Finite Group Databasebeta

← #10740102

Subgroups — #10740102

Order 768.

2018 conjugacy classes of subgroups; only the 363 normal subgroups are shown.

normal subgroups only

This group has 2018 conjugacy classes of subgroups — too many to draw. Only its 363 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×#944C212yesmaximal
2C2×#4656C214yesmaximal
3C3×#1446C211yesmaximal, characteristic
4#69420C214yesmaximal
5#69405C214yesmaximal
6#15762—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C23×Dic6C2212yes—
9C2×#944—28no—
10C2×#944—216no—
11C22×C4⋊Dic3C2214yes—
12C22×Dic3⋊C4C2214yes—
13C22×Dic3⋊C4C2214yes—
14C22×C4×Dic3C2212yes—
15C2×C3×#95C2214yes—
16C2×C3×#95C2212yes—
17C2×C3⋊#95C2218yes—
18C2×C3⋊#95C2214yes—
19#4656—216no—
20C22×C4⋊Q8—32no—
21C2×#395—34no—
22#1446D311yescharacteristic
23#1444—34no—
24#1436—34no—
25C22×C3×Q8—28no—
26C22×C3×Q8—42no—
27C22×C3×Q8D414yes—
28C23×C3×C4C2311yescharacteristic
29C23×C3×C4C2314yes—
30C23×C3×C4C2312yes—
31C23×Dic3C2314yes—
32C23×Dic3C2314yes—
33C22×Dic6—24no—
34C22×Dic6—28no—
35C22×Dic6—28no—
36C22×Dic6—44no—
37#944—264no—
38C2×C4⋊Dic3—216no—
39C2×C4⋊Dic3—44no—
40C2×Dic3⋊C4—28no—
41C2×Dic3⋊C4—28no—
42C2×Dic3⋊C4—28no—
43C2×Dic3⋊C4—28no—
44C2×Dic3⋊C4—28no—
45C2×Dic3⋊C4D4116yes—
46C2×C4×Dic3—24no—
47C2×C4×Dic3—28no—
48C3×#95—28no—
49C3×#95—42no—
50C3⋊#95—216no—
51C3⋊#95Q8116yescomplemented
52C23×Q8—32no—
53C23×Q8C2×D311yescharacteristic
54C2×C4⋊Q8—616no—
55C2×C4⋊Q8—68no—
56C22×C4⋊C4—34no—
57C22×C4⋊C4—34no—
58C22×C4⋊C4—34no—
59C22×C42—32no—
60#395—616no—
61C2×#95—34no—
62C2×#95—38no—
63C2×#95C2×D314yes—
64C2×#95C2×D312yes—
65C24×C3C2411yescharacteristic
66C2×C3×Q8—216no—
67C2×C3×Q8—44no—
68C2×C3×Q8—416no—
69C22×C3×C4—24no—
70C22×C3×C4—24no—
71C22×C3×C4—28no—
72C22×C3×C4—24no—
73C22×C3×C4—24no—
74C22×C3×C4—22no—
75C22×C3×C4—24no—
76C22×C3×C4—28no—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D414yes—
79C22×C3×C4C2×D418yes—
80C22×C3×C4C2×Q818yes—
81C22×Dic3—24no—
82C22×Dic3—24no—
83C22×Dic3—28no—
84C22×Dic3—28no—
85C22×Dic3—24no—
86C22×Dic3—24no—
87C22×Dic3—28no—
88C22×Dic3C2×D418yes—
89C22×Dic3C2×D418yes—
90C22×Dic3C2×Q8116yes—
91C2×Dic6—216no—
92C2×Dic6—432no—
93C2×Dic6—48no—
94C2×Dic6—48no—
95C4⋊Dic3—416no—
96Dic3⋊C4—232no—
97Dic3⋊C4—232no—
98C4×Dic3—216no—
99C22×Q8—28no—
100C22×Q8—42no—
101C22×Q8—68no—
102C22×Q8—64no—
103C22×Q8—68no—
104C22×Q8—124no—
105C22×Q8D1214yes—
106C23×C4—34no—
107C23×C4—34no—
108C23×C4C22×D311yescharacteristic
109C23×C4C22×D314yes—
110C23×C4C22×D312yes—
111C4⋊Q8—664no—
112C2×C4⋊C4—316no—
113C2×C4⋊C4—68no—
114C2×C4⋊C4—68no—
115C2×C4⋊C4—68no—
116C2×C4⋊C4—68no—
117C2×C4⋊C4—68no—
118C2×C4⋊C4—616no—
119C2×C4⋊C4—124no—
120C2×C42—68no—
121C2×C42—64no—
122#95—28no—
123#95—316no—
124#95—42no—
125#95—616no—
126C23×C3C22×D411yescharacteristic
127C23×C3C22×D414yes—
128C23×C3C22×Q814yes—
129C23×C3H(5,𝔽2)12yes—
130C23×C32−1+414yes—
131C3×Q8—416no—
132C2×C3×C4—24no—
133C2×C3×C4—24no—
134C2×C3×C4—216no—
135C2×C3×C4—24no—
136C2×C3×C4—22no—
137C2×C3×C4—28no—
138C2×C3×C4—216no—
139C2×C3×C4—48no—
140C2×C3×C4—44no—
141C2×C3×C4—44no—
142C2×C3×C4—44no—
143C2×C3×C4#12018yes—
144C2×Dic3—28no—
145C2×Dic3—28no—
146C2×Dic3—216no—
147C2×Dic3—216no—
148C2×Dic3—28no—
149C2×Dic3—216no—
150C2×Dic3—44no—
151C2×Dic3—44no—
152C2×Dic3—48no—
153C2×Dic3C4⋊Q8116yescomplemented
154Dic6—432no—
155C24C23×D311yescharacteristic
156C2×Q8—216no—
157C2×Q8—44no—
158C2×Q8—416no—
159C2×Q8—616no—
160C2×Q8—1232no—
161C2×Q8—128no—
162C2×Q8—128no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—38no—
172C22×C4—316no—
173C22×C4—38no—
174C22×C4—64no—
175C22×C4—64no—
176C22×C4—68no—
177C22×C4—64no—
178C22×C4—68no—
179C22×C4—64no—
180C22×C4—68no—
181C22×C4C2×D1212yes—
182C22×C4C2×D1214yes—
183C22×C4D3×D418yes—
184C22×C4D3×Q818yes—
185C4⋊C4—632no—
186C4⋊C4—632no—
187C4⋊C4—1216no—
188C42—616no—
189C22×C3C2×#12012yes—
190C22×C3C2×C4⋊Q814yes—
191C22×C3#53411yescharacteristic
192C22×C3#53614yes—
193C22×C3#54414yes—
194C22×C3D4214yes—
195C22×C3D4×Q818yes—
196C22×C3D4×Q814yes—
197C22×C3Q8212yes—
198C22×C3#56312yes—
199C3×C4—28no—
200C3×C4—42no—
201C3×C4—48no—
202Dic3—216no—
203Dic3—48no—
204C23C22×D1211yescharacteristic
205C23C2×D3×D414yes—
206C23#100412yes—
207C23C2×D3×Q814yes—
208C23#101014yes—
209Q8—416no—
210Q8—1232no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—216no—
214C2×C4—216no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—28no—
218C2×C4—316no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—616no—
224C2×C4—68no—
225C2×C4—68no—
226C2×C4—616no—
227C2×C4—68no—
228C2×C4—616no—
229C2×C4—124no—
230C2×C4—124no—
231C2×C4—128no—
232C2×C4#88218yes—
233C2×C3#240311yescharacteristic
234C2×C3#243414yes—
235C2×C3#255114yes—
236C2×C3#267714yes—
237C2×C3#273412yes—
238C22C2×#88212yes—
239C22#549311yescharacteristic
240C22Q8×D1214yes—
241C22#559014yes—
242C22#562314yes—
243C22#563218yes—
244C22#567212yes—
245C22#567514yes—
246C22#572212yes—
247C22D3×C4⋊Q814yes—
248C4—28no—
249C4—48no—
250C4—42no—
251C4—616no—
252C4—128no—
253C3#1576211yescharacteristic, complemented, Sylow
254C2#7385211yescharacteristic
255C2#7649914yes—
256C2#7657412yes—
257C2#7682314yes—
258C2#7718614yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#15762
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
51. C3⋊#95Q8semidirect
153. C2×Dic3C4⋊Q8semidirect
253. C3#15762semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×#46562
C3×#14462
#694202
#694052
#157623