Finite Group Databasebeta

← #10697925

Subgroups — #10697925

Order 768.

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

normal subgroups only

This group has 528 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
1#79313C212yesmaximal
2#73308C211yesmaximal, characteristic
3#70779C212yesmaximal
4#69396C211yesmaximal, characteristic
5C3×#1276C211yesmaximal, characteristic
6#10297—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C2×#869C2212yes—
9C3×#431C2212yes—
10C3×C43C2211yescharacteristic
11#4936—22no—
12#4929—24no—
13#4924C2211yescharacteristic
14#4849—24no—
15#4739—24no—
16#4699—22no—
17#4681—22no—
18#4523—22no—
19#4481—24no—
20#3209—32no—
21#2862—31no—
22#1994—31no—
23#1548—32no—
24#1276D311yescharacteristic
25C22×Dic6D414yes—
26C2×C3×C42—22no—
27C2×C3×C42D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—21no—
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—22no—
32#924—22no—
33C2×#208—21no—
34C2×Dic12—28no—
35#870—21no—
36#869—22no—
37#869—22no—
38#869—22no—
39#869—22no—
40#869D412yes—
41C4×Dic6—22no—
42C3×C4⋊C8D414yes—
43C3×C22⋊C8—22no—
44C3×C4×C8—22no—
45C3⋊#95—28no—
46#816—28no—
47C4⋊(C3⋊C8)—22no—
48#533—31no—
49C2×C4⋊Q8—32no—
50#494—62no—
51#487—62no—
52#462—64no—
53#451—64no—
54#431C2×D312yes—
55#423—31no—
56#391—64no—
57#382—62no—
58C43C2×D311yescharacteristic
59#339—62no—
60#333—64no—
61C22×C3×C4—22no—
62C22×C3×C4C2×D411yescharacteristic
63C22×C3×C4C2×D412yes—
64C22×Dic3—24no—
65C2×Dic6—24no—
66C2×Dic6—22no—
67C2×Dic6—24no—
68C2×Dic6—48no—
69C2×C3×C8—24no—
70C3×C42—22no—
71C3×C42—44no—
72C3×C42Q1614yescomplemented
73C3×C42C2×D412yes—
74C3×C42C2×D411yescharacteristic
75C3×C42#4111yescharacteristic
76#217—41no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79C4⋊Dic3—42no—
80C4⋊Dic3—42no—
81C4⋊Dic3—42no—
82C4⋊Dic3—42no—
83Dic3⋊C4—42no—
84C4×Dic3—41no—
85#208—42no—
86C2×C3⋊C8—41no—
87Dic12—88no—
88C22×Q8—34no—
89C2×Q16—68no—
90C2×M16—61no—
91C4⋊Q8—32no—
92C4⋊Q8—62no—
93C4⋊Q8—62no—
94C4⋊Q8—62no—
95C4⋊Q8—62no—
96#125—61no—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—61no—
100C2×C4⋊C4—62no—
101C2×C4⋊C4—62no—
102C2×C42—22no—
103C2×C42D1212yes—
104C2×C42C22×D311yescharacteristic
105C4⋊C8—62no—
106C4⋊C8D1214yes—
107#103—68no—
108C22⋊C8—22no—
109C4×C8—22no—
110#95—68no—
111C23×C3#12011yescharacteristic
112C2×C3×C4—24no—
113C2×C3×C4—24no—
114C2×C3×C4—44no—
115C2×C3×C4#12012yes—
116C2×C3×C4#12112yes—
117C2×C3×C4#12111yescharacteristic
118C2×C3×C4#12711yescharacteristic
119C2×C3×C4C2×Q1612yes—
120C2×C3×C4#13712yes—
121C2×Dic3—48no—
122C2×Dic3—42no—
123C2×Dic3—44no—
124Dic6—44no—
125Dic6—48no—
126Dic6—84no—
127C3×C8—44no—
128C3⋊C8—81no—
129C2×Q8—64no—
130C2×Q8—62no—
131C2×Q8—64no—
132C2×Q8—128no—
133C22×C4—22no—
134C22×C4—64no—
135C22×C4C2×D1211yescharacteristic
136C22×C4D3×D412yes—
137Q16—248no—
138M16—122no—
139C2×C8—24no—
140C2×C8—121no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C4⋊C4—122no—
147C4⋊C4—122no—
148C22⋊C4—121no—
149C42—22no—
150C42—44no—
151C42—121no—
152C42Dic1214yescomplemented
153C42C2×D1212yes—
154C42D3×D411yescharacteristic
155C42#23911yescharacteristic
156C22×C3—22no—
157C22×C3#39211yescharacteristic
158C22×C3#45112yes—
159C3×C4—22no—
160C3×C4—44no—
161C3×C4#45514yes—
162C3×C4#46212yes—
163C3×C4#49411yescharacteristic
164C3×C4#49711yescharacteristic
165Dic3—84no—
166Dic3—82no—
167C23#88211yescharacteristic
168Q8—124no—
169Q8—128no—
170Q8—244no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—44no—
174C2×C4—128no—
175C2×C4—122no—
176C2×C4—124no—
177C2×C4#87411yescharacteristic
178C2×C4#88212yes—
179C2×C4#89512yes—
180C2×C4#89511yescharacteristic
181C2×C4C2×Dic1212yes—
182C2×C4#90912yes—
183C8—44no—
184C8—241no—
185C2×C3—22no—
186C2×C3#200412yes—
187C2×C3#201611yescharacteristic
188C22—22no—
189C22#467111yescharacteristic
190C22#473912yes—
191C4—22no—
192C4—44no—
193C4—244no—
194C4—242no—
195C4#469911yescharacteristic
196C4#471511yescharacteristic
197C4#482514yes—
198C4#484912yes—
199C3#1029711yescharacteristic, complemented, Sylow
200C2—22no—
201C2#6868811yescharacteristic
202C2#6876212yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1276D3
hypocenterC3#10297
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
72. C3×C42Q16semidirect
152. C42Dic12semidirect
199. C3#10297semidirect

Maximal subgroups

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

subgroupindex
#793132
#733082
#707792
#693962
C3×#12762
#102973