Finite Group Databasebeta

← #150721

Subgroups — #150721

Order 512.

224 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 52 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#23833C211yesmaximal, characteristic
2#11539C211yesmaximal, characteristic, complemented
3#8631C211yesmaximal, characteristic, complemented
4#7441C212yesmaximal, complemented
5#7415C212yesmaximal
6#2960C2211yescharacteristic
7#2955C411yescharacteristic
8#2166—21no—
9#2103C411yescharacteristic
10#2102C2211yescharacteristic
11#2080—21no—
12#2053C411yescharacteristic
13#2050C2211yescharacteristic
14#2031—21no—
15#1764—21no—
16#1757—21no—
17#1747C411yescharacteristic
18#1568—21no—
19#1537—21no—
20#1328—21no—
21#1327C2212yes—
22#1307C2212yes—
23#575C2×C411yescharacteristic
24C2×#131—21no—
25#533D411yescharacteristic
26#446—41no—
27C8×Q8—41no—
28#444—41no—
29#443—41no—
30#432—22no—
31#430C2×C411yescharacteristic
32#424C2×C411yescharacteristic
33#424D411yescharacteristic
34#423C2311yescharacteristic
35#421—21no—
36C2×#104—21no—
37#416—21no—
38#413—21no—
39#410—21no—
40#405—21no—
41#405C2×C411yescharacteristic
42#405D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#364—41no—
45#349—22no—
46#347—41no—
47#344—21no—
48#344—21no—
49#341—21no—
50#339—21no—
51#338C2×C411yescharacteristic
52H(3,ℤ/4)—21no—
53H(3,ℤ/4)C2×C411yescharacteristic
54#335—41no—
55#334—41no—
56#330—42no—
57C2×#41—21no—
58#137—41no—
59AGL(1,ℤ/8)—41no—
60#135—41no—
61#131—41no—
62#131—41no—
63#131—41no—
64#131—41no—
65C2×M16—21no—
66C2×M16—21no—
67C2×M16—41no—
68C2×M16C22×C411yescharacteristic
69C2×M16C2×D411yescharacteristic
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C4⋊Q8—21no—
74#125—21no—
75C22⋊Q8—41no—
76C4×Q8—41no—
77#117—21no—
78#117C22⋊C411yescharacteristic
79#117C22⋊C411yescharacteristic
80C2×C42C2×D411yescharacteristic
81M32—42no—
82#108—41no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—42no—
87#104—42no—
88#104—41no—
89#104—41no—
90#103—41no—
91#102—41no—
92#101—41no—
93#100—41no—
94#99—42no—
95C22⋊C8—42no—
96#97—21no—
97#97—21no—
98#97—41no—
99#97—42no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—41no—
105#41—41no—
106#41—41no—
107#41—41no—
108#41—41no—
109C2×Q8—21no—
110C2×Q8—41no—
111C2×D4—21no—
112C2×D4—41no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4C2×C22⋊C411yescharacteristic
116Q16—41no—
117SD16—42no—
118D8—41no—
119M16—41no—
120M16—41no—
121M16—41no—
122M16—41no—
123M16—41no—
124M16—41no—
125M16—81no—
126M16—81no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—81no—
136C2×C8—81no—
137C2×C8—82no—
138C2×C8#9911yescharacteristic
139C2×C8#9911yescharacteristic
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—42no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—41no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—41no—
152C42AGL(1,ℤ/8)11yescharacteristic
153C42#13711yescharacteristic
154C16—82no—
155C23—41no—
156C23#42111yescharacteristic
157Q8—41no—
158Q8—41no—
159Q8—82no—
160D4—41no—
161D4—41no—
162D4—81no—
163D4—81no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4C2×#9911yescharacteristic
176C2×C4#41711yescharacteristic
177C8—21no—
178C8—41no—
179C8—41no—
180C8—41no—
181C8—81no—
182C8—81no—
183C8—162no—
184C8#35511yescharacteristic
185C8#35511yescharacteristic
186C22—21no—
187C22—41no—
188C22—81no—
189C22—161no—
190C22#151311yescharacteristic
191C4—21no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—81no—
196C4—81no—
197C4—161no—
198C4#212011yescharacteristic
199C4C2×#35511yescharacteristic
200C2—21no—
201C2—41no—
202C2—161no—
203C2#1182911yescharacteristic
subgroupG/H
center, socleC2#11829
commutatorC2×M16C22×C4
Frattini, cosocle#423C23
prime radical#575C2×C4
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2960 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11539C2semidirect
3. #8631C2semidirect
4. #7441C2semidirect

Maximal subgroups

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

subgroupindex
#238332
#115392
#86312
#74412
#74152