Finite Group Databasebeta

← #11714111

Subgroups — #11714111

Order 768.

449 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 75 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×#5386C211yesmaximal, characteristic
3C42×SL(2,𝔽3)C211yesmaximal, characteristic
4#73064C214yesmaximal
5#33202—31nomaximal, Sylow
6C2×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×#859C2212yes—
9C2×#859C2212yes—
10#5391—22no—
11#5386—22no—
12#4929—41nomaximal
13C2×#477—31no—
14C2×#475—31no—
15#2862—31no—
16C42×Q8D311yescharacteristic
17#1988—31no—
18#1987—31no—
19#1949—34no—
20#1842—31no—
21#1805—34no—
22C22×Dic6—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×C3×C42—41no—
25C2×C4⋊Dic3—41no—
26C4×SL(2,𝔽3)D412yes—
27C4×SL(2,𝔽3)Q814yescomplemented
28#859—22no—
29#859—22no—
30C3⋊#95—44no—
31C2×C4⋊Q8—31no—
32C2×C4⋊Q8—31no—
33C2×C4×Q8C2×D312yes—
34C2×C4×Q8C2×D311yescharacteristic
35#477—62no—
36#477—62no—
37#475—62no—
38#475—62no—
39C2×#107—32no—
40C2×#106—32no—
41C2×C4⋊C8—31no—
42C2×C4⋊C8—31no—
43C2×C4⋊C8—34no—
44C2×#103—32no—
45C2×#103—32no—
46#391—61no—
47#391—62no—
48#391—61no—
49#382—34no—
50C4×C4⋊C4—32no—
51C4×C4⋊C4—31no—
52C43—31no—
53#340—34no—
54#340—34no—
55C22×C3×C4—41no—
56C22×C3×C4—42no—
57C22×Dic3—42no—
58C22×Dic3—42no—
59C2×Dic6—44no—
60C2×Dic6—82no—
61C2×Dic6—82no—
62C2×SL(2,𝔽3)C2×D411yescharacteristic
63C2×SL(2,𝔽3)C2×Q812yes—
64C3×C42—82no—
65C4⋊Dic3—82no—
66C22×Q8—61no—
67C22×Q8—61no—
68C22×Q8C22×D311yescharacteristic
69C22×C8—32no—
70C22×C8—32no—
71C4⋊Q8—62no—
72C4⋊Q8—62no—
73C4⋊Q8—62no—
74C4⋊Q8—62no—
75C4×Q8—34no—
76C4×Q8—32no—
77C4×Q8Dic614yescomplemented
78C4×Q8D1212yes—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—61no—
84C2×C4⋊C4—61no—
85C2×C42—31no—
86C2×C42—32no—
87C2×C42—31no—
88C2×C42—32no—
89C2×C42S411yescharacteristic
90#107—64no—
91#106—64no—
92C4⋊C8—68no—
93C4⋊C8—62no—
94C4⋊C8—62no—
95#103—62no—
96#103—62no—
97#103—62no—
98#103—62no—
99#95—61no—
100#95—64no—
101#95—64no—
102C23×C3—41no—
103C2×C3×C4—44no—
104C2×C3×C4—42no—
105C2×C3×C4—42no—
106C2×C3×C4—42no—
107C2×C3×C4—82no—
108C2×C3×C4—82no—
109C2×Dic3—44no—
110C2×Dic3—82no—
111C2×Dic3—82no—
112C2×Dic3—82no—
113C2×Dic3—84no—
114Dic6—88no—
115SL(2,𝔽3)C4⋊Q811yescharacteristic
116C2×Q8—31no—
117C2×Q8—32no—
118C2×Q8—122no—
119C2×Q8—124no—
120C2×Q8—122no—
121C2×Q8—122no—
122C2×Q8—122no—
123C2×Q8C2×Dic612yes—
124C2×Q8C2×D1211yescharacteristic
125C22×C4—31no—
126C22×C4—32no—
127C22×C4—31no—
128C22×C4—62no—
129C22×C4—62no—
130C22×C4C2×S412yes—
131C22×C4C2×S411yescharacteristic
132C2×C8—62no—
133C2×C8—62no—
134C2×C8—64no—
135C2×C8—64no—
136C4⋊C4—32no—
137C4⋊C4—34no—
138C4⋊C4—62no—
139C4⋊C4—61no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C42—22no—
147C42—34no—
148C42—34no—
149C42—32no—
150C42—32no—
151C42—62no—
152C42—122no—
153C22×C3—42no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—42no—
158C3×C4—44no—
159C3×C4—84no—
160Dic3—84no—
161Dic3—162no—
162C23C22×S411yescharacteristic
163Q8—32no—
164Q8—31no—
165Q8—61no—
166Q8—244no—
167Q8—244no—
168Q8#86911yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—34no—
176C2×C4—32no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—122no—
181C2×C4—122no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—124no—
185C2×C4A4⋊Q814yes—
186C2×C4C4⋊S412yes—
187C2×C4#98312yes—
188C2×C4#98512yes—
189C8—122no—
190C8—122no—
191C2×C3—41no—
192C2×C3—42no—
193C2×C3—41no—
194C2×C3—42no—
195C2×C3—41no—
196C22C2×A4⋊Q812yes—
197C22C2×C4⋊S411yescharacteristic
198C22C2×#98311yescharacteristic
199C22C2×#98511yescharacteristic
200C22#592312yes—
201C4—24no—
202C4—31no—
203C4—31no—
204C4—32no—
205C4—64no—
206C4—62no—
207C4—242no—
208C4—242no—
209C4#539514yes—
210C3—41noSylow
211C2#8528311yescharacteristic
212C2#8531812yes—
213C2C2×#539511yescharacteristic
214C2#8533412yes—
215C2#8534711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)C4⋊Q8
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
27. C4×SL(2,𝔽3)Q8semidirect
77. C4×Q8Dic6semidirect

Maximal subgroups

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

subgroupindex
C2×#53912
C2×#53862
C42×SL(2,𝔽3)2
#730642
#332023
#49294