Finite Group Databasebeta

← C2×#63689⋊C3 #13164409

Subgroups — C2×#63689⋊C3 #13164409

Order 1728.

566 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 71 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×#6204C211yesmaximal, characteristic, complemented
2#63689⋊C3C214yesmaximal, direct product factor
3C2×#3689⋊C3C211yesmaximal, characteristic, complemented
4C2×#3688⋊C3C211yesmaximal, characteristic, complemented
5C2×C3×#942—31nomaximal
6C2×#63689C311yesmaximal, characteristic, complemented
7C2×#611⋊C3—41nomaximal
8C2×#6204C2211yescharacteristic
9C2×#6204C2212yescomplemented
10#3689⋊C3C2212yescomplemented
11#3688⋊C3C2212yescomplemented
12C22×C32×Q8D311yescharacteristic
13C2×C3×#235—31no—
14C2×C3×#216—31no—
15C2×C3×#216—31no—
16C3×#942—34no—
17C3×#942—34no—
18C2×C3×#215—31no—
19C2×C3×#215—31no—
20C2×C3×#208—31no—
21C22×Q8⋊C9—21no—
22C22×Q8⋊C9C2×C311yescharacteristic, complemented
23#63689C2×C314yescomplemented
24C2×#3689C2×C311yescharacteristic, complemented
25C2×#3688C2×C311yescharacteristic, complemented
26C23×3−1+2—41no—
27C22×AGL(1,ℤ/9)—41no—
28#611⋊C3—44no—
29C2×3−1+2⋊C4—41no—
30#6204C2311yescharacteristic
31C2×C3×#137—91no—
32C2×#942—31no—
33C2×C32×Q8—31no—
34C2×C32×Q8—32no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×Q8C2×D312yes—
37C22×C32×C4—31no—
38C2×C3×#67—61no—
39C3×#235—34no—
40C3×#235—62no—
41C2×C3×D12—31no—
42C2×C3×C4×D3—61no—
43C2×C3×Dic6—31no—
44C3×#216—32no—
45C3×#216—34no—
46C3×#216—32no—
47C3×#215—34no—
48C3×#215—32no—
49C3×#215—32no—
50C3×#208—34no—
51C2×C3×C3⋊C8—31no—
52C2×C3×C3⋊C8—31no—
53C2×#611—41no—
54C2×Q8⋊C9—21no—
55C2×Q8⋊C9—22no—
56C2×Q8⋊C9C22×C311yescharacteristic
57C2×Q8⋊C9C22×C312yescomplemented
58#3689C22×C312yescomplemented
59#3688C22×C312yescomplemented
60C22×3−1+2—41no—
61C22×3−1+2—42no—
62C22×3−1+2—82no—
63C2×AGL(1,ℤ/9)—42no—
64C2×AGL(1,ℤ/9)—82no—
653−1+2⋊C4—42no—
66C2×C3×#41—91no—
67C22×C3×Q8—31no—
68C22×C3×Q8C3×D311yescharacteristic
69C2×#235—31no—
70C3×#137—94no—
71C3×#137—94no—
72C2×C3×Q16—91no—
73C2×C3×Q16—91no—
74C2×C3×SD16—91no—
75C2×C3×SD16—91no—
76C2×C3×M16—91no—
77C2×#216—31no—
78C2×#216—31no—
79#942—34no—
80#942—34no—
81C2×#215—31no—
82C2×#215—31no—
83C2×#208—31no—
84C23×C32S411yescharacteristic
85C22×C3×D3—61no—
86C32×Q8—31no—
87C32×Q8—32no—
88C32×Q8—61no—
89C32×Q8C22×D311yescharacteristic
90C2×C32×C4—31no—
91C2×C32×C4—32no—
92C2×C32×C4—31no—
93C2×C32×C4—32no—
94C3×#67—64no—
95C2×C3×Dic3—61no—
96C3×D12—32no—
97C3×D12—61no—
98C3×C4×D3—64no—
99C3×Dic6—32no—
100C3×Dic6—61no—
101C23×C9—41no—
102C23×C9—81no—
103C22×D9—41no—
104C3×C3⋊C8—32no—
105C3×C3⋊C8—32no—
106#611—44no—
107C2×Dic9—41no—
108Q8⋊C9—21no—
109Q8⋊C9C23×C311yescharacteristic
110C2×#137—91noSylow
111C2×3−1+2—42no—
112C2×3−1+2—41no—
113C2×3−1+2—82no—
114AGL(1,ℤ/9)—82no—
115C3×#41—94no—
116C3×#41—182no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8—31no—
120C2×C3×Q8—32no—
121C2×C3×Q8—31no—
122C2×C3×Q8—32no—
123C2×C3×Q8—61no—
124C2×C3×Q8—62no—
125C2×C3×Q8—91no—
126C2×C3×Q8C2×C3×D311yescharacteristic
127C2×C3×Q8C2×C3×D312yes—
128C2×C3×D4—91no—
129C2×C3×D4—181no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C22×C3×C4—181no—
134C2×#67—61no—
135#235—34no—
136#235—62no—
137C2×D12—31no—
138C2×C4×D3—61no—
139C2×Dic6—31no—
140C3×Q16—92no—
141C3×Q16—94no—
142C3×Q16—92no—
143C3×SD16—92no—
144C3×SD16—94no—
145C3×SD16—92no—
146C3×M16—94no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149#216—32no—
150#216—34no—
151#216—32no—
152#215—34no—
153#215—32no—
154#215—32no—
155#208—34no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C22×C32—22no—
159C22×C32C2×S412yes—
160C22×C32C2×S411yescharacteristic
161C2×C3×D3—62no—
162C2×C3×D3—122no—
163C32×C4—31no—
164C32×C4—31no—
165C32×C4—32no—
166C3×Dic3—62no—
167C22×C9—41no—
168C22×C9—42no—
169C22×C9—82no—
170C22×C9—84no—
171C22×C9—82no—
172C22×C9—81no—
173C2×D9—42no—
174C2×D9—82no—
175Dic9—42no—
176C2×#41—91no—
177C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
178#137—94no—
179#137—94no—
180C2×Q16—91no—
181C2×Q16—91no—
182C2×SD16—91no—
183C2×SD16—91no—
184C2×M16—91no—
1853−1+2—41noSylow
186C23×C3—31no—
187C23×C3—181no—
188C23×C3C3×S411yescharacteristic
189C22×D3—61no—
190C3×Q8—31no—
191C3×Q8—32no—
192C3×Q8—31no—
193C3×Q8—32no—
194C3×Q8—31no—
195C3×Q8—61no—
196C3×Q8—62no—
197C3×Q8—61no—
198C3×Q8—62no—
199C3×Q8—61no—
200C3×Q8—92no—
201C3×Q8—181no—
202C3×Q8C22×C3×D311yescharacteristic
203C3×D4—92no—
204C3×D4—181no—
205C3×D4—184no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—31no—
209C2×C3×C4—32no—
210C2×C3×C4—31no—
211C2×C3×C4—32no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—184no—
219C2×C3×C4—181no—
220#67—64no—
221C2×Dic3—61no—
222D12—32no—
223D12—61no—
224C4×D3—64no—
225Dic6—32no—
226Dic6—61no—
227C3×C8—92no—
228C3×C8—92no—
229C3⋊C8—32no—
230C3⋊C8—32no—
231C2×C32—22no—
232C2×C32#98312yes—
233C2×C32C22×S411yescharacteristic
234C3×D3—122no—
235C2×C9—41no—
236C2×C9—42no—
237C2×C9—82no—
238C2×C9—84no—
239C2×C9—81no—
240C2×C9—82no—
241D9—82no—
242#41—94no—
243#41—182no—
244C2×Q8—31no—
245C2×Q8—32no—
246C2×Q8—91no—
247C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
248C2×Q8C2×AGL(1,ℤ/9)12yescomplemented
249C2×D4—91no—
250C2×D4—181no—
251C22×C4—31no—
252C22×C4—181no—
253Q16—92no—
254Q16—94no—
255Q16—92no—
256SD16—92no—
257SD16—94no—
258SD16—92no—
259M16—94no—
260C2×C8—91no—
261C2×C8—91no—
262C22×C3—22no—
263C22×C3—32no—
264C22×C3—31no—
265C22×C3—62no—
266C22×C3—182no—
267C22×C3—362no—
268C22×C3C2×C3×S412yes—
269C22×C3C2×C3×S411yescharacteristic
270C2×D3—62no—
271C2×D3—122no—
272C3×C4—31no—
273C3×C4—32no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—32no—
278C3×C4—61no—
279C3×C4—62no—
280C3×C4—61no—
281C3×C4—182no—
282Dic3—62no—
283C32C2×#98311yescharacteristic
284C9—41no—
285C9—81no—
286C23—181no—
287C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
288Q8—31no—
289Q8—32no—
290Q8—61no—
291Q8—92no—
292Q8—181no—
293Q8C22×AGL(1,ℤ/9)11yescharacteristic
294D4—92no—
295D4—181no—
296D4—184no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—184no—
302C2×C4—181no—
303C8—92no—
304C8—92no—
305C2×C3—22no—
306C2×C3—31no—
307C2×C3—32no—
308C2×C3—62no—
309C2×C3—362no—
310C2×C3C3×#98312yes—
311C2×C3C22×C3×S411yescharacteristic
312D3—122no—
313C22—22no—
314C22—182no—
315C22—362no—
316C22C2×((C22⋊C9)⋊C2)⋊C312yes—
317C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
318C4—31no—
319C4—32no—
320C4—31no—
321C4—182no—
322C3—31no—
323C3C2×C3×#98311yescharacteristic
324C2—22no—
325C2—362no—
326C2#63689⋊C312yesdirect product factor
327C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC22×C3C2×C3×S4
hypercenterC23((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6204C2semidirect
2. #63689⋊C3C2direct (N × H)
3. C2×#3689⋊C3C2semidirect
4. C2×#3688⋊C3C2semidirect
6. C2×#63689C3semidirect
9. C2×#6204C22semidirect
10. #3689⋊C3C22semidirect
11. #3688⋊C3C22semidirect
22. C22×Q8⋊C9C2×C3semidirect
23. #63689C2×C3semidirect
24. C2×#3689C2×C3semidirect
25. C2×#3688C2×C3semidirect
57. C2×Q8⋊C9C22×C3semidirect
58. #3689C22×C3semidirect
59. #3688C22×C3semidirect
177. C22×Q8AGL(1,ℤ/9)semidirect
248. C2×Q8C2×AGL(1,ℤ/9)semidirect
326. C2#63689⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62042
#63689⋊C32
C2×#3689⋊C32
C2×#3688⋊C32
C2×C3×#9423
C2×#636893
C2×#611⋊C34