no—
112C2×C3×C4⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C42—91no—
117C2×(C2×D3)⋊C4—31no—
118C2×(C2×D3)⋊C4—31no—
119C2×(C2×D3)⋊C4—31no—
120C2×C4⋊Dic3—31no—
121C2×C4×Dic3—31no—
122C3×#95—31no—
123C3×#95—91no—
124C3×#95C3×D311yescharacteristic
125C3⋊#95—31no—
126C23×C32C22×D311yescharacteristic
127C22×C3×D3—32no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×C32×C4—21no—
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×C4C4×D312yes—
142C2×C32×C4D1212yes—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C23×C9—21no—
153C23×C9C23×C311yescharacteristic
154C22×D9—21no—
155C22×D9—21no—
156C22×D9—21no—
157C22×D9—21no—
158C22×D9—21no—
159C22×D9—21no—
160C22×D9C2×C3×C412yes—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—22no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9—41no—
172C2×C4×C9—41no—
173C2×C4×C9—41no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9C2×C3×C412yes—
178C2×C4×C9C3×D412yes—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9C2×C3×C412yes—
187C2×Dic9C3×D412yes—
188#388—91noSylow
189C2×3−1+2#11711yescharacteristic
190C2×3−1+2C4×D411yescharacteristic
191C2×3−1+2C4×D411yescharacteristic
192C2×3−1+2#12111yescharacteristic
193C2×3−1+2#12311yescharacteristic
194C2×3−1+2#12411yescharacteristic
195C2×3−1+2#12611yescharacteristic
196AGL(1,ℤ/9)—42no—
197C24×C3—91no—
198C23×D3—31no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4C2×C3×D311yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×Dic3—31no—
209C22×Dic3—31no—
210C22×Dic3—31no—
211C3×C4⋊C4—182no—
212C3×C22⋊C4—94no—
213C3×C22⋊C4—182no—
214C3×C22⋊C4—182no—
215C3×C42—182no—
216(C2×D3)⋊C4—34no—
217(C2×D3)⋊C4—62no—
218(C2×D3)⋊C4—62no—
219C4⋊Dic3—62no—
220C4×Dic3—62no—
221C22×C32C2×C4×D311yescharacteristic
222C22×C32C2×D1211yescharacteristic
223C22×C32#23511yescharacteristic
224C22×C32D3×D411yescharacteristic
225C22×C32#23711yescharacteristic
226C22×C32#23711yescharacteristic
227C22×C32#23911yescharacteristic
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—41no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—121no—
241C22×C9—21no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—21no—
245C22×C9—21no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9C22×C3×C411yescharacteristic
249C22×C9C2×C3×D411yescharacteristic
250C22×C9C2×C3×D411yescharacteristic
251C22×C9C3×#4111yescharacteristic
252C22×C9C3×#4111yescharacteristic
253C22×C9C3×#4111yescharacteristic
254C22×C9C3×#4111yescharacteristic
255C2×D9—22no—
256C2×D9—22no—
257C2×D9—22no—
258C2×D9—41no—
259C2×D9—41no—
260C2×D9—41no—
261C2×D9—41no—
262C4×C9—22no—
263C4×C9—22no—
264C4×C9—42no—
265C4×C9—42no—
266C4×C9—42no—
267C4×C9—41no—
268Dic9—22no—
269Dic9—22no—
270Dic9—41no—
271C2×C4⋊C4—91no—
272C2×C22⋊C4—91no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—91no—
275C2×C42—91no—
276#95—91no—
277#95AGL(1,ℤ/9)11yescharacteristic, complemented
2783−1+2#38811yescharacteristic, complemented, Sylow
279C23×C3—31no—
280C23×C3—92no—
281C23×C3—181no—
282C23×C3—181no—
283C23×C3—181no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3—181no—
287C23×C3C22×C3×D311yescharacteristic
288C22×D3—32no—
289C22×D3—61no—
290C22×D3—61no—
291C22×D3—61no—
292C22×D3—61no—
293C22×D3—61no—
294C22×D3—61no—
295C2×C3×C4—21no—
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—32no—
303C2×C3×C4—32no—
304C2×C3×C4—61no—
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—92no—
312C2×C3×C4—92no—
313C2×C3×C4—181no—
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×C4C3×C4×D312yes—
321C2×C3×C4C3×D1212yes—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×C32C4×D1211yescharacteristic
332C2×C32#67⋊C411yescharacteristic
333C2×C32#88511yescharacteristic
334C2×C32#88611yescharacteristic
335C2×C32#89211yescharacteristic
336C2×C32#89511yescharacteristic
337C2×C32#89811yescharacteristic
338C3×D3—122no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—21no—
346C2×C9C3×#11711yescharacteristic
347C2×C9C3×C4×D411yescharacteristic
348C2×C9C3×C4×D411yescharacteristic
349C2×C9C3×#12111yescharacteristic
350C2×C9C3×#12311yescharacteristic
351C2×C9C3×#12411yescharacteristic
352C2×C9C3×#12611yescharacteristic
353D9—42no—
354C24—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
359C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
360C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
361C4⋊C4—182no—
362C22⋊C4—94no—
363C22⋊C4—182no—
364C22⋊C4—182no—
365C42—182no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3—361no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3—361no—
380C22×C3C2×C3×C4×D311yescharacteristic
381C22×C3C2×C3×D1211yescharacteristic
382C22×C3C3×#23511yescharacteristic
383C22×C3C3×D3×D411yescharacteristic
384C22×C3C3×#23711yescharacteristic
385C22×C3C3×#23711yescharacteristic
386C22×C3C3×#23911yescharacteristic
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—41no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—121no—
400C3×C4—182no—
401C3×C4—182no—
402C3×C4—361no—
403Dic3—62no—
404Dic3—62no—
405Dic3—121no—
406C32#467011yescharacteristic
407C9—21no—
408C9C3×#38811yescharacteristic, complemented
409C23—92no—
410C23—181no—
411C23—181no—
412C23—181no—
413C23—181no—
414C23—181no—
415C23—181no—
416C23C22×AGL(1,ℤ/9)11yescharacteristic
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4C4×AGL(1,ℤ/9)12yes—
434C2×C4C4⋊AGL(1,ℤ/9)12yes—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—362no—
443C2×C3C3×C4×D1211yescharacteristic
444C2×C3C3×#67⋊C411yescharacteristic
445C2×C3C3×#88511yescharacteristic
446C2×C3C3×#88611yescharacteristic
447C2×C3C3×#89211yescharacteristic
448C2×C3C3×#89511yescharacteristic
449C2×C3C3×#89811yescharacteristic
450D3—122no—
451C22—182no—
452C22—182no—
453C22—182no—
454C22—361no—
455C22—361no—
456C22—361no—
457C22—361no—
458C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
459C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
460C22#3697⋊C311yescharacteristic
461C22D4×AGL(1,ℤ/9)11yescharacteristic
462C22#3699⋊C311yescharacteristic
463C22#3699⋊C311yescharacteristic
464C22#8872411yescharacteristic
465C4—22no—
466C4—22no—
467C4—41no—
468C4—182no—
469C4—182no—
470C4—361no—
471C3—31no—
472C3C3×#467011yescharacteristic
473C2—362no—
474C2C4×C4⋊AGL(1,ℤ/9)11yescharacteristic
475C2(#611⋊C3)⋊C411yescharacteristic
476C2#63447⋊C311yescharacteristic
477C2#63448⋊C311yescharacteristic
478C2#63454⋊C311yescharacteristic
479C2#63457⋊C311yescharacteristic
480C2#63460⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×AGL(1,ℤ/9)C22
cosocleC23×C9C23×C3
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#388
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×C4×3−1+2⋊C4C2semidirect
6. 3−1+2×#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588350C3semidirect
19. (C2×AGL(1,ℤ/9))⋊C4C4semidirect
32. C2×C4⋊Dic9C2×C3semidirect
33. C2×C4×Dic9C2×C3semidirect
35. C9×#95C2×C3semidirect
36. C9⋊#95C2×C3semidirect
89. (C2×D9)⋊C4C3×C4semidirect
277. #95AGL(1,ℤ/9)semidirect
278. 3−1+2#388semidirect
408. C9C3×#388semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×C4×3−1+2⋊C42
3−1+2×#952
3−1+2⋊#952
C3×#46703
#105883503