Finite Group Databasebeta

← #11711027

Subgroups — #11711027

Order 768.

363 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 37 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2#73149C212yesmaximal, complemented
3#8361—31nomaximal, Sylow
4C22×#41⋊C3C411yescharacteristic, complemented
5C22×#41⋊C3C2211yescharacteristic
6C4⋊(#41⋊C3)—22no—
7C2×C4×SL(2,𝔽3)C411yescharacteristic, complemented
8C2×GU(2,𝔽3)—22no—
9#4528—41nomaximal
10C2×#550D311yescharacteristic
11#1789—32no—
12C4×#99—31no—
13#1507—32no—
14#1465—31no—
15C2×#41⋊C3—21no—
16C2×#41⋊C3—41no—
17C2×#41⋊C3—41no—
18C2×#41⋊C3D412yes—
19C22×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C3×C4⋊C4—41no—
21C4×SL(2,𝔽3)—21no—
22GU(2,𝔽3)—42no—
23#830—42no—
24C22×#41Dic311yescharacteristic
25C22×#41C2×D311yescharacteristic
26#550—22no—
27#550—62no—
28C2×#127—31no—
29C2×#121—31no—
30C2×#121—31no—
31C2×C4×Q8Dic311yescharacteristic
32C2×C4×D4—31no—
33C2×#104—62no—
34C2×#104—62no—
35C2×#99—61no—
36#407—32no—
37C4×C22⋊C4—32no—
38#342—61no—
39#327—64no—
40#324—61no—
41#323—61no—
42C22×C3×C4—41no—
43C22×C3×C4—41no—
44C22×C3×C4—41no—
45#41⋊C3—22no—
46#41⋊C3—41no—
47C2×SL(2,𝔽3)—21no—
48C2×SL(2,𝔽3)C22⋊C411yescharacteristic
49C3×C4⋊C4—82no—
50C2×C3⋊C8—82no—
51C2×#41—21no—
52C2×#41—32no—
53C2×#41—41no—
54C2×#41—41no—
55C2×#41—61no—
56C2×#41—121no—
57C2×#41—121no—
58C2×#41#6712yes—
59C22×Q8C2×Dic311yescharacteristic
60C22×D4—31no—
61C22×D4—31no—
62C22×D4—31no—
63C22×D4—61no—
64C23×C4—31no—
65C23×C4—31no—
66C2×M16—62no—
67#127—62no—
68#127—62no—
69#121—34no—
70#121—62no—
71#121—62no—
72#121—62no—
73C4×Q8—21no—
74C4×Q8—61no—
75C4×D4—62no—
76C4×D4—62no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4S411yescharacteristic
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—62no—
82C2×C42—31no—
83C2×C42—62no—
84#104—122no—
85#104—122no—
86#104—124no—
87#99—64no—
88C22⋊C8—62no—
89C22⋊C8—62no—
90#95—62no—
91C23×C3—41no—
92C2×C3×C4—42no—
93C2×C3×C4—42no—
94C2×C3×C4—81no—
95C2×C3×C4—81no—
96C2×C3×C4—81no—
97C2×C3×C4—161no—
98C2×C3×C4—161no—
99SL(2,𝔽3)#9911yescharacteristic, complemented
100C3⋊C8—162no—
101C24—31no—
102C24—31no—
103#41—22no—
104#41—41no—
105#41—62no—
106#41—64no—
107#41—122no—
108#41—122no—
109#41—122no—
110#41—121no—
111C2×Q8—21no—
112C2×Q8—31no—
113C2×Q8—61no—
114C2×Q8#21711yescharacteristic
115C2×D4—32no—
116C2×D4—32no—
117C2×D4—62no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—64no—
121C2×D4—62no—
122C2×D4—121no—
123C2×D4—121no—
124C2×D4—121no—
125C2×D4—121no—
126C2×D4—121no—
127C2×D4—242no—
128C22×C4—31no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—31no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4—121no—
135C22×C4—121no—
136C22×C4—121no—
137C22×C4—121no—
138C22×C4—122no—
139C22×C4SL(2,ℤ/4)11yescharacteristic
140C22×C4SL(2,ℤ/4)11yescharacteristic
141C22×C4C2×S411yescharacteristic
142M16—122no—
143M16—122no—
144C2×C8—122no—
145C4⋊C4—22no—
146C4⋊C4—32no—
147C4⋊C4—61no—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C42—61no—
153C42—61no—
154C42—122no—
155C42—122no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—81no—
160C22×C3—81no—
161C3×C4—82no—
162C3×C4—82no—
163C3×C4—161no—
164C23—32no—
165C23—61no—
166C23—61no—
167C23—61no—
168C23—61no—
169C23—61no—
170C23—121no—
171C23—121no—
172C23—121no—
173C23—121no—
174C23C2×SL(2,ℤ/4)11yescharacteristic
175Q8—31no—
176Q8—32no—
177Q8—61no—
178Q8#83411yescharacteristic
179D4—62no—
180D4—62no—
181D4—124no—
182D4—121no—
183D4—121no—
184D4—122no—
185D4—122no—
186D4—244no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—122no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4—124no—
205C2×C4—242no—
206C2×C4—242no—
207C2×C4GU(2,𝔽3)12yescomplemented
208C2×C4GL(2,ℤ/4)12yes—
209C8—242no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—81no—
214C2×C3—81no—
215C22—21no—
216C22—21no—
217C22—62no—
218C22—62no—
219C22—62no—
220C22—121no—
221C22—121no—
222C22—121no—
223C22—241no—
224C22—241no—
225C22—241no—
226C22—241no—
227C22C2×GU(2,𝔽3)11yescharacteristic
228C22#542311yescharacteristic
229C22#543211yescharacteristic
230C4—22no—
231C4—22no—
232C4—31no—
233C4—31no—
234C4—41no—
235C4—61no—
236C4—121no—
237C4—244no—
238C3—41noSylow
239C2—21no—
240C2—21no—
241C2—122no—
242C2—241no—
243C2#7314511yescharacteristic
244C2#7314911yescharacteristic
245C2#7315211yescharacteristic
subgroupG/H
center, socleC22#5432
commutatorC22×SL(2,𝔽3)C2×C4
FrattiniC22×C4C2×S4
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
cosocleC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#99
LCSG ▷ C22×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #73149C2semidirect
4. C22×#41⋊C3C4semidirect
7. C2×C4×SL(2,𝔽3)C4semidirect
99. SL(2,𝔽3)#99semidirect
207. C2×C4GU(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
#731492
#83613
#45284