Finite Group Databasebeta

← #10588566⋊C3 #13142247

Subgroups — #10588566⋊C3 #13142247

Order 1728.

378 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 96 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×#3664⋊C3C211yesmaximal, characteristic, complemented
2C2×#3663⋊C3C211yesmaximal, characteristic, complemented
3#63459⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic
53−1+2×#107C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4886—31nomaximal
9#10588566C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
13C2×3−1+2⋊Q8C2211yescharacteristic
14C2×C8×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16C4⋊C4×3−1+2C2211yescharacteristic
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18(C2×AGL(1,ℤ/9))⋊C4—21no—
19C4⋊(3−1+2⋊C4)—21no—
20C2×3−1+2⋊C8C2211yescharacteristic
21#3664⋊C3—22no—
22#3663⋊C3—22no—
23C2×C3×#204—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C3×#897—31no—
26C3×#895—31no—
27C32×#107D311yescharacteristic
28C3×#810—31no—
29C3×#809—31no—
30C2×#3664C2×C311yescharacteristic, complemented
31C2×#3663C2×C311yescharacteristic, complemented
32#63459C2×C311yescharacteristic, complemented
33#63457C2×C311yescharacteristic
34C9×#107—21no—
35C9×#107C2×C311yescharacteristic, complemented
36#63369C2×C311yescharacteristic, complemented
37#63368C2×C311yescharacteristic, complemented
38C22×AGL(1,ℤ/9)—21no—
39C22×AGL(1,ℤ/9)D411yescharacteristic
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4D411yescharacteristic
45C4⋊AGL(1,ℤ/9)—21no—
46C4⋊AGL(1,ℤ/9)—21no—
47C4⋊AGL(1,ℤ/9)—42no—
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+2D412yescomplemented
533−1+2⋊C8—21no—
54C3×#468—91no—
55#4886—31no—
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—31no—
59C2×C3×Dic6—31no—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C3×#204—62no—
68C3×ΓL(1,𝔽25)—62no—
69C2×D36—21no—
70C2×D36C22×C311yescharacteristic
71C2×C4×D9C22×C311yescharacteristic
72C2×Dic18C22×C311yescharacteristic
73C2×C8×C9—21no—
74C2×C8×C9C22×C311yescharacteristic
75C9×C4⋊C4—21no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4C22×C311yescharacteristic
78C9×C4⋊C4C22×C311yescharacteristic
79(C2×D9)⋊C4—21no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9—21no—
82C2×C9⋊C8C22×C311yescharacteristic
83#3664—22no—
84#3663—22no—
85C22×3−1+2C2×D411yescharacteristic
86C2×AGL(1,ℤ/9)—21no—
87C2×AGL(1,ℤ/9)—21no—
88C2×AGL(1,ℤ/9)—21no—
89C2×AGL(1,ℤ/9)—41no—
90C2×AGL(1,ℤ/9)—41no—
91C2×AGL(1,ℤ/9)—41no—
92C4×3−1+2—41no—
93C4×3−1+2—41no—
94C4×3−1+2C2×D411yescharacteristic
95C4×3−1+2#4111yescharacteristic
963−1+2⋊C4—21no—
973−1+2⋊C4—41no—
98C2×C3×SD16—91no—
99C2×C3×M16—91no—
100C3×C22⋊Q8—91no—
101C3×#121—91no—
102C2×#204—31no—
103C2×ΓL(1,𝔽25)—31no—
104#897—31no—
105#895—31no—
106C3×#107—31no—
107C3×#107C3×D311yescharacteristic
108C3×#103—91no—
109C3×#102—91no—
110#810—31no—
111#809—31no—
112C22×C3×D3—31no—
113C22×C3×D3—61no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C3×D12—61no—
120C3×D12—61no—
121C3×D12—122no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C3×Dic6—61no—
125C3×Dic6—61no—
126C22×D9—21no—
127C22×D9C3×D411yescharacteristic
128C32×C8D1212yes—
129C3×C3⋊C8—61no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—41no—
134C2×C4×C9—41no—
135C2×C4×C9C23×C311yescharacteristic
136C2×Dic9—21no—
137C2×Dic9C3×D411yescharacteristic
138D36—21no—
139D36—21no—
140D36—42no—
141C4×D9—21no—
142C4×D9—21no—
143Dic18—21no—
144Dic18—21no—
145C8×C9—22no—
146C8×C9C3×D412yescomplemented
147C9⋊C8—21no—
148#468—91noSylow
149C2×3−1+2#12111yescharacteristic
150C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
151C2×3−1+2#13711yescharacteristic
152AGL(1,ℤ/9)—41no—
153AGL(1,ℤ/9)—81no—
154C2×C3×Q8—91no—
155C2×C3×D4—91no—
156C2×C3×D4—181no—
157C22×C3×C4—91no—
158C2×D12—31no—
159C2×D12—61no—
160C2×C4×D3—31no—
161C2×Dic6—31no—
162C3×SD16—182no—
163C3×M16—182no—
164C2×C3×C8—31no—
165C2×C3×C8—91no—
166C2×C3×C8C2×C3×D311yescharacteristic
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4C2×C3×D311yescharacteristic
171C3×C4⋊C4C2×C3×D311yescharacteristic
172C3×C22⋊C4—181no—
173C3×C22⋊C4—181no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—61no—
176C4⋊Dic3—61no—
177C2×C3⋊C8—31no—
178#204—62no—
179ΓL(1,𝔽25)—62no—
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—121no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4C2×D1211yescharacteristic
190C32×C4#23911yescharacteristic
191C3×Dic3—61no—
192C3×Dic3—121no—
193C22×C9—21no—
194C22×C9C2×C3×D411yescharacteristic
195C2×D9—21no—
196C2×D9—21no—
197C2×D9—21no—
198C2×D9—41no—
199C2×D9—41no—
200C2×D9—41no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—41no—
204C4×C9—41no—
205C4×C9—81no—
206C4×C9—81no—
207C4×C9C2×C3×D411yescharacteristic
208C4×C9C3×#4111yescharacteristic
209Dic9—21no—
210Dic9—41no—
211C2×SD16—91no—
212C2×M16—91no—
213C22⋊Q8—91no—
214#121—91no—
215#107AGL(1,ℤ/9)11yescharacteristic, complemented
216#103—91no—
217#102—91no—
2183−1+2#46811yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C23×C3—181no—
221C22×D3—31no—
222C22×D3—61no—
223C3×Q8—181no—
224C3×Q8—181no—
225C3×D4—181no—
226C3×D4—181no—
227C3×D4—362no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×Dic3—31no—
239C2×Dic3—61no—
240D12—61no—
241D12—61no—
242D12—122no—
243C4×D3—61no—
244C4×D3—61no—
245Dic6—61no—
246Dic6—61no—
247C3×C8—32no—
248C3×C8—181no—
249C3×C8C3×D1212yes—
250C3⋊C8—61no—
251C2×C32#89511yescharacteristic
252C2×C32#91111yescharacteristic
253C2×C32#91811yescharacteristic
254C3×D3—121no—
255C3×D3—241no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9C3×#12111yescharacteristic
260C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
261C2×C9C3×#13711yescharacteristic
262D9—41no—
263D9—81no—
264C2×Q8—91no—
265C2×D4—91no—
266C2×D4—181no—
267C22×C4—91no—
268SD16—182no—
269M16—182no—
270C2×C8—91no—
271C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
272C4⋊C4—181no—
273C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
274C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
275C22⋊C4—181no—
276C22⋊C4—181no—
277C22×C3—31no—
278C22×C3—181no—
279C22×C3—181no—
280C22×C3—181no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3C3×D3×D411yescharacteristic
285C2×D3—61no—
286C2×D3—61no—
287C2×D3—61no—
288C2×D3—121no—
289C2×D3—121no—
290C2×D3—121no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—181no—
298C3×C4—361no—
299C3×C4C2×C3×D1211yescharacteristic
300C3×C4C3×#23911yescharacteristic
301Dic3—61no—
302Dic3—121no—
303C32#488611yescharacteristic
304C9—21no—
305C9C3×#46811yescharacteristic, complemented
306C23—91no—
307C23—181no—
308Q8—181no—
309Q8—181no—
310D4—181no—
311D4—181no—
312D4—362no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—91no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
320C8—181no—
321C8C4⋊AGL(1,ℤ/9)12yescomplemented
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—361no—
326C2×C3—721no—
327C2×C3C3×#89511yescharacteristic
328C2×C3C3×#91111yescharacteristic
329C2×C3C3×#91811yescharacteristic
330D3—121no—
331D3—241no—
332C22—181no—
333C22—181no—
334C22—181no—
335C22—361no—
336C22—361no—
337C22—361no—
338C22D4×AGL(1,ℤ/9)11yescharacteristic
339C4—41no—
340C4—41no—
341C4—181no—
342C4—361no—
343C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
344C4#8872411yescharacteristic
345C3—31no—
346C3C3×#488611yescharacteristic
347C2—361no—
348C2—721no—
349C2#63457⋊C311yescharacteristic
350C2#63473⋊C311yescharacteristic
351C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radical#63457⋊C3C2
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#468
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×#3664⋊C3C2semidirect
2. C2×#3663⋊C3C2semidirect
3. #63459⋊C3C2semidirect
5. 3−1+2×#107C2semidirect
6. #63369⋊C3C2semidirect
7. #63368⋊C3C2semidirect
9. #10588566C3semidirect
30. C2×#3664C2×C3semidirect
31. C2×#3663C2×C3semidirect
32. #63459C2×C3semidirect
35. C9×#107C2×C3semidirect
36. #63369C2×C3semidirect
37. #63368C2×C3semidirect
52. C8×3−1+2D4semidirect
146. C8×C9C3×D4semidirect
215. #107AGL(1,ℤ/9)semidirect
218. 3−1+2#468semidirect
305. C9C3×#468semidirect
321. C8C4⋊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
C2×#3664⋊C32
C2×#3663⋊C32
#63459⋊C32
#63457⋊C32
3−1+2×#1072
#63369⋊C32
#63368⋊C32
C3×#48863
#105885663