Finite Group Databasebeta

← #11713715

Subgroups — #11713715

Order 768.

409 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 47 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#3504⋊C3C211yesmaximal, characteristic
2#85336C211yesmaximal, characteristic
3#73077C214yesmaximal
4#73067C211yesmaximal, characteristic
5#25927—31nomaximal, Sylow
6C2×#984C2212yes—
7#557⋊C3—22no—
8#520⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10C2×GU(2,𝔽3)C2212yes—
11C4⋊#226—22no—
12#5069—41nomaximal
13#3504D311yescharacteristic
14#3139—31no—
15#3039—31no—
16#2642—31no—
17#2028—34no—
18#1998—31no—
19#1995—31no—
20#1882—34no—
21#1759—31no—
22C2×#41⋊C3C2311yescharacteristic
23#984—24no—
24C2×#226—22no—
25C3×#125—41no—
26#869—41no—
27C4×SL(2,𝔽3)—22no—
28C4×SL(2,𝔽3)D412yes—
29GU(2,𝔽3)—22no—
30#810—44no—
31#803—41no—
32C2×#137—32no—
33#557—22no—
34#554—62no—
35#544—32no—
36#533—31no—
37C2×C4⋊Q8—31no—
38C2×#125—31no—
39#520C2×D312yes—
40C4×#41C2×D311yescharacteristic
41#491—61no—
42#491—61no—
43#487—62no—
44#462—62no—
45#461—62no—
46#423—31no—
47C2×#104—32no—
48#419—34no—
49#419—32no—
50C4×M16—31no—
51#400—64no—
52#391—34no—
53#376—31no—
54#343—34no—
55#339—34no—
56#339—32no—
57C2×Dic6—42no—
58#41⋊C3#4112yes—
59C2×SL(2,𝔽3)C2×D411yescharacteristic
60#226—42no—
61C3×C4⋊C4—42no—
62C3×C4⋊C4—82no—
63C3×C42—41no—
64C4⋊Dic3—82no—
65C2×C3⋊C8—42no—
66C2×#41C22×D311yescharacteristic
67C22×Q8—32no—
68#137—64no—
69#137—64no—
70C2×Q16—62no—
71C2×SD16—62no—
72C2×M16—32no—
73C2×M16—32no—
74C4⋊Q8—32no—
75C4⋊Q8—62no—
76C4⋊Q8—61no—
77C4⋊Q8—62no—
78C4⋊Q8—61no—
79#125—31no—
80#125—61no—
81#125—62no—
82#125—62no—
83#125S411yescharacteristic
84C22⋊Q8—62no—
85C22⋊Q8—64no—
86C4×Q8—22no—
87C4×Q8—62no—
88C4×Q8#6712yes—
89C4×D4—32no—
90C4×D4—62no—
91#117—31no—
92#117—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—34no—
95C2×C4⋊C4—62no—
96C2×C42—31no—
97C2×C42—32no—
98C4⋊C8—62no—
99#104—62no—
100#104—62no—
101#103—64no—
102#103—64no—
103#103—64no—
104#97—61no—
105C4×C8—61no—
106#95—62no—
107#95—68no—
108C2×C3×C4—42no—
109C2×C3×C4—41no—
110C2×C3×C4—82no—
111C2×Dic3—82no—
112Dic6—84no—
113SL(2,𝔽3)#12411yescharacteristic
114C3⋊C8—82no—
115#41—32no—
116#41#23912yes—
117C2×Q8—62no—
118C2×Q8—62no—
119C2×Q8—64no—
120C2×Q8—64no—
121C2×Q8—62no—
122C2×Q8C2×#6711yescharacteristic
123C2×D4—31no—
124C22×C4—31no—
125C22×C4—32no—
126C22×C4—34no—
127C22×C4—62no—
128C22×C4—62no—
129Q16—122no—
130Q16—122no—
131SD16—124no—
132M16—62no—
133M16—62no—
134M16—64no—
135C2×C8—62no—
136C2×C8—62no—
137C4⋊C4—22no—
138C4⋊C4—32no—
139C4⋊C4—32no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—64no—
143C4⋊C4—62no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C4⋊C4C2×S412yes—
147C22⋊C4—62no—
148C22⋊C4—61no—
149C42—32no—
150C42—31no—
151C42—62no—
152C42—62no—
153C42—62no—
154C42C2×S411yescharacteristic
155C22×C3—41no—
156C3×C4—42no—
157C3×C4—82no—
158C3×C4—162no—
159Dic3—162no—
160C23—31no—
161Q8—31no—
162Q8—122no—
163Q8—122no—
164Q8—124no—
165Q8—242no—
166Q8#94711yescharacteristic
167D4—32no—
168D4—61no—
169C2×C4—22no—
170C2×C4—31no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4—32no—
174C2×C4—62no—
175C2×C4—122no—
176C2×C4—124no—
177C2×C4—122no—
178C2×C4—124no—
179C2×C4—124no—
180C2×C4—122no—
181C2×C4GL(2,ℤ/4)12yes—
182C2×C4C22×S411yescharacteristic
183C8—122no—
184C8—122no—
185C2×C3—41no—
186C2×C3—42no—
187C22—32no—
188C22—62no—
189C22C2×GL(2,ℤ/4)11yescharacteristic
190C4—22no—
191C4—31no—
192C4—31no—
193C4—42no—
194C4—62no—
195C4—122no—
196C4—242no—
197C4—242no—
198C4#591912yes—
199C3—41noSylow
200C2—62no—
201C2#8550811yescharacteristic
202C2#8554212yes—
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3504D3
hypercenter#125S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #125

Maximal subgroups

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

subgroupindex
#3504⋊C32
#853362
#730772
#730672
#259273
#50694