Finite Group Databasebeta

← #10692598

Subgroups — #10692598

Order 768.

486 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C3×#2546C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71773C212yesmaximal
4#71637C211yesmaximal, characteristic
5#67620C212yesmaximal
6#9402—31nomaximal, Sylow
7C2×C3×C4⋊Q8—21no—
8C2×C3×C4⋊Q8C2211yescharacteristic
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#398C414yes—
11C3×#391—21no—
12C3×#389—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#836—21no—
15#5024—24no—
16#4926—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4547—24no—
20#4530—21no—
21#2983—31no—
22#2546D311yescharacteristic
23#1801—31no—
24#1546—32no—
25#1303—32no—
26C22×C3×Q8—21no—
27C22×C3×Q8D411yescharacteristic
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—44no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—24no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C412yes—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C42—21no—
40C2×C3×C42D412yes—
41C2×C3×C42C2311yescharacteristic
42C2×#208—22no—
43C3×#95—24no—
44C3×#95C2×C412yes—
45#836—42no—
46#830—22no—
47C4⋊(C3⋊C8)—24no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51#803—24no—
52C4×C3⋊C8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×D311yescharacteristic
55C2×C4×Q8C2×D311yescharacteristic
56#446—64no—
57#432—62no—
58#431—32no—
59#423—32no—
60C2×#101—61no—
61#398Dic314yes—
62#391—21no—
63#389—22no—
64C4×C4⋊C4C2×D311yescharacteristic
65#343—61no—
66#332—64no—
67C2×C3×Q8—22no—
68C2×C3×Q8—21no—
69C2×C3×Q8—22no—
70C2×C3×Q8—21no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C2×C3×Q8—41no—
74C2×C3×Q8—41no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4C22⋊C412yes—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C3×C4⋊C4—24no—
83C3×C4⋊C4—24no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—24no—
86C3×C4⋊C4—48no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—41no—
89C3×C42—22no—
90C3×C42—22no—
91C3×C42—42no—
92C3×C42C2×D412yes—
93C3×C42#4112yes—
94#208—42no—
95#208—42no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C22×Q8—21no—
99C22×Q8#6711yescharacteristic
100C2×M16—62no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—44no—
104C4×Q8—22no—
105C4×Q8—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C2×Dic312yes—
110C2×C4⋊C4C2×Dic312yes—
111C2×C4⋊C4#6711yescharacteristic
112C2×C42—21no—
113C2×C42#6712yes—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—34no—
116C4⋊C8—64no—
117C4⋊C8—62no—
118C4⋊C8—62no—
119#101—122no—
120C22⋊C8—62no—
121#97—64no—
122C4×C8—62no—
123#95—24no—
124#95C2×Dic312yes—
125C23×C3C2×C22⋊C411yescharacteristic
126C3×Q8—42no—
127C3×Q8—42no—
128C3×Q8—82no—
129C3×Q8—82no—
130C2×C3×C4—22no—
131C2×C3×C4—24no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4C4×D412yes—
144C2×C3×C4#12011yescharacteristic
145C2×C3×C4#12112yes—
146C2×C3×C4#12311yescharacteristic
147C2×C3×C4AGL(1,ℤ/8)12yes—
148C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
149C2×C3×C4#13711yescharacteristic
150C3⋊C8—44no—
151C3⋊C8—82no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4#21712yes—
163C22×C4#21712yes—
164C22×C4C22×Dic311yescharacteristic
165C22×C4C2×#6711yescharacteristic
166C22×C4C2×#6711yescharacteristic
167M16—122no—
168M16—122no—
169C2×C8—64no—
170C2×C8—122no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—48no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C42—22no—
179C42—22no—
180C42—42no—
181C42D3×D412yes—
182C42#23712yes—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3#38611yescharacteristic
186C22×C3#41711yescharacteristic
187C22×C3#41811yescharacteristic
188C3×C4—21no—
189C3×C4—22no—
190C3×C4—44no—
191C3×C4—42no—
192C3×C4—42no—
193C3×C4—82no—
194C3×C4#45612yes—
195C3×C4#46812yes—
196C3×C4#46912yes—
197C23C2×#21711yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—82no—
201Q8—82no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4#93212yes—
216C2×C4D4×Dic312yes—
217C2×C4#93511yescharacteristic
218C2×C4#93711yescharacteristic
219C2×C4#93912yes—
220C2×C4#94911yescharacteristic
221C2×C4#95111yescharacteristic
222C8—124no—
223C8—242no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3#188611yescharacteristic
227C2×C3#189511yescharacteristic
228C2×C3#191511yescharacteristic
229C22—21no—
230C22—21no—
231C22#521511yescharacteristic
232C22#522011yescharacteristic
233C22#523211yescharacteristic
234C4—21no—
235C4—22no—
236C4—44no—
237C4—42no—
238C4—42no—
239C4—82no—
240C4#503912yes—
241C4#504212yes—
242C4#509512yes—
243C3#940211yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2#7168211yescharacteristic
247C2#7178411yescharacteristic
248C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2546C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2546D3
hypocenterC3#9402
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
243. C3#9402semidirect

Maximal subgroups

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

subgroupindex
C3×#25462
#824362
#717732
#716372
#676202
#94023