Finite Group Databasebeta

← #10588712⋊C3 #13142815

Subgroups — #10588712⋊C3 #13142815

Order 1728.

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

normal subgroups only

This group has 659 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, complemented
23−1+2×#120C211yesmaximal, characteristic, complemented
3#11721704C212yesmaximal, complemented
4#63393⋊C3C211yesmaximal, characteristic, complemented
5#63365⋊C3C212yesmaximal, complemented
6C3×#5032—31nomaximal
7#10588712C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic, complemented
9C2×D4×3−1+2—21no—
10C2×D4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C2×#611⋊C3—22no—
13C2×#611⋊C3C2212yes—
14#3699⋊C3—22no—
15D4×AGL(1,ℤ/9)—22no—
16#3697⋊C3—21no—
17C22⋊C4×3−1+2—21no—
18C22⋊C4×3−1+2C2211yescharacteristic, complemented
19#3676⋊C3—22no—
20#3676⋊C3C2212yescomplemented
21C3×#1004—31no—
22C32×#120D311yescharacteristic
23C3×#953—32no—
24#63710C2×C311yescharacteristic, complemented
25C3×#834—31no—
26C3×#806—32no—
27C9×#120—21no—
28C9×#120C2×C311yescharacteristic, complemented
29#63515C2×C312yescomplemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×3−1+2—21no—
33C23×3−1+2—22no—
34C23×3−1+2—42no—
35C23×3−1+2D411yescharacteristic, complemented
36C23×3−1+2C2311yescharacteristic
37C22×AGL(1,ℤ/9)—21no—
38C22×AGL(1,ℤ/9)D412yescomplemented
39D4×3−1+2—22no—
40D4×3−1+2—42no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2D411yescharacteristic, complemented
43#611⋊C3—22no—
44#611⋊C3—22no—
45#611⋊C3—22no—
46#611⋊C3—44no—
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4—22no—
49C2×3−1+2⋊C4D412yescomplemented
50C4⋊AGL(1,ℤ/9)—21no—
51C4×AGL(1,ℤ/9)—22no—
523−1+2⋊Q8—21no—
53C3×B(4,𝔽2)—91no—
54#5032—31no—
55C24×C32C2×D311yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C3×#67—32no—
59C2×C3×#67—61no—
60C2×C3×#67—62no—
61C3×#237—62no—
62C3×D3×D4—62no—
63C3×#235—61no—
64C24×C9—21no—
65C24×C9C22×C311yescharacteristic, complemented
66C32×C22⋊C4—21no—
67C32×C22⋊C4C2×D311yescharacteristic
68C3×#217—32no—
69C3×#217—62no—
70C2×D4×C9—22no—
71C2×D4×C9—21no—
72C2×D4×C9—21no—
73C2×D4×C9C22×C311yescharacteristic
74C2×#611—21no—
75C2×#611—22no—
76C2×#611C22×C312yes—
77#3699—22no—
78D4×D9—22no—
79#3697—21no—
80C9×C22⋊C4—22no—
81C9×C22⋊C4—21no—
82C9×C22⋊C4—21no—
83C9×C22⋊C4C22×C311yescharacteristic, complemented
84#3676—22no—
85#3676C22×C312yescomplemented
86C22×3−1+2—22no—
87C22×3−1+2—24no—
88C22×3−1+2—21no—
89C22×3−1+2—41no—
90C22×3−1+2—41no—
91C22×3−1+2—42no—
92C22×3−1+2—41no—
93C22×3−1+2—41no—
94C22×3−1+2C2×D412yes—
95C22×3−1+2C2×D411yescharacteristic
96C2×AGL(1,ℤ/9)—22no—
97C2×AGL(1,ℤ/9)—42no—
98C2×AGL(1,ℤ/9)—81no—
99C4×3−1+2—21no—
100C4×3−1+2—41no—
1013−1+2⋊C4—22no—
1023−1+2⋊C4—42no—
103C3×H(5,𝔽2)—91no—
104#1004—31no—
105C3×#120—31no—
106C3×#120—92no—
107C3×#120C3×D311yescharacteristic
108#953—32no—
109C3×#99—92no—
110C3×#99—91no—
111#834—31no—
112#806—32no—
113C23×C32—22no—
114C23×C32—21no—
115C23×C32—42no—
116C23×C32#6711yescharacteristic
117C23×C32C22×D311yescharacteristic
118C22×C3×D3—32no—
119C22×C3×D3—61no—
120C32×D4—22no—
121C32×D4—42no—
122C2×C32×C4—21no—
123C2×C32×C4#6711yescharacteristic
124C3×#67—62no—
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—124no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C3×D12—61no—
132C3×C4×D3—62no—
133C3×Dic6—61no—
134C23×C9—22no—
135C23×C9—22no—
136C23×C9—21no—
137C23×C9—21no—
138C23×C9—21no—
139C23×C9—42no—
140C23×C9—42no—
141C23×C9—44no—
142C23×C9C3×D411yescharacteristic, complemented
143C23×C9C23×C311yescharacteristic
144C22×D9—21no—
145C22×D9C3×D412yescomplemented
146D4×C9—22no—
147D4×C9—42no—
148D4×C9—42no—
149D4×C9—44no—
150C2×C4×C9—22no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9C3×D411yescharacteristic, complemented
154#611—22no—
155#611—22no—
156#611—22no—
157#611—44no—
158C2×Dic9—21no—
159C2×Dic9—22no—
160C2×Dic9C3×D412yescomplemented
161D36—21no—
162C4×D9—22no—
163Dic18—21no—
164B(4,𝔽2)—91noSylow
165C2×3−1+2—21no—
166C2×3−1+2—22no—
167C2×3−1+2—42no—
168C2×3−1+2—41no—
169C2×3−1+2#12011yescharacteristic
170AGL(1,ℤ/9)—42no—
171C24×C3—31no—
172C24×C3C2×C3×D311yescharacteristic
173C3×#41—181no—
174C3×#41—182no—
175C2×C3×D4—21no—
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×C3×D4—92no—
179C2×C3×D4—182no—
180C2×C3×D4—181no—
181C2×C3×D4—182no—
182C2×C3×D4C2×C3×D311yescharacteristic
183C2×#67—32no—
184C2×#67—61no—
185C2×#67—62no—
186#237—62no—
187D3×D4—62no—
188#235—61no—
189C3×C22⋊C4—21no—
190C3×C22⋊C4—31no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—92no—
193C3×C22⋊C4—182no—
194C3×C22⋊C4C2×C3×D311yescharacteristic
195#217—32no—
196#217—62no—
197C22×C32—24no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—42no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32D3×D412yes—
206C22×C32C2×#6711yescharacteristic
207C2×C3×D3—62no—
208C2×C3×D3—122no—
209C2×C3×D3—241no—
210C32×C4—21no—
211C32×C4—41no—
212C3×Dic3—62no—
213C3×Dic3—122no—
214C22×C9—24no—
215C22×C9—24no—
216C22×C9—22no—
217C22×C9—21no—
218C22×C9—22no—
219C22×C9—21no—
220C22×C9—41no—
221C22×C9—44no—
222C22×C9—41no—
223C22×C9—41no—
224C22×C9—42no—
225C22×C9—42no—
226C22×C9—42no—
227C22×C9—41no—
228C22×C9—42no—
229C22×C9—42no—
230C22×C9—44no—
231C22×C9—41no—
232C22×C9C2×C3×D412yes—
233C22×C9C2×C3×D411yescharacteristic
234C2×D9—22no—
235C2×D9—42no—
236C2×D9—81no—
237C4×C9—21no—
238C4×C9—41no—
239C4×C9—42no—
240C4×C9—41no—
241Dic9—22no—
242Dic9—42no—
243H(5,𝔽2)—91no—
244#120—92no—
245#120AGL(1,ℤ/9)11yescharacteristic, complemented
246#99—91no—
247#99—92no—
2483−1+2B(4,𝔽2)11yescharacteristic, complemented, Sylow
249C23×C3—22no—
250C23×C3—21no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—42no—
254C23×C3—61no—
255C23×C3—62no—
256C23×C3—92no—
257C23×C3—122no—
258C23×C3—181no—
259C23×C3C3×#6711yescharacteristic
260C23×C3C22×C3×D311yescharacteristic
261C22×D3—32no—
262C22×D3—61no—
263C3×Q8—181no—
264C3×D4—22no—
265C3×D4—42no—
266C3×D4—62no—
267C3×D4—122no—
268C3×D4—181no—
269C3×D4—182no—
270C3×D4—182no—
271C3×D4—182no—
272C3×D4—364no—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—92no—
277C2×C3×C4—182no—
278C2×C3×C4—181no—
279C2×C3×C4—182no—
280C2×C3×C4C3×#6711yescharacteristic
281#67—62no—
282#67—62no—
283#67—62no—
284#67—124no—
285C2×Dic3—32no—
286C2×Dic3—61no—
287C2×Dic3—62no—
288D12—61no—
289C4×D3—62no—
290Dic6—61no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32#93711yescharacteristic
296C3×D3—122no—
297C2×C9—22no—
298C2×C9—22no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—42no—
302C2×C9—44no—
303C2×C9—41no—
304C2×C9—42no—
305C2×C9—81no—
306C2×C9C3×#12011yescharacteristic
307D9—42no—
308C24C2×AGL(1,ℤ/9)11yescharacteristic, complemented
309#41—182no—
310#41—181no—
311C2×D4—21no—
312C2×D4—92no—
313C2×D4—182no—
314C2×D4—181no—
315C2×D4—182no—
316C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—92no—
319C22⋊C4—182no—
320C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
321C22×C3—24no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—32no—
325C22×C3—31no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—61no—
332C22×C3—62no—
333C22×C3—64no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—122no—
339C22×C3—182no—
340C22×C3—362no—
341C22×C3—721no—
342C22×C3C3×D3×D412yes—
343C22×C3C2×C3×#6711yescharacteristic
344C2×D3—62no—
345C2×D3—122no—
346C2×D3—241no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—61no—
350C3×C4—121no—
351C3×C4—182no—
352C3×C4—362no—
353Dic3—62no—
354Dic3—122no—
355C32#503211yescharacteristic
356C9—21no—
357C9C3×B(4,𝔽2)11yescharacteristic, complemented
358C23—22no—
359C23—21no—
360C23—42no—
361C23—92no—
362C23—181no—
363C23#611⋊C311yescharacteristic, complemented
364C23C22×AGL(1,ℤ/9)11yescharacteristic
365Q8—181no—
366D4—22no—
367D4—42no—
368D4—181no—
369D4—182no—
370D4—182no—
371D4—182no—
372D4—364no—
373C2×C4—21no—
374C2×C4—92no—
375C2×C4—182no—
376C2×C4—181no—
377C2×C4—182no—
378C2×C4#611⋊C311yescharacteristic, complemented
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—31no—
382C2×C3—41no—
383C2×C3—42no—
384C2×C3—61no—
385C2×C3—62no—
386C2×C3—122no—
387C2×C3—121no—
388C2×C3—362no—
389C2×C3C3×#93711yescharacteristic
390D3—122no—
391C22—22no—
392C22—24no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—182no—
400C22—362no—
401C22—721no—
402C22D4×AGL(1,ℤ/9)12yes—
403C22C2×#611⋊C311yescharacteristic
404C4—21no—
405C4—41no—
406C4—182no—
407C4—362no—
408C3—31no—
409C3C3×#503211yescharacteristic
410C2—21no—
411C2—22no—
412C2—42no—
413C2—41no—
414C2—362no—
415C2#63499⋊C311yescharacteristic
subgroupG/H
centerC2#63499⋊C3
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
Fitting3−1+2×#120C2
socleC2×C3C3×#937
hypercenter#120AGL(1,ℤ/9)
hypocenterC9C3×B(4,𝔽2)
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #63710⋊C3C2semidirect
2. 3−1+2×#120C2semidirect
3. #11721704C2semidirect
4. #63393⋊C3C2semidirect
5. #63365⋊C3C2semidirect
7. #10588712C3semidirect
8. C24×3−1+2C22semidirect
18. C22⋊C4×3−1+2C22semidirect
20. #3676⋊C3C22semidirect
24. #63710C2×C3semidirect
28. C9×#120C2×C3semidirect
29. #63515C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
35. C23×3−1+2D4semidirect
38. C22×AGL(1,ℤ/9)D4semidirect
42. C2×C4×3−1+2D4semidirect
49. C2×3−1+2⋊C4D4semidirect
65. C24×C9C22×C3semidirect
83. C9×C22⋊C4C22×C3semidirect
85. #3676C22×C3semidirect
142. C23×C9C3×D4semidirect
145. C22×D9C3×D4semidirect
153. C2×C4×C9C3×D4semidirect
160. C2×Dic9C3×D4semidirect
245. #120AGL(1,ℤ/9)semidirect
248. 3−1+2B(4,𝔽2)semidirect
308. C24C2×AGL(1,ℤ/9)semidirect
320. C22⋊C4C2×AGL(1,ℤ/9)semidirect
357. C9C3×B(4,𝔽2)semidirect
363. C23#611⋊C3semidirect
378. C2×C4#611⋊C3semidirect

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×#1202
#117217042
#63393⋊C32
#63365⋊C32
C3×#50323
#105887123