Finite Group Databasebeta

← C2×C3⋊#63700 #13164625

Subgroups — C2×C3⋊#63700 #13164625

Order 1728.

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

normal subgroups only

This group has 811 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
2C3⋊#63700C214yesmaximal, direct product factor
3C2×C3⋊(#41⋊C9)C212yesmaximal, complemented
4C2×C3×#1009C311yesmaximal, characteristic
5C2×#63700—31nomaximal
6C2×C9×Dic6—41nomaximal
7C3×#41⋊C9C2212yescomplemented
8C2×C3×Q8⋊C9C2211yescharacteristic, complemented
9C3⋊(#41⋊C9)C2214yescomplemented
10C2×C32×#41C2×C311yescharacteristic
11C3×#1009—34no—
12C3×#1009C2×C314yes—
13C2×C3×#239C2×C312yes—
14C2×C3×D3×D4—32no—
15C2×C3×#235—31no—
16C22×C3×D12—31no—
17#63700—34no—
18C2×#41⋊C9—21no—
19C2×#41⋊C9—32no—
20C2×#41⋊C9D311yescharacteristic, complemented
21C2×C3×C4×C9—41no—
22C2×C9×Dic3—42no—
23C9×Dic6—44no—
24C3×Q8⋊C9C2311yescharacteristic, complemented
25C2×C3×H(5,𝔽2)—31no—
26C2×#1009C911yescharacteristic, complemented
27C23×C3×D3—32no—
28C32×#41—32no—
29C32×#41C22×C312yes—
30C2×C32×Q8C22×C311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×C3×#67—32no—
34C3×#239—34no—
35C3×#239C22×C314yes—
36C3×D3×D4—38no—
37C3×D3×D4—38no—
38C3×#235—34no—
39C3×#235—34no—
40C2×C3×D12—34no—
41C2×C3×D12—32no—
42C2×C3×D12—31no—
43C2×C3×D12—34no—
44C2×C3×C4×D3—32no—
45C2×C3×Dic6A411yescharacteristic
46C2×Q8×C9—121no—
47#41⋊C9—22no—
48#41⋊C9—34no—
49#41⋊C9C2×D312yescomplemented
50C2×Q8⋊C9—21no—
51C2×Q8⋊C9C2×D311yescharacteristic, complemented
52C22×C3×C9—41no—
53C3×C4×C9—42no—
54C9×Dic3—44no—
55C3×H(5,𝔽2)—34no—
56C3×H(5,𝔽2)—94no—
57C2×C3×#41—21no—
58C2×C3×#41—32no—
59C2×C3×#41—91no—
60C2×C3×#41C2×C911yescharacteristic
61C2×C3×#41C3×D311yescharacteristic
62C22×C3×D4—91no—
63C22×C3×D4—92no—
64#1009—34no—
65#1009C2×C914yescomplemented
66C2×#239C2×C912yes—
67C2×D3×D4—32no—
68C2×#235—31no—
69C22×D12—31no—
70C23×C32—31no—
71C22×C3×D3—32no—
72C22×C3×D3—38no—
73C22×C3×D3—124no—
74C32×Q8—31no—
75C32×Q8C23×C311yescharacteristic
76C32×D4—32no—
77C32×D4—32no—
78C2×C32×C4—31no—
79C2×C32×C4—32no—
80C2×C32×C4—32no—
81C2×C32×C4C2×A411yescharacteristic
82C3×#67—38no—
83C2×C3×Dic3C2×A412yes—
84C3×D12—34no—
85C3×D12—34no—
86C3×D12—34no—
87C3×C4×D3—34no—
88C3×C4×D3—34no—
89C3×Dic6C2×A414yes—
90Q8×C9—124no—
91C2×C4×C9—41no—
92C2×C4×C9—81no—
93C2×C4×C9—122no—
94Q8⋊C9—21no—
95Q8⋊C9C22×D311yescharacteristic, complemented
96C2×H(5,𝔽2)—31noSylow
97C2×C3×C9—42no—
98C2×C3×C9—41no—
99C24×C3—92no—
100C23×D3—32no—
101C3×#41—22no—
102C3×#41—32no—
103C3×#41—32no—
104C3×#41—34no—
105C3×#41—62no—
106C3×#41—94no—
107C3×#41—94no—
108C3×#41—94no—
109C3×#41C22×C912yes—
110C3×#41C2×C3×D312yes—
111C2×C3×Q8—21no—
112C2×C3×Q8—31no—
113C2×C3×Q8C22×C911yescharacteristic
114C2×C3×Q8C2×C3×D311yescharacteristic
115C2×C3×D4—31no—
116C2×C3×D4—31no—
117C2×C3×D4—61no—
118C2×C3×D4—92no—
119C2×C3×D4—92no—
120C2×C3×D4—94no—
121C2×C3×D4—94no—
122C2×C3×D4—98no—
123C2×C3×D4—98no—
124C2×C3×D4—91no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—61no—
128C22×C3×C4—92no—
129C2×#67—32no—
130#239—34no—
131#239C22×C914yes—
132D3×D4—38no—
133D3×D4—38no—
134#235—34no—
135#235—34no—
136C2×D12—32no—
137C2×D12—34no—
138C2×D12—31no—
139C2×D12—34no—
140C2×C4×D3—32no—
141C2×Dic6C22⋊C911yescharacteristic
142C22×C32—32no—
143C22×C32—62no—
144C22×C32C22×A411yescharacteristic
145C2×C3×D3—34no—
146C2×C3×D3—64no—
147C2×C3×D3—128no—
148C32×C4—31no—
149C32×C4—31no—
150C32×C4C22×A412yes—
151C3×Dic3C22×A414yes—
152C22×C9—41no—
153C22×C9—81no—
154C4×C9—42no—
155C4×C9—82no—
156C4×C9—124no—
157H(5,𝔽2)—34no—
158H(5,𝔽2)—94no—
159C2×#41—32no—
160C2×#41—91no—
161C2×#41D3×C911yescharacteristic
162C22×D4—91no—
163C22×D4—92no—
164C3×C9—41noSylow
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—61no—
168C23×C3—98no—
169C23×C3—92no—
170C23×C3—364no—
171C22×D3—32no—
172C22×D3—38no—
173C22×D3—124no—
174C3×Q8—21no—
175C3×Q8—31no—
176C3×Q8—31no—
177C3×Q8—34no—
178C3×Q8—61no—
179C3×Q8C23×C911yescharacteristic
180C3×Q8C22×C3×D311yescharacteristic
181C3×D4—32no—
182C3×D4—32no—
183C3×D4—32no—
184C3×D4—32no—
185C3×D4—62no—
186C3×D4—62no—
187C3×D4—98no—
188C3×D4—94no—
189C3×D4—94no—
190C3×D4—94no—
191C2×C3×C4—21no—
192C2×C3×C4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—94no—
203C2×C3×C4—94no—
204C2×C3×C4C2×C22⋊C911yescharacteristic
205C2×C3×C4D3×A411yescharacteristic
206#67—38no—
207C2×Dic3C2×C22⋊C912yes—
208D12—34no—
209D12—34no—
210D12—34no—
211C4×D3—34no—
212C4×D3—34no—
213Dic6C2×C22⋊C914yes—
214C2×C32—62no—
215C2×C32#99412yes—
216C2×C32C23×A411yescharacteristic
217C3×D3—64no—
218C2×C9—42no—
219C2×C9—41no—
220C2×C9—82no—
221C2×C9—81no—
222C24—92no—
223#41—32no—
224#41—34no—
225#41—94no—
226#41—94no—
227#41—94no—
228#41C2×D3×C912yes—
229C2×Q8—31no—
230C2×Q8C2×D3×C911yescharacteristic
231C2×D4—31no—
232C2×D4—92no—
233C2×D4—92no—
234C2×D4—94no—
235C2×D4—94no—
236C2×D4—98no—
237C2×D4—98no—
238C2×D4—91no—
239C22×C4—31no—
240C22×C4—92no—
241C22×C3—21no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—94no—
248C22×C3—122no—
249C22×C3—184no—
250C22×C3—368no—
251C22×C3C22×C22⋊C911yescharacteristic
252C22×C3C2×D3×A411yescharacteristic
253C2×D3—34no—
254C2×D3—64no—
255C2×D3—128no—
256C3×C4—22no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—34no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4C22×C22⋊C912yes—
265C3×C4C2×D3×A412yes—
266Dic3C22×C22⋊C914yes—
267C32C2×#99411yescharacteristic
268C9—41no—
269C9—81no—
270C23—31no—
271C23—98no—
272C23—92no—
273C23—364no—
274Q8—31no—
275Q8—34no—
276Q8C22×D3×C911yescharacteristic
277D4—32no—
278D4—32no—
279D4—98no—
280D4—94no—
281D4—94no—
282D4—94no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—94no—
288C2×C4—94no—
289C2×C4D3×C22⋊C911yescharacteristic
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—62no—
293C2×C3—62no—
294C2×C3—122no—
295C2×C3—184no—
296C2×C3#6370012yescomplemented
297C2×C3C23×C22⋊C911yescharacteristic
298C2×C3C3⋊#99412yes—
299C2×C3C22×D3×A411yescharacteristic
300D3—64no—
301C22—32no—
302C22—62no—
303C22—94no—
304C22—184no—
305C22—368no—
306C22C2×D3×C22⋊C911yescharacteristic
307C4—31no—
308C4—31no—
309C4—34no—
310C4C2×D3×C22⋊C912yes—
311C3—21no—
312C3C2×#6370011yescharacteristic, complemented
313C3C2×C3⋊#99411yescharacteristic
314C2—62no—
315C2—184no—
316C2C3⋊#6370012yesdirect product factor
317C2C22×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×#1009C3
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. C3⋊#63700C2direct (N × H)
3. C2×C3⋊(#41⋊C9)C2semidirect
7. C3×#41⋊C9C22semidirect
8. C2×C3×Q8⋊C9C22semidirect
9. C3⋊(#41⋊C9)C22semidirect
20. C2×#41⋊C9D3semidirect
24. C3×Q8⋊C9C23semidirect
26. C2×#1009C9semidirect
49. #41⋊C9C2×D3semidirect
51. C2×Q8⋊C9C2×D3semidirect
65. #1009C2×C9semidirect
95. Q8⋊C9C22×D3semidirect
296. C2×C3#63700semidirect
312. C3C2×#63700semidirect
316. C2C3⋊#63700direct (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
C3⋊#637002
C2×C3⋊(#41⋊C9)2
C2×C3×#10093
C2×#637003
C2×C9×Dic64