Finite Group Databasebeta

← #10697795

Subgroups — #10697795

Order 768.

574 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#5020C212yesmaximal
2C2×#4699C211yesmaximal, characteristic
3C3×#1984C211yesmaximal, characteristic
4C2×#4482C212yesmaximal
5#67654C211yesmaximal, characteristic
6#10276—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C22×Dic12—22no—
9C2×#869C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4×C8C2211yescharacteristic
12C2×#835—22no—
13#5020—24no—
14#5020—24no—
15C2×#808—22no—
16C2×C4⋊(C3⋊C8)C2212yes—
17#4699—24no—
18C3×#336—22no—
19#4482—24no—
20C2×#494—31no—
21C2×#475—32no—
22#1984D311yescharacteristic
23C2×#332—32no—
24#1277—31no—
25C22×C3×Q8D411yescharacteristic
26C22×Dic6—22no—
27C22×C3×C8—21no—
28C22×C3×C8D412yes—
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C42—21no—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3—22no—
37C22×C3⋊C8—22no—
38C2×Dic12—28no—
39C2×Dic12—44no—
40C2×Dic12—44no—
41#869—22no—
42#869—22no—
43#869—22no—
44#869D414yes—
45C3×C4⋊C8—22no—
46C3×C4×C8—22no—
47#835—28no—
48#808—28no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C22×Q16—62no—
52C2×C4⋊Q8—32no—
53C2×C4×Q8C2×D311yescharacteristic
54#494—64no—
55#475—64no—
56#475—64no—
57C2×#106—62no—
58C2×C4⋊C8—32no—
59C2×C4⋊C8C2×D311yescharacteristic
60C2×#103—62no—
61C2×C4×C8C2×D311yescharacteristic
62#336—22no—
63#332—64no—
64C2×C3×Q8—21no—
65C2×C3×Q8—22no—
66C2×C3×Q8—22no—
67C2×C3×Q8Q1612yescomplemented
68C22×C3×C4—21no—
69C22×C3×C4—21no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4#4111yescharacteristic
73C22×Dic3—42no—
74C2×Dic6—24no—
75C2×Dic6—44no—
76C2×Dic6—44no—
77C2×Dic6—42no—
78C2×C3×C8—22no—
79C2×C3×C8—22no—
80C2×C3×C8—41no—
81C2×C3×C8—42no—
82C2×C3×C8SD1614yes—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—42no—
85C3×C42—41no—
86C3×C42—41no—
87C3×C42C2×D412yes—
88C3×C42C2×D412yes—
89C4⋊Dic3—24no—
90C4⋊Dic3—24no—
91C4⋊Dic3—42no—
92C4⋊Dic3—42no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—42no—
95C2×C3⋊C8—42no—
96Dic12—416no—
97C22×Q8—62no—
98C22×Q8D1211yescharacteristic
99C2×Q16—68no—
100C2×Q16—124no—
101C2×Q16—124no—
102C22×C8—21no—
103C22×C8—62no—
104C22×C8#6712yes—
105C4⋊Q8—34no—
106C4⋊Q8—62no—
107C4⋊Q8—62no—
108C4⋊Q8—62no—
109C4×Q8—22no—
110C4×Q8—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4D1211yescharacteristic
115C2×C42—21no—
116C2×C42C22×D311yescharacteristic
117#106—68no—
118C4⋊C8—22no—
119C4⋊C8—34no—
120C4⋊C8—62no—
121#103—68no—
122C4×C8—22no—
123C23×C3#12111yescharacteristic
124C3×Q8—24no—
125C3×Q8—42no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—41no—
131C2×C3×C4—42no—
132C2×C3×C4#12012yes—
133C2×C3×C4#12011yescharacteristic
134C2×C3×C4#12112yes—
135C2×C3×C4#12711yescharacteristic
136C2×C3×C4C2×SD1612yes—
137C2×C3×C4C2×Q1611yescharacteristic
138C2×C3×C4#13511yescharacteristic
139C2×C3×C4AGL(1,ℤ/8)12yes—
140C2×C3×C4#13712yes—
141C2×Dic3—44no—
142C2×Dic3—84no—
143Dic6—48no—
144Dic6—82no—
145Dic6—82no—
146C3×C8—24no—
147C3×C8—81no—
148C3⋊C8—44no—
149C2×Q8—21no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—64no—
153C2×Q8—122no—
154C2×Q8—124no—
155C2×Q8—124no—
156C2×Q8Dic1212yescomplemented
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—122no—
160C22×C4C2×D1211yescharacteristic
161C22×C4#23511yescharacteristic
162C22×C4C2×#6711yescharacteristic
163Q16—1216no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—42no—
168C2×C8—64no—
169C2×C8—122no—
170C2×C8—122no—
171C2×C8#21514yes—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—64no—
175C4⋊C4—64no—
176C4⋊C4—122no—
177C4⋊C4—122no—
178C42—41no—
179C42—41no—
180C42D3×D412yes—
181C42D3×D412yes—
182C22×C3#39211yescharacteristic
183C22×C3#46012yes—
184C22×C3#46711yescharacteristic
185C22×C3#47011yescharacteristic
186C22×C3#47312yes—
187C3×C4—22no—
188C3×C4—42no—
189C3×C4—81no—
190C3×C4#45012yes—
191C3×C4#45112yes—
192C3×C4#45212yes—
193C3×C4#49211yescharacteristic
194C3×C4#49611yescharacteristic
195Dic3—84no—
196C23#93011yescharacteristic
197Q8—24no—
198Q8—42no—
199Q8—128no—
200Q8—242no—
201Q8—242no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—124no—
209C2×C4—244no—
210C2×C4#88212yes—
211C2×C4#88412yes—
212C2×C4#90411yescharacteristic
213C2×C4C2×Dic1211yescharacteristic
214C2×C4#90912yes—
215C2×C4#93711yescharacteristic
216C2×C4#94011yescharacteristic
217C2×C4C2×#21512yes—
218C2×C4#94912yes—
219C8—24no—
220C8—81no—
221C8—124no—
222C2×C3#167012yes—
223C2×C3#169012yes—
224C2×C3#200111yescharacteristic
225C2×C3#201411yescharacteristic
226C2×C3#202011yescharacteristic
227C22#496011yescharacteristic
228C22#502712yes—
229C22#511611yescharacteristic
230C22#513711yescharacteristic
231C22#514312yes—
232C4—22no—
233C4—42no—
234C4—81no—
235C4—244no—
236C4#473312yes—
237C4#473912yes—
238C4#504612yes—
239C4#505811yescharacteristic
240C4#508511yescharacteristic
241C3#1027611yescharacteristic, complemented, Sylow
242C2#6968811yescharacteristic
243C2#6996411yescharacteristic
244C2#7000811yescharacteristic
245C2#7094212yes—
246C2#7096212yes—
subgroupG/H
centerC23#930
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1984C2
socle, prime radicalC23×C3#121
hypercenter#1984D3
hypocenterC3#10276
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
67. C2×C3×Q8Q16semidirect
156. C2×Q8Dic12semidirect
241. C3#10276semidirect

Maximal subgroups

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

subgroupindex
C2×#50202
C2×#46992
C3×#19842
C2×#44822
#676542
#102763