Finite Group Databasebeta

← #10589727⋊C3 #13152976

Subgroups — #10589727⋊C3 #13152976

Order 1728.

613 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×C22⋊Q8C211yesmaximal, characteristic
2Q8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
3#63506⋊C3C211yesmaximal, characteristic, complemented
4#63488⋊C3C211yesmaximal, characteristic
5#63486⋊C3C211yesmaximal, characteristic
6#63452⋊C3C212yesmaximal, complemented
7#63452⋊C3C211yesmaximal, characteristic, complemented
8#63451⋊C3C212yesmaximal, complemented
9#63449⋊C3C211yesmaximal, characteristic, complemented
10#63441⋊C3C212yesmaximal
11#63441⋊C3C212yesmaximal
12C3×#5625—31nomaximal
13#10589727C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×3−1+2⋊Q8C2211yescharacteristic
17C4⋊C4×3−1+2C2211yescharacteristic
18C4⋊C4×3−1+2C2212yes—
19C22⋊C4×3−1+2C2212yes—
20#3676⋊C3C2212yes—
21#3676⋊C3C2211yescharacteristic
22#3676⋊C3C2211yescharacteristic
23C4⋊(3−1+2⋊C4)C2211yescharacteristic
24C4⋊(3−1+2⋊C4)C2211yescharacteristic
25C4⋊(3−1+2⋊C4)C2212yes—
26C4⋊(3−1+2⋊C4)C2211yescharacteristic
27C4⋊(3−1+2⋊C4)C2212yes—
28(3−1+2⋊C4)⋊C4C2212yes—
29(3−1+2⋊C4)⋊C4C2212yes—
30(3−1+2⋊C4)⋊C4C2212yes—
31(3−1+2⋊C4)⋊C4C2212yes—
32(3−1+2⋊C4)⋊C4C2212yes—
33C4×3−1+2⋊C4C2212yes—
34C4×3−1+2⋊C4C2211yescharacteristic
35C4×3−1+2⋊C4C2212yes—
36C4×3−1+2⋊C4C2211yescharacteristic
37C32×C22⋊Q8D311yescharacteristic
38C3×Q8×Dic3—31no—
39C3×#944—31no—
40C3×#926—31no—
41C3×#924—31no—
42C3×#890—32no—
43C3×#890—31no—
44C3×#889—32no—
45C3×#887—31no—
46C3×#879—32no—
47C3×#879—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic
50Q8×Dic9C2×C311yescharacteristic, complemented
51#63506C2×C311yescharacteristic, complemented
52#63488C2×C311yescharacteristic
53#63486C2×C311yescharacteristic
54#63452C2×C311yescharacteristic, complemented
55#63452C2×C312yescomplemented
56#63451C2×C312yescomplemented
57#63449C2×C311yescharacteristic, complemented
58#63441C2×C312yes—
59#63441C2×C312yes—
60C23×3−1+2C2311yescharacteristic
61Q8×3−1+2—22no—
62C2×C4×3−1+2—22no—
63C2×C4×3−1+2C2312yes—
64C2×C4×3−1+2C2312yes—
65C2×C4×3−1+2C2311yescharacteristic
66C2×C4×3−1+2C2311yescharacteristic
67C2×3−1+2⋊C4C2312yes—
68C2×3−1+2⋊C4C2311yescharacteristic
69C2×3−1+2⋊C4C2311yescharacteristic
70C2×3−1+2⋊C4C2311yescharacteristic
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2311yescharacteristic
733−1+2⋊Q8—22no—
74C3×#541—91no—
75#5625—31no—
76C2×C32×Q8C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C2×C3×Dic6—31no—
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C2×D312yes—
81C32×C22⋊C4C2×D312yes—
82C3×#217—32no—
83C3×#217—31no—
84C3×#217—31no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C2×Q8×C9—21no—
100C2×Q8×C9C22×C311yescharacteristic
101C22×C4×C9—21no—
102C22×C4×C9C22×C311yescharacteristic
103C2×Dic18C22×C311yescharacteristic
104C9×C4⋊C4—21no—
105C9×C4⋊C4—22no—
106C9×C4⋊C4C22×C311yescharacteristic
107C9×C4⋊C4C22×C312yes—
108C9×C22⋊C4—22no—
109C9×C22⋊C4C22×C312yes—
110#3676C22×C311yescharacteristic
111#3676C22×C311yescharacteristic
112#3676C22×C312yes—
113C4⋊Dic9C22×C311yescharacteristic
114C4⋊Dic9C22×C311yescharacteristic
115C4⋊Dic9C22×C312yes—
116C4⋊Dic9C22×C311yescharacteristic
117C4⋊Dic9C22×C312yes—
118Dic9⋊C4C22×C312yes—
119Dic9⋊C4C22×C312yes—
120Dic9⋊C4C22×C312yes—
121Dic9⋊C4C22×C312yes—
122Dic9⋊C4C22×C312yes—
123C4×Dic9C22×C311yescharacteristic
124C4×Dic9C22×C311yescharacteristic
125C4×Dic9C22×C312yes—
126C4×Dic9C22×C312yes—
127C22×3−1+2—21no—
128C22×3−1+2—21no—
129C22×3−1+2—21no—
130C22×3−1+2C2411yescharacteristic
131C4×3−1+2—21no—
132C4×3−1+2—22no—
133C4×3−1+2—22no—
134C4×3−1+2#4112yes—
1353−1+2⋊C4—22no—
1363−1+2⋊C4—21no—
1373−1+2⋊C4—22no—
1383−1+2⋊C4—21no—
1393−1+2⋊C4—21no—
1403−1+2⋊C4—21no—
141C3×C4⋊Q8—91no—
142C3×#126—92no—
143C3×#126—92no—
144C3×#125—92no—
145C3×#125—91no—
146C3×#125—92no—
147C3×C22⋊Q8—31no—
148C3×C22⋊Q8—91no—
149C3×C22⋊Q8C3×D311yescharacteristic
150C3×C4×Q8—91no—
151C3×C4×Q8—91no—
152C3×#117—91no—
153Q8×Dic3—31no—
154#944—31no—
155#926—31no—
156#924—31no—
157#890—32no—
158#890—31no—
159#889—32no—
160#887—31no—
161#879—32no—
162#879—32no—
163C23×C32C22×D311yescharacteristic
164C32×Q8—22no—
165C2×C32×C4—22no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—31no—
176C3×Dic6—62no—
177C23×C9—21no—
178C23×C9C23×C311yescharacteristic
179Q8×C9—22no—
180Q8×C9—42no—
181C2×C4×C9—22no—
182C2×C4×C9—21no—
183C2×C4×C9—22no—
184C2×C4×C9—22no—
185C2×C4×C9—21no—
186C2×C4×C9—42no—
187C2×C4×C9C23×C312yes—
188C2×C4×C9C23×C312yes—
189C2×C4×C9C23×C311yescharacteristic
190C2×C4×C9C23×C311yescharacteristic
191C2×Dic9C23×C311yescharacteristic
192C2×Dic9C23×C311yescharacteristic
193C2×Dic9C23×C311yescharacteristic
194C2×Dic9C23×C312yes—
195C2×Dic9C23×C311yescharacteristic
196C2×Dic9C23×C312yes—
197Dic18—22no—
198#541—91noSylow
199C2×3−1+2—41no—
200C2×3−1+2C2×#4111yescharacteristic
201C2×3−1+22−1+411yescharacteristic
202C2×3−1+22−1+411yescharacteristic
203C2×C3×Q8—31no—
204C2×C3×Q8—91no—
205C2×C3×Q8C2×C3×D311yescharacteristic
206C22×C3×C4—31no—
207C22×C3×C4C2×C3×D311yescharacteristic
208C2×Dic6—31no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—32no—
211C3×C4⋊C4—92no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4C2×C3×D311yescharacteristic
222C3×C4⋊C4C2×C3×D312yes—
223C3×C22⋊C4—32no—
224C3×C22⋊C4—91no—
225C3×C22⋊C4—91no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4C2×C3×D312yes—
228C3×C42—91no—
229C3×C42—91no—
230C3×C42—92no—
231C3×C42—92no—
232#217—32no—
233#217—31no—
234#217—31no—
235C4⋊Dic3—32no—
236C4⋊Dic3—32no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240Dic3⋊C4—32no—
241Dic3⋊C4—32no—
242Dic3⋊C4—32no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245C4×Dic3—32no—
246C4×Dic3—32no—
247C4×Dic3—31no—
248C4×Dic3—31no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C23×D311yescharacteristic
253C32×C4—21no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4#23712yes—
257C3×Dic3—62no—
258C3×Dic3—61no—
259C3×Dic3—62no—
260C3×Dic3—61no—
261C3×Dic3—61no—
262C3×Dic3—61no—
263C22×C9—22no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—21no—
268C22×C9—41no—
269C22×C9—41no—
270C22×C9C24×C311yescharacteristic
271C4×C9—21no—
272C4×C9—22no—
273C4×C9—22no—
274C4×C9—22no—
275C4×C9—42no—
276C4×C9—41no—
277C4×C9—42no—
278C4×C9C3×#4112yes—
279Dic9—22no—
280Dic9—21no—
281Dic9—21no—
282Dic9—22no—
283Dic9—21no—
284Dic9—21no—
285C4⋊Q8—91no—
286#126—92no—
287#126—92no—
288#125—92no—
289#125—91no—
290#125—92no—
291C22⋊Q8—91no—
292C22⋊Q8AGL(1,ℤ/9)11yescharacteristic
293C4×Q8—91no—
294C4×Q8—91no—
295#117—91no—
2963−1+2#54111yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3C22×C3×D311yescharacteristic
299C3×Q8—22no—
300C3×Q8—62no—
301C3×Q8—182no—
302C2×C3×C4—22no—
303C2×C3×C4—31no—
304C2×C3×C4—32no—
305C2×C3×C4—32no—
306C2×C3×C4—31no—
307C2×C3×C4—62no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4C22×C3×D312yes—
315C2×C3×C4C22×C3×D312yes—
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D311yescharacteristic
318C2×Dic3—31no—
319C2×Dic3—31no—
320C2×Dic3—31no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—31no—
324Dic6—62no—
325C2×C32—41no—
326C2×C32C2×#23711yescharacteristic
327C2×C32#100711yescharacteristic
328C2×C32#101011yescharacteristic
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—21no—
332C2×C9—42no—
333C2×C9—41no—
334C2×C9C2×C3×#4111yescharacteristic
335C2×C9C3×2−1+411yescharacteristic
336C2×C9C3×2−1+411yescharacteristic
337C2×Q8—91no—
338C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
339C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
340C4⋊C4—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4—92no—
344C4⋊C4—92no—
345C4⋊C4—92no—
346C4⋊C4—92no—
347C4⋊C4—92no—
348C4⋊C4—92no—
349C4⋊C4—92no—
350C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
351C4⋊C4C2×AGL(1,ℤ/9)12yes—
352C22⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—92no—
355C22⋊C4C2×AGL(1,ℤ/9)12yes—
356C42—91no—
357C42—91no—
358C42—92no—
359C42—92no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—31no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3C23×C3×D311yescharacteristic
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—32no—
372C3×C4—61no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—182no—
376C3×C4—181no—
377C3×C4—181no—
378C3×C4—182no—
379C3×C4—181no—
380C3×C4—181no—
381C3×C4C3×#23712yes—
382Dic3—62no—
383Dic3—61no—
384Dic3—62no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388C32#562511yescharacteristic
389C9—21no—
390C9C3×#54111yescharacteristic, complemented
391C23C22×AGL(1,ℤ/9)11yescharacteristic
392Q8—22no—
393Q8—182no—
394C2×C4—22no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—92no—
399C2×C4—91no—
400C2×C4—92no—
401C2×C4C22×AGL(1,ℤ/9)12yes—
402C2×C4C22×AGL(1,ℤ/9)12yes—
403C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
404C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—41no—
409C2×C3—121no—
410C2×C3C2×C3×#23711yescharacteristic
411C2×C3C3×#100711yescharacteristic
412C2×C3C3×#101011yescharacteristic
413C22—21no—
414C22—21no—
415C22—21no—
416C22C23×AGL(1,ℤ/9)11yescharacteristic
417C4—21no—
418C4—22no—
419C4—22no—
420C4—182no—
421C4—181no—
422C4—181no—
423C4—182no—
424C4—181no—
425C4—181no—
426C4#3699⋊C312yes—
427C3—31no—
428C3C3×#562511yescharacteristic
429C2—41no—
430C2C2×#3699⋊C311yescharacteristic
431C2#1172263011yescharacteristic
432C2#1172264211yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
prime radicalC23×C32C22×D3
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#541
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. Q8×3−1+2⋊C4C2semidirect
3. #63506⋊C3C2semidirect
6. #63452⋊C3C2semidirect
7. #63452⋊C3C2semidirect
8. #63451⋊C3C2semidirect
9. #63449⋊C3C2semidirect
13. #10589727C3semidirect
50. Q8×Dic9C2×C3semidirect
51. #63506C2×C3semidirect
54. #63452C2×C3semidirect
55. #63452C2×C3semidirect
56. #63451C2×C3semidirect
57. #63449C2×C3semidirect
296. 3−1+2#541semidirect
390. C9C3×#541semidirect

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
Q8×3−1+2⋊C42
#63506⋊C32
#63488⋊C32
#63486⋊C32
#63452⋊C32
#63451⋊C32
#63449⋊C32
#63441⋊C32
C3×#56253
#105897273