Finite Group Databasebeta

← #10588898⋊C3 #13143575

Subgroups — #10588898⋊C3 #13143575

Order 1728.

712 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×D4×3−1+2C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C212yesmaximal, complemented
3#63492⋊C3C211yesmaximal, characteristic, complemented
4#63392⋊C3C212yesmaximal, complemented
5#63389⋊C3C211yesmaximal, characteristic, complemented
6C3×#5217—31nomaximal
7#10588898C311yesmaximal, characteristic, complemented
8C24×3−1+2—21no—
9C2×D4×3−1+2—24no—
10C2×D4×3−1+2—21no—
11C2×D4×3−1+2C2212yes—
12C22×C4×3−1+2C2211yescharacteristic
13C2×#611⋊C3—21no—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
15#3673⋊C3—24no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
18C2×3−1+2⋊C8C2212yescomplemented
19C22×C32×D4D311yescharacteristic
20C2×C3×#213—32no—
21C3×#930—31no—
22C22×D4×C9—21no—
23C22×D4×C9C2×C311yescharacteristic, complemented
24C3×#832—32no—
25C3×#830—31no—
26C2×#3673C2×C312yescomplemented
27#63492C2×C311yescharacteristic, complemented
28#63392C2×C312yescomplemented
29#63389C2×C311yescharacteristic, complemented
30C23×3−1+2—24no—
31C23×3−1+2—22no—
32C23×3−1+2—42no—
33C23×3−1+2—42no—
34C23×3−1+2D411yescharacteristic
35C22×AGL(1,ℤ/9)—21no—
36D4×3−1+2—24no—
37D4×3−1+2—22no—
38D4×3−1+2D414yescomplemented
39C2×C4×3−1+2—22no—
40C2×C4×3−1+2D411yescharacteristic
41C2×C4×3−1+2C2311yescharacteristic
42#611⋊C3—42no—
43C2×3−1+2⋊C4—21no—
44C4⋊AGL(1,ℤ/9)—22no—
453−1+2⋊C8—22no—
46C3×#447—91no—
47#5217—31no—
48C24×C32—21no—
49C2×C32×D4—24no—
50C2×C32×D4—21no—
51C2×C32×D4C2×D312yes—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—61no—
54C2×C3×D12—31no—
55C24×C9—22no—
56C24×C9—21no—
57C3×#213—64no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—31no—
60C2×C3×C3⋊C8—32no—
61C2×D4×C9—28no—
62C2×D4×C9—22no—
63C2×D4×C9—22no—
64C2×D4×C9—24no—
65C2×D4×C9—21no—
66C2×D4×C9C22×C312yes—
67C22×C4×C9—21no—
68C22×C4×C9C22×C311yescharacteristic
69C2×#611—21no—
70C2×D36C22×C311yescharacteristic, complemented
71#3673—24no—
72(C2×D9)⋊C4—21no—
73C4⋊Dic9C22×C311yescharacteristic, complemented
74C2×C9⋊C8C22×C312yescomplemented
75C22×3−1+2—21no—
76C22×3−1+2—21no—
77C22×3−1+2—24no—
78C22×3−1+2—48no—
79C22×3−1+2—42no—
80C22×3−1+2—42no—
81C22×3−1+2D812yescomplemented
82C22×3−1+2C2×D411yescharacteristic
83C2×AGL(1,ℤ/9)—41no—
84C2×AGL(1,ℤ/9)—41no—
85C2×AGL(1,ℤ/9)—41no—
86C4×3−1+2—21no—
87C4×3−1+2C2×D412yes—
883−1+2⋊C4—41no—
89C22×C3×D4—31no—
90C22×C3×D4C3×D311yescharacteristic
91C2×C3×D8—92no—
92C3×#121—91no—
93C2×#213—32no—
94#930—31no—
95C3×#102—92no—
96C3×C22⋊C8—91no—
97#832—32no—
98#830—31no—
99C23×C32—24no—
100C23×C32—22no—
101C23×C32—42no—
102C23×C32—42no—
103C23×C32#6711yescharacteristic
104C22×C3×D3—61no—
105C32×D4—24no—
106C32×D4—22no—
107C32×D4#6714yes—
108C2×C32×C4—22no—
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C3×#67—122no—
112C2×C3×Dic3—61no—
113C3×D12—62no—
114C23×C9—28no—
115C23×C9—24no—
116C23×C9—24no—
117C23×C9—22no—
118C23×C9—21no—
119C23×C9—44no—
120C23×C9—42no—
121C23×C9—44no—
122C23×C9—42no—
123C23×C9C3×D411yescharacteristic
124C22×D9—21no—
125C3×C3⋊C8—62no—
126D4×C9—28no—
127D4×C9—24no—
128D4×C9—24no—
129D4×C9—22no—
130D4×C9—24no—
131D4×C9C3×D414yescomplemented
132C2×C4×C9—24no—
133C2×C4×C9—22no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9C3×D411yescharacteristic
137C2×C4×C9C23×C311yescharacteristic
138#611—42no—
139C2×Dic9—21no—
140D36—22no—
141C9⋊C8—22no—
142#447—91noSylow
143C2×3−1+2—22no—
144C2×3−1+2—44no—
145C2×3−1+2#12011yescharacteristic
146C2×3−1+2C2×D811yescharacteristic
147C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
148AGL(1,ℤ/9)—81no—
149C24×C3—21no—
150C24×C3—61no—
151C2×C3×D4—24no—
152C2×C3×D4—21no—
153C2×C3×D4—32no—
154C2×C3×D4—61no—
155C2×C3×D4—64no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C2×C3×D4C2×C3×D312yes—
159C22×C3×C4—31no—
160C22×C3×C4C2×C3×D311yescharacteristic
161C2×#67—61no—
162C2×D12—31no—
163C3×D8—184no—
164C2×C3×C8—92no—
165C3×C4⋊C4—91no—
166C3×C22⋊C4—181no—
167#213—64no—
168(C2×D3)⋊C4—61no—
169C4⋊Dic3—31no—
170C2×C3⋊C8—32no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—24no—
174C22×C32—48no—
175C22×C32—42no—
176C22×C32—42no—
177C22×C32#21312yes—
178C22×C32C2×#6711yescharacteristic
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C32×C4—21no—
183C32×C4C2×#6712yes—
184C3×Dic3—121no—
185C22×C9—22no—
186C22×C9—22no—
187C22×C9—21no—
188C22×C9—21no—
189C22×C9—28no—
190C22×C9—24no—
191C22×C9—22no—
192C22×C9—21no—
193C22×C9—416no—
194C22×C9—48no—
195C22×C9—44no—
196C22×C9—44no—
197C22×C9—42no—
198C22×C9—42no—
199C22×C9C3×D812yescomplemented
200C22×C9C2×C3×D411yescharacteristic
201C2×D9—41no—
202C2×D9—41no—
203C2×D9—41no—
204C4×C9—22no—
205C4×C9—21no—
206C4×C9—22no—
207C4×C9C2×C3×D412yes—
208Dic9—41no—
209C22×D4AGL(1,ℤ/9)11yescharacteristic, complemented
210C2×D8—92no—
211#121—91no—
212#102—92no—
213C22⋊C8—91no—
2143−1+2#44711yescharacteristic, complemented, Sylow
215C23×C3—24no—
216C23×C3—22no—
217C23×C3—31no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—62no—
221C23×C3—64no—
222C23×C3—122no—
223C23×C3—122no—
224C23×C3—181no—
225C23×C3C3×#6711yescharacteristic
226C22×D3—61no—
227C3×D4—24no—
228C3×D4—22no—
229C3×D4—34no—
230C3×D4—62no—
231C3×D4—64no—
232C3×D4—182no—
233C3×D4—362no—
234C3×D4C3×#6714yes—
235C2×C3×C4—22no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—62no—
239C2×C3×C4—181no—
240C2×C3×C4C3×#6711yescharacteristic
241C2×C3×C4C22×C3×D311yescharacteristic
242#67—122no—
243C2×Dic3—61no—
244D12—62no—
245C3×C8—182no—
246C3⋊C8—62no—
247C2×C32—22no—
248C2×C32—44no—
249C2×C32C2×#21311yescharacteristic
250C2×C32#93211yescharacteristic
251C2×C32#95311yescharacteristic
252C3×D3—241no—
253C2×C9—24no—
254C2×C9—22no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—48no—
259C2×C9—44no—
260C2×C9C3×#12011yescharacteristic
261C2×C9C2×C3×D811yescharacteristic
262C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
263D9—81no—
264C24—21no—
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—91no—
268C2×D4—181no—
269C2×D4C2×AGL(1,ℤ/9)12yes—
270C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
271D8—184no—
272C2×C8—92no—
273C4⋊C4—91no—
274C22⋊C4—181no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—24no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—48no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—61no—
284C22×C3—61no—
285C22×C3—64no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3—128no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3C3×#21312yes—
293C22×C3C2×C3×#6711yescharacteristic
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—121no—
297C3×C4—21no—
298C3×C4—32no—
299C3×C4—61no—
300C3×C4—361no—
301C3×C4C2×C3×#6712yes—
302Dic3—121no—
303C32#521711yescharacteristic
304C9—21no—
305C9C3×#44711yescharacteristic, complemented
306C23—24no—
307C23—22no—
308C23—42no—
309C23—42no—
310C23—181no—
311C23#611⋊C311yescharacteristic
312D4—24no—
313D4—22no—
314D4—182no—
315D4—362no—
316D4#611⋊C314yescomplemented
317C2×C4—22no—
318C2×C4—181no—
319C2×C4#611⋊C311yescharacteristic
320C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
321C8—182no—
322C2×C3—22no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—44no—
327C2×C3—62no—
328C2×C3—124no—
329C2×C3—721no—
330C2×C3C2×C3×#21311yescharacteristic
331C2×C3C3×#93211yescharacteristic
332C2×C3C3×#95311yescharacteristic
333D3—241no—
334C22—21no—
335C22—21no—
336C22—24no—
337C22—48no—
338C22—42no—
339C22—42no—
340C22—361no—
341C22—361no—
342C22—361no—
343C22#3673⋊C312yescomplemented
344C22C2×#611⋊C311yescharacteristic
345C4—21no—
346C4—361no—
347C4C2×#611⋊C312yes—
348C3—31no—
349C3C3×#521711yescharacteristic
350C2—22no—
351C2—44no—
352C2—721no—
353C2C2×#3673⋊C311yescharacteristic
354C2#63495⋊C311yescharacteristic
355C2#1172170411yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×D4×3−1+2C2
socleC22×C3C2×C3×#67
hypercenterC22×D4AGL(1,ℤ/9)
hypocenterC9C3×#447
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×3−1+2C2semidirect
2. C2×#3673⋊C3C2semidirect
3. #63492⋊C3C2semidirect
4. #63392⋊C3C2semidirect
5. #63389⋊C3C2semidirect
7. #10588898C3semidirect
14. C2×C4⋊AGL(1,ℤ/9)C22semidirect
17. C4⋊(3−1+2⋊C4)C22semidirect
18. C2×3−1+2⋊C8C22semidirect
23. C22×D4×C9C2×C3semidirect
26. C2×#3673C2×C3semidirect
27. #63492C2×C3semidirect
28. #63392C2×C3semidirect
29. #63389C2×C3semidirect
38. D4×3−1+2D4semidirect
70. C2×D36C22×C3semidirect
73. C4⋊Dic9C22×C3semidirect
74. C2×C9⋊C8C22×C3semidirect
81. C22×3−1+2D8semidirect
131. D4×C9C3×D4semidirect
199. C22×C9C3×D8semidirect
209. C22×D4AGL(1,ℤ/9)semidirect
214. 3−1+2#447semidirect
305. C9C3×#447semidirect
316. D4#611⋊C3semidirect
343. C22#3673⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×D4×3−1+22
C2×#3673⋊C32
#63492⋊C32
#63392⋊C32
#63389⋊C32
C3×#52173
#105888983