Finite Group Databasebeta

← C4×#6200 #11721153

Subgroups — C4×#6200 #11721153

Order 864.

383 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 74 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×C4×#272C211yesmaximal, characteristic, complemented
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#6200C211yesmaximal, characteristic
4C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#88451C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8C4×#626—31nomaximal
9C4×#625—31nomaximal
10C22×#272C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C4×#272—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4×(C3⋊D3)⋊C3—22no—
19#6200C414yesdirect product factor
20C2×C4×C3⋊D3D311yescharacteristic, complemented
21C2×C3×C4×D3—31no—
22C2×C3×C4×D3—31no—
23C2×C3×C4×D3—31no—
24C2×C3×C4×D3—61no—
25C2×#626—31no—
26C2×#625—31no—
27C4×C3⋊Dic3D311yescharacteristic, complemented
28C3×C4×Dic3—31no—
29#3724—31no—
30Dic3⋊Dic3—31no—
31#3722—31no—
32(C2×D3)⋊Dic3—31no—
33(C2×D3)⋊Dic3—31no—
34(C2×D3)⋊Dic3—31no—
35C22×H(3,𝔽3)C2311yescharacteristic
36C2×#272—21no—
37C2×#272—21no—
38C2×#272C2×C412yescomplemented
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3—21no—
41C2×(C3⋊D3)⋊C3C2×C412yescomplemented
42C4×H(3,𝔽3)D412yescomplemented
43#1084—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3—21no—
46(C3⋊Dic3)⋊C3C2×C412yescomplemented
47C4×#67—91no—
48C4×#67—91no—
49C4×D12—91no—
50C22×C3⋊D3C2×D311yescharacteristic
51C22×C3×D3—31no—
52C22×C3×D3—31no—
53C22×C3×D3—31no—
54C22×C3×D3—61no—
55C2×C32×C4—21no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C3⋊Dic3C2×D311yescharacteristic
59C2×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊Dic3C2×D311yescharacteristic
61C4×C3⋊D3—22no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—61no—
68C3×C4×D3—64no—
69C3×C4×D3—62no—
70C3×C4×D3—62no—
71C3×C4×D3—62no—
72#626—34no—
73#625—34no—
74C2×H(3,𝔽3)C22×C411yescharacteristic
75C2×H(3,𝔽3)C2×D411yescharacteristic
76C2×H(3,𝔽3)#4111yescharacteristic
77#272—22no—
78(C3⋊D3)⋊C3—22no—
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C2×#67—91no—
82C2×#67—91no—
83C2×D12—91no—
84C2×C4×D3—31no—
85C2×C4×D3—31no—
86C2×C4×D3—31no—
87C2×C4×D3—61no—
88C2×C4×D3—181no—
89C3×C42—91no—
90#217—91no—
91#217—91no—
92(C2×D3)⋊C4—91no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95(C2×D3)⋊C4—91no—
96C4⋊Dic3—91no—
97Dic3⋊C4—91no—
98Dic3⋊C4—91no—
99C4×Dic3—31no—
100C4×Dic3—181no—
101C22×C32—21no—
102C22×C32C22×D311yescharacteristic
103C22×C32C22×D311yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3C4×D312yescomplemented
107C2×C3×D3—32no—
108C2×C3×D3—32no—
109C2×C3×D3—32no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C2×C3×D3—61no—
116C2×C3×D3—61no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C32×C4—22no—
120C32×C4D1212yescomplemented
121C32×C4#6712yescomplemented
122C3⋊Dic3—21no—
123C3⋊Dic3—21no—
124C3⋊Dic3C4×D312yescomplemented
125C3×Dic3—32no—
126C3×Dic3—62no—
127C3×Dic3—61no—
128C3×Dic3—61no—
129C3×Dic3—61no—
130C3×Dic3—61no—
131C4×D4—271noSylow
132H(3,𝔽3)C4×D411yescharacteristic, complemented, Sylow
133C23×C3—91no—
134C23×C3—91no—
135C22×D3—31no—
136C22×D3—31no—
137C22×D3—31no—
138C22×D3—61no—
139C22×D3—181no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—121no—
147C2×C3×C4—182no—
148C2×C3×C4—182no—
149C2×C3×C4D3211yescharacteristic
150#67—94no—
151#67—94no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C2×Dic3—31no—
155C2×Dic3—31no—
156C2×Dic3—31no—
157C2×Dic3—61no—
158C2×Dic3—181no—
159C2×Dic3—181no—
160C2×Dic3—181no—
161D12—94no—
162C4×D3—62no—
163C4×D3—62no—
164C4×D3—64no—
165C4×D3—62no—
166C4×D3—182no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32C2×C4×D311yescharacteristic
171C2×C32C2×C4×D311yescharacteristic
172C2×C32C2×D1211yescharacteristic
173C2×C32#23511yescharacteristic
174C2×C32#23511yescharacteristic
175C2×C32C2×#6711yescharacteristic
176C3⋊D3—22no—
177C3×D3—64no—
178C3×D3—62no—
179C3×D3—62no—
180C3×D3—62no—
181C2×D4—271no—
182C22×C4—91no—
183C22×C4—91no—
184C4⋊C4—271no—
185C22⋊C4—271no—
186C22⋊C4—271no—
187C42—91no—
188C22×C3—61no—
189C22×C3—61no—
190C22×C3—92no—
191C22×C3—92no—
192C22×C3—121no—
193C22×C3—181no—
194C22×C3—181no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3C2×D3211yescharacteristic
198C2×D3—32no—
199C2×D3—32no—
200C2×D3—32no—
201C2×D3—62no—
202C2×D3—61no—
203C2×D3—61no—
204C2×D3—61no—
205C2×D3—61no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—61no—
209C2×D3—61no—
210C2×D3—182no—
211C2×D3—181no—
212C2×D3—181no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—92no—
216C3×C4—122no—
217C3×C4—181no—
218C3×C4—181no—
219C3×C4—181no—
220C3×C4#62612yes—
221Dic3—32no—
222Dic3—61no—
223Dic3—61no—
224Dic3—62no—
225Dic3—61no—
226Dic3—61no—
227Dic3—182no—
228Dic3—181no—
229Dic3—181no—
230C32—21no—
231C32C4×D1211yescharacteristic, complemented
232C32C4×#6711yescharacteristic, complemented
233C23—91no—
234C23—91no—
235D4—274no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—121no—
250C2×C3—121no—
251C2×C3—121no—
252C2×C3—182no—
253C2×C3—182no—
254C2×C3#379811yescharacteristic
255C2×C3C4×D3211yescharacteristic
256C2×C3C2×#62611yescharacteristic
257D3—62no—
258D3—62no—
259D3—64no—
260D3—62no—
261D3—182no—
262C22—92no—
263C22—92no—
264C22—181no—
265C22—181no—
266C22—181no—
267C22—181no—
268C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4—92no—
270C4—181no—
271C4—181no—
272C4—181no—
273C4#620012yesdirect product factor
274C3—61no—
275C3—61no—
276C3—121no—
277C3C4×#62611yescharacteristic
278C2—182no—
279C2—182no—
280C2#8865011yescharacteristic
281C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C2C2×#620011yescharacteristic
subgroupG/H
center, hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×#6200C2
cosocleC22×H(3,𝔽3)C23
hypocenterH(3,𝔽3)C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×#272C2semidirect
2. C2×C4×(C3⋊D3)⋊C3C2semidirect
4. C4×(C3⋊Dic3)⋊C3C2semidirect
5. #88451C2semidirect
6. #88450C2semidirect
7. #88448C2semidirect
19. #6200C4direct (N × H)
20. C2×C4×C3⋊D3D3semidirect
27. C4×C3⋊Dic3D3semidirect
38. C2×#272C2×C4semidirect
41. C2×(C3⋊D3)⋊C3C2×C4semidirect
42. C4×H(3,𝔽3)D4semidirect
46. (C3⋊Dic3)⋊C3C2×C4semidirect
106. C2×C3⋊D3C4×D3semidirect
120. C32×C4D12semidirect
121. C32×C4#67semidirect
124. C3⋊Dic3C4×D3semidirect
132. H(3,𝔽3)C4×D4semidirect
231. C32C4×D12semidirect
232. C32C4×#67semidirect
273. C4#6200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C2×C4×(C3⋊D3)⋊C32
C2×#62002
C4×(C3⋊Dic3)⋊C32
#884512
#884502
#884482
C4×#6263
C4×#6253