Finite Group Databasebeta

← #10588453⋊C3 #13141822

Subgroups — #10588453⋊C3 #13141822

Order 1728.

619 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×D4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C211yesmaximal, characteristic, complemented
3C2×D72⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#102C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
7#63369⋊C3C211yesmaximal, characteristic, complemented
8C3×#4773—31nomaximal
9#10588453C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)—21no—
11C2×D4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—21no—
13D4×AGL(1,ℤ/9)—22no—
14D4×AGL(1,ℤ/9)—22no—
15C2×C4⋊AGL(1,ℤ/9)—21no—
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
18C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
19C2×C8×3−1+2C2211yescharacteristic, complemented
20C4⋊C4×3−1+2C2211yescharacteristic, complemented
21#3673⋊C3—22no—
22(C2×AGL(1,ℤ/9))⋊C4—21no—
23C2×3−1+2⋊C8C2211yescharacteristic, complemented
24D72⋊C3—22no—
25C2×C3×D3×D4—31no—
26C2×C3×#213—31no—
27C2×C3×D24—31no—
28C3×#895—31no—
29C2×D4×D9C2×C311yescharacteristic, complemented
30C32×#102D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32C3×#809—31no—
33C2×#3673C2×C311yescharacteristic, complemented
34C2×D72C2×C311yescharacteristic, complemented
35#63457C2×C311yescharacteristic, complemented
36C9×#102—21no—
37C9×#102C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic, complemented
39#63369C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C22×AGL(1,ℤ/9)—24no—
42C22×AGL(1,ℤ/9)—21no—
43C22×AGL(1,ℤ/9)—21no—
44C22×AGL(1,ℤ/9)—42no—
45C22×AGL(1,ℤ/9)—42no—
46C22×AGL(1,ℤ/9)D411yescharacteristic
47D4×3−1+2—21no—
48D4×3−1+2D412yescomplemented
49C2×C4×3−1+2—21no—
50C2×C4×3−1+2C2311yescharacteristic
51#611⋊C3—24no—
52C2×3−1+2⋊C4D411yescharacteristic
53C4⋊AGL(1,ℤ/9)—21no—
54C4⋊AGL(1,ℤ/9)—21no—
55C4⋊AGL(1,ℤ/9)—21no—
56C4⋊AGL(1,ℤ/9)—42no—
57C4⋊AGL(1,ℤ/9)D412yescomplemented
58C4×AGL(1,ℤ/9)—21no—
59C4×AGL(1,ℤ/9)—21no—
60C8×3−1+2—21no—
613−1+2⋊C8—21no—
62C3×#447—91no—
63#4773—31no—
64C23×C3×D3—61no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C3×#67—61no—
67C3×D3×D4—62no—
68C3×D3×D4—62no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—31no—
73C23×D9—21no—
74C2×C32×C8C2×D311yescharacteristic
75C32×C4⋊C4C2×D311yescharacteristic
76C3×#213—62no—
77C3×(C2×D3)⋊C4—61no—
78C2×C3×C3⋊C8—31no—
79C3×D24—62no—
80C2×D4×C9—21no—
81C2×D4×C9C22×C311yescharacteristic
82C2×#611—21no—
83D4×D9—22no—
84D4×D9—22no—
85C2×D36—21no—
86C2×D36C22×C311yescharacteristic
87C2×D36C22×C311yescharacteristic, complemented
88C2×C4×D9C22×C311yescharacteristic
89C2×C8×C9—21no—
90C2×C8×C9C22×C311yescharacteristic, complemented
91C9×C4⋊C4—21no—
92C9×C4⋊C4C22×C311yescharacteristic, complemented
93#3673—22no—
94(C2×D9)⋊C4—21no—
95C2×C9⋊C8C22×C311yescharacteristic, complemented
96D72—22no—
97C22×3−1+2—22no—
98C22×3−1+2—41no—
99C22×3−1+2—41no—
100C22×3−1+2C2×D411yescharacteristic
101C2×AGL(1,ℤ/9)—21no—
102C2×AGL(1,ℤ/9)—21no—
103C2×AGL(1,ℤ/9)—22no—
104C2×AGL(1,ℤ/9)—48no—
105C2×AGL(1,ℤ/9)—41no—
106C2×AGL(1,ℤ/9)—41no—
107C2×AGL(1,ℤ/9)—41no—
108C2×AGL(1,ℤ/9)—41no—
109C2×AGL(1,ℤ/9)—41no—
110C2×AGL(1,ℤ/9)D812yescomplemented
111C4×3−1+2—41no—
112C4×3−1+2C2×D411yescharacteristic
113C4×3−1+2C2×D411yescharacteristic
1143−1+2⋊C4—21no—
115C22×C3×D4—91no—
116C2×D3×D4—31no—
117C2×C3×D8—91no—
118C2×C3×D8—91no—
119C3×#121—91no—
120C2×#213—31no—
121C2×D24—31no—
122#895—31no—
123C3×#102—31no—
124C3×#102—91no—
125C3×#102C3×D311yescharacteristic
126C3×C22⋊C8—91no—
127(C2×D3)⋊C8—31no—
128#809—31no—
129C23×C32—21no—
130C22×C3×D3—31no—
131C22×C3×D3—64no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—122no—
135C22×C3×D3—122no—
136C32×D4—21no—
137C32×D4D1212yes—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C3×#67—64no—
141C2×C3×Dic3—31no—
142C3×D12—32no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—122no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C23×C9—21no—
150C23×C9—41no—
151C22×D9—24no—
152C22×D9—21no—
153C22×D9—21no—
154C22×D9—42no—
155C22×D9—42no—
156C22×D9C3×D411yescharacteristic
157C32×C8—21no—
158C3×C3⋊C8—61no—
159D4×C9—21no—
160D4×C9—22no—
161D4×C9—41no—
162D4×C9C3×D412yescomplemented
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—41no—
166C2×C4×C9C23×C311yescharacteristic
167#611—24no—
168C2×Dic9C3×D411yescharacteristic
169D36—21no—
170D36—21no—
171D36—21no—
172D36—42no—
173D36C3×D412yescomplemented
174C4×D9—21no—
175C4×D9—21no—
176C8×C9—21no—
177C8×C9—41no—
178C9⋊C8—21no—
179#447—91noSylow
180C2×3−1+2—42no—
181C2×3−1+2#12011yescharacteristic
182C2×3−1+2C2×D811yescharacteristic
183C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
184AGL(1,ℤ/9)—22no—
185AGL(1,ℤ/9)—42no—
186AGL(1,ℤ/9)—81no—
187C24×C3—181no—
188C23×D3—61no—
189C2×C3×D4—31no—
190C2×C3×D4—91no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C2×C3×D4—181no—
194C2×C3×D4—182no—
195C2×C3×D4—182no—
196C2×C3×D4C2×C3×D311yescharacteristic
197C22×C3×C4—91no—
198C2×#67—61no—
199D3×D4—62no—
200D3×D4—62no—
201C2×D12—31no—
202C2×D12—31no—
203C2×D12—61no—
204C2×C4×D3—31no—
205C3×D8—182no—
206C3×D8—182no—
207C2×C3×C8—31no—
208C2×C3×C8—91no—
209C2×C3×C8C2×C3×D311yescharacteristic
210C3×C4⋊C4—31no—
211C3×C4⋊C4C2×C3×D311yescharacteristic
212C3×C22⋊C4—181no—
213#213—62no—
214(C2×D3)⋊C4—61no—
215C2×C3⋊C8—31no—
216D24—62no—
217C22×C32—22no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32D3×D411yescharacteristic
221C2×C3×D3—32no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—62no—
225C2×C3×D3—128no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—41no—
232C32×C4C2×D1211yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3×Dic3—61no—
235C22×C9—22no—
236C22×C9—21no—
237C22×C9—42no—
238C22×C9—41no—
239C22×C9—41no—
240C22×C9—81no—
241C22×C9—81no—
242C22×C9C2×C3×D411yescharacteristic
243C2×D9—21no—
244C2×D9—21no—
245C2×D9—22no—
246C2×D9—48no—
247C2×D9—41no—
248C2×D9—41no—
249C2×D9—41no—
250C2×D9—41no—
251C2×D9—41no—
252C2×D9C3×D812yescomplemented
253C4×C9—21no—
254C4×C9—21no—
255C4×C9—41no—
256C4×C9—81no—
257C4×C9C2×C3×D411yescharacteristic
258C4×C9C2×C3×D411yescharacteristic
259Dic9—21no—
260C22×D4—91no—
261C2×D8—91no—
262C2×D8—91no—
263#121—91no—
264#102—91no—
265#102AGL(1,ℤ/9)11yescharacteristic, complemented
266C22⋊C8—91no—
2673−1+2#44711yescharacteristic, complemented, Sylow
268C23×C3—21no—
269C23×C3—61no—
270C23×C3—91no—
271C23×C3—181no—
272C23×C3—181no—
273C23×C3—184no—
274C23×C3—362no—
275C23×C3—362no—
276C22×D3—31no—
277C22×D3—64no—
278C22×D3—61no—
279C22×D3—61no—
280C22×D3—122no—
281C22×D3—122no—
282C3×D4—21no—
283C3×D4—32no—
284C3×D4—61no—
285C3×D4—92no—
286C3×D4—181no—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—184no—
290C3×D4—362no—
291C3×D4C3×D1212yes—
292C2×C3×C4—21no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—181no—
297C2×C3×C4—181no—
298C2×C3×C4C22×C3×D311yescharacteristic
299#67—64no—
300C2×Dic3—31no—
301D12—32no—
302D12—61no—
303D12—61no—
304D12—61no—
305D12—122no—
306C4×D3—61no—
307C4×D3—61no—
308C3×C8—21no—
309C3×C8—61no—
310C3×C8—181no—
311C3⋊C8—61no—
312C2×C32—42no—
313C2×C32#88211yescharacteristic
314C2×C32D3×D811yescharacteristic
315C2×C32#91411yescharacteristic
316C3×D3—62no—
317C3×D3—122no—
318C3×D3—241no—
319C2×C9—21no—
320C2×C9—21no—
321C2×C9—21no—
322C2×C9—42no—
323C2×C9—82no—
324C2×C9C3×#12011yescharacteristic
325C2×C9C2×C3×D811yescharacteristic
326C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
327D9—22no—
328D9—42no—
329D9—81no—
330C24—181no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—181no—
334C2×D4—181no—
335C2×D4—182no—
336C2×D4—182no—
337C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
338C22×C4—91no—
339D8—182no—
340D8—182no—
341C2×C8—91no—
342C2×C8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
343C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
344C22⋊C4—181no—
345C22×C3—22no—
346C22×C3—31no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—62no—
350C22×C3—92no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3—182no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3—361no—
360C22×C3—361no—
361C22×C3—368no—
362C22×C3C3×D3×D411yescharacteristic
363C2×D3—32no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—62no—
367C2×D3—128no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—41no—
376C3×C4—121no—
377C3×C4—181no—
378C3×C4C2×C3×D1211yescharacteristic
379C3×C4C3×D3×D411yescharacteristic
380Dic3—61no—
381C32#477311yescharacteristic
382C9—21no—
383C9C3×#44711yescharacteristic, complemented
384C23—21no—
385C23—91no—
386C23—184no—
387C23—181no—
388C23—181no—
389C23—362no—
390C23—362no—
391D4—21no—
392D4—92no—
393D4—184no—
394D4—181no—
395D4—181no—
396D4—181no—
397D4—362no—
398D4C4⋊AGL(1,ℤ/9)12yescomplemented
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
404C8—21no—
405C8—181no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—42no—
410C2×C3—122no—
411C2×C3—182no—
412C2×C3—362no—
413C2×C3—721no—
414C2×C3C3×#88211yescharacteristic
415C2×C3C3×D3×D811yescharacteristic
416C2×C3C3×#91411yescharacteristic
417D3—62no—
418D3—122no—
419D3—241no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—92no—
424C22—181no—
425C22—181no—
426C22—182no—
427C22—368no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22—361no—
433C22D4×AGL(1,ℤ/9)11yescharacteristic
434C4—41no—
435C4—181no—
436C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
437C4D4×AGL(1,ℤ/9)11yescharacteristic
438C3—31no—
439C3C3×#477311yescharacteristic
440C2—42no—
441C2—182no—
442C2—362no—
443C2—721no—
444C2#63444⋊C311yescharacteristic
445C2D8×AGL(1,ℤ/9)11yescharacteristic
446C2#63476⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#102C2
socleC22×C3C3×D3×D4
hypercenter#102AGL(1,ℤ/9)
hypocenterC9C3×#447
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×D4×AGL(1,ℤ/9)C2semidirect
2. C2×#3673⋊C3C2semidirect
3. C2×D72⋊C3C2semidirect
4. #63457⋊C3C2semidirect
5. 3−1+2×#102C2semidirect
6. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
7. #63369⋊C3C2semidirect
9. #10588453C3semidirect
17. C2×C4⋊AGL(1,ℤ/9)C22semidirect
19. C2×C8×3−1+2C22semidirect
20. C4⋊C4×3−1+2C22semidirect
23. C2×3−1+2⋊C8C22semidirect
29. C2×D4×D9C2×C3semidirect
33. C2×#3673C2×C3semidirect
34. C2×D72C2×C3semidirect
35. #63457C2×C3semidirect
37. C9×#102C2×C3semidirect
38. (C2×D9)⋊C8C2×C3semidirect
39. #63369C2×C3semidirect
48. D4×3−1+2D4semidirect
57. C4⋊AGL(1,ℤ/9)D4semidirect
87. C2×D36C22×C3semidirect
90. C2×C8×C9C22×C3semidirect
92. C9×C4⋊C4C22×C3semidirect
95. C2×C9⋊C8C22×C3semidirect
110. C2×AGL(1,ℤ/9)D8semidirect
162. D4×C9C3×D4semidirect
173. D36C3×D4semidirect
252. C2×D9C3×D8semidirect
265. #102AGL(1,ℤ/9)semidirect
267. 3−1+2#447semidirect
342. C2×C8C2×AGL(1,ℤ/9)semidirect
343. C4⋊C4C2×AGL(1,ℤ/9)semidirect
383. C9C3×#447semidirect
398. D4C4⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×D4×AGL(1,ℤ/9)2
C2×#3673⋊C32
C2×D72⋊C32
#63457⋊C32
3−1+2×#1022
(C2×AGL(1,ℤ/9))⋊C82
#63369⋊C32
C3×#47733
#105884533