Finite Group Databasebeta

← #10589747⋊C3 #13152992

Subgroups — #10589747⋊C3 #13152992

Order 1728.

691 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
2#63508⋊C3C212yesmaximal, complemented
3#63492⋊C3C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
5#63460⋊C3C212yesmaximal, complemented
6#63456⋊C3C212yesmaximal, complemented
7#63452⋊C3C212yesmaximal, complemented
8#63448⋊C3C212yesmaximal, complemented
9(#611⋊C3)⋊C4C212yesmaximal, complemented
10C3×#5645—31nomaximal
11#10589747C311yesmaximal, characteristic, complemented
12C2×Q8×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C2×#611⋊C3C2212yes—
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2212yes—
19C4⋊C4×3−1+2C2212yes—
20C4⋊C4×3−1+2C2211yescharacteristic
21C22⋊C4×3−1+2C2212yes—
22(C2×AGL(1,ℤ/9))⋊C4C2212yes—
23(C2×AGL(1,ℤ/9))⋊C4C2212yes—
24(C2×AGL(1,ℤ/9))⋊C4C2212yes—
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26C4⋊(3−1+2⋊C4)C2211yescharacteristic
27C4⋊(3−1+2⋊C4)C2212yes—
28C4⋊(3−1+2⋊C4)C2212yes—
29(3−1+2⋊C4)⋊C4C2212yes—
30(3−1+2⋊C4)⋊C4C2212yes—
31(3−1+2⋊C4)⋊C4C2212yes—
32C4×3−1+2⋊C4C2212yes—
33C32×C22⋊Q8D311yescharacteristic
34C3×#946—32no—
35C3×#930—31no—
36C2×C3×Dic3⋊C4—31no—
37C3×#898—32no—
38C3×#894—32no—
39C3×#890—32no—
40C3×#886—32no—
41C3×#67⋊C4—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63508C2×C312yescomplemented
45#63492C2×C311yescharacteristic
46C2×Dic9⋊C4C2×C311yescharacteristic, complemented
47#63460C2×C312yescomplemented
48#63456C2×C312yescomplemented
49#63452C2×C312yescomplemented
50#63448C2×C312yescomplemented
51#611⋊C4C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C22×AGL(1,ℤ/9)C2312yes—
54Q8×3−1+2—41no—
55C2×C4×3−1+2—41no—
56C2×C4×3−1+2C2312yes—
57C2×C4×3−1+2C2311yescharacteristic
58C2×C4×3−1+2C2312yes—
59C2×C4×3−1+2C2311yescharacteristic
60#611⋊C3—24no—
61C2×3−1+2⋊C4—22no—
62C2×3−1+2⋊C4—41no—
63C2×3−1+2⋊C4C2312yes—
64C2×3−1+2⋊C4C2312yes—
65C2×3−1+2⋊C4C2312yes—
66C4⋊AGL(1,ℤ/9)—41no—
67C4×AGL(1,ℤ/9)—42no—
68C3×#539—91no—
69#5645—31no—
70C2×C32×Q8C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C2×C3×#67—32no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—32no—
77C32×C4⋊C4C2×D312yes—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C22⋊C4C2×D312yes—
80C3×(C2×D3)⋊C4—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—31no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×C4×Dic3—32no—
91C2×Q8×C9—21no—
92C2×Q8×C9C22×C311yescharacteristic
93C22×C4×C9—21no—
94C22×C4×C9C22×C311yescharacteristic
95C2×#611C22×C312yes—
96C22×Dic9C22×C311yescharacteristic
97C22×Dic9C22×C311yescharacteristic
98C2×D36C22×C311yescharacteristic
99C2×C4×D9C22×C312yes—
100C9×C4⋊C4—21no—
101C9×C4⋊C4—22no—
102C9×C4⋊C4C22×C312yes—
103C9×C4⋊C4C22×C311yescharacteristic
104C9×C22⋊C4—22no—
105C9×C22⋊C4C22×C312yes—
106(C2×D9)⋊C4C22×C312yes—
107(C2×D9)⋊C4C22×C312yes—
108(C2×D9)⋊C4C22×C312yes—
109(C2×D9)⋊C4C22×C312yes—
110C4⋊Dic9C22×C311yescharacteristic
111C4⋊Dic9C22×C312yes—
112C4⋊Dic9C22×C312yes—
113Dic9⋊C4C22×C312yes—
114Dic9⋊C4C22×C312yes—
115Dic9⋊C4C22×C312yes—
116C4×Dic9C22×C312yes—
117C22×3−1+2—21no—
118C22×3−1+2—21no—
119C22×3−1+2#4112yes—
120C22×3−1+2C2411yescharacteristic
121C2×AGL(1,ℤ/9)—22no—
122C2×AGL(1,ℤ/9)—22no—
123C2×AGL(1,ℤ/9)—22no—
124C4×3−1+2—21no—
125C4×3−1+2—22no—
126C4×3−1+2—21no—
127C4×3−1+2—22no—
1283−1+2⋊C4—22no—
1293−1+2⋊C4—22no—
1303−1+2⋊C4—22no—
131C3×#126—92no—
132C3×#125—92no—
133C3×#123—92no—
134C3×#123—92no—
135C3×C22⋊Q8—31no—
136C3×C22⋊Q8—92no—
137C3×C22⋊Q8C3×D311yescharacteristic
138C3×#121—91no—
139C3×C4×D4—92no—
140C2×C3×C4⋊C4—91no—
141#946—32no—
142#930—31no—
143C2×Dic3⋊C4—31no—
144#898—32no—
145#894—32no—
146#890—32no—
147#886—32no—
148#67⋊C4—32no—
149C23×C32C22×D311yescharacteristic
150C22×C3×D3—32no—
151C32×Q8—41no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C3×#67—64no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—121no—
163C3×D12—121no—
164C3×C4×D3—122no—
165C23×C9—21no—
166C23×C9C23×C311yescharacteristic
167C22×D9C23×C312yes—
168Q8×C9—41no—
169Q8×C9—42no—
170C2×C4×C9—21no—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—41no—
175C2×C4×C9—42no—
176C2×C4×C9C23×C312yes—
177C2×C4×C9C23×C311yescharacteristic
178C2×C4×C9C23×C311yescharacteristic
179C2×C4×C9C23×C312yes—
180#611—24no—
181C2×Dic9—22no—
182C2×Dic9—41no—
183C2×Dic9C23×C312yes—
184C2×Dic9C23×C312yes—
185C2×Dic9C23×C312yes—
186D36—41no—
187C4×D9—42no—
188#539—91noSylow
189C2×3−1+2—22no—
190C2×3−1+2C2×#4111yescharacteristic
191C2×3−1+2H(5,𝔽2)11yescharacteristic
192C2×3−1+22−1+411yescharacteristic
193AGL(1,ℤ/9)—42no—
194C2×C3×Q8—31no—
195C2×C3×Q8C2×C3×D311yescharacteristic
196C2×C3×D4—91no—
197C2×C3×D4—92no—
198C22×C3×C4—31no—
199C22×C3×C4—92no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4C2×C3×D311yescharacteristic
203C2×#67—32no—
204C22×Dic3—31no—
205C22×Dic3—31no—
206C2×D12—31no—
207C2×C4×D3—32no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4—92no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4C2×C3×D312yes—
217C3×C4⋊C4C2×C3×D311yescharacteristic
218C3×C22⋊C4—32no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C22⋊C4C2×C3×D312yes—
224C3×C42—92no—
225(C2×D3)⋊C4—32no—
226(C2×D3)⋊C4—32no—
227(C2×D3)⋊C4—32no—
228(C2×D3)⋊C4—32no—
229C4⋊Dic3—32no—
230C4⋊Dic3—32no—
231C4⋊Dic3—31no—
232Dic3⋊C4—32no—
233Dic3⋊C4—32no—
234Dic3⋊C4—32no—
235C4×Dic3—32no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32#23912yes—
239C22×C32C23×D311yescharacteristic
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C32×C4—22no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—22no—
247C3×Dic3—62no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—22no—
254C22×C9—41no—
255C22×C9—41no—
256C22×C9C3×#4112yes—
257C22×C9C24×C311yescharacteristic
258C2×D9—22no—
259C2×D9—22no—
260C2×D9—22no—
261C4×C9—22no—
262C4×C9—22no—
263C4×C9—21no—
264C4×C9—21no—
265C4×C9—22no—
266C4×C9—42no—
267C4×C9—41no—
268C4×C9—42no—
269Dic9—22no—
270Dic9—22no—
271Dic9—22no—
272#126—92no—
273#125—92no—
274#123—92no—
275#123—92no—
276C22⋊Q8—92no—
277C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
278#121—91no—
279C4×D4—92no—
280C2×C4⋊C4—91no—
2813−1+2#53911yescharacteristic, complemented, Sylow
282C23×C3—31no—
283C23×C3—92no—
284C23×C3C22×C3×D311yescharacteristic
285C22×D3—32no—
286C3×Q8—41no—
287C3×Q8—121no—
288C3×D4—184no—
289C3×D4—361no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—41no—
295C2×C3×C4—92no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—121no—
299C2×C3×C4—182no—
300C2×C3×C4—362no—
301C2×C3×C4—361no—
302C2×C3×C4C22×C3×D312yes—
303C2×C3×C4C22×C3×D311yescharacteristic
304C2×C3×C4C22×C3×D311yescharacteristic
305C2×C3×C4C22×C3×D312yes—
306#67—64no—
307C2×Dic3—32no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—62no—
311C2×Dic3—121no—
312D12—121no—
313C4×D3—122no—
314C2×C32—22no—
315C2×C32#100411yescharacteristic
316C2×C32C2×#23911yescharacteristic
317C2×C32#101011yescharacteristic
318C3×D3—122no—
319C2×C9—22no—
320C2×C9—21no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—42no—
324C2×C9C2×C3×#4111yescharacteristic
325C2×C9C3×H(5,𝔽2)11yescharacteristic
326C2×C9C3×2−1+411yescharacteristic
327D9—42no—
328C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
329C2×D4—92no—
330C2×D4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4—92no—
334C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
335C4⋊C4—92no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—91no—
339C4⋊C4—92no—
340C4⋊C4—92no—
341C4⋊C4C2×AGL(1,ℤ/9)12yes—
342C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
343C22⋊C4—92no—
344C22⋊C4—92no—
345C22⋊C4—92no—
346C22⋊C4—92no—
347C22⋊C4C2×AGL(1,ℤ/9)12yes—
348C42—92no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—31no—
352C22×C3—32no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—182no—
356C22×C3—182no—
357C22×C3—182no—
358C22×C3C3×#23912yes—
359C22×C3C23×C3×D311yescharacteristic
360C2×D3—62no—
361C2×D3—62no—
362C2×D3—62no—
363C3×C4—22no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4—61no—
368C3×C4—61no—
369C3×C4—62no—
370C3×C4—62no—
371C3×C4—182no—
372C3×C4—182no—
373C3×C4—182no—
374Dic3—62no—
375Dic3—62no—
376Dic3—62no—
377C32#564511yescharacteristic
378C9—21no—
379C9C3×#53911yescharacteristic, complemented
380C23—92no—
381C23C22×AGL(1,ℤ/9)11yescharacteristic
382Q8—41no—
383D4—184no—
384D4—361no—
385C2×C4—41no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—182no—
390C2×C4—362no—
391C2×C4—361no—
392C2×C4C22×AGL(1,ℤ/9)12yes—
393C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
394C2×C4C22×AGL(1,ℤ/9)12yes—
395C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
396C2×C3—22no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—62no—
401C2×C3—362no—
402C2×C3C3×#100411yescharacteristic
403C2×C3C2×C3×#23911yescharacteristic
404C2×C3C3×#101011yescharacteristic
405D3—122no—
406C22—21no—
407C22—21no—
408C22—182no—
409C22—182no—
410C22—182no—
411C22#8872412yes—
412C22C23×AGL(1,ℤ/9)11yescharacteristic
413C4—21no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—182no—
418C4—182no—
419C4—182no—
420C3—31no—
421C3C3×#564511yescharacteristic
422C2—22no—
423C2—362no—
424C2#63710⋊C311yescharacteristic
425C2C2×#8872411yescharacteristic
426C2#1172264211yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
prime radical#63492⋊C3C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#539
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. 3−1+2×C22⋊Q8C2semidirect
2. #63508⋊C3C2semidirect
4. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. #63460⋊C3C2semidirect
6. #63456⋊C3C2semidirect
7. #63452⋊C3C2semidirect
8. #63448⋊C3C2semidirect
9. (#611⋊C3)⋊C4C2semidirect
11. #10589747C3semidirect
43. C9×C22⋊Q8C2×C3semidirect
44. #63508C2×C3semidirect
46. C2×Dic9⋊C4C2×C3semidirect
47. #63460C2×C3semidirect
48. #63456C2×C3semidirect
49. #63452C2×C3semidirect
50. #63448C2×C3semidirect
51. #611⋊C4C2×C3semidirect
277. C22⋊Q8AGL(1,ℤ/9)semidirect
281. 3−1+2#539semidirect
379. C9C3×#539semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
#63508⋊C32
#63492⋊C32
C2×(3−1+2⋊C4)⋊C42
#63460⋊C32
#63456⋊C32
#63452⋊C32
#63448⋊C32
(#611⋊C3)⋊C42
C3×#56453
#105897473