Finite Group Databasebeta

← C2×C3⋊(#120⋊C9) #13147305

Subgroups — C2×C3⋊(#120⋊C9) #13147305

Order 1728.

678 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 43 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×C3×C22⋊(C22⋊C9)C211yesmaximal, characteristic, complemented
2C3⋊(#120⋊C9)C212yesmaximal, direct product factor
3C2×C3×#953C311yesmaximal, characteristic
4C2×#120⋊C9—31nomaximal
5C3×C22⋊(C22⋊C9)C2211yescharacteristic, complemented
6C2×D3×C22⋊C9—41nomaximal
7C25×C32C2×C311yescharacteristic
8C22×C3×#67—31no—
9C2×C22⋊(C22⋊C9)—21no—
10C2×C22⋊(C22⋊C9)D311yescharacteristic, complemented
11C3×#953—32no—
12C3×#953C2×C312yes—
13C2×C3×#217—31no—
14#120⋊C9—32no—
15C2×C3×C22⋊C9—41no—
16C2×C3×C22⋊C9—82no—
17D3×C22⋊C9—42no—
18C2×C3×#120—31no—
19C2×#953C911yescharacteristic, complemented
20C24×C32—31no—
21C24×C32—31no—
22C24×C32—62no—
23C24×C32—62no—
24C24×C32C22×C311yescharacteristic
25C23×C3×D3A411yescharacteristic
26C2×C3×#67—32no—
27C2×C3×#67—32no—
28C2×C3×#67—62no—
29C2×C3×#67—62no—
30C22×C3×Dic3—31no—
31C22⋊(C22⋊C9)—21no—
32C22⋊(C22⋊C9)C2×D311yescharacteristic, complemented
33C22×C22⋊C9—121no—
34C3×#217—32no—
35C3×#217—32no—
36C2×D3×C9—161no—
37C3×C22⋊C9—41no—
38C3×C22⋊C9—82no—
39C25×C3—21no—
40C25×C3C2×C911yescharacteristic, complemented
41C25×C3C3×D311yescharacteristic
42C22×C3×D4—91no—
43C22×#67—31no—
44C3×#120—32no—
45C3×#120—92no—
46C2×C3×C22⋊C4—91no—
47#953—32no—
48#953C2×C912yescomplemented
49C2×#217—31no—
50C23×C32—22no—
51C23×C32—32no—
52C23×C32—32no—
53C23×C32—31no—
54C23×C32—31no—
55C23×C32—62no—
56C23×C32—62no—
57C23×C32—62no—
58C23×C32—62no—
59C23×C32—62no—
60C23×C32—64no—
61C23×C32—62no—
62C23×C32—62no—
63C23×C32—64no—
64C23×C32C2×A411yescharacteristic
65C22×C3×D3—61no—
66C22×C3×D3—61no—
67C22×C3×D3C2×A412yes—
68C3×#67—64no—
69C3×#67—64no—
70C2×C3×Dic3—32no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C22⋊C9—41no—
74C2×C22⋊C9—82no—
75C2×C22⋊C9—84no—
76C2×C22⋊C9—81no—
77C2×C22⋊C9—122no—
78C2×#120—31noSylow
79C2×C3×C9—161no—
80D3×C9—162no—
81C24×C3—21no—
82C24×C3—31no—
83C24×C3—31no—
84C24×C3—31no—
85C24×C3—31no—
86C24×C3—31no—
87C24×C3—61no—
88C24×C3—61no—
89C24×C3—62no—
90C24×C3—62no—
91C24×C3—64no—
92C24×C3—62no—
93C24×C3—62no—
94C24×C3—64no—
95C24×C3C22×C911yescharacteristic, complemented
96C24×C3C2×C3×D311yescharacteristic
97C23×D3C22⋊C911yescharacteristic, complemented
98C2×C3×D4—92no—
99C2×C3×D4—92no—
100C2×C3×D4—182no—
101C2×C3×D4—182no—
102C22×C3×C4—91no—
103C2×#67—32no—
104C2×#67—32no—
105C2×#67—62no—
106C2×#67—62no—
107C22×Dic3—31no—
108C3×C22⋊C4—92no—
109C3×C22⋊C4—92no—
110#217—32no—
111#217—32no—
112C22×C32—22no—
113C22×C32—31no—
114C22×C32—31no—
115C22×C32—32no—
116C22×C32—32no—
117C22×C32—62no—
118C22×C32—62no—
119C22×C32—64no—
120C22×C32—62no—
121C22×C32—62no—
122C22×C32—62no—
123C22×C32—62no—
124C22×C32—64no—
125C22×C32—62no—
126C22×C32C22×A411yescharacteristic
127C2×C3×D3—41no—
128C2×C3×D3—62no—
129C2×C3×D3—121no—
130C3×Dic3—62no—
131C22×C9—481no—
132C22⋊C9—41no—
133C22⋊C9—82no—
134C22⋊C9—84no—
135C22⋊C9—81no—
136C25D3×C911yescharacteristic, complemented
137C22×D4—91no—
138#120—32no—
139#120—92no—
140C2×C22⋊C4—91no—
141C3×C9—161noSylow
142C23×C3—22no—
143C23×C3—24no—
144C23×C3—22no—
145C23×C3—21no—
146C23×C3—32no—
147C23×C3—32no—
148C23×C3—32no—
149C23×C3—32no—
150C23×C3—31no—
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—32no—
155C23×C3—61no—
156C23×C3—61no—
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—62no—
161C23×C3—64no—
162C23×C3—62no—
163C23×C3—62no—
164C23×C3—64no—
165C23×C3—62no—
166C23×C3—62no—
167C23×C3—62no—
168C23×C3—62no—
169C23×C3—62no—
170C23×C3—64no—
171C23×C3—62no—
172C23×C3—64no—
173C23×C3—62no—
174C23×C3—62no—
175C23×C3—62no—
176C23×C3—62no—
177C23×C3—64no—
178C23×C3—64no—
179C23×C3—64no—
180C23×C3—68no—
181C23×C3—64no—
182C23×C3—64no—
183C23×C3—64no—
184C23×C3—64no—
185C23×C3—68no—
186C23×C3—181no—
187C23×C3—181no—
188C23×C3C2×C22⋊C911yescharacteristic
189C23×C3D3×A411yescharacteristic
190C22×D3—61no—
191C22×D3—61no—
192C22×D3C2×C22⋊C912yescomplemented
193C3×D4—184no—
194C3×D4—184no—
195C2×C3×C4—92no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198#67—64no—
199#67—64no—
200C2×Dic3—32no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×C32—31no—
204C2×C32—31no—
205C2×C32—62no—
206C2×C32—62no—
207C2×C32#120⋊C311yescharacteristic
208C3×D3—42no—
209C2×C9—161no—
210C2×C9—321no—
211C2×C9—482no—
212C24—31no—
213C24—31no—
214C24—31no—
215C24—62no—
216C24—62no—
217C24C2×D3×C911yescharacteristic, complemented
218C2×D4—92no—
219C2×D4—92no—
220C2×D4—182no—
221C2×D4—182no—
222C22×C4—91no—
223C22⋊C4—92no—
224C22⋊C4—92no—
225C22×C3—22no—
226C22×C3—24no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—32no—
234C22×C3—32no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—62no—
240C22×C3—62no—
241C22×C3—64no—
242C22×C3—62no—
243C22×C3—62no—
244C22×C3—64no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—62no—
248C22×C3—64no—
249C22×C3—62no—
250C22×C3—64no—
251C22×C3—62no—
252C22×C3—62no—
253C22×C3—62no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—64no—
258C22×C3—64no—
259C22×C3—64no—
260C22×C3—68no—
261C22×C3—64no—
262C22×C3—64no—
263C22×C3—68no—
264C22×C3—64no—
265C22×C3—64no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—121no—
269C22×C3—182no—
270C22×C3—361no—
271C22×C3C22×C22⋊C911yescharacteristic
272C22×C3C2×D3×A411yescharacteristic
273C2×D3—41no—
274C2×D3—62no—
275C2×D3—121no—
276C3×C4—182no—
277Dic3—62no—
278C32C2×#120⋊C311yescharacteristic
279C9—161no—
280C9—321no—
281C23—22no—
282C23—32no—
283C23—32no—
284C23—31no—
285C23—31no—
286C23—32no—
287C23—62no—
288C23—62no—
289C23—62no—
290C23—62no—
291C23—62no—
292C23—64no—
293C23—62no—
294C23—62no—
295C23—64no—
296C23—181no—
297C23—181no—
298C23D3×C22⋊C911yescharacteristic
299D4—184no—
300D4—184no—
301C2×C4—92no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C3—21no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3—64no—
312C2×C3—64no—
313C2×C3—62no—
314C2×C3—62no—
315C2×C3—62no—
316C2×C3—62no—
317C2×C3—122no—
318C2×C3#120⋊C911yescharacteristic, complemented
319C2×C3C3⋊(#120⋊C3)11yescharacteristic
320D3—42no—
321C22—22no—
322C22—31no—
323C22—31no—
324C22—32no—
325C22—32no—
326C22—62no—
327C22—62no—
328C22—64no—
329C22—62no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22—64no—
334C22—62no—
335C22—121no—
336C22—182no—
337C22—361no—
338C22C2×D3×C22⋊C911yescharacteristic
339C4—182no—
340C3—21no—
341C3C2×#120⋊C911yescharacteristic, complemented
342C3C2×C3⋊(#120⋊C3)11yescharacteristic
343C2—31no—
344C2—31no—
345C2—62no—
346C2—62no—
347C2—122no—
348C2C3⋊(#120⋊C9)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊(#120⋊C3)
commutator, hypocenterC24×C3C22×C9
FrattiniC22×C3C2×D3×A4
FittingC25×C32C2×C3
socleC23×C32C2×A4
prime radicalC2×C3×#953C3
cosocleC24×C32C22×C3
LCSG ▷ C24×C3
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊(C22⋊C9)C2semidirect
2. C3⋊(#120⋊C9)C2direct (N × H)
5. C3×C22⋊(C22⋊C9)C22semidirect
10. C2×C22⋊(C22⋊C9)D3semidirect
19. C2×#953C9semidirect
32. C22⋊(C22⋊C9)C2×D3semidirect
40. C25×C3C2×C9semidirect
48. #953C2×C9semidirect
95. C24×C3C22×C9semidirect
97. C23×D3C22⋊C9semidirect
136. C25D3×C9semidirect
192. C22×D3C2×C22⋊C9semidirect
217. C24C2×D3×C9semidirect
318. C2×C3#120⋊C9semidirect
341. C3C2×#120⋊C9semidirect
348. C2C3⋊(#120⋊C9)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊(C22⋊C9)2
C3⋊(#120⋊C9)2
C2×C3×#9533
C2×#120⋊C93
C2×D3×C22⋊C94