Finite Group Databasebeta

← #10699622

Subgroups — #10699622

Order 768.

494 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 135 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#74335C211yesmaximal, characteristic
2C3×#1845C211yesmaximal, characteristic
3#69567C212yesmaximal
4#67655C212yesmaximal
5#67634C211yesmaximal, characteristic
6#10573—31nomaximal, Sylow
7C2×#888—22no—
8C2×C3×#107—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C3×#384C2211yescharacteristic
11C3×#384C2211yescharacteristic
12#4979—22no—
13#4972—22no—
14C2×#807—22no—
15C4×C4⋊Dic3C2212yes—
16C2×C4⋊(C3⋊C8)C2212yes—
17#4529—22no—
18#2551—31no—
19#1947—32no—
20#1845D311yescharacteristic
21#1296—31no—
22#1279—32no—
23C22×Dic6—22no—
24C22×C3×C8—21no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4—21no—
28C2×C3×C4⋊C4D412yes—
29C2×C3×C42C2311yescharacteristic
30C2×C4⋊Dic3D412yes—
31C2×C4⋊Dic3D412yes—
32C2×Dic3⋊C4—24no—
33C2×C4×Dic3—22no—
34C22×C3⋊C8—22no—
35#888—44no—
36#888—44no—
37C3×#107—44no—
38C3×C4×C8—21no—
39C3×C4×C8Q812yescomplemented
40C3⋊#95—22no—
41C3⋊#95—22no—
42#807—44no—
43C4⋊(C3⋊C8)—22no—
44C4⋊(C3⋊C8)Q814yescomplemented
45C2×C4⋊Q8—62no—
46C2×#107—22no—
47C2×#107—62no—
48C2×C4⋊C8—32no—
49C2×C4×C8C2×D311yescharacteristic
50#395—62no—
51#389—62no—
52#384C2×D311yescharacteristic
53#384C2×D311yescharacteristic
54C4×C4⋊C4—32no—
55#340—62no—
56C22×C3×C4—22no—
57C22×C3×C4—41no—
58C22×C3×C4C2×D412yes—
59C22×C3×C4C2×D411yescharacteristic
60C22×Dic3—22no—
61C22×Dic3—22no—
62C2×Dic6—48no—
63C2×Dic6—82no—
64C2×C3×C8—22no—
65C2×C3×C8—22no—
66C2×C3×C8—22no—
67C3×C4⋊C4—42no—
68C3×C4⋊C4—42no—
69C3×C4⋊C4—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—42no—
73C3×C42C2×Q811yescharacteristic
74C3×C42C2×Q812yes—
75C3×C42#4111yescharacteristic
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3D814yes—
79C4⋊Dic3Q1614yes—
80Dic3⋊C4—48no—
81C4×Dic3—42no—
82C4×Dic3—42no—
83C2×C3⋊C8—24no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8—42no—
86C22×Q8—62no—
87C22×C8—21no—
88C22×C8—62no—
89C4⋊Q8—124no—
90C4⋊Q8—124no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—64no—
97C2×C4⋊C4#6712yes—
98C2×C42—62no—
99C2×C42C22×D311yescharacteristic
100#107—44no—
101#107—124no—
102C4⋊C8—34no—
103C4⋊C8—62no—
104C4×C8—21no—
105C4×C8Dic612yescomplemented
106#95—62no—
107#95—62no—
108C23×C3#12011yescharacteristic
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—44no—
112C2×C3×C4—42no—
113C2×C3×C4—82no—
114C2×C3×C4—81no—
115C2×C3×C4C22⋊Q812yes—
116C2×C3×C4C22⋊Q811yescharacteristic
117C2×C3×C4C4⋊Q812yes—
118C2×C3×C4C4⋊Q811yescharacteristic
119C2×C3×C4C2×D812yes—
120C2×C3×C4C2×Q1612yes—
121C2×C3×C4AGL(1,ℤ/8)12yes—
122C2×C3×C4#13712yes—
123C2×Dic3—24no—
124C2×Dic3—24no—
125C2×Dic3—42no—
126C2×Dic3—42no—
127C2×Dic3—42no—
128C2×Dic3—42no—
129Dic6—88no—
130C3×C8—24no—
131C3⋊C8—44no—
132C2×Q8—128no—
133C2×Q8—242no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—62no—
137C22×C4—62no—
138C22×C4D3×D412yes—
139C22×C4C2×#6711yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—64no—
144C2×C8—122no—
145C2×C8—122no—
146C4⋊C4—34no—
147C4⋊C4—34no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—128no—
157C42—122no—
158C42—122no—
159C42C2×Dic611yescharacteristic
160C42#23511yescharacteristic
161C42D3×Q812yes—
162C22×C3#39511yescharacteristic
163C22×C3#44712yes—
164C22×C3#45112yes—
165C22×C3#45311yescharacteristic
166C22×C3#45611yescharacteristic
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—82no—
170C3×C4—161no—
171C3×C4#47412yes—
172C3×C4#47712yes—
173C3×C4#50012yes—
174C3×C4#50112yes—
175Dic3—44no—
176Dic3—44no—
177C23#93711yescharacteristic
178Q8—248no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—64no—
184C2×C4—64no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4#88812yes—
192C2×C4#89612yes—
193C2×C4D3×D812yes—
194C2×C4#91112yes—
195C2×C4D3×Q1612yes—
196C2×C4#91812yes—
197C2×C4#92411yescharacteristic
198C2×C4#94411yescharacteristic
199C8—24no—
200C8—124no—
201C2×C3#161911yescharacteristic
202C2×C3#163311yescharacteristic
203C2×C3#164912yes—
204C2×C3#205712yes—
205C2×C3#206011yescharacteristic
206C22#497911yescharacteristic
207C22#507711yescharacteristic
208C22#509511yescharacteristic
209C22#515612yes—
210C22#518612yes—
211C4—21no—
212C4—21no—
213C4—82no—
214C4—124no—
215C4—124no—
216C4—161no—
217C4#487412yes—
218C4#487512yes—
219C4#487612yes—
220C4#489012yes—
221C3#1057311yescharacteristic, complemented, Sylow
222C2#6957311yescharacteristic
223C2#7061712yes—
224C2#7128111yescharacteristic
225C2#7130711yescharacteristic
226C2#7137612yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1845C2
socle, prime radicalC23×C3#120
hypercenter#1845D3
hypocenterC3#10573
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
39. C3×C4×C8Q8semidirect
44. C4⋊(C3⋊C8)Q8semidirect
105. C4×C8Dic6semidirect
221. C3#10573semidirect

Maximal subgroups

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

subgroupindex
#743352
C3×#18452
#695672
#676552
#676342
#105733