Finite Group Databasebeta

← #11711604

Subgroups — #11711604

Order 768.

239 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 33 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×#5407C211yesmaximal, characteristic
2#3407⋊C3C211yesmaximal, characteristic, complemented
3#73120C211yesmaximal, characteristic, complemented
4#13068—31nomaximal, Sylow
5C2×#986—21no—
6C2×#131⋊C3C2211yescharacteristic
7#5423—21no—
8#5407—21no—
9#5407C412yescomplemented
10#504⋊C3—21no—
11#4516—41nomaximal
12#3407D311yescharacteristic, complemented
13C2×#472—31no—
14C2×#361—31no—
15#2140—31no—
16#1814—31no—
17#1389—31no—
18#1385—31no—
19C2×#41⋊C3D411yescharacteristic
20#986—22no—
21#986—41no—
22C2×GL(2,𝔽3)—41no—
23C2×D24—41no—
24#131⋊C3C2×C411yescharacteristic
25#131⋊C3D411yescharacteristic
26GU(2,𝔽3)—41no—
27C3×M32—41no—
28#819—41no—
29C2×AGL(1,ℤ/8)—61no—
30C22×D8—31no—
31C2×#131C2×D311yescharacteristic
32#504—21no—
33#504—61no—
34C2×M32—31no—
35C2×M32—31no—
36#472—32no—
37#472—32no—
38#472—61no—
39#472—61no—
40C2×#108—31no—
41C2×#108—31no—
42#427—31no—
43#421—61no—
44C2×#100—61no—
45#361—61no—
46#361—61no—
47#357—62no—
48#344—61no—
49C2×D12—81no—
50#41⋊C3C22⋊C411yescharacteristic
51#41⋊C3D811yescharacteristic
52C2×SL(2,𝔽3)SD1611yescharacteristic
53GL(2,𝔽3)—42no—
54C2×C3×C8—41no—
55#208—81no—
56D24—42no—
57D24—81no—
58C3×C16—81no—
59C2×#41D1211yescharacteristic
60C22×D4—61no—
61AGL(1,ℤ/8)—62no—
62AGL(1,ℤ/8)—62no—
63AGL(1,ℤ/8)—121no—
64AGL(1,ℤ/8)—121no—
65C2×SD16—121no—
66C2×D8—32no—
67C2×D8—61no—
68C2×D8—61no—
69C2×D8—61no—
70C2×D8—61no—
71C2×D8—61no—
72C2×D8—121no—
73#131—31no—
74#131—31no—
75#131C4×D311yescharacteristic
76#131#6711yescharacteristic
77C2×M16—31no—
78C2×M16—61no—
79C2×M16—61no—
80C22×C8—31no—
81#117—61no—
82M32—31no—
83M32—61no—
84M32—61no—
85M32S411yescharacteristic
86C2×C16—32no—
87C2×C16—61no—
88#108—32no—
89#108—61no—
90#108—61no—
91#108—61no—
92#107—61no—
93#106—61no—
94#104—121no—
95#104—121no—
96#100—62no—
97#100—62no—
98C22×D3—161no—
99C2×C3×C4—41no—
100D12—82no—
101D12—161no—
102SL(2,𝔽3)#10211yescharacteristic
103C3×C8—41no—
104C3×C8—41no—
105C3⋊C8—161no—
106C24—121no—
107#41—31no—
108#41—31no—
109#41D2411yescharacteristic
110#41(C2×D3)⋊C411yescharacteristic
111C2×Q8#20411yescharacteristic
112C2×D4—31no—
113C2×D4—62no—
114C2×D4—121no—
115C2×D4—121no—
116C2×D4—121no—
117C2×D4—121no—
118C2×D4—121no—
119C22×C4—31no—
120SD16—122no—
121SD16—122no—
122D8—62no—
123D8—62no—
124D8—121no—
125D8—122no—
126D8—122no—
127D8—121no—
128M16—31no—
129M16—31no—
130M16—61no—
131M16—62no—
132M16—121no—
133M16—121no—
134M16—121no—
135C2×C8—31no—
136C2×C8—31no—
137C2×C8—31no—
138C2×C8—31no—
139C2×C8—31no—
140C2×C8—121no—
141C2×C8—121no—
142C2×C8C2×S411yescharacteristic
143C4⋊C4—121no—
144C22⋊C4—121no—
145C42—121no—
146C16—21no—
147C16—61no—
148C22×C3—41no—
149C2×D3—162no—
150C2×D3—322no—
151C3×C4—41no—
152C3×C4—41no—
153C23—31no—
154C23—122no—
155C23—241no—
156C23—241no—
157C23—482no—
158Q8—31no—
159Q8#82111yescharacteristic
160D4—31no—
161D4—31no—
162D4—61no—
163D4—122no—
164D4—122no—
165D4—241no—
166D4—241no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—241no—
173C2×C4C4⋊S411yescharacteristic
174C8—31no—
175C8—31no—
176C8—122no—
177C8—241no—
178C8C4×S411yescharacteristic
179C8GL(2,ℤ/4)11yescharacteristic
180C2×C3—41no—
181C2×C3—81no—
182D3—322no—
183C22—31no—
184C22—31no—
185C22—121no—
186C22—242no—
187C22—482no—
188C22—482no—
189C22—482no—
190C22#540111yescharacteristic
191C4—31no—
192C4—31no—
193C4—481no—
194C4#540211yescharacteristic
195C4(C2×S4)⋊C411yescharacteristic
196C3—41noSylow
197C2—21no—
198C2—61no—
199C2—61no—
200C2—482no—
201C2#7311311yescharacteristic
subgroupG/H
center, socleC2#73113
commutator#131⋊C3C2×C4
FrattiniC2×C8C2×S4
Fitting#3407D3
prime radicalC2×#5407C2
cosocleC2×#131⋊C3C22
hypercenterM32S4
hypocenterSL(2,𝔽3)#102
LCSG ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
2. #3407⋊C3C2semidirect
3. #73120C2semidirect
9. #5407C4semidirect
12. #3407D3semidirect

Maximal subgroups

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

subgroupindex
C2×#54072
#3407⋊C32
#731202
#130683
#45164