Finite Group Databasebeta

← #25691

Subgroups — #25691

Order 256.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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#3509C211yesmaximal, characteristic
2#3166C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic
4#2642C211yesmaximal, characteristic, complemented
5#2038C214yesmaximal
6#1995C211yesmaximal, characteristic, complemented
7#1966C211yesmaximal, characteristic, complemented
8#1857C214yesmaximal, complemented
9#1759C211yesmaximal, characteristic
10#557—22no—
11#556—22no—
12#552—24no—
13#541—22no—
14#536C2212yes—
15#533—22no—
16#533C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19#519C2212yes—
20C4×#41C2211yescharacteristic
21#501—22no—
22#499—22no—
23#479—22no—
24#478—22no—
25#428C2212yes—
26#428C2214yes—
27#423C2211yescharacteristic
28C2×#104C2212yes—
29#417C2212yes—
30C2×#101C2214yes—
31C4×M16C2211yescharacteristic
32#400—24no—
33#391C2214yes—
34#376C2211yescharacteristic
35#339C2212yes—
36#339C2214yes—
37C2×#41C2311yescharacteristic
38C22×Q8C2312yes—
39C2×M16C2312yes—
40C2×M16C2312yes—
41C4⋊Q8—22no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—22no—
45C4⋊Q8D412yes—
46#126—28no—
47#125—24no—
48#125—28no—
49#125—22no—
50#125—21no—
51#125—21no—
52#125D412yes—
53#124—22no—
54#123—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C4×Q8—24no—
59C4×Q8—24no—
60C4×Q8—24no—
61C4×Q8—22no—
62C4×Q8D412yes—
63C4×D4—22no—
64C4×D4D412yes—
65#117—24no—
66#117—24no—
67#117—21no—
68#117C2311yescharacteristic
69#117C2312yes—
70C2×C4⋊C4—22no—
71C2×C4⋊C4D414yes—
72C2×C4⋊C4C2312yes—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—22no—
77#107—24no—
78#106—22no—
79#106—24no—
80C4⋊C8—22no—
81#104—24no—
82#103—22no—
83#102—22no—
84#101—28no—
85#97—21no—
86C4×C8—21no—
87#95—22no—
88#95—28no—
89#41—22no—
90#41—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—24no—
96C2×Q8C2×D411yescharacteristic
97C2×D4—21no—
98C2×D4C2×D411yescharacteristic
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D414yes—
104C22×C4C2×D412yes—
105C22×C4C2411yescharacteristic
106M16—22no—
107M16—24no—
108M16#4114yes—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—28no—
112C4⋊C4—28no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4C2×D412yes—
123C4⋊C4C2×D414yes—
124C22⋊C4—28no—
125C22⋊C4—21no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C42—28no—
129C42—22no—
130C42—21no—
131C42—22no—
132C42—22no—
133C42—22no—
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D412yes—
137C23—21no—
138C23C22×D411yescharacteristic
139Q8—21no—
140Q8—21no—
141Q8—44no—
142Q8—42no—
143Q8—42no—
144Q8—44no—
145D4—21no—
146D4—21no—
147D4—42no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—28no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—44no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4#12012yes—
161C2×C4#12012yes—
162C2×C4C22×D411yescharacteristic
163C2×C4C22×D411yescharacteristic
164C2×C4C2×#4112yes—
165C2×C42−1+412yes—
166C8—24no—
167C8—42no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22C2×#12011yescharacteristic
172C22#54712yes—
173C4—22no—
174C4—22no—
175C4—44no—
176C4—41no—
177C4—42no—
178C4—44no—
179C4—82no—
180C4—82no—
181C4#53211yescharacteristic
182C4#53611yescharacteristic
183C4#54812yes—
184C2—22no—
185C2—81no—
186C2#243211yescharacteristic
187C2#302012yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3166C2semidirect
4. #2642C2semidirect
6. #1995C2semidirect
7. #1966C2semidirect
8. #1857C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#31662
#30802
#26422
#20382
#19952
#19662
#18572
#17592