Finite Group Databasebeta

← C2×#73110 #11712643

Subgroups — C2×#73110 #11712643

Order 768.

721 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 125 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×C8×A4C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C212yesmaximal
3#73110C214yesmaximal, direct product factor
4C2×#1807—31nomaximal, Sylow
5C22×C4×A4C2211yescharacteristic
6C22×SL(2,ℤ/4)—22no—
7C2×C8×A4C414yescomplemented
8C2×C8×A4C2212yes—
9C4⋊SL(2,ℤ/4)—22no—
10C4⋊SL(2,ℤ/4)C2214yes—
11C2×#817—41nomaximal
12C24×C8D311yescharacteristic
13C22×#106—31no—
14C22×#106—31no—
15C2×#380—32no—
16#1807—34no—
17#1807—34no—
18C2×#340—32no—
19C23×A4D411yescharacteristic
20C2×C4×A4C2×C412yes—
21C2×C4×A4D411yescharacteristic
22C2×C4×A4Q812yes—
23C2×C4×A4C2311yescharacteristic
24C2×SL(2,ℤ/4)—24no—
25C2×SL(2,ℤ/4)—44no—
26C22×C3×C8—41no—
27C2×C4⋊Dic3—42no—
28C8×A4C2×C414yescomplemented
29#817—44no—
30C24×C4C2×D311yescharacteristic
31C23×C8—31no—
32C23×C8—32no—
33C23×C8—31no—
34C23×C8—34no—
35C23×C8Dic314yescomplemented
36C23×C8C2×D312yes—
37C22×C4⋊C4—32no—
38C22×C4⋊C4—32no—
39C22×C22⋊C4—62no—
40C2×#106—34no—
41C2×#106—34no—
42C2×#106—64no—
43C2×#106—61no—
44C2×#106—62no—
45C2×#106—61no—
46#380—34no—
47#380—34no—
48#380—62no—
49#380—62no—
50C2×#95—62no—
51#340—38no—
52C22×A4C4⋊C412yes—
53C22×A4SD1614yes—
54C22×A4C2×D411yescharacteristic
55C22×C3×C4—41no—
56C22×Dic3—82no—
57C4×A4C4⋊C412yes—
58C4×A4C22×C411yescharacteristic
59C4×A4C2×Q811yescharacteristic
60SL(2,ℤ/4)—44no—
61C2×C3×C8—44no—
62C2×C3×C8—42no—
63C4⋊Dic3—44no—
64C4⋊Dic3—82no—
65C25D1211yescharacteristic
66C23×C4—31no—
67C23×C4—32no—
68C23×C4—31no—
69C23×C4—32no—
70C23×C4—31no—
71C23×C4—31no—
72C23×C4—62no—
73C23×C4—62no—
74C23×C4Dic612yes—
75C23×C4D1211yescharacteristic
76C23×C4C2×Dic312yes—
77C23×C4C22×D311yescharacteristic
78C22×C8—31no—
79C22×C8—34no—
80C22×C8—34no—
81C22×C8—32no—
82C22×C8—32no—
83C22×C8—34no—
84C22×C8—38no—
85C22×C8—34no—
86C22×C8—32no—
87C22×C8—64no—
88C22×C8—62no—
89C22×C8—61no—
90C22×C8C2×Dic314yescomplemented
91C22×C8S411yescharacteristic
92C2×C4⋊C4—34no—
93C2×C4⋊C4—34no—
94C2×C4⋊C4—62no—
95C2×C4⋊C4—64no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4—62no—
99C2×C4⋊C4—62no—
100C2×C4⋊C4—62no—
101C2×C4⋊C4—62no—
102C2×C22⋊C4—64no—
103C2×C22⋊C4—64no—
104C2×C22⋊C4—124no—
105C2×C22⋊C4—124no—
106#106—68no—
107#106—64no—
108#106—64no—
109#95—64no—
110#95—64no—
111C23×C3—41no—
112C2×A4#10614yescomplemented
113C2×A4C2×C4⋊C411yescharacteristic
114C2×A4C2×SD1612yes—
115C2×C3×C4—42no—
116C2×C3×C4—41no—
117C2×C3×C4—42no—
118C2×C3×C4—41no—
119C2×Dic3—84no—
120C2×Dic3—164no—
121C3×C8—44no—
122C24—31no—
123C24—31no—
124C24—34no—
125C24—32no—
126C24#20414yes—
127C24C4⋊Dic312yes—
128C24C2×D1211yescharacteristic
129C22×C4—31no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—32no—
147C22×C4—32no—
148C22×C4—31no—
149C22×C4—61no—
150C22×C4—61no—
151C22×C4—62no—
152C22×C4—61no—
153C22×C4—62no—
154C22×C4—64no—
155C22×C4—64no—
156C22×C4—124no—
157C22×C4—124no—
158C22×C4—124no—
159C22×C4—122no—
160C22×C4—124no—
161C22×C4—124no—
162C22×C4—122no—
163C22×C4C4⋊Dic312yes—
164C22×C4C2×Dic611yescharacteristic
165C22×C4C22×Dic311yescharacteristic
166C22×C4C2×S411yescharacteristic
167C2×C8—34no—
168C2×C8—32no—
169C2×C8—34no—
170C2×C8—34no—
171C2×C8—38no—
172C2×C8—34no—
173C2×C8—62no—
174C2×C8—64no—
175C2×C8SL(2,ℤ/4)14yescomplemented
176C2×C8C2×S412yes—
177C4⋊C4—64no—
178C4⋊C4—64no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C4⋊C4—122no—
182C4⋊C4—124no—
183C4⋊C4—122no—
184C22⋊C4—128no—
185C22⋊C4—124no—
186C22⋊C4—124no—
187C22×C3—41no—
188C22×C3—44no—
189C22×C3—42no—
190A4C2×#10611yescharacteristic, complemented
191C3×C4—41no—
192C3×C4—42no—
193C3×C4—41no—
194Dic3—164no—
195C23—31no—
196C23—31no—
197C23—34no—
198C23—34no—
199C23—32no—
200C23—32no—
201C23—34no—
202C23—32no—
203C23—32no—
204C23—31no—
205C23—38no—
206C23—34no—
207C23—61no—
208C23—64no—
209C23—62no—
210C23#81714yescomplemented
211C23C2×#20412yes—
212C23C2×C4⋊Dic311yescharacteristic
213C23C4⋊S411yescharacteristic
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—32no—
217C2×C4—32no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—124no—
236C2×C4—124no—
237C2×C4—124no—
238C2×C4—124no—
239C2×C4—124no—
240C2×C4—124no—
241C2×C4—248no—
242C2×C4—244no—
243C2×C4—244no—
244C2×C4A4⋊Q812yes—
245C2×C4C4⋊S411yescharacteristic
246C2×C4C2×SL(2,ℤ/4)12yes—
247C2×C4C22×S411yescharacteristic
248C8—34no—
249C8C2×SL(2,ℤ/4)14yescomplemented
250C2×C3—44no—
251C2×C3—42no—
252C2×C3—41no—
253C22—34no—
254C22—32no—
255C22—31no—
256C22—38no—
257C22—34no—
258C22—34no—
259C22—32no—
260C22—32no—
261C22—34no—
262C22—31no—
263C22—31no—
264C22—32no—
265C22—61no—
266C22—64no—
267C22—62no—
268C22C2×#81711yescharacteristic, complemented
269C22#540114yes—
270C22C4⋊SL(2,ℤ/4)12yes—
271C22C2×C4⋊S411yescharacteristic
272C4—31no—
273C4—31no—
274C4—32no—
275C4—244no—
276C4—244no—
277C4C4⋊SL(2,ℤ/4)12yes—
278C4C2×A4⋊Q811yescharacteristic
279C4C22×SL(2,ℤ/4)11yescharacteristic
280C3—41noSylow
281C2—32no—
282C2—31no—
283C2—34no—
284C2—31no—
285C2#7311014yesdirect product factor
286C2C2×#540112yes—
287C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC23C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC25D12
prime radicalC23×A4D4
cosocleC2×C4×A4C23
hypercenterC22×C8S4
hypocenterA4C2×#106
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. #73110C2direct (N × H)
7. C2×C8×A4C4semidirect
28. C8×A4C2×C4semidirect
35. C23×C8Dic3semidirect
90. C22×C8C2×Dic3semidirect
112. C2×A4#106semidirect
175. C2×C8SL(2,ℤ/4)semidirect
190. A4C2×#106semidirect
210. C23#817semidirect
249. C8C2×SL(2,ℤ/4)semidirect
268. C22C2×#817semidirect
285. C2#73110direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×A42
C2×C4⋊SL(2,ℤ/4)2
#731102
C2×#18073
C2×#8174