Finite Group Databasebeta

← C2×#1010⋊C9 #13164627

Subgroups — C2×#1010⋊C9 #13164627

Order 1728.

779 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 117 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×#41⋊C9C211yesmaximal, characteristic, complemented
2#1010⋊C9C214yesmaximal, direct product factor
3C2×D3×Q8⋊C9C212yesmaximal, complemented
4C2×C3×#1010C311yesmaximal, characteristic
5C2×2−1+4⋊C9—31nomaximal
6C2×C9×D12—41nomaximal
7C3×#41⋊C9C2212yescomplemented
8C2×C3×Q8⋊C9C2211yescharacteristic
9D3×Q8⋊C9C2214yescomplemented
10C2×C32×#41C2×C311yescharacteristic
11C3×#1010—34no—
12C3×#1010C2×C314yes—
13C2×C3×D3×Q8C2×C312yes—
14C2×C3×#237—32no—
15C2×C3×#235—31no—
16C22×C3×Dic6—31no—
172−1+4⋊C9—34no—
18C2×#41⋊C9—21no—
19C2×#41⋊C9D311yescharacteristic, complemented
20C22×Q8⋊C9—32no—
21C22×D3×C9—42no—
22C2×C3×C4×C9—41no—
23C9×D12—44no—
24C3×Q8⋊C9C2311yescharacteristic
25C2×C3×2−1+4—31no—
26C2×#1010C911yescharacteristic, complemented
27C32×#41—32no—
28C32×#41C22×C312yes—
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—32no—
33C22×C3×Dic3—32no—
34C3×D3×Q8—34no—
35C3×D3×Q8C22×C314yes—
36C3×#237—38no—
37C3×#237—38no—
38C3×#235—34no—
39C3×#235—34no—
40C2×C3×D12A411yescharacteristic
41C2×C3×C4×D3—32no—
42C2×C3×Dic6—34no—
43C2×C3×Dic6—31no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—34no—
46C2×D4×C9—121no—
47#41⋊C9—22no—
48#41⋊C9C2×D312yescomplemented
49C2×Q8⋊C9—21no—
50C2×Q8⋊C9—34no—
51C2×Q8⋊C9C2×D311yescharacteristic
52C22×C3×C9—41no—
53C2×D3×C9—44no—
54C2×D3×C9—84no—
55C3×C4×C9—42no—
56C3×2−1+4—34no—
57C3×2−1+4—94no—
58C2×C3×#41—21no—
59C2×C3×#41—91no—
60C2×C3×#41—92no—
61C2×C3×#41C2×C911yescharacteristic, complemented
62C2×C3×#41C3×D311yescharacteristic
63C22×C3×Q8—32no—
64C22×C3×Q8—91no—
65#1010—34no—
66#1010C2×C914yescomplemented
67C2×D3×Q8C2×C912yescomplemented
68C2×#237—32no—
69C2×#235—31no—
70C22×Dic6—31no—
71C23×C32—31no—
72C22×C3×D3C2×A412yes—
73C32×Q8—31no—
74C32×Q8C23×C311yescharacteristic
75C32×D4—32no—
76C32×D4—32no—
77C2×C32×C4—31no—
78C2×C32×C4—32no—
79C2×C32×C4—32no—
80C2×C32×C4C2×A411yescharacteristic
81C3×#67—38no—
82C2×C3×Dic3—38no—
83C2×C3×Dic3—32no—
84C3×D12C2×A414yes—
85C3×C4×D3—34no—
86C3×C4×D3—34no—
87C3×Dic6—34no—
88C3×Dic6—34no—
89C3×Dic6—34no—
90C23×C9—122no—
91D4×C9—124no—
92C2×C4×C9—41no—
93C2×C4×C9—81no—
94Q8⋊C9—21no—
95Q8⋊C9C22×D311yescharacteristic
96C2×2−1+4—31noSylow
97C2×C3×C9—42no—
98C2×C3×C9—41no—
99D3×C9—84no—
100C3×#41—22no—
101C3×#41—32no—
102C3×#41—32no—
103C3×#41—62no—
104C3×#41—94no—
105C3×#41—94no—
106C3×#41—98no—
107C3×#41—98no—
108C3×#41C22×C912yescomplemented
109C3×#41C2×C3×D312yes—
110C2×C3×Q8—21no—
111C2×C3×Q8—34no—
112C2×C3×Q8—94no—
113C2×C3×Q8—94no—
114C2×C3×Q8—92no—
115C2×C3×Q8—91no—
116C2×C3×Q8—94no—
117C2×C3×Q8C22×C911yescharacteristic
118C2×C3×Q8C2×C3×D311yescharacteristic
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—61no—
123C2×C3×D4—92no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—61no—
127C22×C3×C4—92no—
128C22×C3×C4—92no—
129C2×#67—32no—
130C22×Dic3—32no—
131D3×Q8—34no—
132D3×Q8C22×C914yescomplemented
133#237—38no—
134#237—38no—
135#235—34no—
136#235—34no—
137C2×D12C22⋊C911yescharacteristic
138C2×C4×D3—32no—
139C2×Dic6—34no—
140C2×Dic6—31no—
141C2×Dic6—32no—
142C2×Dic6—34no—
143C22×C32—32no—
144C22×C32—62no—
145C22×C32C22×A411yescharacteristic
146C2×C3×D3—24no—
147C2×C3×D3C22×A414yes—
148C32×C4—31no—
149C32×C4—31no—
150C32×C4C22×A412yes—
151C3×Dic3—34no—
152C22×C9—41no—
153C22×C9—81no—
154C22×C9—124no—
155C22×C9—244no—
156C4×C9—42no—
157C4×C9—82no—
1582−1+4—34no—
1592−1+4—94no—
160C2×#41—91no—
161C2×#41—92no—
162C2×#41D3×C911yescharacteristic, complemented
163C22×Q8—32no—
164C22×Q8—91no—
165C3×C9—41noSylow
166C23×C3—32no—
167C23×C3—31no—
168C23×C3—31no—
169C23×C3—61no—
170C22×D3C2×C22⋊C912yes—
171C3×Q8—21no—
172C3×Q8—31no—
173C3×Q8—31no—
174C3×Q8—61no—
175C3×Q8—94no—
176C3×Q8—94no—
177C3×Q8—94no—
178C3×Q8C23×C911yescharacteristic
179C3×Q8C22×C3×D311yescharacteristic
180C3×D4—32no—
181C3×D4—32no—
182C3×D4—32no—
183C3×D4—32no—
184C3×D4—34no—
185C3×D4—62no—
186C3×D4—62no—
187C3×D4—98no—
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—61no—
198C2×C3×C4—94no—
199C2×C3×C4—94no—
200C2×C3×C4—98no—
201C2×C3×C4—92no—
202C2×C3×C4C2×C22⋊C911yescharacteristic
203C2×C3×C4D3×A411yescharacteristic
204#67—38no—
205C2×Dic3—38no—
206C2×Dic3—32no—
207D12C2×C22⋊C914yes—
208C4×D3—34no—
209C4×D3—34no—
210Dic6—34no—
211Dic6—34no—
212Dic6—34no—
213C2×C32—62no—
214C2×C322−1+4⋊C312yes—
215C2×C32C23×A411yescharacteristic
216C3×D3—24no—
217C2×C9—42no—
218C2×C9—41no—
219C2×C9—82no—
220C2×C9—81no—
221C2×C9—244no—
222#41—32no—
223#41—94no—
224#41—94no—
225#41—98no—
226#41—98no—
227#41C2×D3×C912yescomplemented
228C2×Q8—34no—
229C2×Q8—94no—
230C2×Q8—94no—
231C2×Q8—92no—
232C2×Q8—91no—
233C2×Q8—94no—
234C2×Q8C2×D3×C911yescharacteristic
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—92no—
238C22×C4—31no—
239C22×C4—92no—
240C22×C4—92no—
241C22×C3—21no—
242C22×C3—32no—
243C22×C3—34no—
244C22×C3—32no—
245C22×C3—62no—
246C22×C3—64no—
247C22×C3—62no—
248C22×C3—62no—
249C22×C3—122no—
250C22×C3C22×C22⋊C911yescharacteristic
251C22×C3C2×D3×A411yescharacteristic
252C2×D3—24no—
253C2×D3C22×C22⋊C914yes—
254C3×C4—22no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—94no—
262C3×C4C22×C22⋊C912yes—
263C3×C4C2×D3×A412yes—
264Dic3—34no—
265C32C2×2−1+4⋊C311yescharacteristic
266C9—41no—
267C9—81no—
268C23—31no—
269C23—32no—
270Q8—31no—
271Q8—94no—
272Q8—94no—
273Q8—94no—
274Q8C22×D3×C911yescharacteristic
275D4—32no—
276D4—32no—
277D4—34no—
278D4—98no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—94no—
283C2×C4—94no—
284C2×C4—98no—
285C2×C4—92no—
286C2×C4D3×C22⋊C911yescharacteristic
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—64no—
290C2×C3—62no—
291C2×C3—62no—
292C2×C3—122no—
293C2×C32−1+4⋊C912yescomplemented
294C2×C3C23×C22⋊C911yescharacteristic
295C2×C3#1010⋊C312yes—
296C2×C3C22×D3×A411yescharacteristic
297D3—24no—
298C22—32no—
299C22—34no—
300C22—64no—
301C22—62no—
302C22C2×D3×C22⋊C911yescharacteristic
303C4—31no—
304C4—31no—
305C4—94no—
306C4C2×D3×C22⋊C912yes—
307C3—21no—
308C3C2×2−1+4⋊C911yescharacteristic, complemented
309C3C2×#1010⋊C311yescharacteristic
310C2—64no—
311C2—62no—
312C2#1010⋊C912yesdirect product factor
313C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutator, hypocenterC3×Q8C23×C9
FrattiniC2×C3C22×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×#1010C3
cosocleC32×Q8C23×C3
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. #1010⋊C9C2direct (N × H)
3. C2×D3×Q8⋊C9C2semidirect
7. C3×#41⋊C9C22semidirect
9. D3×Q8⋊C9C22semidirect
19. C2×#41⋊C9D3semidirect
26. C2×#1010C9semidirect
48. #41⋊C9C2×D3semidirect
61. C2×C3×#41C2×C9semidirect
66. #1010C2×C9semidirect
67. C2×D3×Q8C2×C9semidirect
108. C3×#41C22×C9semidirect
132. D3×Q8C22×C9semidirect
162. C2×#41D3×C9semidirect
227. #41C2×D3×C9semidirect
293. C2×C32−1+4⋊C9semidirect
308. C3C2×2−1+4⋊C9semidirect
312. C2#1010⋊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×#41⋊C92
#1010⋊C92
C2×D3×Q8⋊C92
C2×C3×#10103
C2×2−1+4⋊C93
C2×C9×D124