Finite Group Databasebeta

← #11714060

Subgroups — #11714060

Order 768.

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

normal subgroups only

This group has 467 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
1C2×C4⋊#226C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic
3C42×SL(2,𝔽3)C211yesmaximal, characteristic
4#73061C214yesmaximal
5#33033—31nomaximal, Sylow
6C22×#226C2212yes—
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C4⋊#226—22no—
11#5386—22no—
12#4929—41nomaximal
13C2×#475—31no—
14C2×#462—31no—
15#2862—31no—
16C42×Q8D311yescharacteristic
17#1988—31no—
18#1981—31no—
19#1927—34no—
20#1878—34no—
21#1831—31no—
22C22×Dic6—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×#226—22no—
25C2×#226—24no—
26C2×C3×C42—41no—
27C2×C4⋊Dic3—41no—
28C4×SL(2,𝔽3)—22no—
29C4×SL(2,𝔽3)D412yes—
30#859—22no—
31C3⋊#95—44no—
32C22×Q16—32no—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8—31no—
35C2×C4×Q8C2×D312yes—
36C2×C4×Q8C2×D311yescharacteristic
37#475—62no—
38#475—62no—
39#462—62no—
40#462—62no—
41C2×#106—32no—
42C2×C4⋊C8—31no—
43C2×C4⋊C8—31no—
44C2×#103—34no—
45C2×#103—32no—
46C2×#103—32no—
47#391—34no—
48#391—61no—
49#391—61no—
50#382—62no—
51C4×C4⋊C4—32no—
52C4×C4⋊C4—31no—
53C43—31no—
54#340—34no—
55#336—34no—
56C22×C3×C4—41no—
57C22×C3×C4—42no—
58C22×Dic3—42no—
59C22×Dic3—42no—
60C2×Dic6—44no—
61C2×Dic6—82no—
62C2×Dic6—82no—
63C2×SL(2,𝔽3)C2×D411yescharacteristic
64C2×SL(2,𝔽3)#4112yes—
65#226—42no—
66C3×C42—82no—
67C4⋊Dic3—82no—
68C22×Q8—32no—
69C22×Q8—61no—
70C22×Q8C22×D311yescharacteristic
71C2×Q16—64no—
72C2×Q16—62no—
73C2×Q16—62no—
74C2×Q16—64no—
75C22×C8—32no—
76C22×C8—32no—
77C4⋊Q8—62no—
78C4⋊Q8—62no—
79C4⋊Q8—62no—
80C4⋊Q8—62no—
81C4×Q8—22no—
82C4×Q8—32no—
83C4×Q8—62no—
84C4×Q8D1212yes—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—61no—
89C2×C4⋊C4—61no—
90C2×C4⋊C4—61no—
91C2×C42—31no—
92C2×C42—32no—
93C2×C42—32no—
94C2×C42—31no—
95C2×C42S411yescharacteristic
96#106—64no—
97C4⋊C8—62no—
98C4⋊C8—62no—
99#103—68no—
100#103—64no—
101#103—62no—
102#103—62no—
103#95—61no—
104#95—64no—
105#95—64no—
106C23×C3—41no—
107C2×C3×C4—42no—
108C2×C3×C4—42no—
109C2×C3×C4—44no—
110C2×C3×C4—42no—
111C2×C3×C4—82no—
112C2×C3×C4—82no—
113C2×Dic3—44no—
114C2×Dic3—82no—
115C2×Dic3—82no—
116C2×Dic3—82no—
117C2×Dic3—84no—
118Dic6—88no—
119SL(2,𝔽3)#12411yescharacteristic
120C2×Q8—31no—
121C2×Q8—32no—
122C2×Q8—62no—
123C2×Q8—64no—
124C2×Q8—62no—
125C2×Q8—64no—
126C2×Q8—122no—
127C2×Q8—122no—
128C2×Q8—122no—
129C2×Q8C2×D1211yescharacteristic
130C2×Q8#23512yes—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4C2×S412yes—
137C22×C4C2×S411yescharacteristic
138Q16—122no—
139Q16—124no—
140Q16—122no—
141C2×C8—62no—
142C2×C8—62no—
143C2×C8—64no—
144C2×C8—64no—
145C4⋊C4—32no—
146C4⋊C4—34no—
147C4⋊C4—61no—
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—122no—
152C4⋊C4—122no—
153C4⋊C4—122no—
154C42—22no—
155C42—32no—
156C42—32no—
157C42—62no—
158C42—62no—
159C42—64no—
160C42—62no—
161C22×C3—41no—
162C22×C3—42no—
163C22×C3—41no—
164C22×C3—42no—
165C22×C3—41no—
166C3×C4—44no—
167C3×C4—84no—
168Dic3—84no—
169Dic3—162no—
170C23C22×S411yescharacteristic
171Q8—31no—
172Q8—32no—
173Q8—61no—
174Q8—122no—
175Q8—122no—
176Q8—124no—
177Q8—244no—
178Q8#87511yescharacteristic
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—124no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4C4⋊S412yes—
196C2×C4#98312yes—
197C2×C4#98414yes—
198C2×C4#98512yes—
199C8—122no—
200C8—122no—
201C2×C3—41no—
202C2×C3—42no—
203C2×C3—41no—
204C2×C3—42no—
205C2×C3—41no—
206C22C2×C4⋊S411yescharacteristic
207C22#591212yes—
208C22C2×#98311yescharacteristic
209C22C2×#98412yes—
210C22C2×#98511yescharacteristic
211C4—24no—
212C4—31no—
213C4—31no—
214C4—32no—
215C4—62no—
216C4—122no—
217C4—242no—
218C4—242no—
219C4#539514yes—
220C3—41noSylow
221C2#8529011yescharacteristic
222C2#8531912yes—
223C2C2×#539511yescharacteristic
224C2#8533512yes—
225C2#8533611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Maximal subgroups

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

subgroupindex
C2×C4⋊#2262
C2×#53862
C42×SL(2,𝔽3)2
#730612
#330333
#49294