Finite Group Databasebeta

← #10588355⋊C3 #13141394

Subgroups — #10588355⋊C3 #13141394

Order 1728.

548 conjugacy classes of subgroups.

The subgroup lattice has 476 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×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
53−1+2×#95C211yesmaximal, characteristic, complemented
63−1+2⋊#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4675—31nomaximal
9#10588355C311yesmaximal, 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))⋊C4—22no—
20(3−1+2⋊C4)⋊C4—22no—
21C2×C3×(C2×D3)⋊C4—31no—
22C2×C3×(C2×D3)⋊C4—31no—
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C32×#95D311yescharacteristic
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C2×(C2×D9)⋊C4C2×C311yescharacteristic
29C2×(C2×D9)⋊C4C2×C311yescharacteristic
30C2×(C2×D9)⋊C4C2×C311yescharacteristic
31C2×Dic9⋊C4C2×C311yescharacteristic, complemented
32C9×#95—21no—
33C9×#95C2×C311yescharacteristic, complemented
34C9⋊#95C2×C311yescharacteristic, complemented
35C9⋊#95C2×C311yescharacteristic, complemented
36C23×3−1+2C2311yescharacteristic
37C22×AGL(1,ℤ/9)—21no—
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)D412yes—
44C2×C4×3−1+2—21no—
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+2—21no—
53C2×3−1+2⋊C4—21no—
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⋊C4D412yes—
62C3×#397—91no—
63#4675—31no—
64C23×C3×D3—31no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C23×D9C22×C311yescharacteristic
72C3×(C2×D3)⋊C4—62no—
73C3×(C2×D3)⋊C4—62no—
74C3×(C2×D3)⋊C4—62no—
75C3×Dic3⋊C4—62no—
76C22×C4×C9—21no—
77C22×C4×C9—21no—
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C22×C4×C9C22×C311yescharacteristic
81C22×C4×C9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85(C2×D9)⋊C4—22no—
86(C2×D9)⋊C4—22no—
87(C2×D9)⋊C4—22no—
88Dic9⋊C4—22no—
89C22×3−1+2C2×D411yescharacteristic
90C22×3−1+2C2×D411yescharacteristic
91C22×3−1+2#4111yescharacteristic
92C22×3−1+2#4111yescharacteristic
93C22×3−1+2#4111yescharacteristic
94C22×3−1+2#4111yescharacteristic
95C22×3−1+2#4111yescharacteristic
96C2×AGL(1,ℤ/9)—22no—
97C2×AGL(1,ℤ/9)—22no—
98C2×AGL(1,ℤ/9)—22no—
99C2×AGL(1,ℤ/9)—41no—
100C2×AGL(1,ℤ/9)—41no—
101C2×AGL(1,ℤ/9)—41no—
102C2×AGL(1,ℤ/9)—41no—
103C4×3−1+2—41no—
104C4×3−1+2—41no—
105C4×3−1+2—41no—
1063−1+2⋊C4—22no—
1073−1+2⋊C4—41no—
1083−1+2⋊C4—41no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C22⋊C4—91no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4—91no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—31no—
116C2×Dic3⋊C4—31no—
117C3×#95—31no—
118C3×#95—91no—
119C3×#95—91no—
120C3×#95C3×D311yescharacteristic
121C3⋊#95—31no—
122C3⋊#95—31no—
123C23×C32C22×D311yescharacteristic
124C22×C3×D3—32no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
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×C3×Dic3—32no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C23×C9—21no—
150C23×C9C23×C311yescharacteristic
151C22×D9—21no—
152C22×D9—21no—
153C22×D9—21no—
154C22×D9—21no—
155C22×D9—21no—
156C22×D9—21no—
157C22×D9C3×D412yes—
158C2×C4×C9—22no—
159C2×C4×C9—22no—
160C2×C4×C9—22no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—41no—
171C2×C4×C9—41no—
172C2×C4×C9—41no—
173C2×C4×C9—41no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×Dic9—21no—
177C2×Dic9—21no—
178C2×Dic9—21no—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9C3×D412yes—
185#397—91noSylow
186C2×3−1+2#12111yescharacteristic
187C2×3−1+2#12311yescharacteristic
188C2×3−1+2#12311yescharacteristic
189C2×3−1+2#12311yescharacteristic
190C2×3−1+2#12411yescharacteristic
191C2×3−1+2#12611yescharacteristic
192C2×3−1+2#12611yescharacteristic
193AGL(1,ℤ/9)—42no—
194C24×C3—91no—
195C23×D3—31no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4C2×C3×D311yescharacteristic
203C22×C3×C4C2×C3×D311yescharacteristic
204C22×C3×C4C2×C3×D311yescharacteristic
205C22×Dic3—31no—
206C22×Dic3—31no—
207C22×Dic3—31no—
208C3×C4⋊C4—182no—
209C3×C22⋊C4—182no—
210C3×C22⋊C4—182no—
211C3×C22⋊C4—182no—
212(C2×D3)⋊C4—62no—
213(C2×D3)⋊C4—62no—
214(C2×D3)⋊C4—62no—
215Dic3⋊C4—62no—
216C22×C32#23511yescharacteristic
217C22×C32#23511yescharacteristic
218C22×C32#23511yescharacteristic
219C22×C32D3×D411yescharacteristic
220C22×C32D3×D411yescharacteristic
221C22×C32#23711yescharacteristic
222C22×C32#23911yescharacteristic
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4—41no—
233C3×Dic3—62no—
234C3×Dic3—121no—
235C3×Dic3—121no—
236C22×C9—21no—
237C22×C9—21no—
238C22×C9—21no—
239C22×C9—21no—
240C22×C9—21no—
241C22×C9—21no—
242C22×C9—21no—
243C22×C9C2×C3×D411yescharacteristic
244C22×C9C2×C3×D411yescharacteristic
245C22×C9C3×#4111yescharacteristic
246C22×C9C3×#4111yescharacteristic
247C22×C9C3×#4111yescharacteristic
248C22×C9C3×#4111yescharacteristic
249C22×C9C3×#4111yescharacteristic
250C2×D9—22no—
251C2×D9—22no—
252C2×D9—22no—
253C2×D9—41no—
254C2×D9—41no—
255C2×D9—41no—
256C2×D9—41no—
257C4×C9—42no—
258C4×C9—42no—
259C4×C9—42no—
260C4×C9—41no—
261C4×C9—41no—
262C4×C9—41no—
263Dic9—22no—
264Dic9—41no—
265Dic9—41no—
266C2×C4⋊C4—91no—
267C2×C22⋊C4—91no—
268C2×C22⋊C4—91no—
269C2×C22⋊C4—91no—
270#95—91no—
271#95—91no—
272#95AGL(1,ℤ/9)11yescharacteristic, complemented
2733−1+2#39711yescharacteristic, complemented, Sylow
274C23×C3—31no—
275C23×C3—92no—
276C23×C3—181no—
277C23×C3—181no—
278C23×C3—181no—
279C23×C3—181no—
280C23×C3—181no—
281C23×C3—181no—
282C23×C3C22×C3×D311yescharacteristic
283C22×D3—32no—
284C22×D3—61no—
285C22×D3—61no—
286C22×D3—61no—
287C22×D3—61no—
288C22×D3—61no—
289C22×D3—61no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—92no—
309C2×C3×C4—181no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×Dic3—32no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×C32#87611yescharacteristic
327C2×C32#88311yescharacteristic
328C2×C32#88411yescharacteristic
329C2×C32#88511yescharacteristic
330C2×C32#89411yescharacteristic
331C2×C32#89411yescharacteristic
332C2×C32#89811yescharacteristic
333C3×D3—122no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9C3×#12111yescharacteristic
342C2×C9C3×#12311yescharacteristic
343C2×C9C3×#12311yescharacteristic
344C2×C9C3×#12311yescharacteristic
345C2×C9C3×#12411yescharacteristic
346C2×C9C3×#12611yescharacteristic
347C2×C9C3×#12611yescharacteristic
348D9—42no—
349C24—91no—
350C22×C4—91no—
351C22×C4—91no—
352C22×C4—91no—
353C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
354C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
355C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
356C4⋊C4—182no—
357C22⋊C4—182no—
358C22⋊C4—182no—
359C22⋊C4—182no—
360C22×C3—31no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—182no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3—361no—
374C22×C3C3×#23511yescharacteristic
375C22×C3C3×#23511yescharacteristic
376C22×C3C3×#23511yescharacteristic
377C22×C3C3×D3×D411yescharacteristic
378C22×C3C3×D3×D411yescharacteristic
379C22×C3C3×#23711yescharacteristic
380C22×C3C3×#23911yescharacteristic
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—121no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4—182no—
395C3×C4—361no—
396C3×C4—361no—
397Dic3—62no—
398Dic3—121no—
399Dic3—121no—
400C32#467511yescharacteristic
401C9—21no—
402C9C3×#39711yescharacteristic, complemented
403C23—92no—
404C23—181no—
405C23—181no—
406C23—181no—
407C23—181no—
408C23—181no—
409C23—181no—
410C23C22×AGL(1,ℤ/9)11yescharacteristic
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—92no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—362no—
437C2×C3C3×#87611yescharacteristic
438C2×C3C3×#88311yescharacteristic
439C2×C3C3×#88411yescharacteristic
440C2×C3C3×#88511yescharacteristic
441C2×C3C3×#89411yescharacteristic
442C2×C3C3×#89411yescharacteristic
443C2×C3C3×#89811yescharacteristic
444D3—122no—
445C22—182no—
446C22—182no—
447C22—182no—
448C22—361no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22#3697⋊C311yescharacteristic
453C22#3697⋊C311yescharacteristic
454C22#3697⋊C311yescharacteristic
455C22D4×AGL(1,ℤ/9)11yescharacteristic
456C22D4×AGL(1,ℤ/9)11yescharacteristic
457C22#3699⋊C311yescharacteristic
458C22#8872411yescharacteristic
459C4—41no—
460C4—41no—
461C4—41no—
462C4—182no—
463C4—361no—
464C4—361no—
465C3—31no—
466C3C3×#467511yescharacteristic
467C2—362no—
468C2#1172139611yescharacteristic
469C2#63445⋊C311yescharacteristic
470C2#63446⋊C311yescharacteristic
471C2#63447⋊C311yescharacteristic
472C2#63456⋊C311yescharacteristic
473C2#63456⋊C311yescharacteristic
474C2#63460⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#397
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. 3−1+2×#95C2semidirect
6. 3−1+2⋊#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588355C3semidirect
31. C2×Dic9⋊C4C2×C3semidirect
33. C9×#95C2×C3semidirect
34. C9⋊#95C2×C3semidirect
35. C9⋊#95C2×C3semidirect
272. #95AGL(1,ℤ/9)semidirect
273. 3−1+2#397semidirect
402. C9C3×#397semidirect

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×(3−1+2⋊C4)⋊C42
3−1+2×#952
3−1+2⋊#952
C3×#46753
#105883553