Finite Group Databasebeta

← C2×Dic6×C22⋊C9 #13164603

Subgroups — C2×Dic6×C22⋊C9 #13164603

Order 1728.

810 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 135 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×C4×C22⋊C9C211yesmaximal, characteristic
2C2×Dic3×C22⋊C9C212yesmaximal
3Dic6×C22⋊C9C214yesmaximal, direct product factor
4C23×C3×Dic6C311yesmaximal, characteristic
5C2×Q8×C22⋊C9—31nomaximal
6C22×C3×C22⋊C9C2211yescharacteristic
7C2×C9×Dic6—41nomaximal
8C3×C4×C22⋊C9C2212yes—
9Dic3×C22⋊C9C2214yes—
10C23×C32×C4C2×C311yescharacteristic
11C23×C3×Dic3C2×C312yes—
12C22×C3×Dic6—31no—
13C22×C3×Dic6—31no—
14C22×C3×Dic6—32no—
15C22×C3×Dic6—34no—
16C22×C3×Dic6C2×C314yes—
17Q8×C22⋊C9—34no—
18C2×C4×C22⋊C9—21no—
19C2×C4×C22⋊C9—32no—
20C2×C4×C22⋊C9D311yescharacteristic
21C2×C3×C22⋊C9Q812yescomplemented
22C2×C3×C22⋊C9C2311yescharacteristic
23C2×C3×C4×C9—41no—
24C2×C9×Dic3—42no—
25C9×Dic6—44no—
26C23×C3×Q8—31no—
27C23×Dic6C911yescharacteristic, complemented
28C24×C32C22×C311yescharacteristic
29C22×C32×C4—31no—
30C22×C32×C4—31no—
31C22×C32×C4—32no—
32C22×C32×C4C22×C312yes—
33C22×C3×Dic3—32no—
34C22×C3×Dic3—32no—
35C22×C3×Dic3—34no—
36C22×C3×Dic3C22×C314yes—
37C2×C3×Dic6—32no—
38C2×C3×Dic6—34no—
39C2×C3×Dic6—38no—
40C2×C3×Dic6—31no—
41C2×C3×Dic6—32no—
42C2×C3×Dic6—34no—
43C2×C3×Dic6—34no—
44C2×C3×Dic6—34no—
45C2×C3×Dic6—38no—
46C2×C3×Dic6A411yescharacteristic
47C22×C22⋊C9—21no—
48C22×C22⋊C9C2×D311yescharacteristic
49C2×Q8×C9—121no—
50C4×C22⋊C9—22no—
51C4×C22⋊C9—34no—
52C4×C22⋊C9C2×D312yes—
53C22×C3×C9—41no—
54C3×C22⋊C9C2×Q811yescharacteristic, complemented
55C3×C4×C9—42no—
56C9×Dic3—44no—
57C22×C3×Q8—34no—
58C22×C3×Q8—91no—
59C22×C3×Q8—91no—
60C22×C3×Q8—92no—
61C22×C3×Q8—94no—
62C23×C3×C4—21no—
63C23×C3×C4—32no—
64C23×C3×C4C2×C911yescharacteristic
65C23×C3×C4C3×D311yescharacteristic
66C23×Dic3C2×C912yes—
67C22×Dic6—31no—
68C22×Dic6—31no—
69C22×Dic6—32no—
70C22×Dic6—34no—
71C22×Dic6C2×C914yescomplemented
72C23×C32—32no—
73C23×C32—31no—
74C23×C32—31no—
75C23×C32C3×Q812yes—
76C23×C32C23×C311yescharacteristic
77C2×C32×C4—32no—
78C2×C32×C4—31no—
79C2×C32×C4—32no—
80C2×C32×C4—32no—
81C2×C32×C4—32no—
82C2×C32×C4C2×A411yescharacteristic
83C2×C3×Dic3—34no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3C2×A412yes—
89C3×Dic6—38no—
90C3×Dic6—34no—
91C3×Dic6—38no—
92C3×Dic6C2×A414yes—
93C2×C22⋊C9—22no—
94C2×C22⋊C9—21no—
95C2×C22⋊C9Dic612yesdirect product factor
96C2×C22⋊C9C22×D311yescharacteristic
97Q8×C9—124no—
98C2×C4×C9—41no—
99C2×C4×C9—81no—
100C2×C4×C9—122no—
101C23×Q8—31noSylow
102C2×C3×C9—42no—
103C2×C3×C9—41no—
104C24×C3—21no—
105C24×C3C22×C911yescharacteristic
106C24×C3C2×C3×D311yescharacteristic
107C2×C3×Q8—31no—
108C2×C3×Q8—92no—
109C2×C3×Q8—94no—
110C2×C3×Q8—94no—
111C2×C3×Q8—94no—
112C2×C3×Q8—98no—
113C2×C3×Q8—98no—
114C2×C3×Q8—92no—
115C2×C3×Q8—94no—
116C2×C3×Q8—91no—
117C22×C3×C4—22no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—32no—
122C22×C3×C4—31no—
123C22×C3×C4—32no—
124C22×C3×C4—34no—
125C22×C3×C4—61no—
126C22×C3×C4—62no—
127C22×C3×C4—61no—
128C22×C3×C4—92no—
129C22×C3×C4—94no—
130C22×C3×C4—92no—
131C22×C3×C4C22×C912yes—
132C22×C3×C4C2×C3×D312yes—
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—34no—
136C22×Dic3C22×C914yes—
137C2×Dic6—32no—
138C2×Dic6—34no—
139C2×Dic6—38no—
140C2×Dic6—31no—
141C2×Dic6—32no—
142C2×Dic6—34no—
143C2×Dic6—34no—
144C2×Dic6—34no—
145C2×Dic6—38no—
146C2×Dic6C22⋊C911yescharacteristic, direct product factor
147C22×C32—32no—
148C22×C32—31no—
149C22×C32—32no—
150C22×C32—32no—
151C22×C32—32no—
152C22×C32—31no—
153C22×C32—31no—
154C22×C32C2×C3×Q811yescharacteristic
155C22×C32C22×A411yescharacteristic
156C32×C4—32no—
157C32×C4C22×A412yes—
158C3×Dic3—34no—
159C3×Dic3C22×A414yes—
160C22×C9—41no—
161C22×C9—81no—
162C22⋊C9—21no—
163C22⋊C9C2×Dic611yescharacteristic, direct product factor
164C4×C9—42no—
165C4×C9—82no—
166C4×C9—124no—
167C22×Q8—34no—
168C22×Q8—91no—
169C22×Q8—91no—
170C22×Q8—92no—
171C22×Q8—94no—
172C23×C4—32no—
173C23×C4D3×C911yescharacteristic
174C3×C9—41noSylow
175C23×C3—22no—
176C23×C3—21no—
177C23×C3—31no—
178C23×C3—31no—
179C23×C3—32no—
180C23×C3—32no—
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3—61no—
185C23×C3—62no—
186C23×C3Q8×C912yescomplemented
187C23×C3C23×C911yescharacteristic
188C23×C3C3×Dic612yes—
189C23×C3C22×C3×D311yescharacteristic
190C3×Q8—34no—
191C3×Q8—98no—
192C3×Q8—98no—
193C3×Q8—94no—
194C2×C3×C4—21no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—32no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—94no—
212C2×C3×C4—94no—
213C2×C3×C4—94no—
214C2×C3×C4—92no—
215C2×C3×C4—94no—
216C2×C3×C4C2×C22⋊C911yescharacteristic
217C2×C3×C4D3×A411yescharacteristic
218C2×Dic3—34no—
219C2×Dic3—32no—
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—34no—
223C2×Dic3C2×C22⋊C912yes—
224Dic6—38no—
225Dic6—34no—
226Dic6—38no—
227Dic6C2×C22⋊C914yesdirect product factor
228C2×C32—32no—
229C2×C32—31no—
230C2×C32—31no—
231C2×C32Q8×A412yes—
232C2×C32C23×A411yescharacteristic
233C2×C9—42no—
234C2×C9—41no—
235C2×C9—82no—
236C2×C9—81no—
237C24C2×D3×C911yescharacteristic
238C2×Q8—31no—
239C2×Q8—92no—
240C2×Q8—94no—
241C2×Q8—94no—
242C2×Q8—94no—
243C2×Q8—98no—
244C2×Q8—98no—
245C2×Q8—92no—
246C2×Q8—94no—
247C2×Q8—91no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—34no—
252C22×C4—92no—
253C22×C4—94no—
254C22×C4—92no—
255C22×C4C2×D3×C912yes—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—32no—
259C22×C3—31no—
260C22×C3—32no—
261C22×C3—31no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—32no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—32no—
271C22×C3—32no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—62no—
275C22×C3—62no—
276C22×C3—62no—
277C22×C3—61no—
278C22×C3—62no—
279C22×C3C2×Q8×C911yescharacteristic, complemented
280C22×C3C22×C22⋊C911yescharacteristic
281C22×C3C2×C3×Dic611yescharacteristic
282C22×C3C2×D3×A411yescharacteristic
283C3×C4—22no—
284C3×C4—32no—
285C3×C4—34no—
286C3×C4—32no—
287C3×C4—62no—
288C3×C4—94no—
289C3×C4C22×C22⋊C912yes—
290C3×C4C2×D3×A412yes—
291Dic3—34no—
292Dic3C22×C22⋊C914yes—
293C32C2×Q8×A411yescharacteristic
294C9—41no—
295C9—81no—
296C23—31no—
297C23—32no—
298C23—31no—
299C23C9×Dic612yescomplemented
300C23C22×D3×C911yescharacteristic
301Q8—34no—
302Q8—98no—
303Q8—98no—
304Q8—94no—
305C2×C4—32no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—32no—
311C2×C4—94no—
312C2×C4—94no—
313C2×C4—94no—
314C2×C4—92no—
315C2×C4—94no—
316C2×C4D3×C22⋊C911yescharacteristic
317C2×C3—21no—
318C2×C3—22no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—32no—
324C2×C3—31no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—62no—
328C2×C3Q8×C22⋊C912yescomplemented
329C2×C3C23×C22⋊C911yescharacteristic
330C2×C3A4×Dic612yes—
331C2×C3C22×D3×A411yescharacteristic
332C22—32no—
333C22—31no—
334C22—32no—
335C22—31no—
336C22—31no—
337C22—32no—
338C22—32no—
339C22C2×C9×Dic611yescharacteristic, complemented
340C22C2×D3×C22⋊C911yescharacteristic
341C4—32no—
342C4—34no—
343C4—94no—
344C4C2×D3×C22⋊C912yes—
345C3—21no—
346C3C2×Q8×C22⋊C911yescharacteristic, complemented
347C3C2×A4×Dic611yescharacteristic
348C2—31no—
349C2—32no—
350C2—31no—
351C2Dic6×C22⋊C912yesdirect product factor
352C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutatorC23×C3C23×C9
FrattiniC2×C3C22×D3×A4
FittingC23×C32×C4C2×C3
socle, prime radicalC24×C32C22×C3
cosocleC23×C32C23×C3
hypercenterC2×C3×C4D3×A4
hypocenterC22×C3C2×Q8×C9
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. Dic6×C22⋊C9C2direct (N × H)
21. C2×C3×C22⋊C9Q8semidirect
27. C23×Dic6C9semidirect
54. C3×C22⋊C9C2×Q8semidirect
71. C22×Dic6C2×C9semidirect
95. C2×C22⋊C9Dic6direct (N × H)
146. C2×Dic6C22⋊C9direct (N × H)
163. C22⋊C9C2×Dic6direct (N × H)
186. C23×C3Q8×C9semidirect
227. Dic6C2×C22⋊C9direct (N × H)
279. C22×C3C2×Q8×C9semidirect
299. C23C9×Dic6semidirect
328. C2×C3Q8×C22⋊C9semidirect
339. C22C2×C9×Dic6semidirect
346. C3C2×Q8×C22⋊C9semidirect
351. C2Dic6×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C2×Dic3×C22⋊C92
Dic6×C22⋊C92
C23×C3×Dic63
C2×Q8×C22⋊C93
C2×C9×Dic64