Finite Group Databasebeta

← C2×#63691⋊C3 #13164422

Subgroups — C2×#63691⋊C3 #13164422

Order 1728.

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

normal subgroups only

This group has 562 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
1C2×#88691C211yesmaximal, characteristic, complemented
2#63691⋊C3C214yesmaximal, direct product factor
3C2×#3689⋊C3C211yesmaximal, characteristic, complemented
4C2×#3688⋊C3C211yesmaximal, characteristic, complemented
5C2×C3×#950—31nomaximal
6C2×#63691C311yesmaximal, characteristic, complemented
7C2×C4×AGL(1,ℤ/9)—41nomaximal
8#88691C2212yescomplemented
9C2×#6204C2211yescharacteristic
10#3689⋊C3C2212yescomplemented
11#3688⋊C3C2212yescomplemented
12C2×C32×#41D311yescharacteristic
13C2×C3×#235—31no—
14C3×#950—34no—
15C3×#950—34no—
16C2×C3×#216—31no—
17C2×C3×#215—31no—
18C2×C3×#214—31no—
19C2×C3×#213—31no—
20C22×C3×C3⋊C8—31no—
21C2×#41⋊C9—21no—
22C2×#41⋊C9C2×C311yescharacteristic, complemented
23#63691C2×C314yescomplemented
24C2×#3689C2×C311yescharacteristic, complemented
25C2×#3688C2×C311yescharacteristic, complemented
26C22×AGL(1,ℤ/9)—41no—
27C2×C4×3−1+2—41no—
28C2×3−1+2⋊C4—41no—
29C4×AGL(1,ℤ/9)—44no—
30#6204C2311yescharacteristic
31C2×C3×#135—91no—
32C2×#950—31no—
33C32×#41—32no—
34C32×#41C2×D312yes—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
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—32no—
46C3×#215—32no—
47C3×#215—32no—
48C3×#214—34no—
49C3×#213—34no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52C2×C3×C3⋊C8—34no—
53C2×C4×D9—41no—
54#41⋊C9—22no—
55#41⋊C9C22×C312yescomplemented
56C2×Q8⋊C9—21no—
57C2×Q8⋊C9C22×C311yescharacteristic
58#3689C22×C312yescomplemented
59#3688C22×C312yescomplemented
60C22×3−1+2—41no—
61C2×AGL(1,ℤ/9)—42no—
62C2×AGL(1,ℤ/9)—44no—
63C4×3−1+2—42no—
643−1+2⋊C4—42no—
65C2×C3×#41—31no—
66C2×C3×#41—91no—
67C2×C3×#41C3×D311yescharacteristic
68C2×#235—31no—
69C3×#135—94no—
70C3×#135—94no—
71C2×C3×Q16—91no—
72C2×C3×SD16—91no—
73C2×C3×SD16—91no—
74C2×C3×D8—91no—
75C22×C3×C8—91no—
76#950—34no—
77#950—34no—
78C2×#216—31no—
79C2×#215—31no—
80C2×#214—31no—
81C2×#213—31no—
82C22×C3⋊C8—31no—
83C23×C32—31no—
84C22×C3×D3—61no—
85C32×Q8—31no—
86C32×Q8C22×D311yescharacteristic
87C32×D4—32no—
88C32×D4—61no—
89C2×C32×C4—31no—
90C2×C32×C4—32no—
91C2×C32×C4—32no—
92C2×C32×C4S411yescharacteristic
93C3×#67—64no—
94C2×C3×Dic3—61no—
95C3×D12—32no—
96C3×D12—61no—
97C3×C4×D3—64no—
98C3×Dic6—32no—
99C3×Dic6—61no—
100C22×D9—41no—
101C3×C3⋊C8—32no—
102C3×C3⋊C8—32no—
103C2×C4×C9—41no—
104C2×C4×C9—81no—
105C2×Dic9—41no—
106C4×D9—44no—
107Q8⋊C9—21no—
108Q8⋊C9C23×C311yescharacteristic
109C2×#135—91noSylow
110C2×3−1+2—42no—
111C2×3−1+2—41no—
112AGL(1,ℤ/9)—44no—
113C3×#41—32no—
114C3×#41—32no—
115C3×#41—32no—
116C3×#41—62no—
117C3×#41—94no—
118C3×#41—182no—
119C3×#41C2×C3×D312yes—
120C2×C3×Q8—31no—
121C2×C3×Q8—91no—
122C2×C3×Q8C2×C3×D311yescharacteristic
123C2×C3×D4—31no—
124C2×C3×D4—31no—
125C2×C3×D4—61no—
126C2×C3×D4—91no—
127C2×C3×D4—181no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—181no—
132C2×#67—61no—
133#235—34no—
134#235—62no—
135C2×D12—31no—
136C2×C4×D3—61no—
137C2×Dic6—31no—
138C3×Q16—92no—
139C3×Q16—92no—
140C3×SD16—94no—
141C3×SD16—92no—
142C3×SD16—92no—
143C3×D8—94no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C2×C3×C8—94no—
147#216—32no—
148#216—32no—
149#215—32no—
150#215—32no—
151#214—34no—
152#213—34no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—34no—
156C22×C32—32no—
157C22×C32—62no—
158C22×C32C2×S411yescharacteristic
159C2×C3×D3—62no—
160C2×C3×D3—122no—
161C32×C4—31no—
162C32×C4—31no—
163C32×C4C2×S412yes—
164C3×Dic3—62no—
165C22×C9—41no—
166C22×C9—81no—
167C2×D9—42no—
168C2×D9—44no—
169C4×C9—42no—
170C4×C9—82no—
171Dic9—42no—
172C2×#41—91no—
173C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
174#135—94no—
175#135—94no—
176C2×Q16—91no—
177C2×SD16—91no—
178C2×SD16—91no—
179C2×D8—91no—
180C22×C8—91no—
1813−1+2—41noSylow
182C23×C3—31no—
183C23×C3—31no—
184C23×C3—61no—
185C23×C3—181no—
186C22×D3—61no—
187C3×Q8—31no—
188C3×Q8—31no—
189C3×Q8—31no—
190C3×Q8—61no—
191C3×Q8—92no—
192C3×Q8—181no—
193C3×Q8C22×C3×D311yescharacteristic
194C3×D4—32no—
195C3×D4—32no—
196C3×D4—62no—
197C3×D4—61no—
198C3×D4—61no—
199C3×D4—62no—
200C3×D4—92no—
201C3×D4—181no—
202C3×D4—184no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—31no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—184no—
214C2×C3×C4—181no—
215C2×C3×C4C3×S411yescharacteristic
216#67—64no—
217C2×Dic3—61no—
218D12—32no—
219D12—61no—
220C4×D3—64no—
221Dic6—32no—
222Dic6—61no—
223C3×C8—92no—
224C3×C8—92no—
225C3⋊C8—32no—
226C3⋊C8—32no—
227C2×C32—62no—
228C2×C32#98512yes—
229C2×C32C22×S411yescharacteristic
230C3×D3—122no—
231C2×C9—41no—
232C2×C9—42no—
233C2×C9—81no—
234C2×C9—82no—
235D9—44no—
236#41—32no—
237#41—94no—
238#41—182no—
239#41C2×AGL(1,ℤ/9)12yescomplemented
240C2×Q8—91no—
241C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
242C2×D4—31no—
243C2×D4—91no—
244C2×D4—181no—
245C22×C4—31no—
246C22×C4—181no—
247Q16—92no—
248Q16—92no—
249SD16—94no—
250SD16—92no—
251SD16—92no—
252D8—94no—
253C2×C8—91no—
254C2×C8—91no—
255C2×C8—94no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—31no—
259C22×C3—62no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3—122no—
263C22×C3—182no—
264C22×C3—362no—
265C22×C3C2×C3×S411yescharacteristic
266C2×D3—62no—
267C2×D3—122no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—32no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—182no—
276C3×C4C2×C3×S412yes—
277Dic3—62no—
278C32C2×#98511yescharacteristic
279C9—41no—
280C9—81no—
281C23—31no—
282C23—181no—
283Q8—31no—
284Q8—92no—
285Q8—181no—
286Q8C22×AGL(1,ℤ/9)11yescharacteristic
287D4—32no—
288D4—61no—
289D4—92no—
290D4—181no—
291D4—184no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—181no—
296C2×C4—184no—
297C2×C4((C22⋊C9)⋊C2)⋊C311yescharacteristic
298C8—92no—
299C8—92no—
300C2×C3—31no—
301C2×C3—32no—
302C2×C3—62no—
303C2×C3—62no—
304C2×C3—122no—
305C2×C3—362no—
306C2×C3C3×#98512yes—
307C2×C3C22×C3×S411yescharacteristic
308D3—122no—
309C22—32no—
310C22—62no—
311C22—182no—
312C22—362no—
313C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
314C4—31no—
315C4—31no—
316C4—182no—
317C4C2×((C22⋊C9)⋊C2)⋊C312yes—
318C3—31no—
319C3C2×C3×#98511yescharacteristic
320C2—62no—
321C2—362no—
322C2#63691⋊C312yesdirect product factor
323C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC2×C32×#41D3
socleC22×C3C2×C3×S4
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88691C2semidirect
2. #63691⋊C3C2direct (N × H)
3. C2×#3689⋊C3C2semidirect
4. C2×#3688⋊C3C2semidirect
6. C2×#63691C3semidirect
8. #88691C22semidirect
10. #3689⋊C3C22semidirect
11. #3688⋊C3C22semidirect
22. C2×#41⋊C9C2×C3semidirect
23. #63691C2×C3semidirect
24. C2×#3689C2×C3semidirect
25. C2×#3688C2×C3semidirect
55. #41⋊C9C22×C3semidirect
58. #3689C22×C3semidirect
59. #3688C22×C3semidirect
173. C2×#41AGL(1,ℤ/9)semidirect
239. #41C2×AGL(1,ℤ/9)semidirect
322. C2#63691⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886912
#63691⋊C32
C2×#3689⋊C32
C2×#3688⋊C32
C2×C3×#9503
C2×#636913
C2×C4×AGL(1,ℤ/9)4