Finite Group Databasebeta

← #11713004

Subgroups — #11713004

Order 768.

531 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 69 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×C8×S4C211yesmaximal, characteristic
2#85307C212yesmaximal
3A4×#106C211yesmaximal, characteristic, complemented
4#73110C211yesmaximal, characteristic, complemented
5#73050C212yesmaximal, complemented
6#22654—31nomaximal, Sylow
7C2×C4×S4C2211yescharacteristic
8C2×C8×A4C2211yescharacteristic
9A4×C4⋊C4C2212yes—
10(C2×S4)⋊C4—22no—
11C4⋊SL(2,ℤ/4)C2212yes—
12C4×SL(2,ℤ/4)—22no—
13C2×A4⋊C8C2211yescharacteristic
14C8×S4C414yescomplemented
15#4860—41nomaximal
16C2×C8×D4—31no—
17C2×#427—31no—
18C22×#106D311yescharacteristic, complemented
19#2352—32no—
20#1949—32no—
21#1854—32no—
22#1852—32no—
23#1846—31no—
24#1807—31no—
25#1807—31no—
26C4×#106—31no—
27C22×S4D411yescharacteristic
28C2×C4×A4—22no—
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)—22no—
31C2×SL(2,ℤ/4)D411yescharacteristic
32C4×S4C2×C412yes—
33C4×S4Q812yes—
34C2×D3×C8—41no—
35#892—42no—
36C8×A4C2×C412yes—
37A4⋊C8C2×C412yes—
38C3×#106—41no—
39#817—41no—
40#807—42no—
41C23×C8—31no—
42C23×C8C2×D311yescharacteristic
43C2×C4×D4—31no—
44C2×#117—32no—
45C22×C4⋊C4C2×D312yes—
46C8×D4—34no—
47C8×D4—34no—
48#427—61no—
49#427—62no—
50#427—61no—
51C2×#107—32no—
52C2×#107—32no—
53C2×#106—31no—
54C2×#106—31no—
55C2×#106—32no—
56C2×#106—31no—
57C2×#106—31no—
58C2×C4⋊C8—31no—
59C2×C22⋊C8—31no—
60C2×C22⋊C8—31no—
61C2×C4×C8—31no—
62#388—62no—
63#387—62no—
64#386—62no—
65#382—32no—
66#380—32no—
67#380—32no—
68C4×C4⋊C4—32no—
69C4×C22⋊C4—62no—
70#340—32no—
71#340—32no—
72#340—32no—
73C22×A4C2×D411yescharacteristic
74C2×S4—22no—
75C2×S4C4⋊C412yes—
76C2×C4×D3—41no—
77C4×A4—42no—
78C4×A4C22×C411yescharacteristic
79C4×A4C2×Q811yescharacteristic
80SL(2,ℤ/4)—42no—
81SL(2,ℤ/4)C4⋊C412yes—
82C2×C3×C8—41no—
83C3×C4⋊C4—42no—
84(C2×D3)⋊C4—82no—
85C4⋊Dic3—42no—
86C4×Dic3—82no—
87C2×C3⋊C8—41no—
88D3×C8—44no—
89C22×D4—31no—
90C23×C4—22no—
91C23×C4—31no—
92C23×C4C22×D311yescharacteristic
93C22×C8—31no—
94C22×C8—31no—
95C22×C8—32no—
96C22×C8—31no—
97C22×C8—34no—
98C22×C8—31no—
99C22×C8—62no—
100C22×C8—61no—
101C22×C8—61no—
102C22×C8C4×D312yes—
103C4×D4—32no—
104C4×D4—32no—
105C4×D4—32no—
106C4×D4—32no—
107#117—62no—
108#117—62no—
109#117—122no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—62no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—31no—
118C2×C22⋊C4—62no—
119C2×C22⋊C4—62no—
120C2×C22⋊C4—62no—
121C2×C42—31no—
122C2×C42—62no—
123C2×C42—62no—
124#107—62no—
125#107—62no—
126#107—122no—
127#106—31no—
128#106—32no—
129#106—61no—
130#106—62no—
131#106—61no—
132#106—61no—
133#106S411yescharacteristic, complemented
134C4⋊C8—34no—
135C22⋊C8—34no—
136C22⋊C8—32no—
137C22⋊C8—32no—
138C4×C8—34no—
139#95—62no—
140#95—62no—
141#95—62no—
142#95—62no—
143#95—62no—
144C22×D3—41no—
145C2×A4C2×C4⋊C411yescharacteristic
146C2×A4#13512yes—
147S4—22no—
148C2×C3×C4—41no—
149C2×C3×C4—82no—
150C2×Dic3—41no—
151C2×Dic3—82no—
152C4×D3—42no—
153C4×D3—42no—
154C3×C8—42no—
155C3⋊C8—42no—
156C24—31no—
157C24D3×D411yescharacteristic
158C2×D4—32no—
159C2×D4—32no—
160C2×D4—62no—
161C2×D4—62no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—32no—
170C22×C4—32no—
171C22×C4—42no—
172C22×C4—62no—
173C22×C4—61no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—122no—
181C22×C4C2×C4×D311yescharacteristic
182C22×C4D3×Q811yescharacteristic
183C2×C8—31no—
184C2×C8—32no—
185C2×C8—32no—
186C2×C8—32no—
187C2×C8—32no—
188C2×C8—32no—
189C2×C8—61no—
190C2×C8—64no—
191C2×C8—61no—
192C2×C8—61no—
193C2×C8—61no—
194C2×C8—61no—
195C2×C8—64no—
196C2×C8C2×S411yescharacteristic
197C4⋊C4—32no—
198C4⋊C4—32no—
199C4⋊C4—32no—
200C4⋊C4—62no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4C2×S412yes—
207C22⋊C4—32no—
208C22⋊C4—32no—
209C22⋊C4—32no—
210C22⋊C4—32no—
211C22⋊C4—122no—
212C22⋊C4—122no—
213C22⋊C4—122no—
214C22⋊C4—122no—
215C22⋊C4—242no—
216C42—32no—
217C42—32no—
218C42—122no—
219C42—122no—
220C42—242no—
221C22×C3—41no—
222C2×D3—42no—
223C2×D3—82no—
224A4#42711yescharacteristic, complemented
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—162no—
228Dic3—42no—
229Dic3—162no—
230C23—31no—
231C23—31no—
232C23—32no—
233C23—32no—
234C23—62no—
235C23—61no—
236C23—62no—
237C23—61no—
238C23D3×C4⋊C411yescharacteristic
239C23#91612yes—
240D4—62no—
241D4—64no—
242D4—62no—
243C2×C4—22no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4C22×S411yescharacteristic
275C8—32no—
276C8—62no—
277C8—62no—
278C8C4×S412yes—
279C2×C3—41no—
280C2×C3—42no—
281D3—82no—
282C22—31no—
283C22—32no—
284C22—31no—
285C22—31no—
286C22—32no—
287C22—32no—
288C22—61no—
289C22—62no—
290C22—62no—
291C22—62no—
292C22—122no—
293C22—122no—
294C22#486011yescharacteristic, complemented
295C22D4×S411yescharacteristic
296C4—31no—
297C4—31no—
298C4—42no—
299C4—62no—
300C4—62no—
301C4—62no—
302C4—122no—
303C4—242no—
304C4—242no—
305C4C2×C4×S411yescharacteristic
306C4Q8×S411yescharacteristic
307C3—41noSylow
308C2—31no—
309C2—31no—
310C2—32no—
311C2—122no—
312C2C4⋊C4×S411yescharacteristic
313C2#8546212yes—
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#106D3
socleC24D3×D4
prime radicalC22×S4D4
cosocleC2×C4×A4C23
hypercenter#106S4
hypocenterA4#427
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. A4×#106C2semidirect
4. #73110C2semidirect
5. #73050C2semidirect
14. C8×S4C4semidirect
18. C22×#106D3semidirect
133. #106S4semidirect
224. A4#427semidirect
294. C22#4860semidirect

Maximal subgroups

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

subgroupindex
C2×C8×S42
#853072
A4×#1062
#731102
#730502
#226543
#48604