Finite Group Databasebeta

← #10588816⋊C3 #13143240

Subgroups — #10588816⋊C3 #13143240

Order 1728.

394 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 102 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×M16×3−1+2C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic
3#63486⋊C3C211yesmaximal, characteristic, complemented
4C2×#3664⋊C3C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic, complemented
6#63378⋊C3C211yesmaximal, characteristic, complemented
7#63376⋊C3C211yesmaximal, characteristic, complemented
8C3×#5134—31nomaximal
9#10588816C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×3−1+2⋊Q8C2211yescharacteristic
14M16×3−1+2—22no—
15C2×C8×3−1+2C2211yescharacteristic
16C2×C8×3−1+2C2211yescharacteristic
17#3676⋊C3—21no—
18(C2×AGL(1,ℤ/9))⋊C4—21no—
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20C4⋊(3−1+2⋊C4)C2211yescharacteristic
21(3−1+2⋊C4)⋊C4—21no—
22#3664⋊C3—22no—
23C2×C32×M16D311yescharacteristic
24C3×#930—31no—
25C3×#924—31no—
26C2×C3×#204—31no—
27C3×#821—31no—
28C3×#818—31no—
29C3×#816—31no—
30C2×C9×M16—21no—
31C2×C9×M16C2×C311yescharacteristic, complemented
32#63492C2×C311yescharacteristic
33#63486C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35#63380C2×C311yescharacteristic, complemented
36#63378C2×C311yescharacteristic, complemented
37#63376C2×C311yescharacteristic, complemented
38C23×3−1+2D411yescharacteristic
39C22×AGL(1,ℤ/9)—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2D411yescharacteristic
43C2×C4×3−1+2C2311yescharacteristic
44#611⋊C3—42no—
45C2×3−1+2⋊C4—21no—
46C2×3−1+2⋊C4—21no—
47C2×3−1+2⋊C4—21no—
48C4⋊AGL(1,ℤ/9)—21no—
49C4⋊AGL(1,ℤ/9)—21no—
503−1+2⋊Q8—21no—
513−1+2⋊Q8—21no—
52C8×3−1+2—21no—
53C8×3−1+2D412yescomplemented
54C3×#468—91no—
55#5134—31no—
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×#67—61no—
58C2×C3×D12—31no—
59C2×C3×Dic6—31no—
60C32×M16—22no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C3×#217—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×Dic3⋊C4—61no—
68C3×#204—62no—
69C22×C4×C9—21no—
70C22×C4×C9C22×C311yescharacteristic
71C2×#611—21no—
72C2×D36C22×C311yescharacteristic
73C2×Dic18C22×C311yescharacteristic
74C9×M16—24no—
75C9×M16—22no—
76C2×C8×C9—21no—
77C2×C8×C9—21no—
78C2×C8×C9C22×C311yescharacteristic
79C2×C8×C9C22×C311yescharacteristic
80#3676—21no—
81(C2×D9)⋊C4—21no—
82C4⋊Dic9C22×C311yescharacteristic
83C4⋊Dic9C22×C311yescharacteristic
84Dic9⋊C4—21no—
85#3664—22no—
86C22×3−1+2—21no—
87C22×3−1+2—21no—
88C22×3−1+2—21no—
89C22×3−1+2C2×D411yescharacteristic
90C2×AGL(1,ℤ/9)—41no—
91C2×AGL(1,ℤ/9)—41no—
92C2×AGL(1,ℤ/9)—41no—
93C4×3−1+2—21no—
94C4×3−1+2C2×D411yescharacteristic
95C4×3−1+2#4111yescharacteristic
963−1+2⋊C4—41no—
973−1+2⋊C4—41no—
983−1+2⋊C4—41no—
99C2×C3×SD16—91no—
100C2×C3×M16—31no—
101C2×C3×M16C3×D311yescharacteristic
102C3×C22⋊Q8—91no—
103C3×#121—91no—
104#930—31no—
105#924—31no—
106C2×#204—31no—
107C3×#107—91no—
108C3×#103—91no—
109C3×#102—91no—
110#821—31no—
111#818—31no—
112#816—31no—
113C23×C32D1211yescharacteristic
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4D1211yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C3×#67—122no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—61no—
124C3×D12—61no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C23×C9—21no—
128C23×C9C3×D411yescharacteristic
129C22×D9—21no—
130C32×C8—21no—
131C32×C8#6712yes—
132C2×C4×C9—22no—
133C2×C4×C9—22no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9C3×D411yescharacteristic
139C2×C4×C9C23×C311yescharacteristic
140#611—42no—
141C2×Dic9—21no—
142C2×Dic9—21no—
143C2×Dic9—21no—
144D36—21no—
145D36—21no—
146Dic18—21no—
147Dic18—21no—
148C8×C9—22no—
149C8×C9—21no—
150C8×C9—22no—
151C8×C9C3×D412yescomplemented
152#468—91noSylow
153C2×3−1+2—41no—
154C2×3−1+2#12111yescharacteristic
155C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
156C2×3−1+2#13711yescharacteristic
157AGL(1,ℤ/9)—81no—
158C2×C3×Q8—91no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C22×C3×C4—31no—
162C22×C3×C4C2×C3×D311yescharacteristic
163C2×#67—61no—
164C2×D12—31no—
165C2×Dic6—31no—
166C3×SD16—182no—
167C3×M16—22no—
168C3×M16—62no—
169C2×C3×C8—31no—
170C2×C3×C8—31no—
171C2×C3×C8C2×C3×D311yescharacteristic
172C2×C3×C8C2×C3×D311yescharacteristic
173C3×C4⋊C4—91no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—181no—
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178#217—61no—
179(C2×D3)⋊C4—61no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182Dic3⋊C4—61no—
183#204—62no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32C2×D1211yescharacteristic
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—21no—
192C32×C4#23511yescharacteristic
193C32×C4C2×#6711yescharacteristic
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C22×C9—22no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—21no—
201C22×C9—21no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9C2×C3×D411yescharacteristic
205C2×D9—41no—
206C2×D9—41no—
207C2×D9—41no—
208C4×C9—22no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9C2×C3×D411yescharacteristic
213C4×C9C3×#4111yescharacteristic
214Dic9—41no—
215Dic9—41no—
216Dic9—41no—
217C2×SD16—91no—
218C2×M16AGL(1,ℤ/9)11yescharacteristic, complemented
219C22⋊Q8—91no—
220#121—91no—
221#107—91no—
222#103—91no—
223#102—91no—
2243−1+2#46811yescharacteristic, complemented, Sylow
225C23×C3—31no—
226C23×C3—181no—
227C23×C3C3×D1211yescharacteristic
228C22×D3—61no—
229C3×Q8—181no—
230C3×Q8—181no—
231C3×D4—181no—
232C3×D4—181no—
233C3×D4—362no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C3×D1211yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245#67—122no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249D12—61no—
250D12—61no—
251Dic6—61no—
252Dic6—61no—
253C3×C8—21no—
254C3×C8—32no—
255C3×C8—61no—
256C3×C8C3×#6712yes—
257C2×C32—41no—
258C2×C32#90811yescharacteristic
259C2×C32#90911yescharacteristic
260C2×C32#93011yescharacteristic
261C3×D3—241no—
262C2×C9—21no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—42no—
266C2×C9—41no—
267C2×C9C3×#12111yescharacteristic
268C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
269C2×C9C3×#13711yescharacteristic
270D9—81no—
271C2×Q8—91no—
272C2×D4—91no—
273C2×D4—181no—
274C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
275SD16—182no—
276M16—22no—
277C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
278C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—181no—
282C22⋊C4—181no—
283C22⋊C4—181no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—31no—
288C22×C3—61no—
289C22×C3—61no—
290C22×C3—61no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3C2×C3×D1211yescharacteristic
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C3×C4—21no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—61no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4C3×#23511yescharacteristic
306C3×C4C2×C3×#6711yescharacteristic
307Dic3—121no—
308Dic3—121no—
309Dic3—121no—
310C32#513411yescharacteristic
311C9—21no—
312C9C3×#46811yescharacteristic, complemented
313C23—181no—
314C23C4⋊AGL(1,ℤ/9)11yescharacteristic
315Q8—181no—
316Q8—181no—
317D4—181no—
318D4—181no—
319D4—362no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
326C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
327C8—21no—
328C8#611⋊C312yescomplemented
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—41no—
333C2×C3—121no—
334C2×C3—721no—
335C2×C3C3×#90811yescharacteristic
336C2×C3C3×#90911yescharacteristic
337C2×C3C3×#93011yescharacteristic
338D3—241no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—361no—
343C22—361no—
344C22—361no—
345C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
346C4—21no—
347C4—361no—
348C4—361no—
349C4—361no—
350C4#3697⋊C311yescharacteristic
351C4C2×#611⋊C311yescharacteristic
352C3—31no—
353C3C3×#513411yescharacteristic
354C2—41no—
355C2—721no—
356C2#63470⋊C311yescharacteristic
357C2#63471⋊C311yescharacteristic
358C2#63492⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×3−1+2C2
socleC22×C3C2×C3×D12
prime radical#63492⋊C3C2
hypercenterC2×M16AGL(1,ℤ/9)
hypocenterC9C3×#468
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×3−1+2C2semidirect
3. #63486⋊C3C2semidirect
4. C2×#3664⋊C3C2semidirect
5. #63380⋊C3C2semidirect
6. #63378⋊C3C2semidirect
7. #63376⋊C3C2semidirect
9. #10588816C3semidirect
31. C2×C9×M16C2×C3semidirect
33. #63486C2×C3semidirect
34. C2×#3664C2×C3semidirect
35. #63380C2×C3semidirect
36. #63378C2×C3semidirect
37. #63376C2×C3semidirect
53. C8×3−1+2D4semidirect
151. C8×C9C3×D4semidirect
218. C2×M16AGL(1,ℤ/9)semidirect
224. 3−1+2#468semidirect
312. C9C3×#468semidirect
328. C8#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×M16×3−1+22
#63492⋊C32
#63486⋊C32
C2×#3664⋊C32
#63380⋊C32
#63378⋊C32
#63376⋊C32
C3×#51343
#105888163