Finite Group Databasebeta

← #10697807

Subgroups — #10697807

Order 768.

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

normal subgroups only

This group has 562 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×#5008C212yesmaximal
2C2×#4682C211yesmaximal, characteristic, complemented
3C3×#1954C211yesmaximal, characteristic
4C2×#4481C212yesmaximal, complemented
5#67654C211yesmaximal, characteristic, complemented
6#10278—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#869C2212yes—
9C2×C3×C4⋊C8C2211yescharacteristic
10C2×C3×C22⋊C8—22no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#832—22no—
13C2×#818—22no—
14#5008—24no—
15#5008—24no—
16C2×#808—22no—
17C2×C4⋊(C3⋊C8)C2212yes—
18#4682—24no—
19#4481—24no—
20C2×#500—31no—
21C2×#476—32no—
22#1954D311yescharacteristic
23C2×#333—32no—
24#1277—31no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4—21no—
27C22×Dic6—22no—
28C22×C3×C8—21no—
29C22×C3×C8D412yes—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4—21no—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3—22no—
37C22×C3⋊C8—22no—
38#869—22no—
39#869—22no—
40#869—22no—
41#869D414yes—
42C3×C4⋊C8—22no—
43C3×C22⋊C8—44no—
44C3×C4×C8—22no—
45#832—28no—
46#818—28no—
47#808—28no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)D414yes—
50C2×C4⋊Q8—32no—
51C2×C4×D4C2×D311yescharacteristic
52#500—64no—
53#476—64no—
54#476—64no—
55C2×#107—62no—
56C2×#106—62no—
57C2×C4⋊C8—32no—
58C2×C4⋊C8C2×D311yescharacteristic
59C2×#102—62no—
60C2×C22⋊C8—22no—
61C2×C4×C8C2×D311yescharacteristic
62#333—64no—
63C24×C3—21no—
64C2×C3×D4—22no—
65C2×C3×D4—21no—
66C2×C3×D4—22no—
67C2×C3×D4Q1612yes—
68C22×C3×C4—21no—
69C22×C3×C4—41no—
70C22×C3×C4—41no—
71C22×C3×C4—42no—
72C22×C3×C4—42no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4111yescharacteristic
76C22×Dic3—42no—
77C2×Dic6—24no—
78C2×Dic6—42no—
79C2×Dic6—84no—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8—41no—
83C2×C3×C8—42no—
84C2×C3×C8SD1614yes—
85C3×C4⋊C4—22no—
86C3×C22⋊C4—42no—
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—
96C22×Q8—62no—
97C22×D4D1211yescharacteristic
98C23×C4—21no—
99C22×C8—21no—
100C22×C8—62no—
101C22×C8#6712yes—
102C4⋊Q8—34no—
103C4⋊Q8—62no—
104C4⋊Q8—62no—
105C4⋊Q8—62no—
106C4×D4—22no—
107C4×D4—22no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4D1211yescharacteristic
111C2×C22⋊C4—21no—
112C2×C42C22×D311yescharacteristic
113#107—68no—
114#106—68no—
115C4⋊C8—22no—
116C4⋊C8—34no—
117C4⋊C8—62no—
118#102—68no—
119C22⋊C8—44no—
120C4×C8—22no—
121C23×C3—22no—
122C23×C3—41no—
123C23×C3—42no—
124C23×C3—41no—
125C23×C3—42no—
126C23×C3#12111yescharacteristic
127C3×D4—24no—
128C3×D4—42no—
129C2×C3×C4—22no—
130C2×C3×C4—41no—
131C2×C3×C4—42no—
132C2×C3×C4—82no—
133C2×C3×C4—81no—
134C2×C3×C4—81no—
135C2×C3×C4#12012yes—
136C2×C3×C4#12011yescharacteristic
137C2×C3×C4#12112yes—
138C2×C3×C4#12711yescharacteristic
139C2×C3×C4C2×SD1612yes—
140C2×C3×C4C2×Q1611yescharacteristic
141C2×C3×C4#13511yescharacteristic
142C2×C3×C4#13712yes—
143C2×C3×C4#13712yes—
144C2×Dic3—44no—
145C2×Dic3—84no—
146Dic6—84no—
147Dic6—84no—
148C3×C8—24no—
149C3×C8—81no—
150C3⋊C8—44no—
151C24—21no—
152C2×Q8—64no—
153C2×Q8—122no—
154C2×Q8—244no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4Dic1212yes—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—122no—
165C22×C4C2×D1211yescharacteristic
166C22×C4#23511yescharacteristic
167C22×C4C2×#6711yescharacteristic
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—64no—
173C2×C8—122no—
174C2×C8—122no—
175C2×C8#21414yes—
176C4⋊C4—22no—
177C4⋊C4—64no—
178C4⋊C4—64no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C22⋊C4—42no—
182C42D3×D412yes—
183C42D3×D412yes—
184C22×C3—44no—
185C22×C3—42no—
186C22×C3—82no—
187C22×C3—81no—
188C22×C3—81no—
189C22×C3#39211yescharacteristic
190C22×C3#46112yes—
191C22×C3#46711yescharacteristic
192C22×C3#47011yescharacteristic
193C22×C3#47212yes—
194C3×C4—22no—
195C3×C4—81no—
196C3×C4#44812yes—
197C3×C4#45112yes—
198C3×C4#45212yes—
199C3×C4#49211yescharacteristic
200C3×C4#49711yescharacteristic
201Dic3—84no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—41no—
206C23—42no—
207C23#93011yescharacteristic
208Q8—244no—
209Q8—244no—
210D4—24no—
211D4—42no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—124no—
219C2×C4—244no—
220C2×C4#88212yes—
221C2×C4#88412yes—
222C2×C4#90411yescharacteristic
223C2×C4C2×Dic1211yescharacteristic
224C2×C4#90912yes—
225C2×C4C2×#21412yes—
226C2×C4#93711yescharacteristic
227C2×C4#94011yescharacteristic
228C2×C4#95112yes—
229C8—24no—
230C8—81no—
231C8—124no—
232C2×C3—82no—
233C2×C3#167512yes—
234C2×C3#169512yes—
235C2×C3#200111yescharacteristic
236C2×C3#201311yescharacteristic
237C2×C3#201811yescharacteristic
238C22—42no—
239C22—44no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22#496011yescharacteristic
244C22#501912yes—
245C22#511611yescharacteristic
246C22#513711yescharacteristic
247C22#514212yes—
248C4—22no—
249C4—81no—
250C4—244no—
251C4#473312yes—
252C4#473912yes—
253C4#504012yes—
254C4#506011yescharacteristic
255C4#507611yescharacteristic
256C3#1027811yescharacteristic, complemented, Sylow
257C2—82no—
258C2#6966111yescharacteristic
259C2#6996411yescharacteristic
260C2#7000611yescharacteristic
261C2#7085312yes—
262C2#7087412yes—
subgroupG/H
centerC23#930
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC23×C3#121
prime radicalC22×C3×D4D4
hypercenter#1954D3
hypocenterC3#10278
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. C2×#4682C2semidirect
4. C2×#4481C2semidirect
5. #67654C2semidirect
256. C3#10278semidirect

Maximal subgroups

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

subgroupindex
C2×#50082
C2×#46822
C3×#19542
C2×#44812
#676542
#102783