Finite Group Databasebeta

← #10696039

Subgroups — #10696039

Order 768.

566 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 103 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#83315C212yesmaximal
2#73397C211yesmaximal, characteristic, complemented
3C3×#1963C211yesmaximal, characteristic
4#70500C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10014—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×C4×Dic6C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×#376C2211yescharacteristic
12#5094—22no—
13#5075—22no—
14#5040—22no—
15#5034—22no—
16C4×#214—24no—
17C4×C4⋊Dic3—22no—
18#4925C2212yes—
19#4661—22no—
20#4656—22no—
21#4484—24no—
22#3103—32no—
23#2399—31no—
24#1963D311yescharacteristic
25#1487—32no—
26#1274—31no—
27C2×C3×#41—21no—
28C22×Dic6D412yes—
29C2×C3×M16—21no—
30C3×C4⋊Q8D411yescharacteristic
31C3×#127D411yescharacteristic
32C3×#124—21no—
33C3×#121—22no—
34C3×C4×D4—22no—
35C2×C3×C42D412yes—
36C2×C3×C42C2311yescharacteristic
37C2×#214—24no—
38C2×C4⋊Dic3—24no—
39C2×C4⋊Dic3D412yes—
40C2×Dic3⋊C4—22no—
41C2×C4×Dic3—22no—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×#104—44no—
48C3×#95—22no—
49#832—24no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52#830—22no—
53#810—24no—
54#808—24no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yes—
57#532C2×D311yescharacteristic
58C2×C4×Q8—32no—
59#481—62no—
60#475—62no—
61#460—62no—
62#448—62no—
63C4×SD16—64no—
64#431—32no—
65#423C2×D311yescharacteristic
66C2×#104—22no—
67#398—62no—
68#395—62no—
69C4×C4⋊C4—62no—
70#376C2×D311yescharacteristic
71Q8⋊C8—64no—
72C3×#41—42no—
73C3×#41—42no—
74C2×C3×Q8—21no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—41no—
78C22×C3×C4—22no—
79C22×C3×C4—41no—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—22no—
84C2×Dic6—22no—
85C2×Dic6—22no—
86C2×Dic6—44no—
87C3×M16—42no—
88C2×C3×C8—41no—
89C3×C4⋊C4—22no—
90C3×C22⋊C4—42no—
91C3×C42—22no—
92C3×C42SD1614yescomplemented
93C3×C42C2×D411yescharacteristic
94C3×C42C2×D412yes—
95C3×C42#4111yescharacteristic
96#214—48no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C4⋊Dic3—48no—
100Dic3⋊C4—42no—
101Dic3⋊C4—42no—
102C4×Dic3—42no—
103C4×Dic3—42no—
104C2×C3⋊C8—24no—
105C2×#41—21no—
106C22×Q8—32no—
107C2×SD16—64no—
108C2×M16—21no—
109C4⋊Q8D1211yescharacteristic
110#127D1211yescharacteristic
111#124—21no—
112#121—22no—
113C4×Q8—62no—
114C4×Q8—62no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×D4—22no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—64no—
120C2×C4⋊C4—62no—
121C2×C42—62no—
122C2×C42#6712yes—
123C2×C42C22×D311yescharacteristic
124#106—64no—
125C4⋊C8—22no—
126C4⋊C8—62no—
127#104—44no—
128#103—64no—
129#102—64no—
130C22⋊C8—62no—
131C4×C8—34no—
132#95—22no—
133#95—62no—
134#95—62no—
135C23×C3—41no—
136C23×C3#12011yescharacteristic
137C3×Q8—42no—
138C3×D4—44no—
139C3×D4—42no—
140C3×D4—82no—
141C2×C3×C4—24no—
142C2×C3×C4—42no—
143C2×C3×C4—44no—
144C2×C3×C4—41no—
145C2×C3×C4—81no—
146C2×C3×C4—81no—
147C2×C3×C4#12012yes—
148C2×C3×C4#12112yes—
149C2×C3×C4#12111yescharacteristic
150C2×C3×C4#12711yescharacteristic
151C2×C3×C4C2×SD1612yes—
152C2×C3×C4#13712yes—
153C2×Dic3—24no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—44no—
157C2×Dic3—42no—
158Dic6—24no—
159Dic6—24no—
160Dic6—42no—
161Dic6—42no—
162C3×C8—81no—
163C3⋊C8—44no—
164#41—42no—
165#41—42no—
166C2×Q8—21no—
167C2×Q8—62no—
168C2×Q8—62no—
169C2×Q8—124no—
170C2×D4—22no—
171C2×D4—21no—
172C2×D4—41no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4D3×D412yes—
178C22×C4C2×#6711yescharacteristic
179SD16—128no—
180M16—42no—
181C2×C8—41no—
182C2×C8—64no—
183C4⋊C4—22no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—128no—
187C4⋊C4—122no—
188C4⋊C4—122no—
189C22⋊C4—42no—
190C42—22no—
191C42—122no—
192C42—122no—
193C42#21414yescomplemented
194C42C2×D1211yescharacteristic
195C42#23511yescharacteristic
196C42D3×D412yes—
197C22×C3—22no—
198C22×C3—82no—
199C22×C3—81no—
200C22×C3#39211yescharacteristic
201C22×C3#44812yes—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—44no—
205C3×C4—81no—
206C3×C4#46112yes—
207C3×C4#49211yescharacteristic
208C3×C4#49711yescharacteristic
209Dic3—44no—
210Dic3—82no—
211C23—41no—
212C23#93711yescharacteristic
213Q8—42no—
214Q8—64no—
215Q8—64no—
216Q8—122no—
217Q8—122no—
218D4—44no—
219D4—42no—
220D4—82no—
221C2×C4—24no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—64no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—124no—
231C2×C4—122no—
232C2×C4#88212yes—
233C2×C4#88412yes—
234C2×C4#93011yescharacteristic
235C2×C4C2×#21412yes—
236C2×C4#94011yescharacteristic
237C2×C4#95112yes—
238C8—81no—
239C8—124no—
240C2×C3—22no—
241C2×C3—161no—
242C2×C3#201012yes—
243C2×C3#201311yescharacteristic
244C22—22no—
245C22—82no—
246C22—81no—
247C22#496011yescharacteristic
248C22#504012yes—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—81no—
253C4—124no—
254C4—242no—
255C4#501912yes—
256C4#506011yescharacteristic
257C4#507611yescharacteristic
258C3#1001411yescharacteristic, complemented, Sylow
259C2—22no—
260C2—161no—
261C2#6966111yescharacteristic
262C2#7040612yes—
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1963C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1963D3
hypocenterC3#10014
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #73397C2semidirect
4. #70500C2semidirect
5. #67614C2semidirect
92. C3×C42SD16semidirect
193. C42#214semidirect
258. C3#10014semidirect

Maximal subgroups

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

subgroupindex
#833152
#733972
C3×#19632
#705002
#676142
#100143