Finite Group Databasebeta

← C2×C8×SL(2,ℤ/4) #11712545

Subgroups — C2×C8×SL(2,ℤ/4) #11712545

Order 768.

881 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 881 conjugacy classes of subgroups — too many to draw. Only its 157 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)C211yesmaximal, characteristic
3C22×A4⋊C8C211yesmaximal, characteristic
4C8×SL(2,ℤ/4)C214yesmaximal, direct product factor
5C2×C8×C22⋊C4—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C22×SL(2,ℤ/4)C412yes—
8C2×C8×A4C414yescomplemented
9C2×C8×A4C2212yes—
10C4×SL(2,ℤ/4)C412yes—
11C4×SL(2,ℤ/4)C2212yes—
12C2×A4⋊C8C414yescomplemented
13C2×A4⋊C8C2212yes—
14C2×C8×Dic3—41nomaximal
15C24×C8D311yescharacteristic
16C22×C22⋊C8—31no—
17C22×C4×C8—31no—
18C22×C4×C8—31no—
19C2×C4×C22⋊C4—31no—
20C8×C22⋊C4—34no—
21C8×C22⋊C4—34no—
22C2×#336—31no—
23C2×#336—31no—
24C23×A4C2×C411yescharacteristic
25C2×C4×A4C2×C411yescharacteristic
26C2×C4×A4C2×C412yes—
27C2×C4×A4C2×C412yes—
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)C818yesdirect product factor
30C2×SL(2,ℤ/4)C2×C414yes—
31C22×C3×C8—41no—
32C2×C4×Dic3—41no—
33C22×C3⋊C8—41no—
34C8×A4C2×C414yescomplemented
35A4⋊C8C2×C414yescomplemented
36C8×Dic3—44no—
37C24×C4C2×D311yescharacteristic
38C23×C8—31no—
39C23×C8—31no—
40C23×C8—32no—
41C23×C8—34no—
42C23×C8—31no—
43C23×C8—31no—
44C23×C8Dic314yescomplemented
45C23×C8C2×D312yes—
46C22×C22⋊C4—32no—
47C22×C42—31no—
48C22×C42—31no—
49C2×C22⋊C8—34no—
50C2×C22⋊C8—32no—
51C2×C22⋊C8—34no—
52C2×C22⋊C8—32no—
53C2×C4×C8—34no—
54C2×C4×C8—34no—
55C2×C4×C8—64no—
56C2×C4×C8—61no—
57C2×C4×C8—62no—
58C2×C4×C8—61no—
59C4×C22⋊C4—32no—
60C4×C22⋊C4—32no—
61C4×C22⋊C4—32no—
62C4×C22⋊C4—32no—
63C2×#95—32no—
64#336—34no—
65#336—34no—
66C22×A4C4212yes—
67C22×A4C2×C814yes—
68C22×A4C22×C411yescharacteristic
69C22×C3×C4—41no—
70C22×Dic3—42no—
71C4×A4C4212yes—
72C4×A4C22×C411yescharacteristic
73C4×A4C22×C411yescharacteristic
74SL(2,ℤ/4)C2×C818yesdirect product factor
75C2×C3×C8—44no—
76C2×C3×C8—42no—
77C4×Dic3—42no—
78C4×Dic3—42no—
79C2×C3⋊C8—44no—
80C2×C3⋊C8—42no—
81C25C4×D311yescharacteristic
82C23×C4—31no—
83C23×C4—32no—
84C23×C4—31no—
85C23×C4—31no—
86C23×C4—31no—
87C23×C4—32no—
88C23×C4—32no—
89C23×C4—32no—
90C23×C4C4×D311yescharacteristic
91C23×C4C4×D312yes—
92C23×C4C2×Dic312yes—
93C23×C4C22×D311yescharacteristic
94C22×C8—31no—
95C22×C8—34no—
96C22×C8—34no—
97C22×C8—32no—
98C22×C8—32no—
99C22×C8—34no—
100C22×C8—38no—
101C22×C8—34no—
102C22×C8—32no—
103C22×C8—32no—
104C22×C8—32no—
105C22×C8—34no—
106C22×C8—34no—
107C22×C8—64no—
108C22×C8—62no—
109C22×C8—61no—
110C22×C8—61no—
111C22×C8—61no—
112C22×C8—62no—
113C22×C8—62no—
114C22×C8—61no—
115C22×C8—61no—
116C22×C8C2×Dic314yescomplemented
117C22×C8S411yescharacteristic
118C2×C22⋊C4—38no—
119C2×C22⋊C4—34no—
120C2×C22⋊C4—38no—
121C2×C22⋊C4—34no—
122C2×C42—32no—
123C2×C42—32no—
124C2×C42—32no—
125C2×C42—32no—
126C2×C42—62no—
127C2×C42—61no—
128C2×C42—61no—
129C2×C42—62no—
130C2×C42—61no—
131C2×C42—61no—
132C22⋊C8—38no—
133C22⋊C8—34no—
134C22⋊C8—34no—
135C4×C8—68no—
136C4×C8—64no—
137C4×C8—64no—
138#95—34no—
139#95—34no—
140C23×C3—41no—
141C2×A4C4×C814yescomplemented
142C2×A4C2×C4211yescharacteristic
143C2×A4C22×C812yes—
144C2×C3×C4—42no—
145C2×C3×C4—41no—
146C2×C3×C4—42no—
147C2×C3×C4—41no—
148C2×Dic3—44no—
149C2×Dic3—48no—
150C3×C8—44no—
151C3⋊C8—44no—
152C24—31no—
153C24—31no—
154C24—34no—
155C24—32no—
156C24D3×C814yes—
157C24C4×Dic312yes—
158C24C2×C4×D311yescharacteristic
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—34no—
180C22×C4—34no—
181C22×C4—38no—
182C22×C4—38no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—62no—
186C22×C4—61no—
187C22×C4—62no—
188C22×C4—64no—
189C22×C4—62no—
190C22×C4—64no—
191C22×C4—64no—
192C22×C4—62no—
193C22×C4C4×Dic312yes—
194C22×C4C2×C4×D311yescharacteristic
195C22×C4C22×Dic311yescharacteristic
196C22×C4C2×S411yescharacteristic
197C2×C8—34no—
198C2×C8—32no—
199C2×C8—34no—
200C2×C8—34no—
201C2×C8—38no—
202C2×C8—34no—
203C2×C8—34no—
204C2×C8—34no—
205C2×C8—62no—
206C2×C8—64no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—64no—
210C2×C8—64no—
211C2×C8—62no—
212C2×C8—64no—
213C2×C8—64no—
214C2×C8—62no—
215C2×C8SL(2,ℤ/4)14yesdirect product factor
216C2×C8C2×S412yes—
217C22⋊C4—316no—
218C22⋊C4—38no—
219C22⋊C4—38no—
220C42—62no—
221C42—62no—
222C42—62no—
223C42—62no—
224C42—64no—
225C42—62no—
226C42—62no—
227C22×C3—41no—
228C22×C3—44no—
229C22×C3—42no—
230A4C2×C4×C811yescharacteristic, complemented
231C3×C4—41no—
232C3×C4—42no—
233C3×C4—41no—
234Dic3—48no—
235C23—31no—
236C23—31no—
237C23—34no—
238C23—34no—
239C23—32no—
240C23—32no—
241C23—34no—
242C23—32no—
243C23—32no—
244C23—31no—
245C23—38no—
246C23—34no—
247C23—61no—
248C23—64no—
249C23—62no—
250C23C8×Dic314yescomplemented
251C23C2×D3×C812yes—
252C23C2×C4×Dic311yescharacteristic
253C23C4×S411yescharacteristic
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—38no—
272C2×C4—38no—
273C2×C4—61no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—62no—
277C2×C4—64no—
278C2×C4—616no—
279C2×C4—64no—
280C2×C4—64no—
281C2×C4—68no—
282C2×C4—64no—
283C2×C4—68no—
284C2×C4C4×S412yes—
285C2×C4C4×S411yescharacteristic
286C2×C4C2×SL(2,ℤ/4)12yes—
287C2×C4C22×S411yescharacteristic
288C8—34no—
289C8—64no—
290C8—64no—
291C8C2×SL(2,ℤ/4)14yesdirect product factor
292C2×C3—44no—
293C2×C3—42no—
294C2×C3—41no—
295C22—34no—
296C22—32no—
297C22—31no—
298C22—38no—
299C22—34no—
300C22—34no—
301C22—32no—
302C22—32no—
303C22—34no—
304C22—31no—
305C22—31no—
306C22—32no—
307C22—61no—
308C22—64no—
309C22—62no—
310C22C2×C8×Dic311yescharacteristic, complemented
311C22C8×S414yes—
312C22C4×SL(2,ℤ/4)12yes—
313C22C2×C4×S411yescharacteristic
314C4—31no—
315C4—31no—
316C4—32no—
317C4—68no—
318C4—68no—
319C4C4×SL(2,ℤ/4)12yes—
320C4C2×C4×S411yescharacteristic
321C4C22×SL(2,ℤ/4)11yescharacteristic
322C3—41noSylow
323C2—32no—
324C2—31no—
325C2—34no—
326C2—31no—
327C2C8×SL(2,ℤ/4)14yesdirect product factor
328C2C2×C8×S412yes—
329C2C2×C4×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C8S4
commutator, hypocenterA4C2×C4×C8
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC25C4×D3
prime radicalC23×A4C2×C4
cosocleC2×C4×A4C23
LCSG ▷ A4
UCSC1 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
4. C8×SL(2,ℤ/4)C2direct (N × H)
8. C2×C8×A4C4semidirect
12. C2×A4⋊C8C4semidirect
29. C2×SL(2,ℤ/4)C8direct (N × H)
34. C8×A4C2×C4semidirect
35. A4⋊C8C2×C4semidirect
44. C23×C8Dic3semidirect
74. SL(2,ℤ/4)C2×C8direct (N × H)
116. C22×C8C2×Dic3semidirect
141. C2×A4C4×C8semidirect
215. C2×C8SL(2,ℤ/4)direct (N × H)
230. A4C2×C4×C8semidirect
250. C23C8×Dic3semidirect
291. C8C2×SL(2,ℤ/4)direct (N × H)
310. C22C2×C8×Dic3semidirect
327. C2C8×SL(2,ℤ/4)direct (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
C22×A4⋊C82
C8×SL(2,ℤ/4)2
C2×C8×C22⋊C43
C2×C8×Dic34