Finite Group Databasebeta

← #123287

Subgroups — #123287

Order 512.

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

normal subgroups only

This group has 284 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#19975C211yesmaximal, characteristic
2#12517C211yesmaximal, characteristic, complemented
3#12448C211yesmaximal, characteristic, complemented
4#12447C211yesmaximal, characteristic, complemented
5#7096C211yesmaximal, characteristic, complemented
6#7069C211yesmaximal, characteristic, complemented
7#7068C211yesmaximal, characteristic, complemented
8#3391—21no—
9#3287—21no—
10#2248—22no—
11#2227—21no—
12#2226—21no—
13#1713C2211yescharacteristic
14#1712C2211yescharacteristic
15#1668—21no—
16#1667—21no—
17#1621—22no—
18C8×Q16C2211yescharacteristic
19C8×D8C2211yescharacteristic
20#1370C2211yescharacteristic
21#1370C2211yescharacteristic
22#1365—21no—
23#1365—21no—
24C8×C16C2211yescharacteristic
25C2×#135—41no—
26#550—22no—
27C2×SD32—44no—
28#500—21no—
29#500—21no—
30#494—21no—
31#493—21no—
32#493—21no—
33#493—41no—
34#493D412yes—
35#466—41no—
36#460—22no—
37#459—22no—
38#459—42no—
39#449—42no—
40C8×Q8—21no—
41C4×Q16D411yescharacteristic
42C4×D8—41no—
43C4×D8D411yescharacteristic
44C8×D4—21no—
45#367—41no—
46#366—41no—
47C4⋊C16—21no—
48C4⋊C16—21no—
49#358—41no—
50#357—41no—
51C4×C16—21no—
52#335D411yescharacteristic
53#335D411yescharacteristic
54#331—22no—
55Q8⋊C8—21no—
56D4⋊C8—21no—
57C82C2311yescharacteristic
58C2×#41—42no—
59#135—82no—
60#135—82no—
61C2×Q16—42no—
62C2×Q16D811yescharacteristic
63C2×SD16—42no—
64C2×D8—42no—
65C2×D8—42no—
66C2×D8—41no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8D811yescharacteristic
71C22×C8—41no—
72C4⋊Q8—41no—
73#127—22no—
74#127—42no—
75#121—42no—
76#121—82no—
77C4×Q8—21no—
78C4×D4—21no—
79C4×D4—42no—
80SD32—88no—
81C2×C16—22no—
82C2×C16—41no—
83C2×C16—41no—
84#107—41no—
85#107—41no—
86#107—41no—
87#107—41no—
88#107D811yescharacteristic
89#107D811yescharacteristic
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8—41no—
93#103—21no—
94#102—21no—
95#102—42no—
96#102—42no—
97C22⋊C8—41no—
98C4×C8—41no—
99C4×C8C2×D411yescharacteristic
100C4×C8C2×D411yescharacteristic
101C4×C8C2×D411yescharacteristic
102#41—84no—
103#41—162no—
104C2×Q8—21no—
105C2×Q8—81no—
106C2×D4—21no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—82no—
111C2×D4—82no—
112C22×C4—41no—
113C22×C4—82no—
114Q16—21no—
115Q16—84no—
116Q16SD3212yescomplemented
117SD16—84no—
118D8—21no—
119D8—21no—
120D8—84no—
121D8—81no—
122D8—81no—
123D8—81no—
124D8—81no—
125D8—161no—
126D8—161no—
127D8—162no—
128C2×C8—21no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—81no—
132C2×C8—81no—
133C2×C8C2×D811yescharacteristic
134C2×C8C2×D811yescharacteristic
135C2×C8AGL(1,ℤ/8)11yescharacteristic
136C2×C8AGL(1,ℤ/8)11yescharacteristic
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—41no—
140C4⋊C4—81no—
141C4⋊C4—81no—
142C22⋊C4—41no—
143C22⋊C4—82no—
144C42—41no—
145C42#12011yescharacteristic
146C16—24no—
147C16—81no—
148C16—81no—
149C23—41no—
150C23—82no—
151Q8—22no—
152Q8—41no—
153Q8—162no—
154D4—41no—
155D4—41no—
156D4—82no—
157D4—82no—
158D4—82no—
159D4—82no—
160D4—162no—
161D4—164no—
162D4—322no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—164no—
169C2×C4—161no—
170C2×C4#44711yescharacteristic
171C2×C4#44711yescharacteristic
172C2×C4#45311yescharacteristic
173C8—22no—
174C8—21no—
175C8—21no—
176C8—81no—
177C8—81no—
178C8C2×SD3211yescharacteristic
179C8#50811yescharacteristic
180C8#50911yescharacteristic
181C8#51011yescharacteristic
182C22—81no—
183C22—81no—
184C22—81no—
185C22—162no—
186C22—162no—
187C22—162no—
188C22#161911yescharacteristic
189C4—21no—
190C4—42no—
191C4—81no—
192C4—81no—
193C4—321no—
194C4#218711yescharacteristic
195C4#218911yescharacteristic
196C4#219411yescharacteristic
197C4#219411yescharacteristic
198C2—161no—
199C2—322no—
200C2#1231011yescharacteristic
201C2#1232011yescharacteristic
202C2#1232111yescharacteristic
subgroupG/H
center, socleC22#1619
commutator, Frattini, cosocleC82C23
prime radical#19975C2
LCSG ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ G
LECSG ▷ C82 ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12517C2semidirect
3. #12448C2semidirect
4. #12447C2semidirect
5. #7096C2semidirect
6. #7069C2semidirect
7. #7068C2semidirect
116. Q16SD32semidirect

Maximal subgroups

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

subgroupindex
#199752
#125172
#124482
#124472
#70962
#70692
#70682