Finite Group Databasebeta

← C2×(C3⋊(C3⋊C8))⋊A4 #13147261

Subgroups — C2×(C3⋊(C3⋊C8))⋊A4 #13147261

Order 1728.

480 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 73 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊(C3⋊C8))⋊A4C212yesmaximal, direct product factor
3C23×C3⋊(C3⋊C8)C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊C8—31nomaximal
5C22×C22⋊H(3,𝔽3)C411yescharacteristic
6C4×C22⋊H(3,𝔽3)C411yescharacteristic
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊(C3⋊C8))⋊C3—41nomaximal
9C23×C32×C4C2×C311yescharacteristic
10C2×C3×C4×A4—21no—
11C2×C3×C4×A4D311yescharacteristic
12C22×C3⋊(C3⋊C8)—31no—
13C22×C3⋊(C3⋊C8)—31no—
14C22×C3⋊(C3⋊C8)—32no—
15C22×C3⋊(C3⋊C8)C2×C312yescomplemented
16A4×C3⋊C8—32no—
17C2×C22⋊H(3,𝔽3)C812yescomplemented
18C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
19C2×C4×H(3,𝔽3)—41no—
20(C3⋊(C3⋊C8))⋊C3—42no—
21C23×C3⋊C8—31no—
22C23×C3⋊C8—91no—
23C2×C8×A4—91no—
24C24×C32C3×C411yescharacteristic
25C22×C3×A4—21no—
26C22×C3×A4Dic311yescharacteristic
27C22×C32×C4—31no—
28C22×C32×C4—31no—
29C22×C32×C4—31no—
30C22×C32×C4—31no—
31C22×C32×C4C3×C411yescharacteristic
32C22×C32×C4C22×C311yescharacteristic
33C3×C4×A4—21no—
34C3×C4×A4—21no—
35C3×C4×A4Dic311yescharacteristic
36C3×C4×A4C2×D311yescharacteristic
37C2×C3⋊(C3⋊C8)—32no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—32no—
40C2×C3⋊(C3⋊C8)—32no—
41C2×C3⋊(C3⋊C8)—32no—
42C2×C3⋊(C3⋊C8)A411yescharacteristic, complemented
43C2×C3×C3⋊C8—121no—
44C22×H(3,𝔽3)—41no—
45C22⋊H(3,𝔽3)C2×C811yescharacteristic, complemented
46C4×H(3,𝔽3)—41no—
47C4×H(3,𝔽3)—41no—
48C23×C3×C4—31no—
49C23×C3×C4C3×D311yescharacteristic
50C2×C4×A4—31no—
51C2×C4×A4—61no—
52C22×C3⋊C8—32no—
53C22×C3⋊C8—92no—
54C22×C3⋊C8—91no—
55C22×C3⋊C8—93no—
56C22×C3⋊C8—96no—
57C22×C3⋊C8—93no—
58C22×C3⋊C8—92no—
59C22×C3⋊C8—91no—
60C8×A4—92no—
61C23×C32—32no—
62C23×C32—31no—
63C23×C32—31no—
64C23×C32C3×C812yescomplemented
65C23×C32C2×C3×C411yescharacteristic
66C2×C3×A4—22no—
67C2×C3×A4—21no—
68C2×C3×A4C3⋊C812yescomplemented
69C2×C3×A4C2×Dic311yescharacteristic
70C2×C32×C4—32no—
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C2×C32×C4—41no—
79C2×C32×C4—81no—
80C2×C32×C4C2×A411yescharacteristic
81C3⋊(C3⋊C8)—32no—
82C3⋊(C3⋊C8)C2×A412yescomplemented
83C3×C3⋊C8—122no—
84C23×C8—91noSylow
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)—41no—
87C24×C3—31no—
88C24×C3C3×Dic311yescharacteristic
89C22×A4—31no—
90C22×A4—61no—
91C22×C3×C4—33no—
92C22×C3×C4—33no—
93C22×C3×C4—33no—
94C22×C3×C4—33no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C22×C3×C4C3×Dic311yescharacteristic
102C22×C3×C4C2×C3×D311yescharacteristic
103C4×A4—31no—
104C4×A4—31no—
105C4×A4—61no—
106C4×A4—61no—
107C2×C3×C8—361no—
108C2×C3⋊C8—31no—
109C2×C3⋊C8—91no—
110C2×C3⋊C8—92no—
111C2×C3⋊C8—92no—
112C2×C3⋊C8—96no—
113C2×C3⋊C8—96no—
114C2×C3⋊C8—96no—
115C2×C3⋊C8—92no—
116C2×C3⋊C8—93no—
117C2×C3⋊C8—96no—
118C2×C3⋊C8—91no—
119C2×C3⋊C8—92no—
120C22×C32—32no—
121C22×C32—31no—
122C22×C32—32no—
123C22×C32—32no—
124C22×C32—32no—
125C22×C32—31no—
126C22×C32—31no—
127C22×C32—41no—
128C22×C32—81no—
129C22×C32C2×C3×C811yescharacteristic, complemented
130C22×C32C4×A411yescharacteristic
131C3×A4—21no—
132C3×A4C2×C3⋊C811yescharacteristic, complemented
133C32×C4—31no—
134C32×C4—31no—
135C32×C4—41no—
136C32×C4—41no—
137C32×C4—81no—
138C32×C4—81no—
139C32×C4C4×A411yescharacteristic
140C32×C4C22×A411yescharacteristic
141C23×C4(C3⋊D3)⋊C311yescharacteristic
142C22×C8—92no—
143C22×C8—271no—
144C22×C8—272no—
145C22×C8—271no—
146H(3,𝔽3)—41noSylow
147C23×C3—31no—
148C23×C3—32no—
149C23×C3—33no—
150C23×C3—33no—
151C23×C3—36no—
152C23×C3—31no—
153C23×C3—32no—
154C23×C3—31no—
155C23×C3C3×C3⋊C812yes—
156C23×C3C2×C3×Dic311yescharacteristic
157C2×A4—32no—
158C2×A4—31no—
159C2×A4—62no—
160C2×A4—61no—
161C2×C3×C4—33no—
162C2×C3×C4—33no—
163C2×C3×C4—36no—
164C2×C3×C4—32no—
165C2×C3×C4—33no—
166C2×C3×C4—33no—
167C2×C3×C4—33no—
168C2×C3×C4—33no—
169C2×C3×C4—33no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—121no—
179C2×C3×C4—241no—
180C2×C3×C4D3×A411yescharacteristic
181C3×C8—362no—
182C3⋊C8—32no—
183C3⋊C8—92no—
184C3⋊C8—96no—
185C3⋊C8—92no—
186C2×C32—32no—
187C2×C32—31no—
188C2×C32—31no—
189C2×C32—42no—
190C2×C32—41no—
191C2×C32—82no—
192C2×C32—81no—
193C2×C32C8×A412yescomplemented
194C2×C32C2×C4×A411yescharacteristic
195C24(C3⋊Dic3)⋊C311yescharacteristic
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4(C3⋊Dic3)⋊C311yescharacteristic
201C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
202C2×C8—91no—
203C2×C8—272no—
204C2×C8—272no—
205C2×C8—272no—
206C2×C8—271no—
207C2×C8—272no—
208C22×C3—32no—
209C22×C3—31no—
210C22×C3—33no—
211C22×C3—33no—
212C22×C3—36no—
213C22×C3—33no—
214C22×C3—36no—
215C22×C3—32no—
216C22×C3—36no—
217C22×C3—36no—
218C22×C3—32no—
219C22×C3—32no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—121no—
225C22×C3—241no—
226C22×C3C2×C3×C3⋊C811yescharacteristic
227C22×C3Dic3×A411yescharacteristic
228A4—31no—
229A4—61no—
230C3×C4—33no—
231C3×C4—33no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—121no—
237C3×C4—121no—
238C3×C4—241no—
239C3×C4—241no—
240C3×C4Dic3×A411yescharacteristic
241C3×C4C2×D3×A411yescharacteristic
242C32—41no—
243C32—81no—
244C32C2×C8×A411yescharacteristic, complemented
245C23—31no—
246C23—32no—
247C23—31no—
248C23(C3⋊(C3⋊C8))⋊C312yescomplemented
249C23C2×(C3⋊Dic3)⋊C311yescharacteristic
250C2×C4—32no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4(C3⋊D3)⋊A411yescharacteristic
259C8—92no—
260C8—272no—
261C2×C3—33no—
262C2×C3—36no—
263C2×C3—33no—
264C2×C3—31no—
265C2×C3—32no—
266C2×C3—31no—
267C2×C3—32no—
268C2×C3—31no—
269C2×C3—122no—
270C2×C3—121no—
271C2×C3—242no—
272C2×C3—241no—
273C2×C3A4×C3⋊C812yes—
274C2×C3C2×Dic3×A411yescharacteristic
275C22—32no—
276C22—31no—
277C22—32no—
278C22—32no—
279C22—32no—
280C22—31no—
281C22—31no—
282C22C2×(C3⋊(C3⋊C8))⋊C311yescharacteristic, complemented
283C22(C3⋊Dic3)⋊A411yescharacteristic
284C4—31no—
285C4—31no—
286C4(C3⋊Dic3)⋊A411yescharacteristic
287C4C2×(C3⋊D3)⋊A411yescharacteristic
288C3—31no—
289C3—121no—
290C3—241no—
291C3C2×A4×C3⋊C811yescharacteristic
292C2—31no—
293C2—32no—
294C2—31no—
295C2(C3⋊(C3⋊C8))⋊A412yesdirect product factor
296C2C2×(C3⋊Dic3)⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊D3)⋊A4
commutator, hypocenterC22×C32C2×C3×C8
FrattiniC3×C4C2×D3×A4
FittingC23×C32×C4C2×C3
socleC24×C3C3×Dic3
prime radicalC22×C22⋊H(3,𝔽3)C4
cosocleC22×C32×C4C22×C3
LCSG ▷ C22×C32
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. (C3⋊(C3⋊C8))⋊A4C2direct (N × H)
3. C23×C3⋊(C3⋊C8)C3semidirect
15. C22×C3⋊(C3⋊C8)C2×C3semidirect
17. C2×C22⋊H(3,𝔽3)C8semidirect
42. C2×C3⋊(C3⋊C8)A4semidirect
45. C22⋊H(3,𝔽3)C2×C8semidirect
64. C23×C32C3×C8semidirect
68. C2×C3×A4C3⋊C8semidirect
82. C3⋊(C3⋊C8)C2×A4semidirect
129. C22×C32C2×C3×C8semidirect
132. C3×A4C2×C3⋊C8semidirect
193. C2×C32C8×A4semidirect
244. C32C2×C8×A4semidirect
248. C23(C3⋊(C3⋊C8))⋊C3semidirect
282. C22C2×(C3⋊(C3⋊C8))⋊C3semidirect
295. C2(C3⋊(C3⋊C8))⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
(C3⋊(C3⋊C8))⋊A42
C23×C3⋊(C3⋊C8)3
C2×A4×C3⋊C83
C2×(C3⋊(C3⋊C8))⋊C34