Finite Group Databasebeta

← C4×#6198 #11721137

Subgroups — C4×#6198 #11721137

Order 864.

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

normal subgroups only

This group has 359 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×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#6198C211yesmaximal, characteristic
4C4×#1084C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88449C211yesmaximal, characteristic, complemented
8C4×#626—31nomaximal
9C4×#626—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4×(C3⋊D3)⋊C3—22no—
18C4×(C3⋊D3)⋊C3—22no—
19#6198C414yesdirect product factor
20C2×C4×C3⋊D3D311yescharacteristic, complemented
21C2×C4×C3⋊D3D311yescharacteristic, complemented
22C2×C3×C4×D3—31no—
23C2×C3×C4×D3—31no—
24C2×#626—31no—
25C2×#626—31no—
26C3×C4×Dic3—31no—
27C3×C4×Dic3—31no—
28C3×C4×Dic3—61no—
29Dic3⋊Dic3—31no—
30Dic3⋊Dic3—31no—
31#3722—31no—
32#3722—31no—
33(C2×D3)⋊Dic3—31no—
34(C2×D3)⋊Dic3—31no—
35C22×H(3,𝔽3)C2311yescharacteristic
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3C2×C412yescomplemented
41C2×(C3⋊D3)⋊C3C2×C412yescomplemented
42C4×H(3,𝔽3)D412yescomplemented
43#1084—21no—
44#1084C2×C412yescomplemented
45(C3⋊Dic3)⋊C3—21no—
46(C3⋊Dic3)⋊C3—21no—
47C4×#67—91no—
48C4×#67—91no—
49C4×D12—91no—
50C22×C3⋊D3C2×D311yescharacteristic
51C22×C3⋊D3C2×D311yescharacteristic
52C22×C3×D3—31no—
53C22×C3×D3—31no—
54C2×C32×C4—21no—
55C2×C32×C4C2×D311yescharacteristic
56C2×C32×C4C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊Dic3C2×D311yescharacteristic
59C4×C3⋊D3—22no—
60C4×C3⋊D3—22no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C3×C4×D3—62no—
70C3×C4×D3—62no—
71#626—34no—
72#626—34no—
73C2×H(3,𝔽3)C22×C411yescharacteristic
74C2×H(3,𝔽3)C2×D411yescharacteristic
75C2×H(3,𝔽3)#4111yescharacteristic
76(C3⋊D3)⋊C3—22no—
77(C3⋊D3)⋊C3—22no—
78C22×C3×C4—91no—
79C22×C3×C4—91no—
80C2×#67—91no—
81C2×#67—91no—
82C2×D12—91no—
83C2×C4×D3—31no—
84C2×C4×D3—31no—
85C2×C4×D3—91no—
86C2×C4×D3—181no—
87C3×C42—91no—
88#217—91no—
89#217—91no—
90(C2×D3)⋊C4—91no—
91(C2×D3)⋊C4—91no—
92(C2×D3)⋊C4—91no—
93(C2×D3)⋊C4—91no—
94C4⋊Dic3—91no—
95Dic3⋊C4—91no—
96Dic3⋊C4—91no—
97C4×Dic3—31no—
98C4×Dic3—31no—
99C4×Dic3—61no—
100C22×C32—21no—
101C22×C32C22×D311yescharacteristic
102C22×C32C22×D311yescharacteristic
103C2×C3⋊D3—21no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C2×C3⋊D3C4×D312yescomplemented
108C2×C3⋊D3C4×D312yescomplemented
109C2×C3×D3—32no—
110C2×C3×D3—32no—
111C2×C3×D3—61no—
112C2×C3×D3—61no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C32×C4—22no—
116C32×C4#6712yescomplemented
117C32×C4#6712yescomplemented
118C3⋊Dic3—21no—
119C3⋊Dic3—21no—
120C3×Dic3—32no—
121C3×Dic3—32no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—61no—
125C3×Dic3—61no—
126C3×Dic3—61no—
127C3×Dic3—61no—
128C4×D4—271noSylow
129H(3,𝔽3)C4×D411yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C23×C3—91no—
132C22×D3—31no—
133C22×D3—31no—
134C22×D3—91no—
135C22×D3—181no—
136C2×C3×C4—31no—
137C2×C3×C4—61no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—121no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4D3211yescharacteristic
146#67—94no—
147#67—94no—
148C2×Dic3—31no—
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—31no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C2×Dic3—61no—
155C2×Dic3—61no—
156C2×Dic3—91no—
157C2×Dic3—181no—
158D12—94no—
159C4×D3—62no—
160C4×D3—62no—
161C4×D3—182no—
162C4×D3—182no—
163C2×C32—21no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32C2×C4×D311yescharacteristic
167C2×C32C2×C4×D311yescharacteristic
168C2×C32#23511yescharacteristic
169C2×C32#23511yescharacteristic
170C2×C32C2×#6711yescharacteristic
171C2×C32C2×#6711yescharacteristic
172C3⋊D3—22no—
173C3⋊D3—22no—
174C3×D3—62no—
175C3×D3—62no—
176C2×D4—271no—
177C22×C4—91no—
178C22×C4—91no—
179C4⋊C4—271no—
180C22⋊C4—271no—
181C22⋊C4—271no—
182C42—91no—
183C22×C3—31no—
184C22×C3—61no—
185C22×C3—92no—
186C22×C3—92no—
187C22×C3—121no—
188C22×C3—181no—
189C22×C3—181no—
190C22×C3—181no—
191C22×C3—181no—
192C22×C3C2×D3211yescharacteristic
193C2×D3—32no—
194C2×D3—32no—
195C2×D3—61no—
196C2×D3—61no—
197C2×D3—61no—
198C2×D3—61no—
199C2×D3—92no—
200C2×D3—182no—
201C2×D3—181no—
202C2×D3—181no—
203C2×D3—181no—
204C2×D3—181no—
205C3×C4—32no—
206C3×C4—62no—
207C3×C4—92no—
208C3×C4—122no—
209C3×C4—181no—
210C3×C4—181no—
211C3×C4—181no—
212C3×C4#62512yes—
213Dic3—32no—
214Dic3—32no—
215Dic3—62no—
216Dic3—61no—
217Dic3—61no—
218Dic3—62no—
219Dic3—61no—
220Dic3—61no—
221Dic3—181no—
222Dic3—181no—
223C32—21no—
224C32C4×#6711yescharacteristic, complemented
225C32C4×#6711yescharacteristic, complemented
226C23—91no—
227C23—91no—
228D4—274no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—61no—
240C2×C3—61no—
241C2×C3—61no—
242C2×C3—121no—
243C2×C3—121no—
244C2×C3—121no—
245C2×C3—182no—
246C2×C3—182no—
247C2×C3#379811yescharacteristic
248C2×C3C4×D3211yescharacteristic
249C2×C3C2×#62511yescharacteristic
250D3—62no—
251D3—62no—
252D3—182no—
253D3—182no—
254C22—92no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22—181no—
259C22—181no—
260C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
261C4—92no—
262C4—181no—
263C4—181no—
264C4—181no—
265C4#619812yesdirect product factor
266C3—31no—
267C3—61no—
268C3—121no—
269C3C4×#62511yescharacteristic
270C2—182no—
271C2—182no—
272C2#8865011yescharacteristic
273C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
274C2C2×#619811yescharacteristic
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×#6198C2
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×(C3⋊D3)⋊C3C2semidirect
2. C2×C4×(C3⋊D3)⋊C3C2semidirect
4. C4×#1084C2semidirect
5. #88450C2semidirect
6. #88450C2semidirect
7. #88449C2semidirect
19. #6198C4direct (N × H)
20. C2×C4×C3⋊D3D3semidirect
21. C2×C4×C3⋊D3D3semidirect
40. C2×(C3⋊D3)⋊C3C2×C4semidirect
41. C2×(C3⋊D3)⋊C3C2×C4semidirect
42. C4×H(3,𝔽3)D4semidirect
44. #1084C2×C4semidirect
107. C2×C3⋊D3C4×D3semidirect
108. C2×C3⋊D3C4×D3semidirect
116. C32×C4#67semidirect
117. C32×C4#67semidirect
129. H(3,𝔽3)C4×D4semidirect
224. C32C4×#67semidirect
225. C32C4×#67semidirect
265. C4#6198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C2×#61982
C4×#10842
#884502
#884492
C4×#6263