Finite Group Databasebeta

← #10696056

Subgroups — #10696056

Order 768.

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

normal subgroups only

This group has 528 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#83368C212yesmaximal
2#73397C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic
4#70500C212yesmaximal
5#67614C211yesmaximal, characteristic
6#10017—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C4×Dic6C2212yes—
9C3×#423C2211yescharacteristic
10C3×#376C2211yescharacteristic
11#5091—22no—
12#5090—22no—
13#5050—22no—
14#5044—22no—
15C4×#216—24no—
16C4×C4⋊Dic3—22no—
17#4925C2212yes—
18#4661—22no—
19#4656—22no—
20C3×#339—22no—
21#4484—24no—
22#3104—32no—
23#2399—31no—
24#1995D311yescharacteristic
25#1487—32no—
26#1274—31no—
27C22×Dic6D412yes—
28C2×C3×M16—21no—
29C3×C4⋊Q8D412yes—
30C3×#125—21no—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117—21no—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#216—24no—
37C2×C4⋊Dic3—24no—
38C2×C4⋊Dic3D412yes—
39C2×Dic3⋊C4—22no—
40C2×C4×Dic3—22no—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×#95—22no—
47#835—24no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50#830—22no—
51#810—24no—
52#807—24no—
53C4⋊(C3⋊C8)—22no—
54C4×C3⋊C8D414yes—
55#533C2×D311yescharacteristic
56C2×C4×Q8—32no—
57#484—62no—
58#477—62no—
59#462—62no—
60#451—62no—
61C4×Q16—64no—
62#431—32no—
63#423C2×D311yescharacteristic
64#398—62no—
65#395—62no—
66C4×C4⋊C4—62no—
67#376C2×D311yescharacteristic
68#339—22no—
69Q8⋊C8—64no—
70C2×C3×Q8—22no—
71C22×C3×C4—22no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C22×Dic3—22no—
75C22×Dic3—22no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—44no—
79C3×M16—42no—
80C2×C3×C8—41no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—42no—
84C3×C22⋊C4—41no—
85C3×C42—22no—
86C3×C42—41no—
87C3×C42Q1614yescomplemented
88C3×C42C2×D411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42#4111yescharacteristic
91#216—48no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—48no—
95Dic3⋊C4—42no—
96Dic3⋊C4—42no—
97C4×Dic3—42no—
98C4×Dic3—42no—
99C2×C3⋊C8—24no—
100C22×Q8—32no—
101C2×Q16—64no—
102C2×M16—21no—
103C4⋊Q8D1212yes—
104#125—21no—
105C22⋊Q8—22no—
106C4×Q8—22no—
107C4×Q8—62no—
108C4×Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—21no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—64no—
114C2×C4⋊C4—62no—
115C2×C42—62no—
116C2×C42#6712yes—
117C2×C42C22×D311yescharacteristic
118#107—64no—
119C4⋊C8—22no—
120C4⋊C8—62no—
121#103—64no—
122#103—64no—
123C22⋊C8—62no—
124C4×C8—34no—
125#95—22no—
126#95—62no—
127#95—62no—
128C23×C3#12011yescharacteristic
129C3×Q8—44no—
130C2×C3×C4—24no—
131C2×C3×C4—42no—
132C2×C3×C4—44no—
133C2×C3×C4—42no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12112yes—
136C2×C3×C4#12111yescharacteristic
137C2×C3×C4#12711yescharacteristic
138C2×C3×C4C2×Q1612yes—
139C2×C3×C4#13712yes—
140C2×Dic3—24no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143C2×Dic3—44no—
144C2×Dic3—42no—
145Dic6—24no—
146Dic6—24no—
147Dic6—42no—
148Dic6—42no—
149C3×C8—81no—
150C3⋊C8—44no—
151C2×Q8—22no—
152C2×Q8—62no—
153C2×Q8—62no—
154C2×Q8—124no—
155C22×C4—22no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4D3×D412yes—
159C22×C4C2×#6711yescharacteristic
160Q16—128no—
161M16—42no—
162C2×C8—41no—
163C2×C8—64no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—128no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C22⋊C4—41no—
173C42—22no—
174C42—41no—
175C42—122no—
176C42—122no—
177C42#21614yescomplemented
178C42C2×D1211yescharacteristic
179C42#23511yescharacteristic
180C42D3×D412yes—
181C22×C3—22no—
182C22×C3#39211yescharacteristic
183C22×C3#45112yes—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—44no—
187C3×C4—82no—
188C3×C4#46212yes—
189C3×C4#49411yescharacteristic
190C3×C4#49711yescharacteristic
191Dic3—44no—
192Dic3—82no—
193C23#93711yescharacteristic
194Q8—44no—
195Q8—64no—
196Q8—64no—
197Q8—122no—
198Q8—122no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—64no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—124no—
207C2×C4—122no—
208C2×C4#88212yes—
209C2×C4#88412yes—
210C2×C4#93011yescharacteristic
211C2×C4#94011yescharacteristic
212C2×C4C2×#21612yes—
213C2×C4#95112yes—
214C8—81no—
215C8—124no—
216C2×C3—22no—
217C2×C3#201012yes—
218C2×C3#201611yescharacteristic
219C22—22no—
220C22#496011yescharacteristic
221C22#505012yes—
222C4—22no—
223C4—22no—
224C4—44no—
225C4—82no—
226C4—124no—
227C4—242no—
228C4#503112yes—
229C4#506011yescharacteristic
230C4#509211yescharacteristic
231C3#1001711yescharacteristic, complemented, Sylow
232C2—22no—
233C2#6969111yescharacteristic
234C2#7040612yes—
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1995D3
hypocenterC3#10017
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
87. C3×C42Q16semidirect
177. C42#216semidirect
231. C3#10017semidirect

Maximal subgroups

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

subgroupindex
#833682
#733972
C3×#19952
#705002
#676142
#100173