Finite Group Databasebeta

← #11712645

Subgroups — #11712645

Order 768.

633 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 101 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
2#85482C212yesmaximal
3#73111C212yesmaximal, complemented
4#73110C212yesmaximal, complemented
5#20658—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C2×C8×A4C414yescomplemented
8C2×C8×A4C2211yescharacteristic
9C2×C8×A4C2211yescharacteristic
10#5432—22no—
11C4⋊SL(2,ℤ/4)C2214yes—
12C4×SL(2,ℤ/4)—22no—
13#5106—41nomaximal
14C24×C8D311yescharacteristic
15C2×#427—31no—
16C2×#427—31no—
17#2280—32no—
18#1808—32no—
19#1808—32no—
20#1807—32no—
21#1807—32no—
22#1735—32no—
23C23×A4D411yescharacteristic
24C2×C4×A4C2×C412yes—
25C2×C4×A4D411yescharacteristic
26C2×C4×A4Q812yes—
27C2×C4×A4C2311yescharacteristic
28C2×SL(2,ℤ/4)—24no—
29C22×C3×C8—41no—
30#926—42no—
31C8×A4C2×C412yes—
32C8×A4C2×C412yes—
33#818—42no—
34#817—42no—
35C24×C4C2×D311yescharacteristic
36C23×C8—31no—
37C23×C8—31no—
38C23×C8—31no—
39C23×C8—31no—
40C23×C8—34no—
41C23×C8Dic314yescomplemented
42C23×C8C2×D311yescharacteristic
43C23×C8C2×D311yescharacteristic
44C2×#117—32no—
45C2×#117—32no—
46#427—64no—
47#427—61no—
48#427—61no—
49#427—61no—
50#427—61no—
51C2×#107—32no—
52C2×#107—32no—
53C2×#106—32no—
54C2×#106—32no—
55#381—62no—
56#380—34no—
57#380—34no—
58#379—62no—
59C4×C22⋊C4—62no—
60C4×C22⋊C4—62no—
61#340—34no—
62#340—34no—
63C22×A4—22no—
64C22×A4C4⋊C412yes—
65C22×A4C2×D411yescharacteristic
66C22×C3×C4—41no—
67C4×A4C4⋊C412yes—
68C4×A4C22×C411yescharacteristic
69C4×A4C2×Q811yescharacteristic
70SL(2,ℤ/4)—44no—
71C2×C3×C8—44no—
72C2×C3×C8—41no—
73C2×C3×C8—41no—
74#217—82no—
75C4⋊Dic3—44no—
76C4×Dic3—82no—
77C25D1211yescharacteristic
78C23×C4—31no—
79C23×C4—32no—
80C23×C4—31no—
81C23×C4—32no—
82C23×C4—31no—
83C23×C4—31no—
84C23×C4Dic612yes—
85C23×C4D1211yescharacteristic
86C23×C4C2×Dic312yes—
87C23×C4C22×D311yescharacteristic
88C22×C8—31no—
89C22×C8—34no—
90C22×C8—34no—
91C22×C8—31no—
92C22×C8—31no—
93C22×C8—31no—
94C22×C8—31no—
95C22×C8—34no—
96C22×C8—34no—
97C22×C8—34no—
98C22×C8—32no—
99C22×C8—32no—
100C22×C8—31no—
101C22×C8—31no—
102C22×C8—62no—
103C22×C8—62no—
104C22×C8—62no—
105C22×C8—61no—
106C22×C8C2×Dic312yes—
107C22×C8C2×Dic312yes—
108C22×C8S411yescharacteristic
109#117—64no—
110#117—62no—
111#117—62no—
112#117—64no—
113#117—62no—
114#117—62no—
115C2×C4⋊C4—34no—
116C2×C4⋊C4—34no—
117C2×C22⋊C4—62no—
118C2×C22⋊C4—62no—
119C2×C22⋊C4—64no—
120C2×C22⋊C4—64no—
121C2×C42—62no—
122C2×C42—62no—
123#107—64no—
124#107—62no—
125#107—62no—
126#106—64no—
127#106—62no—
128#106—62no—
129#95—64no—
130#95—64no—
131C23×C3—41no—
132C2×A4—22no—
133C2×A4C2×C4⋊C411yescharacteristic
134C2×A4#13512yes—
135C2×C3×C4—42no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—41no—
139C2×Dic3—84no—
140C3×C8—42no—
141C3×C8—42no—
142C24—22no—
143C24—31no—
144C24—31no—
145C24—32no—
146C24—62no—
147C24C4⋊Dic312yes—
148C24C2×D1211yescharacteristic
149C22×C4—31no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—61no—
170C22×C4—61no—
171C22×C4—62no—
172C22×C4—61no—
173C22×C4—62no—
174C22×C4—64no—
175C22×C4—64no—
176C22×C4C4⋊Dic312yes—
177C22×C4C2×Dic611yescharacteristic
178C22×C4C22×Dic311yescharacteristic
179C22×C4C2×S411yescharacteristic
180C2×C8—34no—
181C2×C8—31no—
182C2×C8—31no—
183C2×C8—32no—
184C2×C8—32no—
185C2×C8—32no—
186C2×C8—32no—
187C2×C8—34no—
188C2×C8—34no—
189C2×C8—32no—
190C2×C8—32no—
191C2×C8—61no—
192C2×C8—61no—
193C2×C8—64no—
194C2×C8SL(2,ℤ/4)14yescomplemented
195C2×C8C2×S411yescharacteristic
196C2×C8C2×S411yescharacteristic
197C4⋊C4—64no—
198C4⋊C4—64no—
199C4⋊C4—64no—
200C4⋊C4—64no—
201C22⋊C4—124no—
202C22⋊C4—128no—
203C22⋊C4—124no—
204C22⋊C4—122no—
205C22⋊C4—124no—
206C22⋊C4—122no—
207C42—122no—
208C42—124no—
209C42—122no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—82no—
213A4#42711yescharacteristic, complemented
214C3×C4—41no—
215C3×C4—42no—
216C3×C4—41no—
217Dic3—164no—
218C23—22no—
219C23—31no—
220C23—32no—
221C23—32no—
222C23—31no—
223C23—34no—
224C23—31no—
225C23—32no—
226C23—32no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—61no—
231C23—64no—
232C23—62no—
233C23—64no—
234C23#90412yes—
235C23C2×C4⋊Dic311yescharacteristic
236C23C4⋊S411yescharacteristic
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—31no—
242C2×C4—31no—
243C2×C4—32no—
244C2×C4—32no—
245C2×C4—32no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4—124no—
261C2×C4—124no—
262C2×C4—124no—
263C2×C4—124no—
264C2×C4A4⋊Q812yes—
265C2×C4C4⋊S411yescharacteristic
266C2×C4C2×SL(2,ℤ/4)12yes—
267C2×C4C22×S411yescharacteristic
268C8—32no—
269C8—32no—
270C8C2×SL(2,ℤ/4)12yes—
271C8C2×SL(2,ℤ/4)12yes—
272C2×C3—42no—
273C2×C3—41no—
274C2×C3—82no—
275C22—22no—
276C22—32no—
277C22—31no—
278C22—32no—
279C22—32no—
280C22—31no—
281C22—31no—
282C22—34no—
283C22—32no—
284C22—62no—
285C22—64no—
286C22—62no—
287C22—61no—
288C22—62no—
289C22—64no—
290C22—62no—
291C22#510611yescharacteristic, complemented
292C22C4⋊SL(2,ℤ/4)12yes—
293C22C2×C4⋊S411yescharacteristic
294C4—32no—
295C4—31no—
296C4—31no—
297C4—244no—
298C4—244no—
299C4C4⋊SL(2,ℤ/4)12yes—
300C4C2×A4⋊Q811yescharacteristic
301C4C22×SL(2,ℤ/4)11yescharacteristic
302C3—41noSylow
303C2—22no—
304C2—32no—
305C2—31no—
306C2—31no—
307C2—62no—
308C2#8544012yes—
309C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC24C2×D12
prime radicalC23×A4D4
cosocleC2×C4×A4C23
hypercenterC22×C8S4
hypocenterA4#427
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. #73111C2semidirect
4. #73110C2semidirect
7. C2×C8×A4C4semidirect
41. C23×C8Dic3semidirect
194. C2×C8SL(2,ℤ/4)semidirect
213. A4#427semidirect
291. C22#5106semidirect

Maximal subgroups

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

subgroupindex
C22×C8×A42
#854822
#731112
#731102
#206583
#51064