Finite Group Databasebeta

← #11714116

Subgroups — #11714116

Order 768.

469 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 77 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#85283C212yesmaximal
2C4×C8×A4C211yesmaximal, characteristic
3#73111C214yesmaximal
4#33209—31nomaximal, Sylow
5C2×A4⋊Q8—22no—
6C2×C8×A4C2212yes—
7C42×A4C2211yescharacteristic
8C4⋊SL(2,ℤ/4)—22no—
9C4⋊SL(2,ℤ/4)C2214yes—
10#4682—41nomaximal
11C2×#500—31no—
12C22×C4×C8D311yescharacteristic
13#2857—32no—
14#1990—32no—
15#1942—34no—
16#1845—31no—
17#1808—34no—
18C2×C4×A4D411yescharacteristic
19C2×C4×A4D411yescharacteristic
20C2×C4×A4C2311yescharacteristic
21C2×SL(2,ℤ/4)—24no—
22A4⋊Q8—44no—
23#869—42no—
24C8×A4Q814yescomplemented
25C3×C4×C8—41no—
26#818—44no—
27C23×C8C2×D312yes—
28C2×C4⋊Q8—32no—
29C2×C22⋊Q8—62no—
30C22×C42C2×D311yescharacteristic
31#500—62no—
32#500—61no—
33#500—61no—
34C2×#107—34no—
35C2×#107—34no—
36C2×C4×C8—31no—
37C2×C4×C8—32no—
38C2×C4×C8—31no—
39#391—62no—
40#389—34no—
41#389—62no—
42#384—32no—
43#380—34no—
44#380—62no—
45#340—38no—
46C22×A4C2×D411yescharacteristic
47C2×Dic6—82no—
48C4×A4D812yes—
49C4×A4Q1612yes—
50C4×A4C2×Q812yes—
51SL(2,ℤ/4)—44no—
52C2×C3×C8—42no—
53C3×C42—41no—
54C4⋊Dic3—44no—
55C4⋊Dic3—82no—
56C22×Q8—62no—
57C23×C4D1211yescharacteristic
58C23×C4D1211yescharacteristic
59C23×C4C22×D311yescharacteristic
60C22×C8—32no—
61C22×C8—34no—
62C22×C8—32no—
63C22×C8Dic614yescomplemented
64C4⋊Q8—62no—
65C4⋊Q8—62no—
66C4⋊Q8—62no—
67C4⋊Q8—62no—
68C22⋊Q8—124no—
69C22⋊Q8—124no—
70C2×C4⋊C4—34no—
71C2×C4⋊C4—34no—
72C2×C4⋊C4—62no—
73C2×C4⋊C4—64no—
74C2×C4⋊C4—62no—
75C2×C4⋊C4—62no—
76C2×C22⋊C4—64no—
77C2×C42—31no—
78C2×C42—31no—
79C2×C42—31no—
80C2×C42—31no—
81#107—68no—
82#107—64no—
83#107—64no—
84C4×C8—32no—
85C4×C8—31no—
86C4×C8—61no—
87C4×C8S411yescharacteristic
88#95—64no—
89#95—64no—
90C2×A4C4⋊Q811yescharacteristic
91C2×A4C2×D811yescharacteristic
92C2×A4C2×Q1611yescharacteristic
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×Dic3—84no—
97Dic6—164no—
98C3×C8—44no—
99C24C2×D1211yescharacteristic
100C2×Q8—124no—
101C2×Q8—122no—
102C2×Q8—124no—
103C2×Q8—122no—
104C22×C4—31no—
105C22×C4—31no—
106C22×C4—31no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—32no—
111C22×C4—32no—
112C22×C4—32no—
113C22×C4—64no—
114C22×C4—64no—
115C22×C4D2412yes—
116C22×C4Dic1212yes—
117C22×C4C2×Dic612yes—
118C2×C8—32no—
119C2×C8—34no—
120C2×C8—34no—
121C2×C8—34no—
122C2×C8—62no—
123C2×C8C2×S412yes—
124C4⋊C4—68no—
125C4⋊C4—64no—
126C4⋊C4—64no—
127C4⋊C4—124no—
128C4⋊C4—124no—
129C4⋊C4—128no—
130C4⋊C4—122no—
131C4⋊C4—122no—
132C22⋊C4—124no—
133C22⋊C4—124no—
134C42—31no—
135C42—31no—
136C42—31no—
137C42—61no—
138C42C2×S411yescharacteristic
139C22×C3—41no—
140A4#50011yescharacteristic, complemented
141C3×C4—42no—
142C3×C4—42no—
143C3×C4—42no—
144Dic3—164no—
145C23—31no—
146C23—31no—
147C23—31no—
148C23—31no—
149C23#86911yescharacteristic
150C23C2×D2411yescharacteristic
151C23C2×Dic1211yescharacteristic
152Q8—244no—
153Q8—244no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—32no—
158C2×C4—32no—
159C2×C4—32no—
160C2×C4—32no—
161C2×C4—32no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—124no—
170C2×C4—124no—
171C2×C4—124no—
172C2×C4—124no—
173C2×C4—124no—
174C2×C4—124no—
175C2×C4C4⋊S411yescharacteristic
176C2×C4C4⋊S411yescharacteristic
177C2×C4C22×S411yescharacteristic
178C8—34no—
179C8A4⋊Q814yescomplemented
180C2×C3—41no—
181C2×C3—41no—
182C2×C3—41no—
183C22—31no—
184C22—31no—
185C22—31no—
186C22—31no—
187C22—31no—
188C22—31no—
189C22—31no—
190C22—31no—
191C22—31no—
192C22—61no—
193C22#468211yescharacteristic, complemented
194C22C2×C4⋊S411yescharacteristic
195C4—32no—
196C4—32no—
197C4—32no—
198C4—244no—
199C4—244no—
200C4#539812yes—
201C4#540212yes—
202C4C2×A4⋊Q812yes—
203C3—41noSylow
204C2—31no—
205C2—31no—
206C2—31no—
207C2—31no—
208C2#8528311yescharacteristic
209C2C2×#539811yescharacteristic
210C2C2×#540211yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radicalC22×A4C2×D4
hypercenterC4×C8S4
hypocenterA4#500
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
24. C8×A4Q8semidirect
63. C22×C8Dic6semidirect
140. A4#500semidirect
179. C8A4⋊Q8semidirect
193. C22#4682semidirect

Maximal subgroups

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

subgroupindex
#852832
C4×C8×A42
#731112
#332093
#46824