Finite Group Databasebeta

← C4×#5409 #11712543

Subgroups — C4×#5409 #11712543

Order 768.

801 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 129 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×C42×A4C211yesmaximal, characteristic
2C2×#5409C212yesmaximal
3#73044C212yesmaximal, complemented
4C4×A4⋊C8C212yesmaximal, complemented
5C4×#407—31nomaximal, Sylow
6C22×C4×A4C412yes—
7C22×C4×A4C2211yescharacteristic
8C42×A4C412yes—
9C42×A4C2211yescharacteristic
10C42×A4C2211yescharacteristic
11#5409C418yesdirect product factor
12C2×A4⋊C8C2214yes—
13C4×#208—41nomaximal
14C23×C42D311yescharacteristic
15C2×#407—32no—
16C2×C4×M16—31no—
17C2×C4×M16—31no—
18#1749—32no—
19#1749—32no—
20C4×C22⋊C8—32no—
21C4×C22⋊C8—32no—
22#1728—32no—
23C23×A4C2×C411yescharacteristic
24C2×C4×A4—24no—
25C2×C4×A4C2×C414yes—
26C2×C4×A4C2×C412yes—
27C2×C4×A4C2×C411yescharacteristic
28C2×C4×A4C2×C412yes—
29C2×C4×A4C2311yescharacteristic
30C2×C3×C42—41no—
31C2×#208—42no—
32A4⋊C8C2×C418yescomplemented
33#803—42no—
34C4×C3⋊C8—42no—
35C24×C4Dic312yes—
36C24×C4C2×D311yescharacteristic
37C22×M16—32no—
38C22×M16—32no—
39C22×C42—31no—
40C22×C42—32no—
41C22×C42—32no—
42C22×C42—31no—
43C22×C42—31no—
44C22×C42—31no—
45C22×C42Dic312yes—
46C22×C42C2×D311yescharacteristic
47C22×C42C2×D311yescharacteristic
48#407—38no—
49#407—38no—
50C2×C22⋊C8—34no—
51C2×C22⋊C8—34no—
52C4×M16—62no—
53C4×M16—61no—
54C4×M16—62no—
55C4×M16—61no—
56C4×M16—61no—
57C4×M16—61no—
58C2×#97—32no—
59C2×#97—32no—
60C2×C4×C8—32no—
61C2×C4×C8—32no—
62#336—34no—
63#336—34no—
64C22×A4—22no—
65C22×A4C4212yes—
66C22×A4C22×C411yescharacteristic
67C22×C3×C4—42no—
68C22×C3×C4—41no—
69C4×A4—22no—
70C4×A4C4212yes—
71C4×A4M1614yescomplemented
72C4×A4C22×C412yes—
73C3×C42—42no—
74C3×C42—41no—
75C3×C42—41no—
76#208—48no—
77C2×C3⋊C8—44no—
78C25C2×Dic311yescharacteristic
79C23×C4—24no—
80C23×C4—31no—
81C23×C4—32no—
82C23×C4—31no—
83C23×C4—34no—
84C23×C4—31no—
85C23×C4—31no—
86C23×C4—32no—
87C23×C4—32no—
88C23×C4—32no—
89C23×C4—64no—
90C23×C4C4×D314yes—
91C23×C4C2×Dic312yes—
92C23×C4C2×Dic312yes—
93C23×C4C2×Dic311yescharacteristic
94C23×C4C22×D311yescharacteristic
95C2×M16—38no—
96C2×M16—38no—
97C2×M16—64no—
98C2×M16—64no—
99C2×M16—62no—
100C2×M16—62no—
101C2×M16—62no—
102C2×M16—62no—
103C22×C8—34no—
104C22×C8—34no—
105C2×C42—32no—
106C2×C42—31no—
107C2×C42—32no—
108C2×C42—32no—
109C2×C42—31no—
110C2×C42—31no—
111C2×C42—31no—
112C2×C42—31no—
113C2×C42—31no—
114C2×C42—31no—
115C2×C42—32no—
116C2×C42—32no—
117C2×C42—32no—
118C2×C42—32no—
119C2×C42—32no—
120C2×C42—62no—
121C2×C42—62no—
122C2×C42—61no—
123C2×C42—61no—
124C2×C42—61no—
125C2×C42S411yescharacteristic
126C22⋊C8—316no—
127C22⋊C8—38no—
128C22⋊C8—38no—
129#97—64no—
130#97—62no—
131#97—62no—
132C4×C8—64no—
133C4×C8—62no—
134C4×C8—62no—
135C23×C3—41no—
136C2×A4—22no—
137C2×A4C2×C4211yescharacteristic
138C2×A4C2×M1612yes—
139C2×C3×C4—44no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4—42no—
143C2×C3×C4—41no—
144C2×C3×C4—84no—
145C3⋊C8—48no—
146C24—22no—
147C24—31no—
148C24—31no—
149C24—32no—
150C24—62no—
151C24C4×Dic312yes—
152C24C22×Dic311yescharacteristic
153C22×C4—22no—
154C22×C4—31no—
155C22×C4—34no—
156C22×C4—34no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—34no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—31no—
168C22×C4—34no—
169C22×C4—34no—
170C22×C4—32no—
171C22×C4—34no—
172C22×C4—38no—
173C22×C4—32no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—64no—
178C22×C4—64no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—64no—
186C22×C4—64no—
187C22×C4—64no—
188C22×C4—64no—
189C22×C4—64no—
190C22×C4#20814yescomplemented
191C22×C4C4×Dic312yes—
192C22×C4SL(2,ℤ/4)12yes—
193C22×C4C2×C4×D312yes—
194C22×C4C2×S411yescharacteristic
195M16—68no—
196M16—68no—
197M16—68no—
198M16—68no—
199C2×C8—38no—
200C2×C8—38no—
201C2×C8—64no—
202C2×C8—64no—
203C2×C8—64no—
204C2×C8—64no—
205C42—32no—
206C42—32no—
207C42—32no—
208C42—32no—
209C42—32no—
210C42—31no—
211C42—31no—
212C42—61no—
213C42—61no—
214C42—62no—
215C42SL(2,ℤ/4)12yes—
216C42C2×S411yescharacteristic
217C42C2×S411yescharacteristic
218C22×C3—42no—
219C22×C3—41no—
220C22×C3—82no—
221A4C4×M1611yescharacteristic, complemented
222C3×C4—42no—
223C3×C4—44no—
224C3×C4—42no—
225C3×C4—82no—
226C23—22no—
227C23—31no—
228C23—32no—
229C23—31no—
230C23—32no—
231C23—34no—
232C23—32no—
233C23—32no—
234C23—31no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—61no—
239C23—62no—
240C23—64no—
241C23—64no—
242C23C2×#20812yes—
243C23C2×C4×Dic311yescharacteristic
244C23C2×SL(2,ℤ/4)11yescharacteristic
245C2×C4—24no—
246C2×C4—34no—
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—34no—
253C2×C4—34no—
254C2×C4—34no—
255C2×C4—34no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—34no—
261C2×C4—34no—
262C2×C4—32no—
263C2×C4—64no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—64no—
269C2×C4—61no—
270C2×C4—64no—
271C2×C4—68no—
272C2×C4—64no—
273C2×C4—62no—
274C2×C4C4×S414yes—
275C2×C4C2×SL(2,ℤ/4)12yes—
276C2×C4C2×SL(2,ℤ/4)12yes—
277C2×C4C2×SL(2,ℤ/4)11yescharacteristic
278C2×C4C22×S411yescharacteristic
279C8—68no—
280C8—68no—
281C2×C3—42no—
282C2×C3—41no—
283C2×C3—82no—
284C22—22no—
285C22—32no—
286C22—31no—
287C22—32no—
288C22—32no—
289C22—31no—
290C22—31no—
291C22—34no—
292C22—32no—
293C22—62no—
294C22—62no—
295C22—64no—
296C22—61no—
297C22—62no—
298C22—64no—
299C22—62no—
300C22C4×#20811yescharacteristic, complemented
301C22C4×SL(2,ℤ/4)12yes—
302C22C22×SL(2,ℤ/4)11yescharacteristic
303C4—22no—
304C4—34no—
305C4—32no—
306C4—32no—
307C4—62no—
308C4#540914yesdirect product factor
309C4C4×SL(2,ℤ/4)12yes—
310C4C2×C4×S412yes—
311C3—41noSylow
312C2—22no—
313C2—31no—
314C2—31no—
315C2—32no—
316C2—62no—
317C2C2×#540912yes—
318C2C2×C4×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC42C2×S4
commutatorC2×A4C2×C42
FrattiniC2×C4C22×S4
FittingC23×C42D3
socleC24C22×Dic3
prime radicalC23×A4C2×C4
cosocleC2×C4×A4C23
hypercenterC2×C42S4
hypocenterA4C4×M16
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C42 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. #73044C2semidirect
4. C4×A4⋊C8C2semidirect
11. #5409C4direct (N × H)
32. A4⋊C8C2×C4semidirect
71. C4×A4M16semidirect
190. C22×C4#208semidirect
221. A4C4×M16semidirect
300. C22C4×#208semidirect
308. C4#5409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42×A42
C2×#54092
#730442
C4×A4⋊C82
C4×#4073
C4×#2084