Finite Group Databasebeta

← #10589753⋊C3 #13153062

Subgroups — #10589753⋊C3 #13153062

Order 1728.

627 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#123C211yesmaximal, characteristic
2#63498⋊C3C211yesmaximal, characteristic
3#63497⋊C3C211yesmaximal, characteristic
4#63486⋊C3C212yesmaximal, complemented
5#63451⋊C3C212yesmaximal, complemented
6#63450⋊C3C212yesmaximal, complemented
7#63441⋊C3C212yesmaximal, complemented
8#63441⋊C3C212yesmaximal, complemented
9#63440⋊C3C212yesmaximal, complemented
10C3×#5651—31nomaximal
11#10589753C311yesmaximal, characteristic, complemented
12C2×D4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C2×3−1+2⋊Q8C2211yescharacteristic
16C2×3−1+2⋊Q8C2212yes—
17C4⋊C4×3−1+2C2212yes—
18C22⋊C4×3−1+2C2212yes—
19C22⋊C4×3−1+2C2211yescharacteristic
20#3676⋊C3C2212yes—
21#3676⋊C3C2212yes—
22#3676⋊C3C2211yescharacteristic
23#3676⋊C3C2212yes—
24C4⋊(3−1+2⋊C4)C2212yes—
25C4⋊(3−1+2⋊C4)C2212yes—
26(3−1+2⋊C4)⋊C4C2212yes—
27(3−1+2⋊C4)⋊C4C2212yes—
28(3−1+2⋊C4)⋊C4C2212yes—
29(3−1+2⋊C4)⋊C4C2212yes—
30(3−1+2⋊C4)⋊C4C2212yes—
31C4×3−1+2⋊C4C2212yes—
32C4×3−1+2⋊C4C2211yescharacteristic
33C32×#123D311yescharacteristic
34C3×#936—31no—
35C3×#935—31no—
36C3×#924—32no—
37C3×#889—32no—
38C3×#888—32no—
39C3×#879—32no—
40C3×#879—32no—
41C3×#878—32no—
42C9×#123—21no—
43C9×#123C2×C311yescharacteristic
44#63498C2×C311yescharacteristic
45#63497C2×C311yescharacteristic
46#63486C2×C312yescomplemented
47#63451C2×C312yescomplemented
48#63450C2×C312yescomplemented
49#63441C2×C312yescomplemented
50#63441C2×C312yescomplemented
51#63440C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C23×3−1+2C2311yescharacteristic
54D4×3−1+2—41no—
55C2×C4×3−1+2—41no—
56C2×C4×3−1+2C2311yescharacteristic
57C2×C4×3−1+2C2312yes—
58C2×C4×3−1+2C2312yes—
59C2×3−1+2⋊C4—41no—
60C2×3−1+2⋊C4C2312yes—
61C2×3−1+2⋊C4C2312yes—
62C2×3−1+2⋊C4C2312yes—
63C2×3−1+2⋊C4C2312yes—
643−1+2⋊Q8—41no—
653−1+2⋊Q8—42no—
66C3×#563—91no—
67#5651—31no—
68C2×C32×D4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3×Dic3—31no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—31no—
73C32×C4⋊C4C2×D312yes—
74C32×C22⋊C4C2×D312yes—
75C32×C22⋊C4C2×D311yescharacteristic
76C3×#217—32no—
77C3×#217—31no—
78C3×#217—32no—
79C3×#217—32no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—32no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—31no—
89C2×D4×C9—21no—
90C2×D4×C9C22×C311yescharacteristic
91C22×C4×C9—21no—
92C22×C4×C9C22×C311yescharacteristic
93C22×Dic9C22×C311yescharacteristic
94C2×Dic18C22×C311yescharacteristic
95C2×Dic18C22×C312yes—
96C9×C4⋊C4—22no—
97C9×C4⋊C4C22×C312yes—
98C9×C22⋊C4—21no—
99C9×C22⋊C4—22no—
100C9×C22⋊C4C22×C312yes—
101C9×C22⋊C4C22×C311yescharacteristic
102#3676C22×C311yescharacteristic
103#3676C22×C312yes—
104#3676C22×C312yes—
105#3676C22×C312yes—
106C4⋊Dic9C22×C312yes—
107C4⋊Dic9C22×C312yes—
108Dic9⋊C4C22×C312yes—
109Dic9⋊C4C22×C312yes—
110Dic9⋊C4C22×C312yes—
111Dic9⋊C4C22×C312yes—
112Dic9⋊C4C22×C312yes—
113C4×Dic9C22×C311yescharacteristic
114C4×Dic9C22×C312yes—
115C22×3−1+2—21no—
116C22×3−1+2—21no—
117C22×3−1+2—22no—
118C22×3−1+2—22no—
119C22×3−1+2C2411yescharacteristic
120C4×3−1+2—22no—
121C4×3−1+2—21no—
122C4×3−1+2—22no—
1233−1+2⋊C4—22no—
1243−1+2⋊C4—22no—
1253−1+2⋊C4—22no—
1263−1+2⋊C4—22no—
127C3×C4⋊Q8—92no—
128C3×#126—92no—
129C3×#126—92no—
130C3×#125—92no—
131C3×#124—91no—
132C3×#123—31no—
133C3×#123—91no—
134C3×#123C3×D311yescharacteristic
135C3×C22⋊Q8—92no—
136C3×C22⋊Q8—92no—
137#936—31no—
138#935—31no—
139#924—32no—
140#889—32no—
141#888—32no—
142#879—32no—
143#879—32no—
144#878—32no—
145C23×C32C22×D311yescharacteristic
146C23×C32C22×D311yescharacteristic
147C32×D4—41no—
148C2×C32×C4—41no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D312yes—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—121no—
157C3×Dic6—121no—
158C3×Dic6—122no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9C23×C311yescharacteristic
162C23×C9C23×C311yescharacteristic
163D4×C9—41no—
164D4×C9—42no—
165C2×C4×C9—22no—
166C2×C4×C9—22no—
167C2×C4×C9—21no—
168C2×C4×C9—41no—
169C2×C4×C9—42no—
170C2×C4×C9C23×C312yes—
171C2×C4×C9C23×C311yescharacteristic
172C2×C4×C9C23×C312yes—
173C2×Dic9—41no—
174C2×Dic9C23×C312yes—
175C2×Dic9C23×C312yes—
176C2×Dic9C23×C312yes—
177C2×Dic9C23×C312yes—
178Dic18—41no—
179Dic18—42no—
180#563—91noSylow
181C2×3−1+2—41no—
182C2×3−1+2—41no—
183C2×3−1+2H(5,𝔽2)11yescharacteristic
184C2×3−1+22−1+412yes—
185C2×C3×Q8—92no—
186C2×C3×Q8—91no—
187C2×C3×D4—31no—
188C2×C3×D4C2×C3×D311yescharacteristic
189C22×C3×C4—31no—
190C22×C3×C4—91no—
191C22×C3×C4C2×C3×D311yescharacteristic
192C22×Dic3—31no—
193C2×Dic6—32no—
194C2×Dic6—31no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4—92no—
198C3×C4⋊C4—92no—
199C3×C4⋊C4—92no—
200C3×C4⋊C4—92no—
201C3×C4⋊C4—92no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4C2×C3×D312yes—
204C3×C22⋊C4—32no—
205C3×C22⋊C4—31no—
206C3×C22⋊C4—92no—
207C3×C22⋊C4—92no—
208C3×C22⋊C4—92no—
209C3×C22⋊C4—91no—
210C3×C22⋊C4C2×C3×D312yes—
211C3×C22⋊C4C2×C3×D311yescharacteristic
212C3×C42—91no—
213C3×C42—92no—
214#217—32no—
215#217—31no—
216#217—32no—
217#217—32no—
218C4⋊Dic3—32no—
219C4⋊Dic3—32no—
220Dic3⋊C4—32no—
221Dic3⋊C4—32no—
222Dic3⋊C4—32no—
223Dic3⋊C4—32no—
224Dic3⋊C4—32no—
225C4×Dic3—32no—
226C4×Dic3—31no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32C23×D311yescharacteristic
232C32×C4—22no—
233C32×C4—21no—
234C32×C4—22no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—62no—
239C22×C9—22no—
240C22×C9—21no—
241C22×C9—21no—
242C22×C9—22no—
243C22×C9—22no—
244C22×C9—21no—
245C22×C9—42no—
246C22×C9—41no—
247C22×C9—42no—
248C22×C9C24×C311yescharacteristic
249C4×C9—22no—
250C4×C9—21no—
251C4×C9—22no—
252C4×C9—42no—
253C4×C9—42no—
254C4×C9—41no—
255Dic9—22no—
256Dic9—22no—
257Dic9—22no—
258Dic9—22no—
259C4⋊Q8—92no—
260#126—92no—
261#126—92no—
262#125—92no—
263#124—91no—
264#123—91no—
265#123AGL(1,ℤ/9)11yescharacteristic
266C22⋊Q8—92no—
267C22⋊Q8—92no—
2683−1+2#56311yescharacteristic, complemented, Sylow
269C23×C3—31no—
270C23×C3—31no—
271C23×C3C22×C3×D311yescharacteristic
272C23×C3C22×C3×D311yescharacteristic
273C3×Q8—362no—
274C3×Q8—361no—
275C3×D4—41no—
276C3×D4—121no—
277C2×C3×C4—32no—
278C2×C3×C4—31no—
279C2×C3×C4—32no—
280C2×C3×C4—41no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—121no—
286C2×C3×C4—361no—
287C2×C3×C4C22×C3×D312yes—
288C2×C3×C4C22×C3×D311yescharacteristic
289C2×C3×C4C22×C3×D312yes—
290C2×Dic3—32no—
291C2×Dic3—32no—
292C2×Dic3—32no—
293C2×Dic3—32no—
294C2×Dic3—121no—
295Dic6—121no—
296Dic6—122no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32#100411yescharacteristic
300C2×C32#101012yes—
301C2×C9—21no—
302C2×C9—22no—
303C2×C9—42no—
304C2×C9—41no—
305C2×C9—41no—
306C2×C9—81no—
307C2×C9C3×H(5,𝔽2)11yescharacteristic
308C2×C9C3×2−1+412yes—
309C2×Q8—92no—
310C2×Q8—91no—
311C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
312C22×C4—91no—
313C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
314C4⋊C4—92no—
315C4⋊C4—92no—
316C4⋊C4—92no—
317C4⋊C4—92no—
318C4⋊C4—92no—
319C4⋊C4—92no—
320C4⋊C4—92no—
321C4⋊C4C2×AGL(1,ℤ/9)12yes—
322C22⋊C4—92no—
323C22⋊C4—92no—
324C22⋊C4—92no—
325C22⋊C4—91no—
326C22⋊C4C2×AGL(1,ℤ/9)12yes—
327C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
328C42—91no—
329C42—92no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—62no—
339C22×C3C23×C3×D311yescharacteristic
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—22no—
343C3×C4—62no—
344C3×C4—61no—
345C3×C4—62no—
346C3×C4—182no—
347C3×C4—182no—
348C3×C4—182no—
349C3×C4—182no—
350Dic3—62no—
351Dic3—62no—
352Dic3—62no—
353Dic3—62no—
354C32#565111yescharacteristic
355C9—21no—
356C9C3×#56311yescharacteristic, complemented
357C23C22×AGL(1,ℤ/9)11yescharacteristic
358C23C22×AGL(1,ℤ/9)11yescharacteristic
359Q8—362no—
360Q8—361no—
361D4—41no—
362C2×C4—41no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—92no—
366C2×C4—92no—
367C2×C4—361no—
368C2×C4C22×AGL(1,ℤ/9)12yes—
369C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
370C2×C4C22×AGL(1,ℤ/9)12yes—
371C2×C3—31no—
372C2×C3—32no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—121no—
376C2×C3—121no—
377C2×C3C3×#100411yescharacteristic
378C2×C3C3×#101012yes—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22C23×AGL(1,ℤ/9)11yescharacteristic
384C4—22no—
385C4—21no—
386C4—22no—
387C4—182no—
388C4—182no—
389C4—182no—
390C4—182no—
391C3—31no—
392C3C3×#565111yescharacteristic
393C2—41no—
394C2—41no—
395C2#63710⋊C311yescharacteristic
396C2#1172264212yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#123C2
prime radicalC2×C32×D4C2×D3
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#563
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #63486⋊C3C2semidirect
5. #63451⋊C3C2semidirect
6. #63450⋊C3C2semidirect
7. #63441⋊C3C2semidirect
8. #63441⋊C3C2semidirect
9. #63440⋊C3C2semidirect
11. #10589753C3semidirect
46. #63486C2×C3semidirect
47. #63451C2×C3semidirect
48. #63450C2×C3semidirect
49. #63441C2×C3semidirect
50. #63441C2×C3semidirect
51. #63440C2×C3semidirect
268. 3−1+2#563semidirect
356. C9C3×#563semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1232
#63498⋊C32
#63497⋊C32
#63486⋊C32
#63451⋊C32
#63450⋊C32
#63441⋊C32
#63440⋊C32
C3×#56513
#105897533