Finite Group Databasebeta

← C2×#85283 #11714468

Subgroups — C2×#85283 #11714468

Order 768.

1721 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1721 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C22×A4⋊Q8C212yesmaximal
2C2×C42×A4C211yesmaximal, characteristic
3C2×C4⋊SL(2,ℤ/4)C214yesmaximal
4#85283C218yesmaximal, direct product factor
5C2×#2857—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C22×C4×A4C2212yes—
8C22×SL(2,ℤ/4)C2214yes—
9C2×A4⋊Q8—28no—
10C2×A4⋊Q8C2218yes—
11C2×#869—41nomaximal
12C42×A4C2214yes—
13C4⋊SL(2,ℤ/4)C22116yes—
14C22×C4⋊Q8—31no—
15C22×C22⋊Q8—32no—
16C23×C42D311yescharacteristic
17#2857—38no—
18#2857—38no—
19C2×#391—32no—
20C2×#389—34no—
21C2×#384—31no—
22C2×#380—34no—
23C23×A4C2311yescharacteristic
24C22×Dic6—42no—
25C2×C4×A4D414yes—
26C2×C4×A4Q818yescomplemented
27C2×C4×A4C2312yes—
28C2×C4×A4C2314yes—
29C2×SL(2,ℤ/4)—28no—
30C2×SL(2,ℤ/4)C2318yes—
31A4⋊Q8—216no—
32C2×C3×C42—41no—
33C2×C4⋊Dic3—44no—
34#869—48no—
35C23×Q8—32no—
36C24×C4C2×D312yes—
37C24×C4C2×D311yescharacteristic
38C2×C4⋊Q8—38no—
39C2×C4⋊Q8—61no—
40C2×C4⋊Q8—62no—
41C2×C4⋊Q8—64no—
42C2×C4⋊Q8—61no—
43C2×C22⋊Q8—38no—
44C2×C22⋊Q8—38no—
45C2×C22⋊Q8—68no—
46C2×C22⋊Q8—68no—
47C22×C4⋊C4—32no—
48C22×C4⋊C4—34no—
49C22×C4⋊C4—34no—
50C22×C4⋊C4—34no—
51C22×C22⋊C4—34no—
52C22×C42—31no—
53C22×C42—34no—
54C22×C42—32no—
55C22×C42—31no—
56C22×C42C2×D314yes—
57#391—316no—
58#389—332no—
59#384—38no—
60#380—316no—
61#380—316no—
62C2×#95—38no—
63C22×A4C2×D412yes—
64C22×A4C2×Q814yes—
65C22×A4C2411yescharacteristic
66C22×C3×C4—42no—
67C22×C3×C4—41no—
68C22×Dic3—44no—
69C2×Dic6—48no—
70C2×Dic6—88no—
71C4×A4C2×D414yes—
72C4×A4C2×Q818yescomplemented
73SL(2,ℤ/4)—28no—
74C3×C42—44no—
75C4⋊Dic3—416no—
76C25C22×D311yescharacteristic
77C22×Q8—38no—
78C22×Q8—68no—
79C22×Q8—62no—
80C22×Q8—64no—
81C22×Q8—68no—
82C22×Q8—62no—
83C23×C4—32no—
84C23×C4—31no—
85C23×C4—31no—
86C23×C4—32no—
87C23×C4—34no—
88C23×C4—38no—
89C23×C4—32no—
90C23×C4—34no—
91C23×C4—34no—
92C23×C4—34no—
93C23×C4Dic618yescomplemented
94C23×C4D1214yes—
95C23×C4C22×D314yes—
96C23×C4C22×D312yes—
97C4⋊Q8—68no—
98C4⋊Q8—616no—
99C4⋊Q8—68no—
100C22⋊Q8—632no—
101C22⋊Q8—616no—
102C22⋊Q8—616no—
103C2×C4⋊C4—38no—
104C2×C4⋊C4—316no—
105C2×C4⋊C4—316no—
106C2×C4⋊C4—316no—
107C2×C4⋊C4—616no—
108C2×C4⋊C4—68no—
109C2×C4⋊C4—616no—
110C2×C4⋊C4—64no—
111C2×C4⋊C4—68no—
112C2×C4⋊C4—64no—
113C2×C22⋊C4—38no—
114C2×C22⋊C4—38no—
115C2×C22⋊C4—68no—
116C2×C22⋊C4—68no—
117C2×C42—31no—
118C2×C42—32no—
119C2×C42—34no—
120C2×C42—34no—
121C2×C42—38no—
122C2×C42—34no—
123C2×C42—64no—
124C2×C42—62no—
125C2×C42—61no—
126C2×C42S411yescharacteristic
127#95—332no—
128C23×C3—41no—
129C2×A4C4⋊Q814yescomplemented
130C2×A4C22×D411yescharacteristic
131C2×A4C22×Q812yes—
132C2×C3×C4—48no—
133C2×C3×C4—44no—
134C2×C3×C4—44no—
135C2×C3×C4—42no—
136C2×Dic3—48no—
137C2×Dic3—88no—
138Dic6—816no—
139C24—31no—
140C24—31no—
141C24—32no—
142C24—34no—
143C24C2×Dic614yes—
144C24C2×D1212yes—
145C24C23×D311yescharacteristic
146C2×Q8—616no—
147C2×Q8—68no—
148C2×Q8—616no—
149C2×Q8—68no—
150C2×Q8—128no—
151C2×Q8—1216no—
152C2×Q8—128no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4—38no—
156C22×C4—38no—
157C22×C4—34no—
158C22×C4—34no—
159C22×C4—34no—
160C22×C4—32no—
161C22×C4—34no—
162C22×C4—32no—
163C22×C4—34no—
164C22×C4—32no—
165C22×C4—34no—
166C22×C4—38no—
167C22×C4—34no—
168C22×C4—38no—
169C22×C4—316no—
170C22×C4—38no—
171C22×C4—38no—
172C22×C4—38no—
173C22×C4—68no—
174C22×C4—64no—
175C22×C4—62no—
176C22×C4—61no—
177C22×C4—64no—
178C22×C4—62no—
179C22×C4—68no—
180C22×C4—68no—
181C22×C4—64no—
182C22×C4—616no—
183C22×C4—68no—
184C22×C4—64no—
185C22×C4C2×Dic618yescomplemented
186C22×C4C2×D1214yes—
187C22×C4C2×S412yes—
188C22×C4C2×S411yescharacteristic
189C4⋊C4—632no—
190C4⋊C4—616no—
191C4⋊C4—632no—
192C4⋊C4—616no—
193C4⋊C4—616no—
194C22⋊C4—616no—
195C22⋊C4—68no—
196C22⋊C4—68no—
197C42—34no—
198C42—38no—
199C42—64no—
200C42C2×S414yes—
201C22×C3—44no—
202C22×C3—42no—
203C22×C3—41no—
204A4C2×C4⋊Q811yescharacteristic, complemented
205C3×C4—48no—
206C3×C4—44no—
207Dic3—88no—
208C23—34no—
209C23—34no—
210C23—32no—
211C23—32no—
212C23—31no—
213C23—31no—
214C23—32no—
215C23—31no—
216C23—34no—
217C23—32no—
218C23—38no—
219C23—34no—
220C23—62no—
221C23—61no—
222C23—64no—
223C23#86914yescomplemented
224C23C22×Dic612yes—
225C23C22×D1211yescharacteristic
226C23C22×S411yescharacteristic
227Q8—1216no—
228Q8—1216no—
229C2×C4—38no—
230C2×C4—34no—
231C2×C4—34no—
232C2×C4—32no—
233C2×C4—38no—
234C2×C4—38no—
235C2×C4—316no—
236C2×C4—34no—
237C2×C4—34no—
238C2×C4—38no—
239C2×C4—34no—
240C2×C4—38no—
241C2×C4—64no—
242C2×C4—68no—
243C2×C4—62no—
244C2×C4—64no—
245C2×C4—68no—
246C2×C4—68no—
247C2×C4—616no—
248C2×C4—68no—
249C2×C4—68no—
250C2×C4—1216no—
251C2×C4—128no—
252C2×C4—128no—
253C2×C4A4⋊Q818yescomplemented
254C2×C4C4⋊S414yes—
255C2×C4C22×S414yes—
256C2×C4C22×S412yes—
257C2×C3—44no—
258C2×C3—42no—
259C2×C3—41no—
260C22—34no—
261C22—32no—
262C22—31no—
263C22—34no—
264C22—34no—
265C22—38no—
266C22—32no—
267C22—32no—
268C22—34no—
269C22—31no—
270C22—31no—
271C22—32no—
272C22—62no—
273C22—61no—
274C22—64no—
275C22C2×#86911yescharacteristic, complemented
276C22C2×A4⋊Q814yes—
277C22C2×C4⋊S412yes—
278C22C23×S411yescharacteristic
279C4—38no—
280C4—34no—
281C4—128no—
282C4—128no—
283C4C2×A4⋊Q818yescomplemented
284C4C2×C4⋊S414yes—
285C3—41noSylow
286C2—31no—
287C2—31no—
288C2—34no—
289C2—32no—
290C2#8528314yesdirect product factor
291C2C22×A4⋊Q812yes—
292C2C22×C4⋊S411yescharacteristic
subgroupG/H
centerC23C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC25C22×D3
prime radicalC23×A4C23
hypercenterC2×C42S4
hypocenterA4C2×C4⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. #85283C2direct (N × H)
26. C2×C4×A4Q8semidirect
72. C4×A4C2×Q8semidirect
93. C23×C4Dic6semidirect
129. C2×A4C4⋊Q8semidirect
185. C22×C4C2×Dic6semidirect
204. A4C2×C4⋊Q8semidirect
223. C23#869semidirect
253. C2×C4A4⋊Q8semidirect
275. C22C2×#869semidirect
283. C4C2×A4⋊Q8semidirect
290. C2#85283direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4⋊Q82
C2×C42×A42
C2×C4⋊SL(2,ℤ/4)2
#852832
C2×#28573
C2×#8694