Finite Group Databasebeta

← #10588733⋊C3 #13142899

Subgroups — #10588733⋊C3 #13142899

Order 1728.

519 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#63710⋊C3C211yesmaximal, characteristic
23−1+2×#123C211yesmaximal, characteristic, complemented
3#63491⋊C3C212yesmaximal, complemented
4#63393⋊C3C211yesmaximal, characteristic, complemented
5#63365⋊C3C212yesmaximal, complemented
6C3×#5053—31nomaximal
7#10588733C311yesmaximal, characteristic, complemented
8C2×D4×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic, complemented
10C2×#611⋊C3—21no—
11C2×#611⋊C3C2212yes—
12#3699⋊C3—22no—
13D4×AGL(1,ℤ/9)—22no—
14#3697⋊C3—21no—
15C4⋊C4×3−1+2—21no—
16C22⋊C4×3−1+2—21no—
17C22⋊C4×3−1+2C2211yescharacteristic, complemented
18#3676⋊C3C2212yescomplemented
19(C2×AGL(1,ℤ/9))⋊C4—22no—
20(3−1+2⋊C4)⋊C4—22no—
21C3×#1004—31no—
22C32×#123D311yescharacteristic
23C3×#929—32no—
24#63710C2×C311yescharacteristic
25C3×#834—31no—
26C3×#806—32no—
27C9×#123—21no—
28C9×#123C2×C311yescharacteristic, complemented
29#63491C2×C312yescomplemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×3−1+2D411yescharacteristic
33C23×3−1+2C2311yescharacteristic
34C22×AGL(1,ℤ/9)—21no—
35C22×AGL(1,ℤ/9)D412yes—
36D4×3−1+2—22no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—22no—
40C2×C4×3−1+2D411yescharacteristic
41#611⋊C3—22no—
42#611⋊C3—22no—
43#611⋊C3—22no—
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4—22no—
46C2×3−1+2⋊C4D412yes—
47C4⋊AGL(1,ℤ/9)—21no—
48C4×AGL(1,ℤ/9)—22no—
493−1+2⋊Q8—21no—
50C3×#458—91no—
51#5053—31no—
52C2×C32×D4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C2×C3×#67—32no—
55C2×C3×#67—61no—
56C3×#237—62no—
57C3×D3×D4—62no—
58C3×#235—61no—
59C32×C4⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4C2×D311yescharacteristic
62C3×#217—32no—
63C3×(C2×D3)⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C2×D4×C9—21no—
66C2×D4×C9C22×C311yescharacteristic
67C22×C4×C9—21no—
68C22×C4×C9C22×C311yescharacteristic, complemented
69C2×#611—21no—
70C2×#611C22×C312yes—
71#3699—22no—
72D4×D9—22no—
73#3697—21no—
74C9×C4⋊C4—22no—
75C9×C4⋊C4—21no—
76C9×C22⋊C4—22no—
77C9×C22⋊C4—21no—
78C9×C22⋊C4—21no—
79C9×C22⋊C4C22×C311yescharacteristic, complemented
80#3676C22×C312yescomplemented
81(C2×D9)⋊C4—22no—
82Dic9⋊C4—22no—
83C22×3−1+2—21no—
84C22×3−1+2—41no—
85C22×3−1+2—41no—
86C22×3−1+2C2×D412yes—
87C22×3−1+2C2×D411yescharacteristic
88C2×AGL(1,ℤ/9)—22no—
89C2×AGL(1,ℤ/9)—42no—
90C2×AGL(1,ℤ/9)—81no—
91C4×3−1+2—21no—
92C4×3−1+2—41no—
93C4×3−1+2—41no—
943−1+2⋊C4—22no—
953−1+2⋊C4—42no—
96C3×H(5,𝔽2)—91no—
97#1004—31no—
98C3×#123—31no—
99C3×#123—92no—
100C3×#123C3×D311yescharacteristic
101#929—32no—
102C3×#99—92no—
103C3×#99—91no—
104#834—31no—
105#806—32no—
106C23×C32#6711yescharacteristic
107C23×C32C22×D311yescharacteristic
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C32×D4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4#6711yescharacteristic
115C3×#67—62no—
116C3×#67—62no—
117C3×#67—62no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C3×D12—61no—
122C3×C4×D3—62no—
123C3×Dic6—61no—
124C23×C9—21no—
125C23×C9—21no—
126C23×C9C3×D411yescharacteristic
127C23×C9C23×C311yescharacteristic
128C22×D9—21no—
129C22×D9C3×D412yes—
130D4×C9—22no—
131D4×C9—42no—
132C2×C4×C9—22no—
133C2×C4×C9—22no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—42no—
139C2×C4×C9C3×D411yescharacteristic
140#611—22no—
141#611—22no—
142#611—22no—
143C2×Dic9—21no—
144C2×Dic9—22no—
145C2×Dic9C3×D412yes—
146D36—21no—
147C4×D9—22no—
148Dic18—21no—
149#458—91noSylow
150C2×3−1+2—21no—
151C2×3−1+2—22no—
152C2×3−1+2—41no—
153C2×3−1+2#12011yescharacteristic
154AGL(1,ℤ/9)—42no—
155C3×#41—181no—
156C3×#41—182no—
157C2×C3×D4—31no—
158C2×C3×D4—92no—
159C2×C3×D4—182no—
160C2×C3×D4—181no—
161C2×C3×D4C2×C3×D311yescharacteristic
162C22×C3×C4—31no—
163C22×C3×C4C2×C3×D311yescharacteristic
164C2×#67—32no—
165C2×#67—61no—
166#237—62no—
167D3×D4—62no—
168#235—61no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—182no—
172C3×C22⋊C4—21no—
173C3×C22⋊C4—31no—
174C3×C22⋊C4—61no—
175C3×C22⋊C4—92no—
176C3×C22⋊C4—182no—
177C3×C22⋊C4C2×C3×D311yescharacteristic
178#217—32no—
179(C2×D3)⋊C4—62no—
180Dic3⋊C4—62no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32D3×D412yes—
185C22×C32C2×#6711yescharacteristic
186C2×C3×D3—62no—
187C2×C3×D3—122no—
188C2×C3×D3—241no—
189C32×C4—21no—
190C32×C4—41no—
191C32×C4—41no—
192C3×Dic3—62no—
193C3×Dic3—122no—
194C22×C9—21no—
195C22×C9—22no—
196C22×C9—21no—
197C22×C9—41no—
198C22×C9—41no—
199C22×C9—41no—
200C22×C9—42no—
201C22×C9—42no—
202C22×C9C2×C3×D412yes—
203C22×C9C2×C3×D411yescharacteristic
204C2×D9—22no—
205C2×D9—42no—
206C2×D9—81no—
207C4×C9—21no—
208C4×C9—41no—
209C4×C9—42no—
210C4×C9—41no—
211C4×C9—42no—
212C4×C9—41no—
213Dic9—22no—
214Dic9—42no—
215H(5,𝔽2)—91no—
216#123—92no—
217#123AGL(1,ℤ/9)11yescharacteristic, complemented
218#99—91no—
219#99—92no—
2203−1+2#45811yescharacteristic, complemented, Sylow
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—92no—
224C23×C3—181no—
225C23×C3C3×#6711yescharacteristic
226C23×C3C22×C3×D311yescharacteristic
227C22×D3—32no—
228C22×D3—61no—
229C3×Q8—181no—
230C3×D4—22no—
231C3×D4—62no—
232C3×D4—181no—
233C3×D4—182no—
234C3×D4—182no—
235C3×D4—182no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—62no—
243C2×C3×C4—92no—
244C2×C3×C4—182no—
245C2×C3×C4—181no—
246C2×C3×C4—182no—
247C2×C3×C4C3×#6711yescharacteristic
248#67—62no—
249#67—62no—
250#67—62no—
251C2×Dic3—32no—
252C2×Dic3—61no—
253C2×Dic3—62no—
254D12—61no—
255C4×D3—62no—
256Dic6—61no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32#93711yescharacteristic
261C3×D3—122no—
262C2×C9—22no—
263C2×C9—22no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—42no—
267C2×C9—41no—
268C2×C9—81no—
269C2×C9C3×#12011yescharacteristic
270D9—42no—
271#41—182no—
272#41—181no—
273C2×D4—92no—
274C2×D4—182no—
275C2×D4—181no—
276C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
277C22×C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
278C4⋊C4—21no—
279C4⋊C4—182no—
280C22⋊C4—21no—
281C22⋊C4—92no—
282C22⋊C4—182no—
283C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
284C22×C3—21no—
285C22×C3—32no—
286C22×C3—31no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—61no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—182no—
293C22×C3—362no—
294C22×C3—721no—
295C22×C3C3×D3×D412yes—
296C22×C3C2×C3×#6711yescharacteristic
297C2×D3—62no—
298C2×D3—122no—
299C2×D3—241no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—61no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—182no—
307C3×C4—362no—
308Dic3—62no—
309Dic3—122no—
310C32#505311yescharacteristic
311C9—21no—
312C9C3×#45811yescharacteristic, complemented
313C23—92no—
314C23—181no—
315C23#611⋊C311yescharacteristic
316C23C22×AGL(1,ℤ/9)11yescharacteristic
317Q8—181no—
318D4—22no—
319D4—181no—
320D4—182no—
321D4—182no—
322D4—182no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—92no—
327C2×C4—182no—
328C2×C4—181no—
329C2×C4—182no—
330C2×C4#611⋊C311yescharacteristic
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—31no—
334C2×C3—41no—
335C2×C3—61no—
336C2×C3—62no—
337C2×C3—121no—
338C2×C3—362no—
339C2×C3C3×#93711yescharacteristic
340D3—122no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—182no—
345C22—362no—
346C22—721no—
347C22D4×AGL(1,ℤ/9)12yes—
348C22C2×#611⋊C311yescharacteristic
349C4—21no—
350C4—41no—
351C4—41no—
352C4—182no—
353C4—362no—
354C3—31no—
355C3C3×#505311yescharacteristic
356C2—21no—
357C2—22no—
358C2—41no—
359C2—362no—
360C2#63499⋊C311yescharacteristic
subgroupG/H
centerC2#63499⋊C3
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
Fitting3−1+2×#123C2
socleC2×C3C3×#937
prime radical#63710⋊C3C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#458
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#123C2semidirect
3. #63491⋊C3C2semidirect
4. #63393⋊C3C2semidirect
5. #63365⋊C3C2semidirect
7. #10588733C3semidirect
9. C22×C4×3−1+2C22semidirect
17. C22⋊C4×3−1+2C22semidirect
18. #3676⋊C3C22semidirect
28. C9×#123C2×C3semidirect
29. #63491C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
68. C22×C4×C9C22×C3semidirect
79. C9×C22⋊C4C22×C3semidirect
80. #3676C22×C3semidirect
217. #123AGL(1,ℤ/9)semidirect
220. 3−1+2#458semidirect
277. C22×C4C2×AGL(1,ℤ/9)semidirect
283. C22⋊C4C2×AGL(1,ℤ/9)semidirect
312. C9C3×#458semidirect

Maximal subgroups

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

subgroupindex
#63710⋊C32
3−1+2×#1232
#63491⋊C32
#63393⋊C32
#63365⋊C32
C3×#50533
#105887333