Finite Group Databasebeta

← #72213835

Subgroups — #72213835

Order 1536.

977 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 977 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#14711⋊C3C213yesmaximal
2#14710⋊C3C213yesmaximal, complemented
3C42×C42⋊C3C211yesmaximal, characteristic, complemented
4#6342420C311yesmaximal, characteristic, complemented, Sylow
5A4×#126—41nomaximal
6C22×#126⋊C3C2211yescharacteristic
7C2×#383⋊C3C2213yes—
8C2×C4×C42⋊C3C2213yes—
9#14711—33no—
10#14711—31no—
11#14711C2×C313yes—
12#14710—31no—
13#14710—33no—
14#14710C2×C313yescomplemented
15C44C2×C311yescharacteristic, complemented
16C2×#126⋊C3—23no—
17A4×C4⋊C4—43no—
18A4×C22⋊C4—43no—
19C42×A4—41no—
20C22×C42⋊C3C2311yescharacteristic
21#383⋊C3—23no—
22C4×C42⋊C3—23no—
23C22×#126—31no—
24C22×#126A411yescharacteristic
25C22×#126C22×C311yescharacteristic
26C2×#383—33no—
27C2×#383—31no—
28C2×#383C22×C313yes—
29C2×C43—33no—
30C2×C43—31no—
31C2×C43C22×C313yes—
32#1437—31no—
33#1437—33no—
34#1437—33no—
35#1437—33no—
36#1436—33no—
37#1436—33no—
38#1436—33no—
39#1436—33no—
40#1436—33no—
41#1436—33no—
42#1436—33no—
43#1436—33no—
44#1436—33no—
45#1436—33no—
46C23×A4—41no—
47C2×C4×A4—43no—
48C2×C4×A4—43no—
49C3×#126—161no—
50#126⋊C3—41no—
51C2×C42⋊C3#4113yes—
52C2×#126—23no—
53C2×#126—61no—
54C2×#126—63no—
55C2×#126—612no—
56C2×#126—63no—
57C22×C4⋊C4—31no—
58C22×C4⋊C4—33no—
59C22×C4⋊C4C2×A413yes—
60C22×C22⋊C4—33no—
61C22×C22⋊C4—31no—
62C22×C22⋊C4C2×A413yes—
63C22×C42—31no—
64C22×C42—33no—
65C22×C42—31no—
66C22×C42—33no—
67C22×C42—33no—
68C22×C42C2×A411yescharacteristic
69C22×C42C23×C311yescharacteristic
70#383—23no—
71#383—612no—
72#383—61no—
73#383—63no—
74#383—63no—
75C2×#95—33no—
76C2×#95—33no—
77C2×#95—33no—
78C2×#95—33no—
79C2×#95—33no—
80C2×#95—33no—
81C2×#95—33no—
82C2×#95—33no—
83C2×#95—33no—
84C2×#95—33no—
85C43—23no—
86C43—63no—
87C43—612no—
88C43—61no—
89C43—63no—
90C22×A4—41no—
91C22×A4—83no—
92C4×A4—83no—
93C4×A4—83no—
94C3×C4⋊C4—163no—
95C3×C22⋊C4—163no—
96C3×C42—161no—
97C42⋊C3#12611yescharacteristic, complemented
98C25C22×A411yescharacteristic
99C23×C4—33no—
100C23×C4—31no—
101C23×C4—33no—
102C23×C4—31no—
103C23×C4C22×A413yes—
104C23×C4C22×A413yes—
105#126—41no—
106#126—41no—
107#126—123no—
108#126—122no—
109#126—122no—
110#126—1216no—
111#126—123no—
112C2×C4⋊C4—63no—
113C2×C4⋊C4—63no—
114C2×C4⋊C4—63no—
115C2×C4⋊C4—66no—
116C2×C4⋊C4—66no—
117C2×C4⋊C4—66no—
118C2×C4⋊C4—63no—
119C2×C22⋊C4—63no—
120C2×C22⋊C4—63no—
121C2×C22⋊C4—63no—
122C2×C22⋊C4—66no—
123C2×C22⋊C4—66no—
124C2×C22⋊C4—66no—
125C2×C22⋊C4—63no—
126C2×C42—33no—
127C2×C42—312no—
128C2×C42—31no—
129C2×C42—33no—
130C2×C42—63no—
131C2×C42—63no—
132C2×C42—63no—
133C2×C42—63no—
134C2×C42—66no—
135C2×C42—66no—
136C2×C42—66no—
137C2×C42—66no—
138C2×C42—66no—
139C2×C42—66no—
140C2×C42—63no—
141C2×C42—63no—
142C2×C42—63no—
143C2×C42—63no—
144C2×C42C3×#4113yes—
145#95—63no—
146#95—63no—
147#95—63no—
148#95—63no—
149#95—66no—
150#95—66no—
151#95—66no—
152#95—66no—
153#95—66no—
154#95—66no—
155#95—63no—
156#95—63no—
157#95—63no—
158#95—63no—
159C23×C3—161no—
160C2×A4—43no—
161C2×A4—161no—
162C2×C3×C4—163no—
163C2×C3×C4—163no—
164C24—23no—
165C24—61no—
166C24—63no—
167C24C23×A411yescharacteristic
168C22×C4—23no—
169C22×C4—23no—
170C22×C4—36no—
171C22×C4—36no—
172C22×C4—36no—
173C22×C4—33no—
174C22×C4—33no—
175C22×C4—33no—
176C22×C4—33no—
177C22×C4—63no—
178C22×C4—63no—
179C22×C4—63no—
180C22×C4—63no—
181C22×C4—63no—
182C22×C4—61no—
183C22×C4—63no—
184C22×C4—63no—
185C22×C4—63no—
186C22×C4—63no—
187C22×C4—63no—
188C22×C4—63no—
189C22×C4—61no—
190C22×C4—63no—
191C22×C4—63no—
192C22×C4—63no—
193C22×C4—63no—
194C22×C4—63no—
195C22×C4—63no—
196C4⋊C4—43no—
197C4⋊C4—123no—
198C4⋊C4—121no—
199C4⋊C4—123no—
200C4⋊C4—1212no—
201C22⋊C4—43no—
202C22⋊C4—123no—
203C22⋊C4—121no—
204C22⋊C4—123no—
205C22⋊C4—1212no—
206C42—33no—
207C42—316no—
208C42—32no—
209C42—32no—
210C42—33no—
211C42—66no—
212C42—66no—
213C42—612no—
214C42—612no—
215C42—612no—
216C42—68no—
217C42—66no—
218C42—66no—
219C42—612no—
220C42#126⋊C311yescharacteristic, complemented
221C42C3×#12611yescharacteristic, complemented
222C22×C3—161no—
223C22×C3—323no—
224A4—41no—
225C3×C4—323no—
226C3×C4—323no—
227C23—33no—
228C23—31no—
229C23—41no—
230C23—41no—
231C23—123no—
232C23—123no—
233C23—123no—
234C23—121no—
235C23—121no—
236C23A4×#4113yes—
237C2×C4—312no—
238C2×C4—33no—
239C2×C4—31no—
240C2×C4—33no—
241C2×C4—43no—
242C2×C4—63no—
243C2×C4—63no—
244C2×C4—66no—
245C2×C4—66no—
246C2×C4—63no—
247C2×C4—63no—
248C2×C4—66no—
249C2×C4—66no—
250C2×C4—66no—
251C2×C4—66no—
252C2×C4—63no—
253C2×C4—63no—
254C2×C4—63no—
255C2×C4—63no—
256C2×C4—123no—
257C2×C4—123no—
258C2×C4—123no—
259C2×C4—123no—
260C2×C4—123no—
261C2×C4—123no—
262C2×C4—123no—
263C2×C4—123no—
264C2×C4—123no—
265C2×C4—123no—
266C2×C4—123no—
267C2×C4—121no—
268C2×C4C2×#126⋊C313yes—
269C2×C3—163no—
270C2×C3—641no—
271C22—33no—
272C22—31no—
273C22—31no—
274C22—33no—
275C22—33no—
276C22—83no—
277C22—241no—
278C22—243no—
279C22C22×#126⋊C311yescharacteristic
280C22A4×#12611yescharacteristic
281C4—23no—
282C4—612no—
283C4—61no—
284C4—63no—
285C4—63no—
286C4—83no—
287C4—243no—
288C4—241no—
289C3—161noSylow
290C2—33no—
291C2—31no—
292C2—161no—
293C2#36641⋊C313yes—
subgroupG/H
centerC22C22×#126⋊C3
commutator, cosocleC22×C42C23×C3
Frattini, socleC24C23×A4
Fitting#6342420C3
prime radicalC22×#126⋊C3C22
hypercenterC42#126⋊C3
hypocenterC42C3×#126
LCSG ▷ C22×C42 ▷ C42
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #14710⋊C3C2semidirect
3. C42×C42⋊C3C2semidirect
4. #6342420C3semidirect
14. #14710C2×C3semidirect
15. C44C2×C3semidirect
97. C42⋊C3#126semidirect
220. C42#126⋊C3semidirect
221. C42C3×#126semidirect

Maximal subgroups

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

subgroupindex
#14711⋊C32
#14710⋊C32
C42×C42⋊C32
#63424203
A4×#1264