Finite Group Databasebeta

← #10588502⋊C3 #13141990

Subgroups — #10588502⋊C3 #13141990

Order 1728.

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

normal subgroups only

This group has 526 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, complemented
2#63512⋊C3C211yesmaximal, characteristic, complemented
3#63470⋊C3C211yesmaximal, characteristic, complemented
4#11721394C211yesmaximal, characteristic, complemented
53−1+2×#104C211yesmaximal, characteristic, complemented
6#63383⋊C3C211yesmaximal, characteristic, complemented
7#63364⋊C3C211yesmaximal, characteristic, complemented
8C3×#4822—31nomaximal
9#10588502C311yesmaximal, 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)—22no—
18C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
19C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
20M16×3−1+2C2211yescharacteristic, complemented
21C42×3−1+2C2211yescharacteristic, complemented
22#3675⋊C3—21no—
23#3673⋊C3—21no—
24#3667⋊C3C2211yescharacteristic, complemented
25D72⋊C3—21no—
26#3664⋊C3—21no—
27C3×#1009—31no—
28C3×#949—31no—
29C3×#908—31no—
30C3×#874—31no—
31#63715C2×C311yescharacteristic, complemented
32C32×#104D311yescharacteristic
33C3×#823—31no—
34C3×#805—31no—
35#63512C2×C311yescharacteristic, complemented
36#63470C2×C311yescharacteristic, complemented
37#63436C2×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)—21no—
44C22×AGL(1,ℤ/9)D412yes—
45Q8×3−1+2D411yescharacteristic, complemented
46D4×3−1+2—21no—
47D4×3−1+2D411yescharacteristic, complemented
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2—21no—
50C2×C4×3−1+2C2311yescharacteristic
51#611⋊C3—22no—
52#611⋊C3—21no—
53C4⋊AGL(1,ℤ/9)—22no—
54C4⋊AGL(1,ℤ/9)—21no—
55C4⋊AGL(1,ℤ/9)—21no—
56C4⋊AGL(1,ℤ/9)—21no—
57C4⋊AGL(1,ℤ/9)—21no—
58C4⋊AGL(1,ℤ/9)—44no—
59C4⋊AGL(1,ℤ/9)D411yescharacteristic, complemented
60C4×AGL(1,ℤ/9)—22no—
61C4×AGL(1,ℤ/9)—21no—
623−1+2⋊Q8D411yescharacteristic, complemented
63C8×3−1+2—21no—
643−1+2⋊C8—21no—
65C3×#453—91no—
66#4822—31no—
67C32×#41C2×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—31no—
75C2×C3×D12—61no—
76C2×C3×D12—62no—
77C32×M16C2×D311yescharacteristic
78C32×C42C2×D311yescharacteristic
79C3×#215—61no—
80C3×#213—61no—
81C3×#208—31no—
82C3×D24—61no—
83C3×#204—61no—
84C9×#41—21no—
85C9×#41C22×C311yescharacteristic
86#3701—21no—
87D4×D9—22no—
88D4×D9—21no—
89#3697—21no—
90#3697C22×C311yescharacteristic
91C2×D36—21no—
92C2×D36—22no—
93C2×D36C22×C311yescharacteristic
94C2×D36C22×C311yescharacteristic, complemented
95C9×M16—21no—
96C9×M16C22×C311yescharacteristic, complemented
97C42×C9—21no—
98C42×C9C22×C311yescharacteristic, complemented
99#3675—21no—
100#3673—21no—
101#3667C22×C311yescharacteristic, complemented
102D72—21no—
103#3664—21no—
104C22×3−1+2—21no—
105C22×3−1+2C2×D411yescharacteristic
106C2×AGL(1,ℤ/9)—22no—
107C2×AGL(1,ℤ/9)—21no—
108C2×AGL(1,ℤ/9)—42no—
109C2×AGL(1,ℤ/9)—42no—
110C2×AGL(1,ℤ/9)—41no—
111C2×AGL(1,ℤ/9)—82no—
112C4×3−1+2—22no—
113C4×3−1+2—21no—
114C4×3−1+2C2×D411yescharacteristic
115C4×3−1+2C2×D411yescharacteristic
1163−1+2⋊C4—21no—
117C3×H(5,𝔽2)—91no—
118#1009—31no—
119C3×AGL(1,ℤ/8)—91no—
120C3×AGL(1,ℤ/8)—91no—
121C3×#127—91no—
122#949—31no—
123#908—31no—
124#874—31no—
125C3×#104—31no—
126C3×#104—91no—
127C3×#104C3×D311yescharacteristic
128C3×#100—91no—
129#823—31no—
130#805—31no—
131C22×C3×D3—32no—
132C22×C3×D3—62no—
133C22×C3×D3—61no—
134C32×Q8D1211yescharacteristic
135C32×D4—21no—
136C32×D4D1211yescharacteristic
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C3×#67—62no—
141C3×#67—61no—
142C3×D12—31no—
143C3×D12—62no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—124no—
149C3×C4×D3—62no—
150C3×C4×D3—61no—
151C3×Dic6—31no—
152C22×D9—22no—
153C22×D9—21no—
154C22×D9C3×D412yes—
155C32×C8—21no—
156C3×C3⋊C8—61no—
157Q8×C9—21no—
158Q8×C9C3×D411yescharacteristic, complemented
159D4×C9—21no—
160D4×C9—21no—
161D4×C9—41no—
162D4×C9C3×D411yescharacteristic, complemented
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—41no—
167C2×C4×C9—41no—
168C2×C4×C9C23×C311yescharacteristic
169#611—22no—
170#611—21no—
171D36—22no—
172D36—21no—
173D36—21no—
174D36—21no—
175D36—21no—
176D36—44no—
177D36C3×D411yescharacteristic, complemented
178C4×D9—22no—
179C4×D9—21no—
180Dic18C3×D411yescharacteristic, complemented
181C8×C9—21no—
182C8×C9—41no—
183C9⋊C8—21no—
184#453—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—
190AGL(1,ℤ/9)—81no—
191C3×#41—31no—
192C3×#41—91no—
193C3×#41—181no—
194C3×#41—181no—
195C3×#41C2×C3×D311yescharacteristic
196C2×C3×D4—91no—
197C2×C3×D4—91no—
198C2×C3×D4—181no—
199C2×C3×D4—182no—
200C2×C3×D4—181no—
201C2×C3×D4—182no—
202#239—61no—
203D3×D4—62no—
204D3×D4—61no—
205#235—31no—
206#235—61no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×D12—62no—
211C3×SD16—181no—
212C3×SD16—181no—
213C3×D8—181no—
214C3×D8—181no—
215C3×M16—31no—
216C3×M16—91no—
217C3×M16C2×C3×D311yescharacteristic
218C3×C42—31no—
219C3×C42C2×C3×D311yescharacteristic
220#215—61no—
221#213—61no—
222#208—31no—
223D24—61no—
224#204—61no—
225C22×C32—21no—
226C22×C32D3×D411yescharacteristic
227C2×C3×D3—61no—
228C2×C3×D3—62no—
229C2×C3×D3—122no—
230C2×C3×D3—122no—
231C2×C3×D3—121no—
232C2×C3×D3—242no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4C2×D1211yescharacteristic
236C32×C4D3×D411yescharacteristic
237C3×Dic3—61no—
238C22×C9—21no—
239C22×C9—21no—
240C22×C9—41no—
241C22×C9C2×C3×D411yescharacteristic
242C2×D9—22no—
243C2×D9—21no—
244C2×D9—42no—
245C2×D9—42no—
246C2×D9—41no—
247C2×D9—82no—
248C4×C9—22no—
249C4×C9—21no—
250C4×C9—21no—
251C4×C9—21no—
252C4×C9—41no—
253C4×C9—42no—
254C4×C9C2×C3×D411yescharacteristic
255C4×C9C2×C3×D411yescharacteristic
256Dic9—21no—
257H(5,𝔽2)—91no—
258AGL(1,ℤ/8)—91no—
259AGL(1,ℤ/8)—91no—
260#127—91no—
261#104—91no—
262#104AGL(1,ℤ/9)11yescharacteristic, complemented
263#100—91no—
2643−1+2#45311yescharacteristic, complemented, Sylow
265C23×C3—92no—
266C23×C3—181no—
267C23×C3—182no—
268C22×D3—32no—
269C22×D3—62no—
270C22×D3—61no—
271C3×Q8—31no—
272C3×Q8—91no—
273C3×Q8C3×D1211yescharacteristic
274C3×D4—21no—
275C3×D4—31no—
276C3×D4—61no—
277C3×D4—91no—
278C3×D4—181no—
279C3×D4—181no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4—182no—
283C3×D4—181no—
284C3×D4—182no—
285C3×D4—364no—
286C3×D4C3×D1211yescharacteristic
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—31no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—181no—
293C2×C3×C4—182no—
294C2×C3×C4C22×C3×D311yescharacteristic
295#67—62no—
296#67—61no—
297D12—31no—
298D12—62no—
299D12—61no—
300D12—61no—
301D12—61no—
302D12—61no—
303D12—124no—
304C4×D3—62no—
305C4×D3—61no—
306Dic6—31no—
307C3×C8—21no—
308C3×C8—61no—
309C3×C8—181no—
310C3⋊C8—61no—
311C2×C32—21no—
312C2×C32—41no—
313C2×C32#88211yescharacteristic
314C3×D3—122no—
315C3×D3—121no—
316C3×D3—241no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—41no—
320C2×C9—41no—
321C2×C9—81no—
322C2×C9C3×#12011yescharacteristic
323D9—42no—
324D9—41no—
325D9—81no—
326#41—91no—
327#41—181no—
328#41—181no—
329#41C2×AGL(1,ℤ/9)11yescharacteristic
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—181no—
333C2×D4—182no—
334C2×D4—182no—
335C2×D4—181no—
336SD16—181no—
337SD16—181no—
338D8—181no—
339D8—181no—
340M16—91no—
341M16C2×AGL(1,ℤ/9)11yescharacteristic, complemented
342C42C2×AGL(1,ℤ/9)11yescharacteristic, complemented
343C22×C3—21no—
344C22×C3—31no—
345C22×C3—61no—
346C22×C3—181no—
347C22×C3—182no—
348C22×C3—361no—
349C22×C3—362no—
350C22×C3—362no—
351C22×C3—722no—
352C22×C3C3×D3×D411yescharacteristic
353C2×D3—61no—
354C2×D3—62no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—121no—
358C2×D3—242no—
359C3×C4—22no—
360C3×C4—21no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—62no—
365C3×C4—181no—
366C3×C4C2×C3×D1211yescharacteristic
367C3×C4C3×D3×D411yescharacteristic
368Dic3—61no—
369C32#482211yescharacteristic
370C9—21no—
371C9C3×#45311yescharacteristic, complemented
372C23—92no—
373C23—182no—
374C23—181no—
375Q8—91no—
376Q8C4⋊AGL(1,ℤ/9)11yescharacteristic, complemented
377D4—21no—
378D4—91no—
379D4—182no—
380D4—182no—
381D4—181no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—181no—
386D4—364no—
387D4C4⋊AGL(1,ℤ/9)11yescharacteristic, complemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—182no—
391C2×C4—181no—
392C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
393C8—21no—
394C8—181no—
395C2×C3—21no—
396C2×C3—31no—
397C2×C3—41no—
398C2×C3—61no—
399C2×C3—121no—
400C2×C3—362no—
401C2×C3—361no—
402C2×C3—721no—
403C2×C3C3×#88211yescharacteristic
404D3—122no—
405D3—121no—
406D3—241no—
407C22—21no—
408C22—181no—
409C22—182no—
410C22—362no—
411C22—362no—
412C22—361no—
413C22—722no—
414C22D4×AGL(1,ℤ/9)11yescharacteristic
415C4—22no—
416C4—21no—
417C4—181no—
418C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
419C4D4×AGL(1,ℤ/9)11yescharacteristic
420C3—31no—
421C3C3×#482211yescharacteristic
422C2—21no—
423C2—41no—
424C2—362no—
425C2—361no—
426C2—721no—
427C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#104C2
socleC2×C3C3×#882
hypercenter#104AGL(1,ℤ/9)
hypocenterC9C3×#453
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
1. #11722641C2semidirect
2. #63512⋊C3C2semidirect
3. #63470⋊C3C2semidirect
4. #11721394C2semidirect
5. 3−1+2×#104C2semidirect
6. #63383⋊C3C2semidirect
7. #63364⋊C3C2semidirect
9. #10588502C3semidirect
19. C2×C4⋊AGL(1,ℤ/9)C22semidirect
20. M16×3−1+2C22semidirect
21. C42×3−1+2C22semidirect
24. #3667⋊C3C22semidirect
31. #63715C2×C3semidirect
35. #63512C2×C3semidirect
36. #63470C2×C3semidirect
37. #63436C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63383C2×C3semidirect
41. #63364C2×C3semidirect
45. Q8×3−1+2D4semidirect
47. D4×3−1+2D4semidirect
59. C4⋊AGL(1,ℤ/9)D4semidirect
62. 3−1+2⋊Q8D4semidirect
94. C2×D36C22×C3semidirect
96. C9×M16C22×C3semidirect
98. C42×C9C22×C3semidirect
101. #3667C22×C3semidirect
158. Q8×C9C3×D4semidirect
162. D4×C9C3×D4semidirect
177. D36C3×D4semidirect
180. Dic18C3×D4semidirect
262. #104AGL(1,ℤ/9)semidirect
264. 3−1+2#453semidirect
341. M16C2×AGL(1,ℤ/9)semidirect
342. C42C2×AGL(1,ℤ/9)semidirect
371. C9C3×#453semidirect
376. Q8C4⋊AGL(1,ℤ/9)semidirect
387. 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
#117226412
#63512⋊C32
#63470⋊C32
#117213942
3−1+2×#1042
#63383⋊C32
#63364⋊C32
C3×#48223
#105885023