Finite Group Databasebeta

← #11712657

Subgroups — #11712657

Order 768.

517 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 77 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
1A4×#117C211yesmaximal, characteristic
2#73141C212yesmaximal
3C4⋊(A4⋊C8)C212yesmaximal, complemented
4#73044C211yesmaximal, characteristic, complemented
5C4×A4⋊C8C211yesmaximal, characteristic, complemented
6#20716—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8A4×C4⋊C4C412yes—
9A4×C22⋊C4C412yes—
10C42×A4C2211yescharacteristic
11C42×A4C2211yescharacteristic
12C2×A4⋊C8C2212yes—
13C2×A4⋊C8C2212yes—
14#5000—41nomaximal
15C22×#117D311yescharacteristic
16C2×#433—31no—
17C2×#433—31no—
18#1818—32no—
19#1798—32no—
20#1797—32no—
21#1779—32no—
22#1750—31no—
23#1750—31no—
24#1749—31no—
25C4×C22⋊C8—31no—
26C23×A4C2×C411yescharacteristic
27C2×C4×A4—22no—
28C2×C4×A4C2×C411yescharacteristic
29C2×C4×A4C2×C412yes—
30C2×C4×A4C2×C412yes—
31C2×C4×A4C2311yescharacteristic
32C3×#117—41no—
33A4⋊C8—22no—
34A4⋊C8—22no—
35#830—42no—
36C4⋊(C3⋊C8)—42no—
37#803—41no—
38C4×C3⋊C8—41no—
39C24×C4C2×D311yescharacteristic
40C2×#117—31no—
41C2×#117—31no—
42C2×#117—31no—
43C2×#117—32no—
44C2×#117—32no—
45C2×#117—31no—
46C22×C4⋊C4Dic312yes—
47C22×C22⋊C4Dic312yes—
48C22×C42C2×D311yescharacteristic
49C22×C42C2×D311yescharacteristic
50#433—62no—
51#433—62no—
52#433—61no—
53#433—61no—
54#433—61no—
55#433—61no—
56C2×C4⋊C8—32no—
57C2×C4⋊C8—32no—
58C2×C22⋊C8—32no—
59C2×C22⋊C8—32no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—32no—
62C2×C22⋊C8—32no—
63C2×C22⋊C8—32no—
64C2×#97—31no—
65C2×#97—31no—
66C2×C4×C8—31no—
67C2×C4×C8—31no—
68#336—32no—
69#336—32no—
70#336—32no—
71#336—32no—
72C22×A4—21no—
73C22×A4—22no—
74C22×A4C22×C411yescharacteristic
75C22×C3×C4—41no—
76C4×A4—21no—
77C4×A4—22no—
78C4×A4—22no—
79C4×A4#4112yes—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—42no—
82C3×C42—41no—
83C3×C42—41no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8—42no—
86C25C2×Dic311yescharacteristic
87C23×C4—22no—
88C23×C4—31no—
89C23×C4—31no—
90C23×C4—31no—
91C23×C4—31no—
92C23×C4—62no—
93C23×C4C2×Dic312yes—
94C23×C4C2×Dic312yes—
95C23×C4C2×Dic311yescharacteristic
96C23×C4C22×D311yescharacteristic
97C22×C8—32no—
98C22×C8—32no—
99C22×C8—32no—
100C22×C8—32no—
101#117—31no—
102#117—32no—
103#117—32no—
104#117—31no—
105#117—31no—
106#117—61no—
107#117—62no—
108#117—62no—
109#117—61no—
110#117—61no—
111#117S411yescharacteristic
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C42—31no—
121C2×C42—31no—
122C2×C42—31no—
123C2×C42—31no—
124C2×C42—32no—
125C2×C42—32no—
126C4⋊C8—64no—
127C4⋊C8—62no—
128C4⋊C8—62no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131C22⋊C8—64no—
132C22⋊C8—64no—
133C22⋊C8—62no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#97—62no—
140#97—61no—
141#97—61no—
142C4×C8—62no—
143C4×C8—61no—
144C4×C8—61no—
145C23×C3—41no—
146C2×A4—41no—
147C2×A4#11711yescharacteristic
148C2×A4#13112yes—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—82no—
154C3⋊C8—82no—
155C3⋊C8—82no—
156C24—21no—
157C24—22no—
158C24—31no—
159C24—31no—
160C24—61no—
161C24—62no—
162C24C22×Dic311yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—34no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—32no—
174C22×C4—32no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4#23512yes—
191C22×C4C2×S411yescharacteristic
192C2×C8—62no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—62no—
197C2×C8—62no—
198C2×C8—62no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C2×C8—62no—
203C2×C8—62no—
204C4⋊C4—32no—
205C4⋊C4—32no—
206C4⋊C4—32no—
207C4⋊C4—62no—
208C4⋊C4SL(2,ℤ/4)12yes—
209C22⋊C4—32no—
210C22⋊C4—32no—
211C22⋊C4—32no—
212C22⋊C4—62no—
213C22⋊C4SL(2,ℤ/4)12yes—
214C42—32no—
215C42—32no—
216C42—31no—
217C42—31no—
218C42—61no—
219C42—61no—
220C42C2×S411yescharacteristic
221C42C2×S411yescharacteristic
222C22×C3—41no—
223C22×C3—81no—
224C22×C3—82no—
225A4#43311yescharacteristic, complemented
226C3×C4—42no—
227C3×C4—81no—
228C3×C4—82no—
229C3×C4—82no—
230C23—31no—
231C23—31no—
232C23—32no—
233C23—32no—
234C23—31no—
235C23—41no—
236C23—61no—
237C23—61no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—61no—
243C23—122no—
244C23—121no—
245C23—122no—
246C23#92611yescharacteristic
247C23#94812yes—
248C23C2×SL(2,ℤ/4)11yescharacteristic
249C2×C4—22no—
250C2×C4—31no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—31no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—64no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4C2×SL(2,ℤ/4)12yes—
275C2×C4C2×SL(2,ℤ/4)12yes—
276C2×C4C2×SL(2,ℤ/4)11yescharacteristic
277C2×C4C22×S411yescharacteristic
278C8—122no—
279C8—122no—
280C8—122no—
281C8—122no—
282C2×C3—42no—
283C2×C3—41no—
284C2×C3—161no—
285C22—22no—
286C22—21no—
287C22—31no—
288C22—32no—
289C22—32no—
290C22—32no—
291C22—31no—
292C22—31no—
293C22—61no—
294C22—64no—
295C22—62no—
296C22—61no—
297C22—121no—
298C22—122no—
299C22—121no—
300C22—121no—
301C22—121no—
302C22#500011yescharacteristic, complemented
303C22C22×SL(2,ℤ/4)11yescharacteristic
304C4—22no—
305C4—22no—
306C4—21no—
307C4—32no—
308C4—62no—
309C4—62no—
310C4—61no—
311C4#591212yes—
312C3—41noSylow
313C2—31no—
314C2—32no—
315C2—31no—
316C2—41no—
317C2—121no—
318C2#8548211yescharacteristic
319C2#8551112yes—
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×#117D3
socleC24C22×Dic3
prime radicalC23×A4C2×C4
cosocleC2×C4×A4C23
hypercenter#117S4
hypocenterA4#433
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. C4⋊(A4⋊C8)C2semidirect
4. #73044C2semidirect
5. C4×A4⋊C8C2semidirect
225. A4#433semidirect
302. C22#5000semidirect

Maximal subgroups

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

subgroupindex
A4×#1172
#731412
C4⋊(A4⋊C8)2
#730442
C4×A4⋊C82
#207163
#50004