Finite Group Databasebeta

← #10588834⋊C3 #13143321

Subgroups — #10588834⋊C3 #13143321

Order 1728.

470 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 100 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×D8×3−1+2C211yesmaximal, characteristic, complemented
2#63502⋊C3C211yesmaximal, characteristic, complemented
3#63498⋊C3C211yesmaximal, characteristic, complemented
4C2×#3673⋊C3C211yesmaximal, characteristic, complemented
5C2×#3672⋊C3C211yesmaximal, characteristic, complemented
6C2×#3664⋊C3C211yesmaximal, characteristic, complemented
7#63375⋊C3C211yesmaximal, characteristic, complemented
8C3×#5154—31nomaximal
9#10588834C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C2×D4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—22no—
13C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
14C2×3−1+2⋊Q8C2211yescharacteristic
15D8×3−1+2—22no—
16C2×C8×3−1+2C2211yescharacteristic
17#3676⋊C3—22no—
18#3673⋊C3—22no—
19#3672⋊C3—22no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22#3664⋊C3—22no—
23C2×C32×D8D311yescharacteristic
24C3×#940—31no—
25C3×#936—31no—
26C2×C3×#214—31no—
27C2×C3×#213—31no—
28C2×C3×#204—31no—
29C3×#815—31no—
30C2×C9×D8—21no—
31C2×C9×D8C2×C311yescharacteristic, complemented
32#63502C2×C311yescharacteristic, complemented
33#63498C2×C311yescharacteristic, complemented
34C2×#3673C2×C311yescharacteristic, complemented
35C2×#3672C2×C311yescharacteristic, complemented
36C2×#3664C2×C311yescharacteristic, complemented
37#63375C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39C23×3−1+2—21no—
40C22×AGL(1,ℤ/9)—21no—
41D4×3−1+2—21no—
42D4×3−1+2—21no—
43D4×3−1+2—21no—
44D4×3−1+2—21no—
45C2×C4×3−1+2C2311yescharacteristic
46#611⋊C3—44no—
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4D412yes—
49C4⋊AGL(1,ℤ/9)—21no—
50C4⋊AGL(1,ℤ/9)—21no—
513−1+2⋊Q8—21no—
523−1+2⋊Q8—21no—
53C8×3−1+2D412yescomplemented
543−1+2⋊C8D412yescomplemented
55C3×#496—91no—
56#5154—31no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C32×D4C2×D311yescharacteristic
59C2×C3×#67—62no—
60C2×C3×D12—31no—
61C2×C3×Dic6—31no—
62C32×D8—22no—
63C2×C32×C8C2×D311yescharacteristic
64C3×#217—62no—
65C3×#214—62no—
66C3×#213—62no—
67C3×C4×Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C3×#204—62no—
70C2×D4×C9—21no—
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C2×D4×C9C22×C311yescharacteristic
74C2×#611—22no—
75C2×D36C22×C311yescharacteristic
76C2×Dic18C22×C311yescharacteristic
77C9×D8—24no—
78C9×D8—22no—
79C2×C8×C9—21no—
80C2×C8×C9C22×C311yescharacteristic
81#3676—22no—
82#3673—22no—
83#3672—22no—
84C4×Dic9C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86#3664—22no—
87C22×3−1+2—41no—
88C22×3−1+2—41no—
89C22×3−1+2—42no—
90C22×3−1+2—42no—
91C22×3−1+2C2×D411yescharacteristic
92C2×AGL(1,ℤ/9)—41no—
93C2×AGL(1,ℤ/9)—42no—
94C4×3−1+2C2×D411yescharacteristic
95C4×3−1+2C2×D411yescharacteristic
963−1+2⋊C4—22no—
973−1+2⋊C4—41no—
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C2×C3×D8—31no—
101C2×C3×D8—91no—
102C2×C3×D8C3×D311yescharacteristic
103C3×#127—91no—
104C3×#124—91no—
105#940—31no—
106#936—31no—
107C2×#214—31no—
108C2×#213—31no—
109C2×#204—31no—
110C3×#97—91no—
111#815—31no—
112C23×C32—21no—
113C23×C32—21no—
114C22×C3×D3—61no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—21no—
118C32×D4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C3×#67—124no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C3×D12—61no—
124C3×D12—61no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C23×C9—21no—
128C23×C9—21no—
129C23×C9—41no—
130C23×C9—41no—
131C22×D9—21no—
132C32×C8#6712yes—
133C3×C3⋊C8—32no—
134D4×C9—22no—
135D4×C9—22no—
136D4×C9—21no—
137D4×C9—21no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—41no—
141D4×C9—41no—
142C2×C4×C9—21no—
143C2×C4×C9C23×C311yescharacteristic
144#611—44no—
145C2×Dic9—21no—
146C2×Dic9C3×D412yes—
147D36—21no—
148D36—21no—
149Dic18—21no—
150Dic18—21no—
151C8×C9—22no—
152C8×C9C3×D412yescomplemented
153C9⋊C8C3×D412yescomplemented
154#496—91noSylow
155C2×3−1+2—81no—
156C2×3−1+2—81no—
157C2×3−1+2#12711yescharacteristic
158C2×3−1+2AGL(1,ℤ/8)12yes—
159AGL(1,ℤ/9)—81no—
160C2×C3×Q8—91no—
161C2×C3×D4—31no—
162C2×C3×D4—31no—
163C2×C3×D4—91no—
164C2×C3×D4—182no—
165C2×C3×D4C2×C3×D311yescharacteristic
166C2×C3×D4C2×C3×D311yescharacteristic
167C2×#67—62no—
168C2×D12—31no—
169C2×Dic6—31no—
170C3×SD16—182no—
171C3×SD16—182no—
172C3×D8—22no—
173C3×D8—62no—
174C3×D8—182no—
175C2×C3×C8—31no—
176C2×C3×C8—91no—
177C2×C3×C8C2×C3×D311yescharacteristic
178C3×C22⋊C4—182no—
179C3×C42—91no—
180#217—62no—
181#214—62no—
182#213—62no—
183C4×Dic3—31no—
184C2×C3⋊C8—31no—
185#204—62no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—122no—
192C2×C3×D3—121no—
193C32×C4D3×D411yescharacteristic
194C32×C4C2×#6711yescharacteristic
195C3×Dic3—62no—
196C3×Dic3—121no—
197C22×C9—21no—
198C22×C9—42no—
199C22×C9—42no—
200C22×C9—41no—
201C22×C9—41no—
202C22×C9—42no—
203C22×C9—42no—
204C22×C9—82no—
205C22×C9—82no—
206C22×C9C2×C3×D411yescharacteristic
207C2×D9—41no—
208C2×D9—42no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9C2×C3×D411yescharacteristic
212C4×C9C2×C3×D411yescharacteristic
213Dic9—22no—
214Dic9—41no—
215C2×SD16—91no—
216C2×SD16—91no—
217C2×D8—91no—
218C2×D8AGL(1,ℤ/9)11yescharacteristic, complemented
219#127—91no—
220#124—91no—
221#97—91no—
2223−1+2#49611yescharacteristic, complemented, Sylow
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—181no—
228C22×D3—61no—
229C3×Q8—181no—
230C3×Q8—181no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—61no—
238C3×D4—61no—
239C3×D4—181no—
240C3×D4—181no—
241C3×D4—364no—
242C2×C3×C4—31no—
243C2×C3×C4—92no—
244C2×C3×C4—181no—
245C2×C3×C4C22×C3×D311yescharacteristic
246#67—124no—
247C2×Dic3—32no—
248C2×Dic3—61no—
249D12—61no—
250D12—61no—
251Dic6—61no—
252Dic6—61no—
253C3×C8—32no—
254C3×C8—92no—
255C3×C8C3×#6712yes—
256C3⋊C8—32no—
257C2×C32—81no—
258C2×C32—81no—
259C2×C32#91112yes—
260C2×C32#94011yescharacteristic
261C3×D3—241no—
262C2×C9—22no—
263C2×C9—21no—
264C2×C9—82no—
265C2×C9—82no—
266C2×C9—81no—
267C2×C9—81no—
268C2×C9C3×#12711yescharacteristic
269C2×C9C3×AGL(1,ℤ/8)12yes—
270D9—81no—
271C2×Q8—91no—
272C2×D4—91no—
273C2×D4—182no—
274C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
275C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
276SD16—182no—
277SD16—182no—
278D8—22no—
279D8—182no—
280C2×C8—91no—
281C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
282C22⋊C4—182no—
283C42—91no—
284C22×C3—31no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—121no—
290C22×C3—121no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—361no—
294C22×C3—362no—
295C22×C3C3×D3×D411yescharacteristic
296C2×D3—122no—
297C2×D3—121no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—182no—
301C3×C4—361no—
302C3×C4C3×D3×D411yescharacteristic
303C3×C4C2×C3×#6711yescharacteristic
304Dic3—62no—
305Dic3—121no—
306C32#515411yescharacteristic
307C9—21no—
308C9C3×#49611yescharacteristic, complemented
309C23—21no—
310C23—21no—
311C23—181no—
312Q8—181no—
313Q8—181no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—181no—
319D4—181no—
320D4—364no—
321C2×C4—92no—
322C2×C4—181no—
323C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
324C8—92no—
325C8#611⋊C312yescomplemented
326C2×C3—32no—
327C2×C3—31no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—241no—
331C2×C3—241no—
332C2×C3—721no—
333C2×C3C3×#91112yes—
334C2×C3C3×#94011yescharacteristic
335D3—241no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—362no—
341C22—361no—
342C22D4×AGL(1,ℤ/9)11yescharacteristic
343C4—182no—
344C4—361no—
345C4D4×AGL(1,ℤ/9)11yescharacteristic
346C4C2×#611⋊C311yescharacteristic
347C3—31no—
348C3C3×#515411yescharacteristic
349C2—81no—
350C2—81no—
351C2—721no—
352C2#63473⋊C312yes—
353C2#63502⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×D8×3−1+2C2
socleC22×C3C3×D3×D4
hypercenterC2×D8AGL(1,ℤ/9)
hypocenterC9C3×#496
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×3−1+2C2semidirect
2. #63502⋊C3C2semidirect
3. #63498⋊C3C2semidirect
4. C2×#3673⋊C3C2semidirect
5. C2×#3672⋊C3C2semidirect
6. C2×#3664⋊C3C2semidirect
7. #63375⋊C3C2semidirect
9. #10588834C3semidirect
31. C2×C9×D8C2×C3semidirect
32. #63502C2×C3semidirect
33. #63498C2×C3semidirect
34. C2×#3673C2×C3semidirect
35. C2×#3672C2×C3semidirect
36. C2×#3664C2×C3semidirect
37. #63375C2×C3semidirect
53. C8×3−1+2D4semidirect
54. 3−1+2⋊C8D4semidirect
152. C8×C9C3×D4semidirect
153. C9⋊C8C3×D4semidirect
218. C2×D8AGL(1,ℤ/9)semidirect
222. 3−1+2#496semidirect
308. C9C3×#496semidirect
325. C8#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D8×3−1+22
#63502⋊C32
#63498⋊C32
C2×#3673⋊C32
C2×#3672⋊C32
C2×#3664⋊C32
#63375⋊C32
C3×#51543
#105888343