Finite Group Databasebeta

← #10588883⋊C3 #13143506

Subgroups — #10588883⋊C3 #13143506

Order 1728.

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

normal subgroups only

This group has 486 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#11722641C211yesmaximal, characteristic
23−1+2×#137C211yesmaximal, characteristic, complemented
3#63509⋊C3C211yesmaximal, characteristic, complemented
4#63505⋊C3C211yesmaximal, characteristic, complemented
5#63396⋊C3C211yesmaximal, characteristic, complemented
6#63384⋊C3C211yesmaximal, characteristic, complemented
7#63383⋊C3C211yesmaximal, characteristic, complemented
8C3×#5202—31nomaximal
9#10588883C311yesmaximal, characteristic, complemented
10#41×3−1+2C2211yescharacteristic
11C2×Q8×3−1+2C2211yescharacteristic, complemented
12#88724—21no—
13D4×AGL(1,ℤ/9)—22no—
14D4×AGL(1,ℤ/9)—21no—
15#3697⋊C3—21no—
16#3697⋊C3C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)—21no—
18C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
19Q16×3−1+2—21no—
20SD16×3−1+2—21no—
21M16×3−1+2C2211yescharacteristic, complemented
22#3675⋊C3—21no—
23#3674⋊C3—21no—
24(C2×AGL(1,ℤ/9))⋊C4—22no—
25C4×3−1+2⋊C4C2211yescharacteristic, complemented
26#3667⋊C3C2211yescharacteristic, complemented
27C3×#1009—31no—
28C32×#137D311yescharacteristic
29C3×#947—31no—
30C3×#942—31no—
31#63715C2×C311yescharacteristic
32C3×#837—31no—
33C3×#825—31no—
34C3×#823—31no—
35C9×#137—21no—
36C9×#137C2×C311yescharacteristic, complemented
37#63509C2×C311yescharacteristic, complemented
38#63505C2×C311yescharacteristic, complemented
39#63396C2×C311yescharacteristic, complemented
40#63384C2×C311yescharacteristic, complemented
41#63383C2×C311yescharacteristic, complemented
42C22×AGL(1,ℤ/9)—22no—
43C22×AGL(1,ℤ/9)D412yes—
44Q8×3−1+2—21no—
45Q8×3−1+2—21no—
46Q8×3−1+2D411yescharacteristic
47D4×3−1+2—21no—
48D4×3−1+2D411yescharacteristic
49C2×C4×3−1+2—21no—
50C2×C4×3−1+2—21no—
51C2×C4×3−1+2C2311yescharacteristic
52#611⋊C3—22no—
53#611⋊C3—21no—
54C2×3−1+2⋊C4—21no—
55C4⋊AGL(1,ℤ/9)—22no—
56C4⋊AGL(1,ℤ/9)—21no—
57C4⋊AGL(1,ℤ/9)—21no—
58C4⋊AGL(1,ℤ/9)D411yescharacteristic
59C4×AGL(1,ℤ/9)—22no—
60C4×AGL(1,ℤ/9)—21no—
613−1+2⋊Q8D411yescharacteristic
62C8×3−1+2—21no—
633−1+2⋊C8—21no—
64C3×#455—91no—
65#5202—31no—
66C32×#41C2×D311yescharacteristic
67C2×C32×Q8C2×D311yescharacteristic
68C3×#239—61no—
69C3×D3×D4—62no—
70C3×D3×D4—61no—
71C3×#235—31no—
72C3×#235—61no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C32×Q16—21no—
76C32×SD16—21no—
77C32×M16C2×D311yescharacteristic
78C3×#216—61no—
79C3×#215—61no—
80C3×(C2×D3)⋊C4—62no—
81C3×C4×Dic3—31no—
82C3×#208—31no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85C2×Q8×C9—21no—
86C2×Q8×C9C22×C311yescharacteristic, complemented
87#3701—21no—
88D4×D9—22no—
89D4×D9—21no—
90#3697—21no—
91#3697C22×C311yescharacteristic
92C2×D36—21no—
93C2×D36C22×C311yescharacteristic
94C9×Q16—22no—
95C9×Q16—21no—
96C9×SD16—22no—
97C9×SD16—21no—
98C9×M16—21no—
99C9×M16C22×C311yescharacteristic, complemented
100#3675—21no—
101#3674—21no—
102(C2×D9)⋊C4—22no—
103C4×Dic9C22×C311yescharacteristic, complemented
104#3667C22×C311yescharacteristic, complemented
105C22×3−1+2—21no—
106C22×3−1+2C2×D411yescharacteristic
107C2×AGL(1,ℤ/9)—22no—
108C2×AGL(1,ℤ/9)—21no—
109C2×AGL(1,ℤ/9)—42no—
110C2×AGL(1,ℤ/9)—82no—
111C4×3−1+2—21no—
112C4×3−1+2—41no—
113C4×3−1+2C2×D411yescharacteristic
114C4×3−1+2C2×D411yescharacteristic
1153−1+2⋊C4—21no—
1163−1+2⋊C4—41no—
117C3×H(5,𝔽2)—91no—
118#1009—31no—
119C3×#137—31no—
120C3×#137—91no—
121C3×#137C3×D311yescharacteristic
122C3×#124—91no—
123#947—31no—
124#942—31no—
125C3×#104—91no—
126C3×#104—91no—
127C3×#100—91no—
128#837—31no—
129#825—31no—
130#823—31no—
131C22×C3×D3—32no—
132C22×C3×D3—62no—
133C32×Q8—21no—
134C32×Q8—21no—
135C32×Q8#6711yescharacteristic
136C32×D4—21no—
137C32×D4#6711yescharacteristic
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4C22×D311yescharacteristic
141C3×#67—62no—
142C3×#67—61no—
143C2×C3×Dic3—61no—
144C3×D12—31no—
145C3×D12—62no—
146C3×D12—61no—
147C3×D12—61no—
148C3×C4×D3—62no—
149C3×C4×D3—61no—
150C3×Dic6—31no—
151C22×D9—22no—
152C22×D9C3×D412yes—
153C32×C8—21no—
154C3×C3⋊C8—61no—
155Q8×C9—22no—
156Q8×C9—21no—
157Q8×C9—21no—
158Q8×C9—21no—
159Q8×C9—41no—
160Q8×C9C3×D411yescharacteristic
161D4×C9—21no—
162D4×C9—21no—
163D4×C9—41no—
164D4×C9C3×D411yescharacteristic
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—41no—
169C2×C4×C9—41no—
170C2×C4×C9C23×C311yescharacteristic
171#611—22no—
172#611—21no—
173C2×Dic9—21no—
174D36—22no—
175D36—21no—
176D36—21no—
177D36C3×D411yescharacteristic
178C4×D9—22no—
179C4×D9—21no—
180Dic18C3×D411yescharacteristic
181C8×C9—22no—
182C8×C9—21no—
183C9⋊C8—21no—
184#455—91noSylow
185C2×3−1+2—21no—
186C2×3−1+2—41no—
187C2×3−1+2#12011yescharacteristic
188AGL(1,ℤ/9)—42no—
189AGL(1,ℤ/9)—41no—
190C3×#41—31no—
191C3×#41—91no—
192C3×#41—181no—
193C3×#41—181no—
194C3×#41C2×C3×D311yescharacteristic
195C2×C3×Q8—31no—
196C2×C3×Q8C2×C3×D311yescharacteristic
197C2×C3×D4—91no—
198C2×C3×D4—181no—
199C2×C3×D4—181no—
200C2×C3×D4—182no—
201#239—61no—
202D3×D4—62no—
203D3×D4—61no—
204#235—31no—
205#235—61no—
206C2×D12—31no—
207C2×D12—61no—
208C3×Q16—21no—
209C3×Q16—61no—
210C3×Q16—181no—
211C3×SD16—21no—
212C3×SD16—61no—
213C3×SD16—181no—
214C3×M16—31no—
215C3×M16—91no—
216C3×M16C2×C3×D311yescharacteristic
217C3×C22⋊C4—182no—
218C3×C42—91no—
219#216—61no—
220#215—61no—
221(C2×D3)⋊C4—62no—
222C4×Dic3—31no—
223#208—31no—
224C22×C32—21no—
225C22×C32D3×D411yescharacteristic
226C2×C3×D3—61no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C2×C3×D3—242no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4D3×D411yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3×Dic3—61no—
235C3×Dic3—121no—
236C22×C9—21no—
237C22×C9—21no—
238C22×C9—41no—
239C22×C9C2×C3×D411yescharacteristic
240C2×D9—22no—
241C2×D9—21no—
242C2×D9—42no—
243C2×D9—82no—
244C4×C9—21no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—41no—
248C4×C9—42no—
249C4×C9—41no—
250C4×C9C2×C3×D411yescharacteristic
251C4×C9C2×C3×D411yescharacteristic
252Dic9—21no—
253Dic9—41no—
254H(5,𝔽2)—91no—
255#137—91no—
256#137AGL(1,ℤ/9)11yescharacteristic, complemented
257#124—91no—
258#104—91no—
259#104—91no—
260#100—91no—
2613−1+2#45511yescharacteristic, complemented, Sylow
262C23×C3—92no—
263C23×C3—182no—
264C22×D3—32no—
265C22×D3—62no—
266C3×Q8—21no—
267C3×Q8—21no—
268C3×Q8—31no—
269C3×Q8—61no—
270C3×Q8—61no—
271C3×Q8—91no—
272C3×Q8C3×#6711yescharacteristic
273C3×D4—21no—
274C3×D4—31no—
275C3×D4—61no—
276C3×D4—91no—
277C3×D4—181no—
278C3×D4—181no—
279C3×D4—182no—
280C3×D4—181no—
281C3×D4—182no—
282C3×D4C3×#6711yescharacteristic
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—31no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—181no—
289C2×C3×C4—182no—
290C2×C3×C4—181no—
291C2×C3×C4C22×C3×D311yescharacteristic
292#67—62no—
293#67—61no—
294C2×Dic3—61no—
295D12—31no—
296D12—62no—
297D12—61no—
298D12—61no—
299C4×D3—62no—
300C4×D3—61no—
301Dic6—31no—
302C3×C8—21no—
303C3×C8—61no—
304C3×C8—181no—
305C3⋊C8—61no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32#93711yescharacteristic
309C3×D3—122no—
310C3×D3—121no—
311C2×C9—21no—
312C2×C9—21no—
313C2×C9—41no—
314C2×C9—41no—
315C2×C9—81no—
316C2×C9C3×#12011yescharacteristic
317D9—42no—
318D9—41no—
319#41—91no—
320#41—181no—
321#41—181no—
322#41C2×AGL(1,ℤ/9)11yescharacteristic
323C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
324C2×D4—91no—
325C2×D4—181no—
326C2×D4—182no—
327C2×D4—181no—
328Q16—21no—
329Q16—181no—
330SD16—21no—
331SD16—181no—
332M16—91no—
333M16C2×AGL(1,ℤ/9)11yescharacteristic, complemented
334C22⋊C4—182no—
335C42—91no—
336C22×C3—21no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—181no—
340C22×C3—182no—
341C22×C3—362no—
342C22×C3—722no—
343C22×C3C3×D3×D411yescharacteristic
344C2×D3—61no—
345C2×D3—62no—
346C2×D3—122no—
347C2×D3—242no—
348C3×C4—21no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—41no—
352C3×C4—61no—
353C3×C4—121no—
354C3×C4—181no—
355C3×C4—361no—
356C3×C4C3×D3×D411yescharacteristic
357C3×C4C2×C3×#6711yescharacteristic
358Dic3—61no—
359Dic3—121no—
360C32#520211yescharacteristic
361C9—21no—
362C9C3×#45511yescharacteristic, complemented
363C23—92no—
364C23—182no—
365Q8—21no—
366Q8—21no—
367Q8—91no—
368Q8#611⋊C311yescharacteristic
369D4—21no—
370D4—91no—
371D4—182no—
372D4—182no—
373D4—181no—
374D4—181no—
375D4—181no—
376D4#611⋊C311yescharacteristic
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—182no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
383C8—21no—
384C8—181no—
385C2×C3—21no—
386C2×C3—31no—
387C2×C3—41no—
388C2×C3—61no—
389C2×C3—121no—
390C2×C3—362no—
391C2×C3—361no—
392C2×C3C3×#93711yescharacteristic
393D3—122no—
394D3—121no—
395C22—21no—
396C22—181no—
397C22—182no—
398C22—362no—
399C22—722no—
400C22D4×AGL(1,ℤ/9)11yescharacteristic
401C4—21no—
402C4—41no—
403C4—181no—
404C4—361no—
405C4D4×AGL(1,ℤ/9)11yescharacteristic
406C4C2×#611⋊C311yescharacteristic
407C3—31no—
408C3C3×#520211yescharacteristic
409C2—21no—
410C2—41no—
411C2—362no—
412C2—361no—
413C2#63499⋊C311yescharacteristic
subgroupG/H
centerC2#63499⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#137C2
socleC2×C3C3×#937
prime radical#11722641C2
hypercenter#137AGL(1,ℤ/9)
hypocenterC9C3×#455
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#137C2semidirect
3. #63509⋊C3C2semidirect
4. #63505⋊C3C2semidirect
5. #63396⋊C3C2semidirect
6. #63384⋊C3C2semidirect
7. #63383⋊C3C2semidirect
9. #10588883C3semidirect
11. C2×Q8×3−1+2C22semidirect
21. M16×3−1+2C22semidirect
25. C4×3−1+2⋊C4C22semidirect
26. #3667⋊C3C22semidirect
36. C9×#137C2×C3semidirect
37. #63509C2×C3semidirect
38. #63505C2×C3semidirect
39. #63396C2×C3semidirect
40. #63384C2×C3semidirect
41. #63383C2×C3semidirect
86. C2×Q8×C9C22×C3semidirect
99. C9×M16C22×C3semidirect
103. C4×Dic9C22×C3semidirect
104. #3667C22×C3semidirect
256. #137AGL(1,ℤ/9)semidirect
261. 3−1+2#455semidirect
323. C2×Q8C2×AGL(1,ℤ/9)semidirect
333. M16C2×AGL(1,ℤ/9)semidirect
362. C9C3×#455semidirect

Maximal subgroups

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

subgroupindex
#117226412
3−1+2×#1372
#63509⋊C32
#63505⋊C32
#63396⋊C32
#63384⋊C32
#63383⋊C32
C3×#52023
#105888833