Finite Group Databasebeta

← #134012

Subgroups — #134012

Order 512.

1112 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2658C211yesmaximal, characteristic
2#12694C214yesmaximal
3#8544C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic, complemented
5C22×C4⋊Q8C412yescomplemented
6C22×#124C2212yes—
7C22×#124C2211yescharacteristic
8#2658—28no—
9C2×#399—24no—
10C2×#376C411yescharacteristic, complemented
11C2×#354—28no—
12#1819—24no—
13#1781C2214yes—
14C23×Q8C2×C412yes—
15C23×Q8C2×C411yescharacteristic
16C23×D4C2311yescharacteristic
17C2×C4⋊Q8—24no—
18C2×C4⋊Q8—24no—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—48no—
21C2×C4⋊Q8—44no—
22C2×#124—24no—
23C2×#124—24no—
24C2×#124—44no—
25C2×#124—48no—
26C2×#124D418yes—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—22no—
29C22×C22⋊C4—24no—
30C22×C22⋊C4D414yes—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33C2×#99—28no—
34#399—232no—
35#379—24no—
36#376—24no—
37C2×#95—22no—
38#354—416no—
39#342—48no—
40C25C2×D412yes—
41C22×Q8—22no—
42C22×Q8—22no—
43C22×Q8—42no—
44C22×Q8—44no—
45C22×Q8—44no—
46C22×Q8—44no—
47C22×Q8—44no—
48C22×Q8C22⋊C414yes—
49C22×D4—44no—
50C22×D4—42no—
51C22×D4C2×D414yes—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4C22⋊C412yes—
55C23×C4C22⋊C412yes—
56C23×C4C22×C411yescharacteristic
57C4⋊Q8—416no—
58C4⋊Q8—416no—
59C4⋊Q8—48no—
60C4⋊Q8—48no—
61#124—216no—
62#124—48no—
63#124—416no—
64#124—416no—
65C2×C4⋊C4—28no—
66C2×C4⋊C4—216no—
67C2×C4⋊C4—416no—
68C2×C4⋊C4—44no—
69C2×C4⋊C4—44no—
70C2×C22⋊C4—216no—
71C2×C22⋊C4—28no—
72C2×C22⋊C4—28no—
73C2×C22⋊C4—44no—
74C2×C22⋊C4—416no—
75C2×C22⋊C4—48no—
76C2×C22⋊C4—44no—
77C2×C42—24no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—28no—
81C2×C42C22⋊C414yes—
82#99—416no—
83#95—28no—
84C24—24no—
85C24—44no—
86C24—48no—
87C24C2×C22⋊C411yescharacteristic
88C24#12014yes—
89C2×Q8—28no—
90C2×Q8—44no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—816no—
99C2×Q8—88no—
100C2×D4—28no—
101C2×D4—44no—
102C2×D4—44no—
103C2×D4—88no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—28no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—48no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—84no—
120C22×C4#9914yes—
121C22×C4C2×C22⋊C412yes—
122C4⋊C4—432no—
123C4⋊C4—416no—
124C22⋊C4—416no—
125C22⋊C4—432no—
126C22⋊C4—88no—
127C22⋊C4—816no—
128C42—28no—
129C42—28no—
130C42—24no—
131C42#9918yes—
132C23—24no—
133C23—22no—
134C23—24no—
135C23—28no—
136C23—48no—
137C23—48no—
138C23—84no—
139C23—84no—
140C23—816no—
141C23#37911yescharacteristic
142C23C2×#9912yes—
143Q8—88no—
144Q8—88no—
145Q8—88no—
146D4—88no—
147C2×C4—28no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—28no—
151C2×C4—24no—
152C2×C4—416no—
153C2×C4—48no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—88no—
163C2×C4—84no—
164C2×C4—164no—
165C2×C4—168no—
166C2×C4C2×#9914yes—
167C22—28no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—44no—
172C22—48no—
173C22—88no—
174C22—168no—
175C22#178112yes—
176C22#178111yescharacteristic
177C22#212914yes—
178C4—24no—
179C4—48no—
180C4—44no—
181C4—84no—
182C4—82no—
183C4—324no—
184C2—22no—
185C2—24no—
186C2—84no—
187C2#854611yescharacteristic
188C2#1269812yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Q8C4semidirect
5. C22×C4⋊Q8C4semidirect
10. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26582
#126942
#85442