Finite Group Databasebeta

← #11711271

Subgroups — #11711271

Order 768.

243 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 20 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#85562C211yesmaximal, characteristic
2#85562C211yesmaximal, characteristic
3#1446⋊C3C211yesmaximal, characteristic
4#10345—31nomaximal, Sylow
5C2×C22⋊SL(2,𝔽3)C2211yescharacteristic
6SL(2,ℤ/4)⋊C4—41nomaximal
7#2646—31no—
8#1787—31no—
9#1787—31no—
10#1446D311yescharacteristic
11#1294—31no—
12#1294—31no—
13#1280—31no—
14C22⋊SL(2,𝔽3)Q811yescharacteristic
15C2×C4×A4—41no—
16C2×SL(2,ℤ/4)—41no—
17C2×SL(2,ℤ/4)—41no—
18#859—81no—
19#859—81no—
20C23×Q8C2×D311yescharacteristic
21C2×#103—61no—
22C2×#103—61no—
23C2×#103—61no—
24C2×#103—61no—
25C2×C22⋊C8—31no—
26C2×C22⋊C8—31no—
27#400—61no—
28#400—61no—
29#398—61no—
30#398—61no—
31#385—61no—
32#385—61no—
33#380—31no—
34#380—31no—
35C2×#95—31no—
36C2×#95—31no—
37C22×A4—41no—
38C2×SL(2,𝔽3)—81no—
39C2×SL(2,𝔽3)—81no—
40C4×A4—81no—
41SL(2,ℤ/4)—42no—
42SL(2,ℤ/4)—81no—
43Dic3⋊C4—161no—
44C22×Q8—31no—
45C22×Q8—61no—
46C22×Q8—61no—
47C22×Q8—61no—
48C22×Q8—61no—
49C22×Q8Dic611yescharacteristic
50C23×C4—31no—
51C23×C4—31no—
52C23×C4S411yescharacteristic
53C22×C8—61no—
54C22×C8—61no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—31no—
57C2×C4⋊C4—31no—
58C2×C4⋊C4—31no—
59C2×C4⋊C4—61no—
60C2×C4⋊C4—61no—
61C2×C4⋊C4—61no—
62C2×C4⋊C4—61no—
63C2×C22⋊C4—61no—
64C2×C22⋊C4—61no—
65#103—121no—
66#103—121no—
67#103—121no—
68#103—121no—
69#103—121no—
70#103—121no—
71#103—121no—
72#103—121no—
73C22⋊C8—61no—
74C22⋊C8—61no—
75C22⋊C8—61no—
76C22⋊C8—61no—
77#95—61no—
78#95—61no—
79#95—61no—
80#95—61no—
81#95—61no—
82#95—61no—
83#95—61no—
84#95—61no—
85#95—61no—
86#95—61no—
87C2×A4—41no—
88C2×A4—41no—
89C2×A4—41no—
90C2×C3×C4—161no—
91C2×Dic3—161no—
92C2×Dic3—161no—
93SL(2,𝔽3)—81no—
94SL(2,𝔽3)—81no—
95C24C2×S411yescharacteristic
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—61no—
99C2×Q8—61no—
100C2×Q8—61no—
101C2×Q8—61no—
102C2×Q8—61no—
103C2×Q8—61no—
104C2×Q8—61no—
105C2×Q8—61no—
106C2×Q8—61no—
107C2×Q8—61no—
108C2×Q8—121no—
109C2×Q8—121no—
110C2×Q8—121no—
111C2×Q8—121no—
112C22×C4—21no—
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—61no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4—61no—
125C22×C4—61no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4—121no—
129C22×C4—121no—
130C22×C4—121no—
131C22×C4—121no—
132C2×C8—121no—
133C2×C8—121no—
134C2×C8—121no—
135C2×C8—121no—
136C2×C8—121no—
137C2×C8—121no—
138C4⋊C4—121no—
139C4⋊C4—121no—
140C4⋊C4—121no—
141C4⋊C4—121no—
142C4⋊C4—241no—
143C4⋊C4—241no—
144C4⋊C4—241no—
145C4⋊C4—241no—
146C22⋊C4—121no—
147C22⋊C4—121no—
148C22⋊C4—121no—
149C22⋊C4—121no—
150C22×C3—161no—
151A4—41no—
152C3×C4—321no—
153Dic3—162no—
154Dic3—321no—
155C23—31no—
156C23—31no—
157C23—31no—
158C23—31no—
159C23A4⋊Q811yescharacteristic
160C23#98311yescharacteristic
161C23#98511yescharacteristic
162Q8—21no—
163Q8—21no—
164Q8—61no—
165Q8—61no—
166Q8—61no—
167Q8—61no—
168Q8—61no—
169Q8—61no—
170Q8—121no—
171Q8—121no—
172C2×C4—31no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—41no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4—121no—
194C2×C4—121no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—122no—
201C2×C4—121no—
202C2×C4—121no—
203C2×C4—241no—
204C8—241no—
205C8—241no—
206C2×C3—161no—
207C2×C3—161no—
208C2×C3—161no—
209C22—31no—
210C22—31no—
211C22—31no—
212C22—31no—
213C22—31no—
214C22—31no—
215C22—31no—
216C22—31no—
217C22—31no—
218C22—61no—
219C22#538911yescharacteristic
220C22#1020⋊C211yescharacteristic
221C4—61no—
222C4—61no—
223C4—61no—
224C4—61no—
225C4—81no—
226C4—241no—
227C4—241no—
228C4—241no—
229C4—241no—
230C4—241no—
231C3—161noSylow
232C2—31no—
233C2—31no—
234C2—31no—
235C2—31no—
236C2#7298911yescharacteristic
237C2#7300011yescharacteristic
238C2#7300211yescharacteristic
subgroupG/H
center, hypercenterC22#1020⋊C2
commutator, prime radical, cosocleC2×C22⋊SL(2,𝔽3)C22
Frattini, socleC24C2×S4
Fitting#1446D3
hypocenterC22⋊SL(2,𝔽3)Q8
LCSG ▷ C2×C22⋊SL(2,𝔽3) ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22

Maximal subgroups

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

subgroupindex
#855622
#1446⋊C32
#103453
SL(2,ℤ/4)⋊C44