Finite Group Databasebeta

← #10588505⋊C3 #13141993

Subgroups — #10588505⋊C3 #13141993

Order 1728.

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

normal subgroups only

This group has 478 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
2#63511⋊C3C211yesmaximal, characteristic, complemented
3#63471⋊C3C211yesmaximal, characteristic, complemented
4#63437⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#104C211yesmaximal, characteristic, complemented
6#63383⋊C3C211yesmaximal, characteristic, complemented
7#63364⋊C3C211yesmaximal, characteristic, complemented
8C3×#4825—31nomaximal
9#10588505C311yesmaximal, characteristic, complemented
10#41×3−1+2C2211yescharacteristic
11#88724—21no—
12D4×AGL(1,ℤ/9)—22no—
13D4×AGL(1,ℤ/9)—21no—
14#3697⋊C3—21no—
15#3697⋊C3C2211yescharacteristic
16C2×C4⋊AGL(1,ℤ/9)—21no—
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×3−1+2⋊Q8C2211yescharacteristic, complemented
19M16×3−1+2C2211yescharacteristic, complemented
20C42×3−1+2C2211yescharacteristic, complemented
21#3674⋊C3—21no—
22#3672⋊C3—21no—
23(C2×AGL(1,ℤ/9))⋊C4—22no—
24#3667⋊C3C2211yescharacteristic, complemented
25#3664⋊C3—21no—
26Dic36⋊C3—21no—
27C3×#1009—31no—
28C3×#951—31no—
29C3×#909—31no—
30C3×#875—31no—
31#63715C2×C311yescharacteristic
32C32×#104D311yescharacteristic
33C3×#823—31no—
34C3×#805—31no—
35#63511C2×C311yescharacteristic, complemented
36#63471C2×C311yescharacteristic, complemented
37#63437C2×C311yescharacteristic, complemented
38C9×#104—21no—
39C9×#104C2×C311yescharacteristic, complemented
40#63383C2×C311yescharacteristic, complemented
41#63364C2×C311yescharacteristic, complemented
42C22×AGL(1,ℤ/9)—22no—
43C22×AGL(1,ℤ/9)D412yes—
44Q8×3−1+2D411yescharacteristic
45D4×3−1+2—21no—
46D4×3−1+2D411yescharacteristic
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2C2311yescharacteristic
50#611⋊C3—22no—
51#611⋊C3—21no—
52C2×3−1+2⋊C4—21no—
53C4⋊AGL(1,ℤ/9)—22no—
54C4⋊AGL(1,ℤ/9)—21no—
55C4⋊AGL(1,ℤ/9)—21no—
56C4⋊AGL(1,ℤ/9)D411yescharacteristic
57C4×AGL(1,ℤ/9)—22no—
58C4×AGL(1,ℤ/9)—21no—
593−1+2⋊Q8—21no—
603−1+2⋊Q8—21no—
613−1+2⋊Q8D411yescharacteristic
62C8×3−1+2—21no—
633−1+2⋊C8—21no—
64C3×#455—91no—
65#4825—31no—
66C32×#41C2×D311yescharacteristic
67C3×#239—61no—
68C3×D3×D4—62no—
69C3×D3×D4—61no—
70C3×#235—31no—
71C3×#235—61no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—31no—
75C32×M16C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C3×#216—61no—
78C3×#214—61no—
79C3×(C2×D3)⋊C4—62no—
80C3×#208—31no—
81C3×Dic12—61no—
82C3×#204—61no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85#3701—21no—
86D4×D9—22no—
87D4×D9—21no—
88#3697—21no—
89#3697C22×C311yescharacteristic
90C2×D36—21no—
91C2×D36C22×C311yescharacteristic
92C2×Dic18C22×C311yescharacteristic, complemented
93C9×M16—21no—
94C9×M16C22×C311yescharacteristic, complemented
95C42×C9—21no—
96C42×C9C22×C311yescharacteristic, complemented
97#3674—21no—
98#3672—21no—
99(C2×D9)⋊C4—22no—
100#3667C22×C311yescharacteristic, complemented
101#3664—21no—
102Dic36—21no—
103C22×3−1+2—21no—
104C22×3−1+2C2×D411yescharacteristic
105C2×AGL(1,ℤ/9)—22no—
106C2×AGL(1,ℤ/9)—21no—
107C2×AGL(1,ℤ/9)—42no—
108C2×AGL(1,ℤ/9)—82no—
109C4×3−1+2—21no—
110C4×3−1+2—41no—
111C4×3−1+2C2×D411yescharacteristic
112C4×3−1+2C2×D411yescharacteristic
1133−1+2⋊C4—21no—
1143−1+2⋊C4—41no—
115C3×H(5,𝔽2)—91no—
116#1009—31no—
117C3×#137—91no—
118C3×#137—91no—
119C3×#124—91no—
120#951—31no—
121#909—31no—
122#875—31no—
123C3×#104—31no—
124C3×#104—91no—
125C3×#104C3×D311yescharacteristic
126C3×#100—91no—
127#823—31no—
128#805—31no—
129C22×C3×D3—32no—
130C22×C3×D3—62no—
131C32×Q8D1211yescharacteristic
132C32×D4—21no—
133C32×D4D1211yescharacteristic
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C3×#67—62no—
138C3×#67—61no—
139C2×C3×Dic3—61no—
140C3×D12—31no—
141C3×D12—62no—
142C3×D12—61no—
143C3×D12—61no—
144C3×C4×D3—62no—
145C3×C4×D3—61no—
146C3×Dic6—31no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C22×D9—22no—
150C22×D9C3×D412yes—
151C32×C8—21no—
152C3×C3⋊C8—61no—
153Q8×C9—21no—
154Q8×C9C3×D411yescharacteristic
155D4×C9—21no—
156D4×C9—21no—
157D4×C9—41no—
158D4×C9C3×D411yescharacteristic
159C2×C4×C9—21no—
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9—41no—
163C2×C4×C9—41no—
164C2×C4×C9C23×C311yescharacteristic
165#611—22no—
166#611—21no—
167C2×Dic9—21no—
168D36—22no—
169D36—21no—
170D36—21no—
171D36C3×D411yescharacteristic
172C4×D9—22no—
173C4×D9—21no—
174Dic18—21no—
175Dic18—21no—
176Dic18C3×D411yescharacteristic
177C8×C9—21no—
178C8×C9—41no—
179C9⋊C8—21no—
180#455—91noSylow
181C2×3−1+2—21no—
182C2×3−1+2—41no—
183C2×3−1+2#12011yescharacteristic
184AGL(1,ℤ/9)—42no—
185AGL(1,ℤ/9)—41no—
186C3×#41—31no—
187C3×#41—91no—
188C3×#41—181no—
189C3×#41—181no—
190C3×#41C2×C3×D311yescharacteristic
191C2×C3×Q8—91no—
192C2×C3×D4—91no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4—182no—
196#239—61no—
197D3×D4—62no—
198D3×D4—61no—
199#235—31no—
200#235—61no—
201C2×D12—31no—
202C2×D12—61no—
203C2×Dic6—31no—
204C3×Q16—181no—
205C3×Q16—181no—
206C3×SD16—181no—
207C3×SD16—181no—
208C3×M16—31no—
209C3×M16—91no—
210C3×M16C2×C3×D311yescharacteristic
211C3×C22⋊C4—182no—
212C3×C42—31no—
213C3×C42C2×C3×D311yescharacteristic
214#216—61no—
215#214—61no—
216(C2×D3)⋊C4—62no—
217#208—31no—
218Dic12—61no—
219#204—61no—
220C22×C32—21no—
221C22×C32D3×D411yescharacteristic
222C2×C3×D3—61no—
223C2×C3×D3—62no—
224C2×C3×D3—122no—
225C2×C3×D3—242no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4C2×D1211yescharacteristic
229C32×C4D3×D411yescharacteristic
230C3×Dic3—61no—
231C3×Dic3—121no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—41no—
235C22×C9C2×C3×D411yescharacteristic
236C2×D9—22no—
237C2×D9—21no—
238C2×D9—42no—
239C2×D9—82no—
240C4×C9—21no—
241C4×C9—21no—
242C4×C9—21no—
243C4×C9—41no—
244C4×C9—41no—
245C4×C9—42no—
246C4×C9C2×C3×D411yescharacteristic
247C4×C9C2×C3×D411yescharacteristic
248Dic9—21no—
249Dic9—41no—
250H(5,𝔽2)—91no—
251#137—91no—
252#137—91no—
253#124—91no—
254#104—91no—
255#104AGL(1,ℤ/9)11yescharacteristic, complemented
256#100—91no—
2573−1+2#45511yescharacteristic, complemented, Sylow
258C23×C3—92no—
259C23×C3—182no—
260C22×D3—32no—
261C22×D3—62no—
262C3×Q8—31no—
263C3×Q8—91no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×Q8C3×D1211yescharacteristic
267C3×D4—21no—
268C3×D4—31no—
269C3×D4—61no—
270C3×D4—91no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—182no—
274C3×D4—181no—
275C3×D4—182no—
276C3×D4C3×D1211yescharacteristic
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—181no—
283C2×C3×C4—182no—
284C2×C3×C4—181no—
285C2×C3×C4C22×C3×D311yescharacteristic
286#67—62no—
287#67—61no—
288C2×Dic3—61no—
289D12—31no—
290D12—62no—
291D12—61no—
292D12—61no—
293C4×D3—62no—
294C4×D3—61no—
295Dic6—31no—
296Dic6—61no—
297Dic6—61no—
298C3×C8—21no—
299C3×C8—61no—
300C3×C8—181no—
301C3⋊C8—61no—
302C2×C32—21no—
303C2×C32—41no—
304C2×C32#88211yescharacteristic
305C3×D3—122no—
306C3×D3—121no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—41no—
310C2×C9—41no—
311C2×C9—81no—
312C2×C9C3×#12011yescharacteristic
313D9—42no—
314D9—41no—
315#41—91no—
316#41—181no—
317#41—181no—
318#41C2×AGL(1,ℤ/9)11yescharacteristic
319C2×Q8—91no—
320C2×D4—91no—
321C2×D4—181no—
322C2×D4—182no—
323C2×D4—181no—
324Q16—181no—
325Q16—181no—
326SD16—181no—
327SD16—181no—
328M16—91no—
329M16C2×AGL(1,ℤ/9)11yescharacteristic, complemented
330C22⋊C4—182no—
331C42C2×AGL(1,ℤ/9)11yescharacteristic, complemented
332C22×C3—21no—
333C22×C3—31no—
334C22×C3—61no—
335C22×C3—181no—
336C22×C3—182no—
337C22×C3—362no—
338C22×C3—722no—
339C22×C3C3×D3×D411yescharacteristic
340C2×D3—61no—
341C2×D3—62no—
342C2×D3—122no—
343C2×D3—242no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—41no—
348C3×C4—61no—
349C3×C4—121no—
350C3×C4—181no—
351C3×C4—361no—
352C3×C4C2×C3×D1211yescharacteristic
353C3×C4C3×D3×D411yescharacteristic
354Dic3—61no—
355Dic3—121no—
356C32#482511yescharacteristic
357C9—21no—
358C9C3×#45511yescharacteristic, complemented
359C23—92no—
360C23—182no—
361Q8—91no—
362Q8—181no—
363Q8—181no—
364Q8C4⋊AGL(1,ℤ/9)11yescharacteristic
365D4—21no—
366D4—91no—
367D4—182no—
368D4—182no—
369D4—181no—
370D4—181no—
371D4—181no—
372D4C4⋊AGL(1,ℤ/9)11yescharacteristic
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—182no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
379C8—21no—
380C8—181no—
381C2×C3—21no—
382C2×C3—31no—
383C2×C3—41no—
384C2×C3—61no—
385C2×C3—121no—
386C2×C3—362no—
387C2×C3—361no—
388C2×C3C3×#88211yescharacteristic
389D3—122no—
390D3—121no—
391C22—21no—
392C22—181no—
393C22—182no—
394C22—362no—
395C22—722no—
396C22D4×AGL(1,ℤ/9)11yescharacteristic
397C4—21no—
398C4—41no—
399C4—181no—
400C4—361no—
401C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
402C4D4×AGL(1,ℤ/9)11yescharacteristic
403C3—31no—
404C3C3×#482511yescharacteristic
405C2—21no—
406C2—41no—
407C2—362no—
408C2—361no—
409C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#104C2
socleC2×C3C3×#882
prime radical#11722641C2
hypercenter#104AGL(1,ℤ/9)
hypocenterC9C3×#455
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #63511⋊C3C2semidirect
3. #63471⋊C3C2semidirect
4. #63437⋊C3C2semidirect
5. 3−1+2×#104C2semidirect
6. #63383⋊C3C2semidirect
7. #63364⋊C3C2semidirect
9. #10588505C3semidirect
18. C2×3−1+2⋊Q8C22semidirect
19. M16×3−1+2C22semidirect
20. C42×3−1+2C22semidirect
24. #3667⋊C3C22semidirect
35. #63511C2×C3semidirect
36. #63471C2×C3semidirect
37. #63437C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63383C2×C3semidirect
41. #63364C2×C3semidirect
92. C2×Dic18C22×C3semidirect
94. C9×M16C22×C3semidirect
96. C42×C9C22×C3semidirect
100. #3667C22×C3semidirect
255. #104AGL(1,ℤ/9)semidirect
257. 3−1+2#455semidirect
329. M16C2×AGL(1,ℤ/9)semidirect
331. C42C2×AGL(1,ℤ/9)semidirect
358. C9C3×#455semidirect

Maximal subgroups

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

subgroupindex
#117226412
#63511⋊C32
#63471⋊C32
#63437⋊C32
3−1+2×#1042
#63383⋊C32
#63364⋊C32
C3×#48253
#105885053