Finite Group Databasebeta

← #10693052

Subgroups — #10693052

Order 768.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2581C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#71622C212yesmaximal
5#67631C212yesmaximal
6#9488—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#400C412yes—
10C3×#395C412yes—
11C3×#391—21no—
12C3×#389—21no—
13C3×#387—22no—
14C3×#382C2211yescharacteristic
15C2×#836—21no—
16#5024—24no—
17#4926—22no—
18#4925C2212yes—
19#4924C2212yes—
20#4535—24no—
21#4530—21no—
22#2983—31no—
23#2581D311yescharacteristic
24#1801—31no—
25#1487—32no—
26#1286—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×C3×Q8D411yescharacteristic
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C42—21no—
40C2×C3×C42D412yes—
41C2×C3×C42C2311yescharacteristic
42C2×#208—22no—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95—22no—
46C3×#95—21no—
47C3×#95C2×C412yes—
48#836—42no—
49#830—22no—
50C4⋊(C3⋊C8)—24no—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54#803—24no—
55C4×C3⋊C8D414yes—
56C2×C4×Q8C2×D311yescharacteristic
57C2×C4×Q8C2×D311yescharacteristic
58#446—64no—
59#432—62no—
60#431—32no—
61#423—32no—
62C2×#101—61no—
63#400Dic312yes—
64#395Dic312yes—
65#391—21no—
66#389—21no—
67#387—22no—
68#382C2×D311yescharacteristic
69#343—61no—
70Q8⋊C8—64no—
71C2×C3×Q8—22no—
72C2×C3×Q8—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—21no—
75C2×C3×Q8—22no—
76C2×C3×Q8—42no—
77C22×C3×C4—21no—
78C22×C3×C4—22no—
79C22×C3×C4—21no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C3×C4⋊C4—24no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—44no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—41no—
93C3×C42—22no—
94C3×C42—22no—
95C3×C42—42no—
96C3×C42C2×D412yes—
97C3×C42#4112yes—
98#208—42no—
99#208—42no—
100C2×C3⋊C8—24no—
101C2×C3⋊C8—42no—
102C22×Q8C2×Dic311yescharacteristic
103C22×Q8#6711yescharacteristic
104C2×M16—62no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2×Dic311yescharacteristic
113C2×C4⋊C4C2×Dic312yes—
114C2×C4⋊C4#6711yescharacteristic
115C2×C42—21no—
116C2×C42#6712yes—
117C2×C42C22×D311yescharacteristic
118C4⋊C8—64no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121C4⋊C8—62no—
122#101—122no—
123C22⋊C8—62no—
124#97—64no—
125C4×C8—34no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—21no—
130#95C2×Dic312yes—
131C23×C3C2×C22⋊C411yescharacteristic
132C3×Q8—24no—
133C3×Q8—42no—
134C3×Q8—42no—
135C3×Q8—42no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—41no—
150C2×C3×C4—42no—
151C2×C3×C4C4×D412yes—
152C2×C3×C4#12011yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4#12311yescharacteristic
155C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
156C2×C3×C4#13712yes—
157C2×C3×C4#13711yescharacteristic
158C3⋊C8—44no—
159C3⋊C8—82no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—22no—
165C2×Q8—42no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4#21712yes—
170C22×C4#21712yes—
171C22×C4C22×Dic311yescharacteristic
172C22×C4C2×#6711yescharacteristic
173C22×C4C2×#6711yescharacteristic
174M16—122no—
175M16—122no—
176C2×C8—64no—
177C2×C8—122no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—44no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C42—22no—
187C42—22no—
188C42—42no—
189C42D3×D412yes—
190C42#23712yes—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3#38611yescharacteristic
194C22×C3#41711yescharacteristic
195C22×C3#41911yescharacteristic
196C3×C4—22no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—42no—
202C3×C4—82no—
203C3×C4—81no—
204C3×C4#46812yes—
205C3×C4#47012yes—
206C23C2×#21711yescharacteristic
207Q8—24no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4D4×Dic312yes—
227C2×C4#93511yescharacteristic
228C2×C4#93711yescharacteristic
229C2×C4#93912yes—
230C2×C4#94212yes—
231C2×C4#94911yescharacteristic
232C2×C4#95111yescharacteristic
233C8—124no—
234C8—242no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#189111yescharacteristic
238C2×C3#189411yescharacteristic
239C2×C3#191211yescharacteristic
240C22—21no—
241C22—21no—
242C22#522011yescharacteristic
243C22#522511yescharacteristic
244C22#523211yescharacteristic
245C4—22no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—81no—
253C4#504912yes—
254C4#505212yes—
255C3#948811yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#7169711yescharacteristic
259C2#7171411yescharacteristic
260C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2581C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2581D3
hypocenterC3#9488
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
255. C3#9488semidirect

Maximal subgroups

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

subgroupindex
C3×#25812
#824362
#716372
#716222
#676312
#94883