Finite Group Databasebeta

← #11713717

Subgroups — #11713717

Order 768.

463 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 63 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×#557⋊C3C211yesmaximal, characteristic
2C2×C4⋊#226C211yesmaximal, characteristic
3#73061C214yesmaximal
4C4×#859C211yesmaximal, characteristic
5#25931—31nomaximal, Sylow
6C22×#226C2212yes—
7#557⋊C3—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C4⋊#226—22no—
12#4977—41nomaximal
13C2×#557D311yescharacteristic
14C2×#487—31no—
15C2×#462—31no—
16#2568—31no—
17#2026—34no—
18#1988—31no—
19#1984—31no—
20#1878—34no—
21C4×#103—31no—
22C22×SL(2,𝔽3)C2311yescharacteristic
23C2×#226—22no—
24C2×#226—24no—
25C2×C3×C4⋊C4—41no—
26C2×C4⋊Dic3—41no—
27C2×C4×Dic3—41no—
28C4×SL(2,𝔽3)—22no—
29C4×SL(2,𝔽3)D412yes—
30#859—22no—
31C3⋊#95—44no—
32C22×Q16—32no—
33#557—22no—
34#557—62no—
35C2×C4⋊Q8—31no—
36C2×C4⋊Q8—31no—
37C2×#125—32no—
38C2×C4×Q8—31no—
39C2×C4×Q8C2×D312yes—
40C2×C4×Q8C2×D311yescharacteristic
41#487—64no—
42#462—62no—
43#462—62no—
44C2×C4⋊C8—31no—
45C2×#103—34no—
46C2×#103—32no—
47C2×#103—32no—
48C2×C4×C8—31no—
49#400—62no—
50#391—34no—
51#391—61no—
52#382—61no—
53C4×C4⋊C4—31no—
54#340—34no—
55#340—32no—
56#336—34no—
57C22×C3×C4—41no—
58C22×C3×C4—42no—
59C22×Dic3—42no—
60C22×Dic3—42no—
61C2×SL(2,𝔽3)C2×D411yescharacteristic
62C2×SL(2,𝔽3)#4112yes—
63#226—42no—
64C3×C4⋊C4—82no—
65C4⋊Dic3—82no—
66C4×Dic3—44no—
67C22×Q8—31no—
68C22×Q8—32no—
69C22×Q8C22×D311yescharacteristic
70C2×Q16—64no—
71C2×Q16—62no—
72C2×Q16—64no—
73C2×Q16—62no—
74C22×C8—32no—
75C22×C8—32no—
76C4⋊Q8—34no—
77C4⋊Q8—62no—
78C4⋊Q8—62no—
79C4⋊Q8—62no—
80#125—64no—
81#125—64no—
82C4×Q8—22no—
83C4×Q8—32no—
84C4×Q8—62no—
85C4×Q8—62no—
86C4×Q8—62no—
87C4×Q8#6712yes—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—61no—
93C2×C4⋊C4—61no—
94C2×C4⋊C4—61no—
95C2×C4⋊C4S411yescharacteristic
96C2×C42—31no—
97C2×C42—32no—
98C2×C42—61no—
99C4⋊C8—62no—
100#103—64no—
101#103—68no—
102#103—62no—
103#103—62no—
104C4×C8—62no—
105#95—61no—
106#95—64no—
107#95—64no—
108C23×C3—41no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4—82no—
113C2×C3×C4—84no—
114C2×Dic3—44no—
115C2×Dic3—82no—
116C2×Dic3—82no—
117C2×Dic3—82no—
118C2×Dic3—84no—
119SL(2,𝔽3)#12411yescharacteristic
120C2×Q8—32no—
121C2×Q8—31no—
122C2×Q8—62no—
123C2×Q8—62no—
124C2×Q8—64no—
125C2×Q8—64no—
126C2×Q8—62no—
127C2×Q8—62no—
128C2×Q8—62no—
129C2×Q8#23912yes—
130C2×Q8C2×#6711yescharacteristic
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4C2×S412yes—
137C22×C4C2×S411yescharacteristic
138Q16—124no—
139Q16—122no—
140Q16—122no—
141C2×C8—62no—
142C2×C8—62no—
143C2×C8—62no—
144C2×C8—62no—
145C2×C8—64no—
146C4⋊C4—22no—
147C4⋊C4—32no—
148C4⋊C4—34no—
149C4⋊C4—64no—
150C4⋊C4—61no—
151C4⋊C4—62no—
152C4⋊C4—64no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—122no—
156C4⋊C4—122no—
157C42—32no—
158C42—32no—
159C42—62no—
160C42—62no—
161C42—122no—
162C22×C3—42no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—42no—
166C22×C3—41no—
167C3×C4—44no—
168C3×C4—162no—
169Dic3—84no—
170Dic3—162no—
171C23C22×S411yescharacteristic
172Q8—32no—
173Q8—31no—
174Q8—61no—
175Q8—122no—
176Q8—124no—
177Q8—122no—
178Q8—124no—
179Q8#94711yescharacteristic
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—32no—
183C2×C4—32no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—32no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—122no—
192C2×C4—124no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4#98412yes—
197C2×C4#98512yes—
198C2×C4GL(2,ℤ/4)12yes—
199C8—122no—
200C8—122no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—42no—
205C2×C3—42no—
206C22#591912yes—
207C22C2×#98411yescharacteristic
208C22C2×#98511yescharacteristic
209C22#592712yes—
210C22C2×GL(2,ℤ/4)11yescharacteristic
211C4—31no—
212C4—32no—
213C4—31no—
214C4—42no—
215C4—62no—
216C4—122no—
217C4—242no—
218C4—242no—
219C4#542514yes—
220C3—41noSylow
221C2#8541712yes—
222C2#8541912yes—
223C2#8550811yescharacteristic
224C2C2×#542511yescharacteristic
225C2#8553611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×C4⋊#2262
#730612
C4×#8592
#259313
#49774