Finite Group Databasebeta

← C2×C32⋊#997 #13164887

Subgroups — C2×C32⋊#997 #13164887

Order 1728.

1132 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C32⋊#997C211yesmaximal, characteristic, direct product factor
2C2×C32×H(5,𝔽2)C311yesmaximal, characteristic, complemented
3C2×C3×#997C313yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—43nomaximal
5C2×Q8⋊H(3,𝔽3)—42nomaximal
6C32×H(5,𝔽2)—32no—
7C32×H(5,𝔽2)—33no—
8C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
9C2×C32×#41—32no—
10C22×C32×D4—33no—
11C3×#997C2×C313yescomplemented
12C2×C22⋊H(3,𝔽3)—46no—
13C2×C22⋊H(3,𝔽3)—43no—
14Q8⋊H(3,𝔽3)—42no—
15C2×C3×H(5,𝔽2)—31no—
16C2×C3×H(5,𝔽2)C3211yescharacteristic
17C2×#997—33no—
18C24×C32—31no—
19C24×C32A413yescomplemented
20C22×C3×A4—49no—
21C32×#41—36no—
22C32×#41—36no—
23C32×#41—32no—
24C32×#41—32no—
25C2×C32×Q8A412yescomplemented
26C2×C32×D4—36no—
27C2×C32×D4—33no—
28C2×C32×D4—312no—
29C2×C32×D4—33no—
30C2×C32×D4—33no—
31C2×C32×D4—33no—
32C22×C32×C4—33no—
33C2×C3×SL(2,𝔽3)—46no—
34C22×H(3,𝔽3)—161no—
35C22⋊H(3,𝔽3)—43no—
36C3×H(5,𝔽2)—32no—
37C3×H(5,𝔽2)—36no—
38C3×H(5,𝔽2)—39no—
39C3×H(5,𝔽2)—33no—
40C3×H(5,𝔽2)—31no—
41C3×H(5,𝔽2)C2×C3211yescharacteristic
42C2×C3×#41—36no—
43C2×C3×#41—32no—
44C22×C3×D4—39no—
45C22×C3×D4—33no—
46#997—33no—
47C23×C32—33no—
48C23×C32—31no—
49C23×C32—33no—
50C23×C32—33no—
51C23×C32—46no—
52C23×C32—122no—
53C23×C32C2×A413yescomplemented
54C2×C3×A4—418no—
55C2×C3×A4—49no—
56C32×Q8—32no—
57C32×Q8C2×A412yescomplemented
58C32×D4—312no—
59C32×D4—36no—
60C32×D4—36no—
61C2×C32×C4—33no—
62C2×C32×C4—36no—
63C2×C32×C4—33no—
64C2×C32×C4—32no—
65C3×SL(2,𝔽3)—46no—
66C2×H(5,𝔽2)H(3,𝔽3)11yescharacteristic, complemented, Sylow
67C2×H(3,𝔽3)—162no—
68C2×H(3,𝔽3)—161no—
69C24×C3—33no—
70C24×C3—31no—
71C24×C3—33no—
72C24×C3C3×A413yes—
73C22×A4—129no—
74C3×#41—36no—
75C3×#41—36no—
76C3×#41—318no—
77C3×#41—318no—
78C3×#41—36no—
79C3×#41—36no—
80C3×#41—32no—
81C3×#41—32no—
82C2×C3×Q8—32no—
83C2×C3×Q8C3×A412yes—
84C2×C3×D4—318no—
85C2×C3×D4—36no—
86C2×C3×D4—39no—
87C2×C3×D4—336no—
88C2×C3×D4—39no—
89C2×C3×D4—33no—
90C2×C3×D4—312no—
91C2×C3×D4—33no—
92C2×C3×D4—39no—
93C2×C3×D4—39no—
94C2×C3×D4—33no—
95C2×C3×D4—33no—
96C22×C3×C4—39no—
97C22×C3×C4—33no—
98C2×SL(2,𝔽3)—126no—
99C22×C32—33no—
100C22×C32—33no—
101C22×C32—43no—
102C22×C32—66no—
103C22×C32—123no—
104C22×C32—121no—
105C22×C32—123no—
106C22×C32—163no—
107C22×C32C22⋊A411yescharacteristic
108C3×A4—49no—
109C32×C4—32no—
110C32×C4—32no—
111H(5,𝔽2)—32no—
112H(5,𝔽2)—33no—
113H(5,𝔽2)C2×H(3,𝔽3)11yescharacteristic, complemented
114C2×#41—32no—
115C22×D4—33no—
116H(3,𝔽3)—161noSylow
117C23×C3—39no—
118C23×C3—33no—
119C23×C3—39no—
120C23×C3—31no—
121C23×C3—33no—
122C23×C3—39no—
123C23×C3—33no—
124C23×C3—33no—
125C23×C3—33no—
126C23×C3—46no—
127C23×C3—126no—
128C23×C3—126no—
129C23×C3—122no—
130C23×C3C2×C3×A413yes—
131C2×A4—1218no—
132C2×A4—129no—
133C3×Q8—32no—
134C3×Q8—36no—
135C3×Q8—32no—
136C3×Q8C2×C3×A412yes—
137C3×D4—318no—
138C3×D4—318no—
139C3×D4—336no—
140C3×D4—36no—
141C3×D4—36no—
142C3×D4—312no—
143C2×C3×C4—36no—
144C2×C3×C4—39no—
145C2×C3×C4—318no—
146C2×C3×C4—39no—
147C2×C3×C4—33no—
148C2×C3×C4—36no—
149C2×C3×C4—33no—
150C2×C3×C4—32no—
151SL(2,𝔽3)—126no—
152C2×C32—63no—
153C2×C32—63no—
154C2×C32—166no—
155C2×C32—163no—
156C2×C32#99712yescomplemented
157C2×C32C2×C22⋊A411yescharacteristic
158C24—31no—
159C24C22⋊H(3,𝔽3)13yescomplemented
160#41—36no—
161#41—36no—
162#41—32no—
163#41—32no—
164C2×Q8C22⋊H(3,𝔽3)12yescomplemented
165C2×D4—36no—
166C2×D4—33no—
167C2×D4—312no—
168C2×D4—33no—
169C2×D4—33no—
170C2×D4—33no—
171C22×C4—33no—
172C22×C3—39no—
173C22×C3—39no—
174C22×C3—33no—
175C22×C3—33no—
176C22×C3—31no—
177C22×C3—43no—
178C22×C3—618no—
179C22×C3—66no—
180C22×C3—123no—
181C22×C3—123no—
182C22×C3—129no—
183C22×C3—123no—
184C22×C3—129no—
185C22×C3—121no—
186C22×C3—123no—
187C22×C3—483no—
188C22×C3C3×C22⋊A411yescharacteristic
189A4—129no—
190C3×C4—36no—
191C3×C4—36no—
192C3×C4—32no—
193C3×C4—32no—
194C32—163no—
195C32C2×#99711yescharacteristic, complemented
196C23—31no—
197C23—33no—
198C23—33no—
199C23—33no—
200C23—46no—
201C23—122no—
202C23C2×C22⋊H(3,𝔽3)13yescomplemented
203Q8—32no—
204Q8C2×C22⋊H(3,𝔽3)12yescomplemented
205D4—36no—
206D4—36no—
207D4—312no—
208C2×C4—33no—
209C2×C4—36no—
210C2×C4—33no—
211C2×C4—32no—
212C2×C3—31no—
213C2×C3—32no—
214C2×C3—69no—
215C2×C3—69no—
216C2×C3—63no—
217C2×C3—63no—
218C2×C3—483no—
219C2×C3—486no—
220C2×C3C3×#99712yes—
221C2×C3C2×C3×C22⋊A411yescharacteristic
222C22—33no—
223C22—33no—
224C22—43no—
225C22—66no—
226C22—123no—
227C22—121no—
228C22—123no—
229C22C32⋊(C22⋊A4)11yescharacteristic
230C4—32no—
231C4—32no—
232C3—31no—
233C3—483no—
234C3C2×C3×#99711yescharacteristic
235C2—63no—
236C2—63no—
237C2C32⋊#99712yesdirect product factor
238C2C2×C32⋊(C22⋊A4)11yescharacteristic
subgroupG/H
center, socleC22×C3C3×C22⋊A4
commutator, cosocleC3×H(5,𝔽2)C2×C32
FrattiniC2×C3C2×C3×C22⋊A4
FittingC2×C32×H(5,𝔽2)C3
hypercenterC22×C32C22⋊A4
hypocenterH(5,𝔽2)C2×H(3,𝔽3)
LCSG ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C22×C3 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C32⋊#997C2direct (N × H)
2. C2×C32×H(5,𝔽2)C3semidirect
3. C2×C3×#997C3semidirect
8. C32×H(5,𝔽2)C2×C3semidirect
11. C3×#997C2×C3semidirect
19. C24×C32A4semidirect
25. C2×C32×Q8A4semidirect
53. C23×C32C2×A4semidirect
57. C32×Q8C2×A4semidirect
66. C2×H(5,𝔽2)H(3,𝔽3)semidirect
113. H(5,𝔽2)C2×H(3,𝔽3)semidirect
156. C2×C32#997semidirect
159. C24C22⋊H(3,𝔽3)semidirect
164. C2×Q8C22⋊H(3,𝔽3)semidirect
195. C32C2×#997semidirect
202. C23C2×C22⋊H(3,𝔽3)semidirect
204. Q8C2×C22⋊H(3,𝔽3)semidirect
237. C2C32⋊#997direct (N × H)

Maximal subgroups

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

subgroupindex
C32⋊#9972
C2×C32×H(5,𝔽2)3
C2×C3×#9973
C22×C22⋊H(3,𝔽3)4
C2×Q8⋊H(3,𝔽3)4