Finite Group Databasebeta

← C2×#85284 #11714464

Subgroups — C2×#85284 #11714464

Order 768.

1353 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 165 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×C4⋊SL(2,ℤ/4)C212yesmaximal
3C2×SL(2,ℤ/4)⋊C4C214yesmaximal
4#85284C218yesmaximal, direct product factor
5C2×#2851—31nomaximal, Sylow
6C22×C4×A4C2212yes—
7C22×C4×A4C2211yescharacteristic
8C22×SL(2,ℤ/4)C2214yes—
9C2×#870—41nomaximal
10C42×A4C2214yes—
11C4⋊SL(2,ℤ/4)C2218yes—
12SL(2,ℤ/4)⋊C4C22116yes—
13C22×#125—31no—
14C23×C42D311yescharacteristic
15#2851—38no—
16#2851—38no—
17C2×#389—32no—
18C2×#387—34no—
19C2×#385—34no—
20C2×#382—31no—
21C2×#380—32no—
22C23×A4C2311yescharacteristic
23C2×C4×A4—24no—
24C2×C4×A4Q814yes—
25C2×C4×A4C2314yes—
26C2×C4×A4C2312yes—
27C2×SL(2,ℤ/4)—28no—
28C2×SL(2,ℤ/4)C2318yes—
29C2×C3×C42—41no—
30C2×C4⋊Dic3—42no—
31C2×Dic3⋊C4—44no—
32#870—48no—
33C24×C4C2×D312yes—
34C24×C4C2×D311yescharacteristic
35C2×#125—38no—
36C2×#125—62no—
37C2×#125—61no—
38C2×#125—64no—
39C2×#125—61no—
40C22×C4⋊C4—32no—
41C22×C4⋊C4—34no—
42C22×C4⋊C4—32no—
43C22×C4⋊C4—34no—
44C22×C22⋊C4—34no—
45C22×C42—31no—
46C22×C42—34no—
47C22×C42—32no—
48C22×C42—31no—
49C22×C42C2×D314yes—
50#389—316no—
51#387—332no—
52#385—316no—
53#385—316no—
54#382—38no—
55#380—38no—
56#380—38no—
57C2×#95—34no—
58C2×#95—34no—
59C2×#95—34no—
60C22×A4C2×Q812yes—
61C22×A4#4114yes—
62C22×A4C2411yescharacteristic
63C22×C3×C4—41no—
64C22×C3×C4—42no—
65C22×Dic3—44no—
66C4×A4—24no—
67C4×A4C2×Q814yes—
68SL(2,ℤ/4)—28no—
69C3×C42—44no—
70C4⋊Dic3—48no—
71Dic3⋊C4—416no—
72C25C22×D311yescharacteristic
73C23×C4—24no—
74C23×C4—31no—
75C23×C4—32no—
76C23×C4—34no—
77C23×C4—31no—
78C23×C4—32no—
79C23×C4—32no—
80C23×C4—34no—
81C23×C4—34no—
82C23×C4—34no—
83C23×C4—64no—
84C23×C4Dic614yes—
85C23×C4C22×D314yes—
86C23×C4C22×D312yes—
87#125—616no—
88#125—68no—
89#125—68no—
90C2×C4⋊C4—38no—
91C2×C4⋊C4—316no—
92C2×C4⋊C4—38no—
93C2×C4⋊C4—316no—
94C2×C4⋊C4—68no—
95C2×C4⋊C4—616no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4—68no—
99C2×C4⋊C4—64no—
100C2×C4⋊C4—64no—
101C2×C4⋊C4—62no—
102C2×C22⋊C4—38no—
103C2×C22⋊C4—38no—
104C2×C22⋊C4—68no—
105C2×C22⋊C4—68no—
106C2×C42—32no—
107C2×C42—31no—
108C2×C42—34no—
109C2×C42—34no—
110C2×C42—38no—
111C2×C42—34no—
112C2×C42—62no—
113C2×C42—64no—
114C2×C42—61no—
115C2×C42S411yescharacteristic
116#95—316no—
117#95—316no—
118#95—316no—
119C23×C3—41no—
120C2×A4#12514yescomplemented
121C2×A4C22×Q811yescharacteristic
122C2×A4C2×#4112yes—
123C2×C3×C4—44no—
124C2×C3×C4—42no—
125C2×C3×C4—44no—
126C2×C3×C4—84no—
127C2×Dic3—48no—
128C2×Dic3—88no—
129C24—31no—
130C24—31no—
131C24—34no—
132C24—32no—
133C24C2×Dic612yes—
134C24#23514yes—
135C24C23×D311yescharacteristic
136C22×C4—24no—
137C22×C4—31no—
138C22×C4—34no—
139C22×C4—34no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—34no—
144C22×C4—34no—
145C22×C4—38no—
146C22×C4—34no—
147C22×C4—38no—
148C22×C4—32no—
149C22×C4—34no—
150C22×C4—34no—
151C22×C4—38no—
152C22×C4—38no—
153C22×C4—64no—
154C22×C4—64no—
155C22×C4—64no—
156C22×C4—61no—
157C22×C4—62no—
158C22×C4—62no—
159C22×C4—64no—
160C22×C4—68no—
161C22×C4—68no—
162C22×C4—64no—
163C22×C4—68no—
164C22×C4—68no—
165C22×C4—64no—
166C22×C4—68no—
167C22×C4—68no—
168C22×C4—68no—
169C22×C4—64no—
170C22×C4C2×Dic614yes—
171C22×C4C2×S412yes—
172C22×C4C2×S411yescharacteristic
173C4⋊C4—616no—
174C4⋊C4—632no—
175C4⋊C4—68no—
176C4⋊C4—616no—
177C4⋊C4—616no—
178C4⋊C4—68no—
179C22⋊C4—616no—
180C22⋊C4—68no—
181C22⋊C4—68no—
182C42—38no—
183C42—34no—
184C42—64no—
185C42C2×S414yes—
186C22×C3—42no—
187C22×C3—44no—
188C22×C3—41no—
189A4C2×#12511yescharacteristic, complemented
190C3×C4—44no—
191C3×C4—84no—
192Dic3—88no—
193C23—32no—
194C23—32no—
195C23—34no—
196C23—34no—
197C23—31no—
198C23—31no—
199C23—34no—
200C23—32no—
201C23—32no—
202C23—31no—
203C23—34no—
204C23—38no—
205C23—61no—
206C23—62no—
207C23—64no—
208C23#87014yescomplemented
209C23C22×Dic611yescharacteristic
210C23C2×#23512yes—
211C23C22×S411yescharacteristic
212C2×C4—24no—
213C2×C4—34no—
214C2×C4—32no—
215C2×C4—34no—
216C2×C4—34no—
217C2×C4—38no—
218C2×C4—34no—
219C2×C4—38no—
220C2×C4—34no—
221C2×C4—64no—
222C2×C4—64no—
223C2×C4—68no—
224C2×C4—62no—
225C2×C4—64no—
226C2×C4—68no—
227C2×C4—64no—
228C2×C4—64no—
229C2×C4—68no—
230C2×C4—68no—
231C2×C4—68no—
232C2×C4—68no—
233C2×C4—68no—
234C2×C4—68no—
235C2×C4—1216no—
236C2×C4—128no—
237C2×C4—128no—
238C2×C4A4⋊Q814yes—
239C2×C4C22×S414yes—
240C2×C4C22×S412yes—
241C2×C3—44no—
242C2×C3—41no—
243C2×C3—42no—
244C22—34no—
245C22—31no—
246C22—32no—
247C22—38no—
248C22—34no—
249C22—34no—
250C22—31no—
251C22—31no—
252C22—32no—
253C22—32no—
254C22—32no—
255C22—34no—
256C22—61no—
257C22—64no—
258C22—62no—
259C22C2×#87011yescharacteristic, complemented
260C22C2×A4⋊Q812yes—
261C22#591214yes—
262C22C23×S411yescharacteristic
263C4—24no—
264C4—34no—
265C4—64no—
266C4—128no—
267C4—128no—
268C4C2×A4⋊Q814yes—
269C3—41noSylow
270C2—32no—
271C2—31no—
272C2—34no—
273C2—31no—
274C2#8528414yesdirect product factor
275C2C22×A4⋊Q811yescharacteristic
276C2C2×#591212yes—
subgroupG/H
centerC23C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC25C22×D3
prime radicalC23×A4C23
hypercenterC2×C42S4
hypocenterA4C2×#125
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. #85284C2direct (N × H)
120. C2×A4#125semidirect
189. A4C2×#125semidirect
208. C23#870semidirect
259. C22C2×#870semidirect
274. C2#85284direct (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×C4⋊SL(2,ℤ/4)2
C2×SL(2,ℤ/4)⋊C42
#852842
C2×#28513
C2×#8704