Finite Group Databasebeta

← #13164560

Subgroups — #13164560

Order 1728.

771 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 771 conjugacy classes of subgroups — too many to draw. Only its 39 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×C32⋊(C22⋊A4)C211yesmaximal, characteristic
2C3×#5936—33nomaximal
3C3×#5247—31nomaximal
4C32⋊(C22⋊A4)C411yescharacteristic, complemented
5#88613—43nomaximal
6C25×C32D311yescharacteristic
7C2×C3×C22⋊A4D313yes—
8C2×C3×#217—33no—
9C2×C3×#217—33no—
10C2×C22⋊H(3,𝔽3)—43no—
11C2×C22⋊H(3,𝔽3)—81no—
12#5936—33no—
13C3×#379—91no—
14#5247—31no—
15C24×C32—33no—
16C24×C32—33no—
17C24×C32—61no—
18C24×C32—61no—
19C24×C32Dic311yescharacteristic, complemented
20C3×C22⋊A4Dic313yescomplemented
21C22×C3×Dic3—33no—
22C22×C3×Dic3—31no—
23C3×SL(2,ℤ/4)—129no—
24C3×#217—63no—
25C3×#217—63no—
26C3×#217—63no—
27C3×#217—63no—
28C22⋊H(3,𝔽3)—43no—
29C22⋊H(3,𝔽3)—81no—
30#1084—161no—
31C25×C3—31no—
32C25×C3C3⋊D311yescharacteristic
33C2×C22⋊A4—33no—
34C2×C3×C22⋊C4—93no—
35C2×C3×C22⋊C4—93no—
36C2×#217—33no—
37C2×#217—33no—
38C23×C32—21no—
39C23×C32—31no—
40C23×C32—33no—
41C23×C32—33no—
42C23×C32—33no—
43C23×C32—33no—
44C23×C32—33no—
45C23×C32—61no—
46C23×C32—61no—
47C23×C32—66no—
48C23×C32—63no—
49C23×C32—63no—
50C23×C32—63no—
51C23×C32S413yes—
52C2×C3×A4—49no—
53C2×C3×A4—83no—
54C2×C3×Dic3—66no—
55C2×C3×Dic3—63no—
56C2×C3×Dic3—63no—
57#379—91noSylow
58C2×H(3,𝔽3)—161no—
59C24×C3—33no—
60C24×C3—33no—
61C24×C3—33no—
62C24×C3—33no—
63C24×C3—31no—
64C24×C3—63no—
65C24×C3—63no—
66C24×C3—61no—
67C24×C3—61no—
68C24×C3—63no—
69C24×C3—63no—
70C24×C3C3⋊Dic311yescharacteristic
71C22⋊A4—33no—
72C22×C3×C4—93no—
73C22×C3×C4—91no—
74C22×Dic3—33no—
75C22×Dic3—31no—
76SL(2,ℤ/4)—129no—
77C3×C22⋊C4—183no—
78C3×C22⋊C4—183no—
79C3×C22⋊C4—183no—
80C3×C22⋊C4—183no—
81#217—63no—
82#217—63no—
83#217—63no—
84#217—63no—
85C22×C32—21no—
86C22×C32—33no—
87C22×C32—33no—
88C22×C32—31no—
89C22×C32—33no—
90C22×C32—33no—
91C22×C32—33no—
92C22×C32—63no—
93C22×C32—66no—
94C22×C32—63no—
95C22×C32—63no—
96C22×C32—61no—
97C22×C32—61no—
98C22×C32SL(2,ℤ/4)13yescomplemented
99C3×A4—49no—
100C3×A4—83no—
101C3×Dic3—123no—
102C3×Dic3—121no—
103C3×Dic3—483no—
104C25#27211yescharacteristic
105C2×C22⋊C4—93no—
106C2×C22⋊C4—93no—
107H(3,𝔽3)—161noSylow
108C23×C3—21no—
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—33no—
112C23×C3—33no—
113C23×C3—33no—
114C23×C3—33no—
115C23×C3—33no—
116C23×C3—33no—
117C23×C3—33no—
118C23×C3—33no—
119C23×C3—33no—
120C23×C3—33no—
121C23×C3—33no—
122C23×C3—61no—
123C23×C3—63no—
124C23×C3—63no—
125C23×C3—61no—
126C23×C3—63no—
127C23×C3—63no—
128C23×C3—63no—
129C23×C3—61no—
130C23×C3—61no—
131C23×C3—66no—
132C23×C3—63no—
133C23×C3—63no—
134C23×C3—63no—
135C23×C3—66no—
136C23×C3—66no—
137C23×C3—66no—
138C23×C3—66no—
139C23×C3—63no—
140C23×C3—63no—
141C23×C3—63no—
142C23×C3—63no—
143C23×C3—66no—
144C23×C3—63no—
145C23×C3—66no—
146C23×C3C3⋊S413yes—
147C2×A4—129no—
148C2×A4—243no—
149C2×C3×C4—186no—
150C2×C3×C4—183no—
151C2×C3×C4—183no—
152C2×Dic3—66no—
153C2×Dic3—63no—
154C2×Dic3—63no—
155C2×C32—33no—
156C2×C32—33no—
157C2×C32—61no—
158C2×C32—61no—
159C2×C32—163no—
160C2×C32#102011yescharacteristic
161C24—33no—
162C24—33no—
163C24—61no—
164C24—61no—
165C24#108411yescharacteristic, complemented
166C22×C4—93no—
167C22×C4—91no—
168C22⋊C4—183no—
169C22⋊C4—183no—
170C22⋊C4—183no—
171C22⋊C4—183no—
172C22×C3—21no—
173C22×C3—33no—
174C22×C3—33no—
175C22×C3—33no—
176C22×C3—33no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—33no—
180C22×C3—33no—
181C22×C3—33no—
182C22×C3—33no—
183C22×C3—33no—
184C22×C3—33no—
185C22×C3—33no—
186C22×C3—61no—
187C22×C3—63no—
188C22×C3—63no—
189C22×C3—63no—
190C22×C3—61no—
191C22×C3—63no—
192C22×C3—63no—
193C22×C3—63no—
194C22×C3—66no—
195C22×C3—63no—
196C22×C3—66no—
197C22×C3—66no—
198C22×C3—66no—
199C22×C3—66no—
200C22×C3—63no—
201C22×C3—63no—
202C22×C3—63no—
203C22×C3—61no—
204C22×C3—63no—
205C22×C3—66no—
206C22×C3—63no—
207C22×C3—66no—
208C22×C3—63no—
209C22×C3—61no—
210C22×C3C3⋊SL(2,ℤ/4)13yes—
211A4—129no—
212A4—243no—
213C3×C4—363no—
214C3×C4—361no—
215Dic3—123no—
216Dic3—121no—
217Dic3—483no—
218C32—163no—
219C32#593611yescharacteristic, complemented
220C23—21no—
221C23—31no—
222C23—33no—
223C23—33no—
224C23—33no—
225C23—33no—
226C23—33no—
227C23—61no—
228C23—61no—
229C23—66no—
230C23—63no—
231C23—63no—
232C23—63no—
233C23#625813yes—
234C2×C4—186no—
235C2×C4—183no—
236C2×C4—183no—
237C2×C3—33no—
238C2×C3—33no—
239C2×C3—33no—
240C2×C3—33no—
241C2×C3—31no—
242C2×C3—63no—
243C2×C3—63no—
244C2×C3—63no—
245C2×C3—63no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—483no—
249C2×C3C3⋊#102011yescharacteristic
250C22—21no—
251C22—33no—
252C22—33no—
253C22—31no—
254C22—33no—
255C22—33no—
256C22—33no—
257C22—63no—
258C22—66no—
259C22—63no—
260C22—63no—
261C22—61no—
262C22—61no—
263C22#8861313yescomplemented
264C4—363no—
265C4—361no—
266C3—31no—
267C3—483no—
268C3C3⋊#593611yescharacteristic
269C2—33no—
270C2—33no—
271C2—61no—
272C2—61no—
273C2#1172405511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3C3⋊#1020
commutator, hypocenterC32⋊(C22⋊A4)C4
FittingC25×C32D3
socleC25×C3C3⋊D3
prime radical, cosocleC2×C32⋊(C22⋊A4)C2
LCSG ▷ C32⋊(C22⋊A4)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
4. C32⋊(C22⋊A4)C4semidirect
19. C24×C32Dic3semidirect
20. C3×C22⋊A4Dic3semidirect
98. C22×C32SL(2,ℤ/4)semidirect
165. C24#1084semidirect
219. C32#5936semidirect
263. C22#88613semidirect

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
C3×#59363
C3×#52473
#886134