Finite Group Databasebeta

← #11713685

Subgroups — #11713685

Order 768.

519 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 65 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×C4⋊Q8C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3#73147C214yesmaximal
4C4×A4⋊C8C211yesmaximal, characteristic
5#25856—31nomaximal, Sylow
6C2×Q8×A4C2212yes—
7C2×A4⋊Q8—21no—
8A4×C4⋊C4—21no—
9A4×C4⋊C4—21no—
10C42×A4C2211yescharacteristic
11C4⋊SL(2,ℤ/4)—21no—
12C4⋊SL(2,ℤ/4)C2212yes—
13C2×A4⋊C8C2212yes—
14#5079—41nomaximal
15C22×C4⋊Q8D311yescharacteristic
16C2×#487—31no—
17C2×#487—31no—
18#2857—31no—
19#2043—34no—
20#1956—31no—
21#1956—31no—
22#1787—34no—
23C4×C22⋊C8—31no—
24Q8×A4—24no—
25C2×C4×A4—22no—
26C2×C4×A4D411yescharacteristic
27C2×C4×A4D411yescharacteristic
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)—22no—
30A4⋊Q8—42no—
31C3×C4⋊Q8—41no—
32#869—41no—
33A4⋊C8—22no—
34#835—44no—
35C4×C3⋊C8—41no—
36C23×Q8C2×D312yes—
37C2×C4⋊Q8—31no—
38C2×C4⋊Q8—31no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—32no—
41C2×C4⋊Q8—61no—
42C2×C4⋊Q8—61no—
43C2×C22⋊Q8—61no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C42C2×D311yescharacteristic
47#487—62no—
48#487—62no—
49#487—62no—
50#487—61no—
51#487—61no—
52C2×#103—34no—
53C2×#103—34no—
54C2×#103—34no—
55C2×#103—34no—
56C2×C22⋊C8—32no—
57C2×C4×C8—31no—
58C2×C4×C8—31no—
59#391—61no—
60#389—32no—
61#389—61no—
62#384—31no—
63#380—32no—
64#380—61no—
65#336—32no—
66C22×A4C2×D411yescharacteristic
67C2×C3×Q8—42no—
68C2×Dic6—81no—
69C4×A4—42no—
70C4×A4SD1612yes—
71C4×A4Q1612yes—
72C4×A4#4112yes—
73SL(2,ℤ/4)—42no—
74C3×C4⋊C4—81no—
75C3×C4⋊C4—81no—
76C3×C42—41no—
77C4⋊Dic3—42no—
78C4⋊Dic3—81no—
79C2×C3⋊C8—42no—
80C22×Q8—24no—
81C22×Q8—32no—
82C22×Q8—32no—
83C22×Q8—62no—
84C22×Q8—61no—
85C22×Q8—64no—
86C23×C4—22no—
87C23×C4#6711yescharacteristic
88C23×C4#6711yescharacteristic
89C23×C4C22×D311yescharacteristic
90C22×C8—32no—
91C22×C8—32no—
92C4⋊Q8—32no—
93C4⋊Q8—32no—
94C4⋊Q8—32no—
95C4⋊Q8—31no—
96C4⋊Q8—61no—
97C4⋊Q8—61no—
98C4⋊Q8—61no—
99C4⋊Q8—61no—
100C4⋊Q8—61no—
101C4⋊Q8—61no—
102C4⋊Q8—64no—
103C4⋊Q8—61no—
104C4⋊Q8S411yescharacteristic
105C22⋊Q8—122no—
106C22⋊Q8—122no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C22⋊C4—62no—
120C2×C42—31no—
121C2×C42—31no—
122C2×C42—31no—
123C2×C42—31no—
124#103—68no—
125#103—64no—
126#103—64no—
127#103—124no—
128#103—124no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131C4×C8—61no—
132C4×C8—61no—
133C4×C8—61no—
134C4×C8—61no—
135#95—62no—
136#95—62no—
137C2×A4#12411yescharacteristic
138C2×A4C2×SD1611yescharacteristic
139C2×A4C2×Q1611yescharacteristic
140C3×Q8—84no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—82no—
145C2×Dic3—82no—
146Dic6—162no—
147C3⋊C8—82no—
148C24C2×#6711yescharacteristic
149C2×Q8—34no—
150C2×Q8—32no—
151C2×Q8—62no—
152C2×Q8—64no—
153C2×Q8—64no—
154C2×Q8—64no—
155C2×Q8—64no—
156C2×Q8—124no—
157C2×Q8—122no—
158C2×Q8—121no—
159C2×Q8—122no—
160C2×Q8—124no—
161C2×Q8—121no—
162C2×Q8C2×S412yes—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—32no—
172C22×C4—42no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4—122no—
178C22×C4#21512yes—
179C22×C4#21612yes—
180C22×C4#23712yes—
181C2×C8—62no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—62no—
185C2×C8—62no—
186C2×C8—62no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—64no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4—62no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C4⋊C4—124no—
201C4⋊C4—121no—
202C4⋊C4—121no—
203C22⋊C4—122no—
204C22⋊C4—122no—
205C42—31no—
206C42—31no—
207C42—31no—
208C42—61no—
209C42C2×S411yescharacteristic
210C22×C3—41no—
211A4#48711yescharacteristic, complemented
212C3×C4—42no—
213C3×C4—42no—
214C3×C4—42no—
215C3×C4—162no—
216Dic3—162no—
217C23—31no—
218C23—31no—
219C23—31no—
220C23—31no—
221C23#93611yescharacteristic
222C23C2×#21511yescharacteristic
223C23C2×#21611yescharacteristic
224Q8—24no—
225Q8—68no—
226Q8—64no—
227Q8—124no—
228Q8—242no—
229Q8—242no—
230C2×C4—22no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—32no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4GL(2,ℤ/4)11yescharacteristic
258C2×C4GL(2,ℤ/4)11yescharacteristic
259C2×C4C22×S411yescharacteristic
260C8—122no—
261C8—122no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—41no—
265C22—31no—
266C22—31no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—31no—
271C22—31no—
272C22—31no—
273C22—31no—
274C22—61no—
275C22#507911yescharacteristic, complemented
276C22C2×GL(2,ℤ/4)11yescharacteristic
277C4—32no—
278C4—32no—
279C4—32no—
280C4—42no—
281C4—122no—
282C4—242no—
283C4—242no—
284C4#541612yes—
285C4#541712yes—
286C4#591412yes—
287C3—41noSylow
288C2—31no—
289C2—31no—
290C2—31no—
291C2—31no—
292C2#8549211yescharacteristic
293C2C2×#541611yescharacteristic
294C2C2×#541711yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4⋊Q8S4
hypocenterA4#487
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
211. A4#487semidirect
275. C22#5079semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852832
#731472
C4×A4⋊C82
#258563
#50794