Finite Group Databasebeta

← #10697935

Subgroups — #10697935

Order 768.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2868C211yesmaximal, characteristic
2#83223C212yesmaximal
3#71773C212yesmaximal
4#69396C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic
6#10299—31nomaximal, Sylow
7C2×C3×C4⋊Q8—22no—
8C2×C3×C4⋊Q8C2212yes—
9#5470C2211yescharacteristic
10C3×#384—24no—
11C3×C43C2211yescharacteristic
12#5094—24no—
13#5093—22no—
14#5080—22no—
15#4971—24no—
16#4925C2212yes—
17#4924C2211yescharacteristic
18#4547—24no—
19#4523—22no—
20#3229—32no—
21#2868D311yescharacteristic
22#1994—31no—
23#1546—32no—
24#1276—31no—
25C22×C3×Q8—22no—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—44no—
31C3×C4⋊Q8—44no—
32C2×C3×C4⋊C4—28no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4D414yes—
35C2×C3×C42—22no—
36C2×C3×C42D412yes—
37C2×C3×C42C2311yescharacteristic
38#926—21no—
39#924—22no—
40C2×#208—21no—
41#870—21no—
42#869D412yes—
43C4×Dic6—22no—
44#830—22no—
45#810—28no—
46#808—28no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#533—31no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×D312yes—
53#498—62no—
54#487—62no—
55#483—64no—
56#475—64no—
57#431—32no—
58#423—31no—
59#384—24no—
60C43C2×D311yescharacteristic
61#339—62no—
62#332—64no—
63C2×C3×Q8—42no—
64C2×C3×Q8—42no—
65C2×C3×Q8—44no—
66C2×C3×Q8—44no—
67C22×C3×C4—22no—
68C22×C3×C4—24no—
69C22×C3×C4C2×D412yes—
70C22×C3×C4C2×D411yescharacteristic
71C2×Dic6—22no—
72C3×C4⋊C4—24no—
73C3×C4⋊C4—24no—
74C3×C4⋊C4—416no—
75C3×C4⋊C4—42no—
76C3×C4⋊C4—42no—
77C3×C42—28no—
78C3×C42—22no—
79C3×C42SD1614yes—
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D412yes—
82C3×C42#4111yescharacteristic
83#217—41no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87C4×Dic3—41no—
88#208—42no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—41no—
91C22×Q8—22no—
92C2×M16—61no—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—32no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100#125—61no—
101C22⋊Q8—62no—
102C4×Q8—62no—
103#117—61no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4#6714yes—
107C2×C42—22no—
108C2×C42#6712yes—
109C2×C42C22×D311yescharacteristic
110#106—68no—
111C4⋊C8—34no—
112C4⋊C8—62no—
113#103—68no—
114C22⋊C8—62no—
115C4×C8—62no—
116C23×C3#12011yescharacteristic
117C3×Q8—84no—
118C3×Q8—82no—
119C3×Q8—82no—
120C2×C3×C4—28no—
121C2×C3×C4—24no—
122C2×C3×C4—24no—
123C2×C3×C4—44no—
124C2×C3×C4—48no—
125C2×C3×C4#12012yes—
126C2×C3×C4#12111yescharacteristic
127C2×C3×C4#12112yes—
128C2×C3×C4#12711yescharacteristic
129C2×C3×C4C2×SD1612yes—
130C2×C3×C4AGL(1,ℤ/8)12yes—
131C2×Dic3—42no—
132Dic6—44no—
133C3⋊C8—44no—
134C3⋊C8—81no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—62no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4C2×#6712yes—
143C22×C4C2×#6711yescharacteristic
144M16—122no—
145C2×C8—64no—
146C2×C8—121no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—416no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—122no—
155C22⋊C4—121no—
156C42—28no—
157C42—22no—
158C42—121no—
159C42#21414yes—
160C42D3×D411yescharacteristic
161C42D3×D412yes—
162C42#23711yescharacteristic
163C22×C3—22no—
164C22×C3#39211yescharacteristic
165C22×C3#45012yes—
166C3×C4—28no—
167C3×C4—22no—
168C3×C4—84no—
169C3×C4#45614yes—
170C3×C4#46012yes—
171C3×C4#49211yescharacteristic
172C3×C4#49611yescharacteristic
173Dic3—82no—
174C23#95311yescharacteristic
175Q8—82no—
176Q8—84no—
177Q8—82no—
178Q8—124no—
179C2×C4—28no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—122no—
185C2×C4#93212yes—
186C2×C4C2×#21412yes—
187C2×C4#93712yes—
188C2×C4#93811yescharacteristic
189C2×C4#93912yes—
190C2×C4#94011yescharacteristic
191C8—124no—
192C8—241no—
193C2×C3—22no—
194C2×C3#200512yes—
195C2×C3#201411yescharacteristic
196C22—22no—
197C22#521812yes—
198C22#522311yescharacteristic
199C4—28no—
200C4—22no—
201C4—84no—
202C4—242no—
203C4#507411yescharacteristic
204C4#507512yes—
205C4#507611yescharacteristic
206C4#509514yes—
207C3#1029911yescharacteristic, complemented, Sylow
208C2—22no—
209C2#7176811yescharacteristic
210C2#7179512yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2868C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2868D3
hypocenterC3#10299
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
207. C3#10299semidirect

Maximal subgroups

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

subgroupindex
C3×#28682
#832232
#717732
#693962
#676112
#102993