Finite Group Databasebeta

← #11714066

Subgroups — #11714066

Order 768.

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

normal subgroups only

This group has 469 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
1#85283C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3C4×C8×A4C211yesmaximal, characteristic
4#73109C214yesmaximal
5#33043—31nomaximal, Sylow
6C2×A4⋊Q8—21no—
7C2×A4⋊Q8C2212yes—
8C2×C8×A4C2212yes—
9C42×A4C2211yescharacteristic
10C4⋊SL(2,ℤ/4)—21no—
11C4⋊SL(2,ℤ/4)—21no—
12C4⋊SL(2,ℤ/4)—21no—
13C4⋊SL(2,ℤ/4)C2212yes—
14#4681—41nomaximal
15C2×#487—31no—
16C22×C4×C8D311yescharacteristic
17#2857—31no—
18#2857—31no—
19#1990—31no—
20#1956—31no—
21#1933—34no—
22#1892—34no—
23#1834—31no—
24C2×C4×A4D411yescharacteristic
25C2×C4×A4D411yescharacteristic
26C2×C4×A4C2311yescharacteristic
27C2×SL(2,ℤ/4)—22no—
28C2×SL(2,ℤ/4)—22no—
29A4⋊Q8—24no—
30A4⋊Q8—42no—
31#869—41no—
32#869—41no—
33C8×A4—22no—
34C3×C4×C8—41no—
35#816—44no—
36C23×C8C2×D312yes—
37C2×C4⋊Q8—31no—
38C2×C4⋊Q8—31no—
39C2×C22⋊Q8—32no—
40C2×C22⋊Q8—61no—
41C22×C42C2×D311yescharacteristic
42#487—62no—
43#487—61no—
44#487—61no—
45C2×#103—34no—
46C2×#103—34no—
47C2×C4×C8—31no—
48C2×C4×C8—32no—
49C2×C4×C8—31no—
50#391—32no—
51#391—61no—
52#389—32no—
53#389—61no—
54#389—61no—
55#389—61no—
56#384—31no—
57#384—31no—
58#380—32no—
59#380—61no—
60#380—61no—
61#380—61no—
62#340—34no—
63#340—34no—
64C22×A4C2×D411yescharacteristic
65C2×Dic6—42no—
66C2×Dic6—81no—
67C4×A4SD1612yes—
68C4×A4Q1612yes—
69C4×A4#4112yes—
70SL(2,ℤ/4)—42no—
71SL(2,ℤ/4)—42no—
72C2×C3×C8—42no—
73C3×C42—41no—
74C4⋊Dic3—42no—
75C4⋊Dic3—81no—
76C4⋊Dic3—81no—
77C4⋊Dic3—81no—
78C22×Q8—32no—
79C22×Q8—61no—
80C23×C4D1211yescharacteristic
81C23×C4D1211yescharacteristic
82C23×C4C22×D311yescharacteristic
83C22×C8—22no—
84C22×C8—32no—
85C22×C8—32no—
86C22×C8—62no—
87C4⋊Q8—61no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—61no—
93C4⋊Q8—61no—
94C4⋊Q8—61no—
95C22⋊Q8—64no—
96C22⋊Q8—64no—
97C22⋊Q8—122no—
98C22⋊Q8—122no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4—61no—
111C2×C22⋊C4—62no—
112C2×C22⋊C4—62no—
113C2×C42—31no—
114C2×C42—31no—
115C2×C42—31no—
116C2×C42—31no—
117#103—68no—
118#103—64no—
119#103—64no—
120C4×C8—32no—
121C4×C8—31no—
122C4×C8—61no—
123C4×C8S411yescharacteristic
124#95—62no—
125#95—62no—
126#95—62no—
127#95—62no—
128C2×A4#12411yescharacteristic
129C2×A4C2×SD1611yescharacteristic
130C2×A4C2×Q1611yescharacteristic
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×Dic3—82no—
135C2×Dic3—82no—
136Dic6—84no—
137Dic6—162no—
138C3×C8—82no—
139C24C2×D1211yescharacteristic
140C2×Q8—64no—
141C2×Q8—64no—
142C2×Q8—62no—
143C2×Q8—62no—
144C2×Q8—122no—
145C2×Q8—121no—
146C2×Q8—122no—
147C2×Q8—121no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—32no—
157C22×C4—62no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4#20412yes—
162C22×C4Dic1212yes—
163C22×C4#23512yes—
164C2×C8—34no—
165C2×C8—32no—
166C2×C8—62no—
167C2×C8—62no—
168C2×C8—62no—
169C2×C8C2×S412yes—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C4⋊C4—122no—
177C4⋊C4—124no—
178C4⋊C4—122no—
179C4⋊C4—124no—
180C4⋊C4—121no—
181C4⋊C4—121no—
182C4⋊C4—121no—
183C4⋊C4—121no—
184C4⋊C4—121no—
185C4⋊C4—121no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42—61no—
194C42C2×S411yescharacteristic
195C22×C3—41no—
196A4#48711yescharacteristic, complemented
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—42no—
200Dic3—162no—
201Dic3—162no—
202C23—31no—
203C23—31no—
204C23—31no—
205C23—31no—
206C23#87511yescharacteristic
207C23C2×#20411yescharacteristic
208C23C2×Dic1211yescharacteristic
209Q8—124no—
210Q8—124no—
211Q8—242no—
212Q8—242no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—32no—
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4C4⋊S411yescharacteristic
241C2×C4C4⋊S411yescharacteristic
242C2×C4C22×S411yescharacteristic
243C8—22no—
244C8—62no—
245C2×C3—41no—
246C2×C3—41no—
247C2×C3—41no—
248C22—31no—
249C22—31no—
250C22—31no—
251C22—31no—
252C22—31no—
253C22—31no—
254C22—31no—
255C22—31no—
256C22—31no—
257C22—61no—
258C22#468111yescharacteristic, complemented
259C22C2×C4⋊S411yescharacteristic
260C4—32no—
261C4—32no—
262C4—32no—
263C4—242no—
264C4—242no—
265C4—242no—
266C4—242no—
267C4#539812yes—
268C4#540112yes—
269C4#591212yes—
270C3—41noSylow
271C2—31no—
272C2—31no—
273C2—31no—
274C2—31no—
275C2#8529011yescharacteristic
276C2C2×#539811yescharacteristic
277C2C2×#540111yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radicalC22×A4C2×D4
hypercenterC4×C8S4
hypocenterA4#487
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
196. A4#487semidirect
258. C22#4681semidirect

Maximal subgroups

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

subgroupindex
#852832
C4×C8×A42
#731092
#330433
#46814