Finite Group Databasebeta

← #11714057

Subgroups — #11714057

Order 768.

451 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 71 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×#5391C211yesmaximal, characteristic
2C2×C4⋊#226C211yesmaximal, characteristic
3C42×SL(2,𝔽3)C211yesmaximal, characteristic
4#73061C214yesmaximal
5#33027—31nomaximal, Sylow
6C22×#226C2212yes—
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10#5391—22no—
11C4⋊#226—22no—
12#4936—41nomaximal
13C2×#477—31no—
14C2×#462—31no—
15#2862—31no—
16C42×Q8D311yescharacteristic
17#1987—31no—
18#1981—31no—
19#1928—34no—
20#1878—34no—
21#1831—31no—
22C22×SL(2,𝔽3)C2311yescharacteristic
23C2×#226—22no—
24C2×#226—24no—
25C2×C3×C42—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—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8—31no—
35C2×C4×Q8C2×D312yes—
36C2×C4×Q8C2×D311yescharacteristic
37#477—62no—
38#477—62no—
39#462—62no—
40#462—62no—
41C2×#107—32no—
42C2×C4⋊C8—31no—
43C2×C4⋊C8—31no—
44C2×#103—34no—
45C2×#103—32no—
46C2×#103—32no—
47#391—34no—
48#391—62no—
49#382—61no—
50#382—61no—
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×SL(2,𝔽3)C2×D411yescharacteristic
61C2×SL(2,𝔽3)#4112yes—
62#226—42no—
63C3×C42—44no—
64C4⋊Dic3—82no—
65C4⋊Dic3—82no—
66C22×Q8—32no—
67C22×Q8—61no—
68C22×Q8C22×D311yescharacteristic
69C2×Q16—64no—
70C2×Q16—62no—
71C2×Q16—62no—
72C2×Q16—64no—
73C22×C8—32no—
74C22×C8—32no—
75C4⋊Q8—62no—
76C4⋊Q8—62no—
77C4⋊Q8—62no—
78C4⋊Q8—62no—
79C4×Q8—22no—
80C4×Q8—32no—
81C4×Q8—62no—
82C4×Q8D1212yes—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—61no—
87C2×C4⋊C4—61no—
88C2×C4⋊C4—61no—
89C2×C42—31no—
90C2×C42—32no—
91C2×C42—31no—
92C2×C42—32no—
93C2×C42S411yescharacteristic
94#107—64no—
95C4⋊C8—62no—
96C4⋊C8—62no—
97#103—68no—
98#103—64no—
99#103—62no—
100#103—62no—
101#95—61no—
102#95—64no—
103#95—64no—
104C23×C3—41no—
105C2×C3×C4—42no—
106C2×C3×C4—42no—
107C2×C3×C4—44no—
108C2×C3×C4—42no—
109C2×C3×C4—82no—
110C2×C3×C4—82no—
111C2×Dic3—82no—
112C2×Dic3—82no—
113C2×Dic3—84no—
114C2×Dic3—84no—
115SL(2,𝔽3)#12411yescharacteristic
116C2×Q8—31no—
117C2×Q8—32no—
118C2×Q8—62no—
119C2×Q8—64no—
120C2×Q8—62no—
121C2×Q8—64no—
122C2×Q8—122no—
123C2×Q8—124no—
124C2×Q8C2×D1211yescharacteristic
125C2×Q8#23512yes—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—32no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4C2×S412yes—
132C22×C4C2×S411yescharacteristic
133Q16—122no—
134Q16—124no—
135Q16—122no—
136C2×C8—62no—
137C2×C8—62no—
138C2×C8—64no—
139C2×C8—64no—
140C4⋊C4—32no—
141C4⋊C4—34no—
142C4⋊C4—61no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—122no—
147C4⋊C4—122no—
148C4⋊C4—122no—
149C42—32no—
150C42—32no—
151C42—62no—
152C42—62no—
153C42—62no—
154C42—122no—
155C42#22614yescomplemented
156C22×C3—41no—
157C22×C3—42no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—42no—
161C3×C4—44no—
162C3×C4—84no—
163Dic3—162no—
164Dic3—162no—
165C23C22×S411yescharacteristic
166Q8—31no—
167Q8—32no—
168Q8—61no—
169Q8—122no—
170Q8—122no—
171Q8—124no—
172Q8—244no—
173Q8#87511yescharacteristic
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—32no—
179C2×C4—32no—
180C2×C4—32no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—122no—
186C2×C4—122no—
187C2×C4—124no—
188C2×C4—124no—
189C2×C4C4⋊S412yes—
190C2×C4C2×#22614yes—
191C2×C4C2×#22612yes—
192C2×C4#98412yes—
193C8—122no—
194C8—122no—
195C2×C3—41no—
196C2×C3—41no—
197C2×C3—42no—
198C2×C3—42no—
199C2×C3—41no—
200C22C2×C4⋊S411yescharacteristic
201C22#591212yes—
202C22C22×#22611yescharacteristic
203C22C2×#98411yescharacteristic
204C22#592312yes—
205C4—24no—
206C4—31no—
207C4—31no—
208C4—32no—
209C4—62no—
210C4—122no—
211C4—242no—
212C4—242no—
213C4C4⋊#22614yes—
214C3—41noSylow
215C2#8529011yescharacteristic
216C2C2×C4⋊#22611yescharacteristic
217C2#8532212yes—
218C2#8533812yes—
219C2#8534711yescharacteristic
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

Complements

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

NH ≅ G/Ntype
155. C42#226semidirect

Maximal subgroups

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

subgroupindex
C2×#53912
C2×C4⋊#2262
C42×SL(2,𝔽3)2
#730612
#330273
#49364