Finite Group Databasebeta

← #10694855

Subgroups — #10694855

Order 768.

466 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×C3×#494C211yesmaximal, characteristic
2#71790C212yesmaximal
3#71774C212yesmaximal
4#69730C211yesmaximal, characteristic
5#69725C211yesmaximal, characteristic
6#9807—31nomaximal, Sylow
7C22×C3×Q16C414yes—
8C2×C3×C4⋊Q8C2212yes—
9C3×#494—21no—
10C3×#494—21no—
11C3×#494—22no—
12C2×C3×C4×C8C2211yescharacteristic
13#4925C2211yescharacteristic
14#4924C2212yes—
15C4×#208C2211yescharacteristic
16#4546—24no—
17#4530—24no—
18#4457—22no—
19#4455—21no—
20#4454—21no—
21C2×#494D311yescharacteristic
22#1997—32no—
23#1566—31no—
24#1548—32no—
25#1451—31no—
26C22×C3×Q8C2×C414yes—
27C2×C3×Q16—24no—
28C2×C3×Q16—24no—
29C2×C3×Q16—48no—
30C22×C3×C8C2×C412yes—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#208—22no—
39C2×#208D412yes—
40C3×C4×C8D411yescharacteristic
41C3×C4×C8D411yescharacteristic
42C3×C4×C8D412yes—
43#830—21no—
44C4⋊(C3⋊C8)—22no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)D412yes—
47#803—21no—
48C4×C3⋊C8—21no—
49C4×C3⋊C8—21no—
50C22×Q16Dic314yes—
51C2×C4⋊Q8C2×D312yes—
52#494—21no—
53#494—21no—
54#494—22no—
55#431—31no—
56#423—32no—
57C4×M16—31no—
58C2×C4×C8C2×D311yescharacteristic
59#343—64no—
60#334—62no—
61#333—64no—
62#322—61no—
63#322—61no—
64C2×C3×Q8—24no—
65C2×C3×Q8—24no—
66C2×C3×Q8—48no—
67C22×C3×C4—24no—
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4#4111yescharacteristic
71C3×Q16—48no—
72C3×Q16—48no—
73C2×C3×C8—24no—
74C2×C3×C8C22⋊C412yes—
75C2×C3×C8C22⋊C412yes—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—42no—
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84#208—24no—
85#208—44no—
86C2×C3⋊C8—22no—
87C2×C3⋊C8—22no—
88C2×C3⋊C8—42no—
89C22×Q8C2×Dic314yes—
90C2×Q16—24no—
91C2×Q16—24no—
92C2×Q16—48no—
93C2×M16—32no—
94C2×M16—62no—
95C22×C8C2×Dic312yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C42C22×D311yescharacteristic
103C4⋊C8—32no—
104C4⋊C8—62no—
105C4⋊C8—62no—
106C22⋊C8—61no—
107#97—61no—
108C4×C8—61no—
109C4×C8—61no—
110C4×C8#6711yescharacteristic
111C4×C8#6711yescharacteristic
112C4×C8#6712yes—
113C23×C3C4×D411yescharacteristic
114C3×Q8—48no—
115C3×Q8—84no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—44no—
119C2×C3×C4—84no—
120C2×C3×C4C2×C22⋊C411yescharacteristic
121C2×C3×C4C4×D411yescharacteristic
122C2×C3×C4#12111yescharacteristic
123C2×C3×C4#12111yescharacteristic
124C2×C3×C4#12411yescharacteristic
125C2×C3×C4#12711yescharacteristic
126C2×C3×C4#13714yes—
127C3×C8—22no—
128C3×C8#10014yes—
129C3⋊C8—42no—
130C3⋊C8—42no—
131C3⋊C8—82no—
132C2×Q8—24no—
133C2×Q8—24no—
134C2×Q8—48no—
135C22×C4—24no—
136C22×C4D3×D411yescharacteristic
137C22×C4#23711yescharacteristic
138C22×C4C22×Dic311yescharacteristic
139Q16—48no—
140Q16—48no—
141M16—64no—
142M16—124no—
143C2×C8—24no—
144C2×C8—62no—
145C2×C8—62no—
146C2×C8—122no—
147C2×C8#21712yes—
148C2×C8#21712yes—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C42D3×D411yescharacteristic
154C42#23711yescharacteristic
155C42C2×#6711yescharacteristic
156C42C2×#6711yescharacteristic
157C22×C3—21no—
158C22×C3—21no—
159C22×C3#39011yescharacteristic
160C22×C3#44112yes—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—84no—
164C3×C4C2×#10012yes—
165C3×C4#47012yes—
166C3×C4#47312yes—
167C3×C4#49712yes—
168C23D4×Dic311yescharacteristic
169Q8—48no—
170Q8—84no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—44no—
174C2×C4—84no—
175C2×C4#91814yes—
176C2×C4D4×Dic311yescharacteristic
177C2×C4#93611yescharacteristic
178C2×C4#93811yescharacteristic
179C2×C4#93811yescharacteristic
180C2×C4#94011yescharacteristic
181C2×C4C2×#21711yescharacteristic
182C8—22no—
183C8—122no—
184C8—122no—
185C8—242no—
186C8#83314yes—
187C2×C3—22no—
188C2×C3#196511yescharacteristic
189C2×C3#197111yescharacteristic
190C2×C3#197911yescharacteristic
191C22—21no—
192C22—21no—
193C22#518412yes—
194C22#522111yescharacteristic
195C4—21no—
196C4—21no—
197C4—84no—
198C4#518912yes—
199C4#519012yes—
200C4#519212yes—
201C4C2×#83312yes—
202C3#980711yescharacteristic, complemented, Sylow
203C2—22no—
204C2#7176611yescharacteristic
205C2#7206711yescharacteristic
206C2#7206911yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#494C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenterC2×#494D3
hypocenterC3#9807
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
202. C3#9807semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
#717902
#717742
#697302
#697252
#98073