Finite Group Databasebeta

← #10697722

Subgroups — #10697722

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×#5031C212yesmaximal
2C2×#4682C211yesmaximal, characteristic
3C3×#1984C211yesmaximal, characteristic
4C2×#4486C212yesmaximal
5#67654C211yesmaximal, characteristic
6#10265—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C22×#216—22no—
9C2×#869C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4×C8C2211yescharacteristic
12C2×#818—22no—
13#5031—24no—
14#5031—24no—
15C2×#810—22no—
16C2×C4⋊(C3⋊C8)C2212yes—
17#4682—24no—
18C3×#336—22no—
19#4486—24no—
20C2×#500—31no—
21C2×#462—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×#216—28no—
36C2×#216—44no—
37C2×#216—44no—
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—22no—
40C22×C3⋊C8—22no—
41#869—22no—
42#869—22no—
43#869—22no—
44#869D414yes—
45C3×C4⋊C8—22no—
46C3×C4×C8—22no—
47#818—28no—
48#810—28no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C22×Q16—62no—
52C2×C4⋊Q8—32no—
53C2×C4×Q8C2×D311yescharacteristic
54#500—64no—
55#462—64no—
56#462—64no—
57C2×#107—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×Q8D812yes—
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—42no—
76C2×Dic6—42no—
77C2×Dic6—42no—
78C2×Dic6—42no—
79C2×Dic6—42no—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8—41no—
83C2×C3×C8—42no—
84C2×C3×C8Q1614yescomplemented
85C3×C4⋊C4—22no—
86C3×C4⋊C4—42no—
87C3×C42—41no—
88C3×C42—41no—
89C3×C42C2×D412yes—
90C3×C42C2×D412yes—
91#216—416no—
92C4⋊Dic3—24no—
93C4⋊Dic3—24no—
94C4⋊Dic3—42no—
95C4⋊Dic3—42no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C2×C3⋊C8—42no—
99C22×Q8—62no—
100C22×Q8D1211yescharacteristic
101C2×Q16—68no—
102C2×Q16—124no—
103C2×Q16—124no—
104C22×C8—21no—
105C22×C8—62no—
106C22×C8#6712yes—
107C4⋊Q8—34no—
108C4⋊Q8—62no—
109C4⋊Q8—62no—
110C4⋊Q8—62no—
111C4×Q8—22no—
112C4×Q8—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4D1211yescharacteristic
117C2×C42—21no—
118C2×C42C22×D311yescharacteristic
119#107—68no—
120C4⋊C8—22no—
121C4⋊C8—34no—
122C4⋊C8—62no—
123#103—68no—
124C4×C8—22no—
125C23×C3#12111yescharacteristic
126C3×Q8—24no—
127C3×Q8—42no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—42no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12011yescharacteristic
136C2×C3×C4#12112yes—
137C2×C3×C4#12711yescharacteristic
138C2×C3×C4C2×D811yescharacteristic
139C2×C3×C4C2×Q1612yes—
140C2×C3×C4#13511yescharacteristic
141C2×C3×C4AGL(1,ℤ/8)12yes—
142C2×C3×C4#13712yes—
143C2×Dic3—44no—
144C2×Dic3—82no—
145C2×Dic3—82no—
146Dic6—44no—
147Dic6—44no—
148Dic6—82no—
149Dic6—82no—
150C3×C8—24no—
151C3×C8—81no—
152C3⋊C8—44no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—64no—
157C2×Q8—122no—
158C2×Q8—122no—
159C2×Q8—122no—
160C2×Q8—122no—
161C2×Q8—122no—
162C2×Q8D2412yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—122no—
166C22×C4C2×D1211yescharacteristic
167C22×C4#23511yescharacteristic
168C22×C4C2×#6711yescharacteristic
169Q16—1216no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—64no—
175C2×C8—122no—
176C2×C8—122no—
177C2×C8#21614yescomplemented
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—64no—
181C4⋊C4—64no—
182C4⋊C4—122no—
183C4⋊C4—122no—
184C42—41no—
185C42—41no—
186C42D3×D412yes—
187C42D3×D412yes—
188C22×C3#39211yescharacteristic
189C22×C3#46212yes—
190C22×C3#46611yescharacteristic
191C22×C3#46911yescharacteristic
192C22×C3#47312yes—
193C3×C4—22no—
194C3×C4—42no—
195C3×C4—81no—
196C3×C4#44712yes—
197C3×C4#44912yes—
198C3×C4#45112yes—
199C3×C4#49411yescharacteristic
200C3×C4#49711yescharacteristic
201Dic3—84no—
202C23#93011yescharacteristic
203Q8—24no—
204Q8—42no—
205Q8—124no—
206Q8—124no—
207Q8—242no—
208Q8—242no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—124no—
216C2×C4—242no—
217C2×C4—242no—
218C2×C4#88212yes—
219C2×C4#88412yes—
220C2×C4C2×D2411yescharacteristic
221C2×C4#90411yescharacteristic
222C2×C4#90812yes—
223C2×C4#93711yescharacteristic
224C2×C4#94011yescharacteristic
225C2×C4C2×#21612yes—
226C2×C4#95112yes—
227C8—24no—
228C8—81no—
229C8—124no—
230C2×C3#167412yes—
231C2×C3#169412yes—
232C2×C3#199911yescharacteristic
233C2×C3#201611yescharacteristic
234C2×C3#202011yescharacteristic
235C22#496011yescharacteristic
236C22#503112yes—
237C22#511511yescharacteristic
238C22#513511yescharacteristic
239C22#514312yes—
240C4—22no—
241C4—42no—
242C4—81no—
243C4—244no—
244C4#473212yes—
245C4#473412yes—
246C4#505012yes—
247C4#506011yescharacteristic
248C4#509211yescharacteristic
249C3#1026511yescharacteristic, complemented, Sylow
250C2#6969111yescharacteristic
251C2#6996111yescharacteristic
252C2#7000811yescharacteristic
253C2#7094412yes—
254C2#7096412yes—
subgroupG/H
centerC23#930
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1984C2
socle, prime radicalC23×C3#121
hypercenter#1984D3
hypocenterC3#10265
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
84. C2×C3×C8Q16semidirect
177. C2×C8#216semidirect
249. C3#10265semidirect

Maximal subgroups

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

subgroupindex
C2×#50312
C2×#46822
C3×#19842
C2×#44862
#676542
#102653