Finite Group Databasebeta

← #10696057

Subgroups — #10696057

Order 768.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2399C211yesmaximal, characteristic
2#82447C212yesmaximal
3#71622C212yesmaximal
4#69441C211yesmaximal, characteristic
5#67614C211yesmaximal, characteristic
6#10017—31nomaximal, Sylow
7C2×C3×C4×Q8C2212yes—
8#5470C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×C4×C4⋊C4—22no—
12C3×#376C2211yescharacteristic
13#5228—22no—
14#5031—22no—
15C4×#216—24no—
16#5021—22no—
17#4970—22no—
18#4925C2212yes—
19#4924C2211yescharacteristic
20#4535—24no—
21#4523—22no—
22#3104—32no—
23#2399D311yescharacteristic
24#1995—31no—
25#1487—32no—
26#1274—31no—
27C22×C3×Q8D412yes—
28C3×C4×Q8—22no—
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4—24no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C42—22no—
36C2×C3×C42D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×#216—24no—
39#926—21no—
40#924—22no—
41C2×#208—21no—
42#870—21no—
43#869D412yes—
44C4×Dic6—22no—
45C3×#95—22no—
46C3×#95—22no—
47C3×#95—22no—
48#835—24no—
49#830—22no—
50#810—24no—
51#807—24no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4×C3⋊C8D414yes—
55#533—31no—
56C2×C4×Q8C2×D312yes—
57#484—62no—
58#477—62no—
59#462—62no—
60#451—62no—
61C4×Q16—64no—
62#431—32no—
63#423—31no—
64#398—22no—
65#395—22no—
66C4×C4⋊C4—22no—
67#376C2×D311yescharacteristic
68#339—62no—
69Q8⋊C8—64no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—44no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D411yescharacteristic
78C2×Dic6—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—48no—
82C3×C4⋊C4—42no—
83C3×C4⋊C4—42no—
84C3×C42—22no—
85C3×C42—42no—
86C3×C42—42no—
87C3×C42Q1614yescomplemented
88C3×C42C2×D411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42#4111yescharacteristic
91#217—41no—
92#216—48no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95Dic3⋊C4—42no—
96C4×Dic3—41no—
97#208—42no—
98C2×C3⋊C8—24no—
99C2×C3⋊C8—41no—
100C22×Q8#6712yes—
101C2×Q16—64no—
102C2×M16—61no—
103C4⋊Q8—32no—
104#125—61no—
105C22⋊Q8—62no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×Q8—62no—
111#117—61no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4#6712yes—
115C2×C42—22no—
116C2×C42#6712yes—
117C2×C42C22×D311yescharacteristic
118#107—64no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121#103—64no—
122#103—64no—
123C22⋊C8—62no—
124C4×C8—34no—
125#95—22no—
126#95—22no—
127#95—22no—
128C23×C3#12011yescharacteristic
129C3×Q8—24no—
130C3×Q8—24no—
131C3×Q8—42no—
132C3×Q8—42no—
133C2×C3×C4—24no—
134C2×C3×C4—24no—
135C2×C3×C4—42no—
136C2×C3×C4—44no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—44no—
140C2×C3×C4—42no—
141C2×C3×C4#12012yes—
142C2×C3×C4#12111yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4#12711yescharacteristic
145C2×C3×C4C2×Q1612yes—
146C2×C3×C4#13712yes—
147C2×Dic3—42no—
148Dic6—44no—
149C3⋊C8—44no—
150C3⋊C8—81no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—44no—
154C2×Q8—62no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4C2×#6712yes—
159C22×C4C2×#6711yescharacteristic
160Q16—128no—
161M16—122no—
162C2×C8—64no—
163C2×C8—121no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—48no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—122no—
172C22⋊C4—121no—
173C42—22no—
174C42—42no—
175C42—42no—
176C42—121no—
177C42#21614yescomplemented
178C42D3×D411yescharacteristic
179C42D3×D412yes—
180C42#23711yescharacteristic
181C22×C3—22no—
182C22×C3#39211yescharacteristic
183C22×C3#45112yes—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—44no—
187C3×C4—44no—
188C3×C4—82no—
189C3×C4#46212yes—
190C3×C4#49411yescharacteristic
191C3×C4#49711yescharacteristic
192Dic3—82no—
193C23#95311yescharacteristic
194Q8—24no—
195Q8—24no—
196Q8—42no—
197Q8—42no—
198Q8—124no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—122no—
208C2×C4#93712yes—
209C2×C4#93811yescharacteristic
210C2×C4#93912yes—
211C2×C4#94011yescharacteristic
212C2×C4#94212yes—
213C2×C4C2×#21612yes—
214C8—124no—
215C8—241no—
216C2×C3—22no—
217C2×C3#201012yes—
218C2×C3#201611yescharacteristic
219C22—22no—
220C22#522311yescharacteristic
221C22#522812yes—
222C4—22no—
223C4—22no—
224C4—44no—
225C4—44no—
226C4—82no—
227C4—242no—
228C4#508611yescharacteristic
229C4#509012yes—
230C4#509211yescharacteristic
231C3#1001711yescharacteristic, complemented, Sylow
232C2—22no—
233C2#7171512yes—
234C2#7179311yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2399C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2399D3
hypocenterC3#10017
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

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
C3×#23992
#824472
#716222
#694412
#676142
#100173