Finite Group Databasebeta

← #11711876

Subgroups — #11711876

Order 768.

449 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 67 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#85325C211yesmaximal, characteristic
2M16×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73018C212yesmaximal, complemented
5C8×GL(2,𝔽3)C212yesmaximal, complemented
6#17997—31nomaximal, Sylow
7C2×#907—41nomaximal
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9#5421C412yes—
10C4⋊GL(2,𝔽3)C411yescharacteristic
11C4×GL(2,𝔽3)C2212yes—
12#5386C411yescharacteristic
13C8×SL(2,𝔽3)C2212yes—
14SL(2,𝔽3)⋊C8C2211yescharacteristic
15SL(2,𝔽3)⋊C8C2211yescharacteristic
16#3103—31no—
17Q8×M16D311yescharacteristic, complemented
18#2935—31no—
19#1590—32no—
20#1583—32no—
21#1582—32no—
22C8×SD16—32no—
23#1570—31no—
24#1492—31no—
25C2×Q8⋊C8—31no—
26C8×M16—31no—
27C2×#235—41no—
28C22×SL(2,𝔽3)C2×C411yescharacteristic
29C2×GL(2,𝔽3)C2×C412yes—
30C2×C3×M16—41no—
31C22×C3⋊C8—41no—
32#907—48no—
33C2×ΓL(1,𝔽25)—42no—
34C2×D3×C8—42no—
35C4×SL(2,𝔽3)C2×C411yescharacteristic
36C4×SL(2,𝔽3)C2311yescharacteristic
37#859C2×C412yes—
38C2×#131—61no—
39#532—31no—
40C2×C4×Q8C2×D311yescharacteristic
41#481—32no—
42#475—31no—
43#475—31no—
44#460—31no—
45#460—31no—
46#448—32no—
47#446—32no—
48C8×Q8C2×D312yes—
49C4×SD16—32no—
50C4×SD16—32no—
51#436—32no—
52#435—62no—
53C8×D4—32no—
54#431—31no—
55#423—31no—
56C2×C4⋊C8—31no—
57#408—61no—
58C4×M16—31no—
59C2×C4×C8—31no—
60#334—32no—
61#334—32no—
62Q8⋊C8—31no—
63Q8⋊C8—31no—
64Q8⋊C8—31no—
65Q8⋊C8—31no—
66D4⋊C8—32no—
67D4⋊C8—32no—
68#322—32no—
69C82—32no—
70C22×C3×C4—41no—
71C2×#67—42no—
72#235—44no—
73#235—44no—
74C2×D12—41no—
75C2×C4×D3—42no—
76C2×Dic6—41no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)C22×C411yescharacteristic
79GL(2,𝔽3)—22no—
80C3×M16—44no—
81C2×C3×C8—42no—
82C2×C3⋊C8—41no—
83C2×C3⋊C8—41no—
84C2×C3⋊C8—44no—
85ΓL(1,𝔽25)—48no—
86D3×C8—48no—
87C2×#41—61no—
88C22×Q8C4×D311yescharacteristic
89C2×SD16—32no—
90C2×SD16—32no—
91#131—124no—
92C2×M16—31no—
93C2×M16—62no—
94C2×M16S411yescharacteristic
95C22×C8—61no—
96C22×C8—61no—
97C22×C8—62no—
98C4⋊Q8—31no—
99#127—31no—
100#124—61no—
101#121—32no—
102#121—61no—
103C4×Q8—31no—
104C4×Q8—31no—
105C4×Q8C4×D311yescharacteristic
106C4×Q8C22×D311yescharacteristic
107C4×D4—32no—
108C4×D4—61no—
109C2×C4⋊C4—31no—
110C2×C42—31no—
111#106—32no—
112#106—32no—
113C4⋊C8—32no—
114C4⋊C8—31no—
115C4⋊C8—31no—
116C4⋊C8—31no—
117C4⋊C8—31no—
118C4⋊C8—61no—
119C4⋊C8—61no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—32no—
125C22⋊C8—62no—
126C22⋊C8—62no—
127#97—32no—
128C4×C8—32no—
129C4×C8—32no—
130C4×C8—31no—
131C4×C8—31no—
132C4×C8—61no—
133C23×C3—41no—
134C22×D3—42no—
135C2×C3×C4—41no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—41no—
139#67—48no—
140C2×Dic3—42no—
141D12—44no—
142C4×D3—44no—
143C4×D3—44no—
144Dic6—44no—
145SL(2,𝔽3)#13111yescharacteristic, complemented
146C3×C8—44no—
147C3⋊C8—42no—
148C3⋊C8—42no—
149#41—122no—
150#41—122no—
151C2×Q8—21no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×Q8—61no—
155C2×Q8C2×C4×D311yescharacteristic
156C2×D4—32no—
157C2×D4—61no—
158C2×D4—62no—
159C2×D4—61no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—62no—
163C22×C4C2×S411yescharacteristic
164SD16—62no—
165SD16—62no—
166SD16—122no—
167M16—62no—
168M16—124no—
169M16GL(2,𝔽3)14yescomplemented
170C2×C8—32no—
171C2×C8—32no—
172C2×C8—32no—
173C2×C8—61no—
174C2×C8—61no—
175C2×C8—61no—
176C2×C8—61no—
177C2×C8—122no—
178C2×C8—122no—
179C2×C8—124no—
180C2×C8C2×S412yes—
181C4⋊C4—31no—
182C4⋊C4—31no—
183C4⋊C4—32no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—31no—
189C42—31no—
190C42—31no—
191C42—31no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—81no—
195C22×C3—81no—
196C2×D3—44no—
197C2×D3—82no—
198C2×D3—82no—
199C3×C4—42no—
200C3×C4—41no—
201C3×C4—41no—
202Dic3—44no—
203C23—62no—
204C23C4×S411yescharacteristic
205Q8—31no—
206Q8—61no—
207Q8—61no—
208Q8—122no—
209Q8#90711yescharacteristic, complemented
210D4—62no—
211D4—62no—
212D4—124no—
213D4—122no—
214D4—241no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4C4×S411yescharacteristic
226C2×C4C2×GL(2,𝔽3)12yes—
227C2×C4#98512yes—
228C2×C4C22×S411yescharacteristic
229C8—62no—
230C8—62no—
231C8—62no—
232C8—62no—
233C8—121no—
234C8—121no—
235C8—121no—
236C8C2×GL(2,𝔽3)14yes—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—82no—
241D3—84no—
242C22—21no—
243C22—21no—
244C22—122no—
245C22—122no—
246C22—122no—
247C22C4×GL(2,𝔽3)12yes—
248C22C2×C4×S411yescharacteristic
249C4—31no—
250C4—31no—
251C4—61no—
252C4—61no—
253C4—61no—
254C4—122no—
255C4C4×GL(2,𝔽3)12yes—
256C4C22×GL(2,𝔽3)11yescharacteristic
257C4C2×#98511yescharacteristic
258C3—41noSylow
259C2—22no—
260C2—242no—
261C2C2×C4×GL(2,𝔽3)11yescharacteristic
262C2#8544211yescharacteristic
263C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingQ8×M16D3
socleC22C2×C4×S4
prime radical#85325C2
cosocleC4×SL(2,𝔽3)C23
hypercenterC2×M16S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. M16×SL(2,𝔽3)C2semidirect
3. C2×SL(2,𝔽3)⋊C8C2semidirect
4. #73018C2semidirect
5. C8×GL(2,𝔽3)C2semidirect
17. Q8×M16D3semidirect
145. SL(2,𝔽3)#131semidirect
169. M16GL(2,𝔽3)semidirect
209. Q8#907semidirect

Maximal subgroups

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

subgroupindex
#853252
M16×SL(2,𝔽3)2
C2×SL(2,𝔽3)⋊C82
#730182
C8×GL(2,𝔽3)2
#179973
C2×#9074