Finite Group Databasebeta

← #8496

Subgroups — #8496

Order 256.

338 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 71 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#3550C211yesmaximal, characteristic
2#1995C211yesmaximal, characteristic, complemented
3#1559C211yesmaximal, characteristic, complemented
4#1557C211yesmaximal, characteristic
5#1537C211yesmaximal, characteristic, complemented
6#1506C211yesmaximal, characteristic, complemented
7#1505C211yesmaximal, characteristic
8#563—22no—
9#559—21no—
10Q82—21no—
11#556—21no—
12#556C412yescomplemented
13D4×Q8—21no—
14D4×Q8C412yescomplemented
15#542—22no—
16#542—22no—
17#536—22no—
18#533C2211yescharacteristic
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#423C2211yescharacteristic
24#376C2211yescharacteristic
25#339—22no—
26#333—21no—
27#332—21no—
28#332—21no—
29#331—21no—
30#330—22no—
31#328—22no—
32#327—22no—
33C2×#41—21no—
34C22×Q8—22no—
35C2×M16—21no—
36C4⋊Q8—24no—
37C4⋊Q8—22no—
38C4⋊Q8—22no—
39C4⋊Q8C2×C411yescharacteristic
40C4⋊Q8C2×C411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#126—24no—
46#126—24no—
47#125—21no—
48#125D412yes—
49#124—24no—
50#124—22no—
51#124—22no—
52#124—21no—
53#123—24no—
54#123—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—21no—
58C22⋊Q8—21no—
59C22⋊Q8—21no—
60C22⋊Q8—22no—
61C22⋊Q8—24no—
62C22⋊Q8D412yes—
63#121—21no—
64#121D412yes—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8C2×C412yes—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—21no—
74C4×D4C2×C412yes—
75#117—21no—
76#117—22no—
77#117D412yes—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C22⋊C8—21no—
85C22⋊C8—21no—
86#97—21no—
87#97—21no—
88#95—21no—
89#95—21no—
90#41—82no—
91C2×Q8—24no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—84no—
97C2×Q8C22⋊C412yes—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4C22⋊C412yes—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108M16—42no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4C22⋊C412yes—
125C4⋊C4C22⋊C412yes—
126C4⋊C4C22⋊C412yes—
127C4⋊C4C22⋊C412yes—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—24no—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C42—24no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—41no—
140C42C22×C411yescharacteristic
141C42C2×D411yescharacteristic
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C23—22no—
145C23#12011yescharacteristic
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—41no—
150Q8—41no—
151Q8—81no—
152Q8—84no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—81no—
157D4—82no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4C2×C22⋊C411yescharacteristic
173C2×C4C2×C22⋊C411yescharacteristic
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4#12011yescharacteristic
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C8—81no—
179C8—81no—
180C8—81no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22#37911yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—42no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4#45411yescharacteristic
200C4#45611yescharacteristic
201C2—41no—
202C2—82no—
203C2#179311yescharacteristic
204C2#179511yescharacteristic
205C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1995C2semidirect
3. #1559C2semidirect
5. #1537C2semidirect
6. #1506C2semidirect
12. #556C4semidirect
14. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#35502
#19952
#15592
#15572
#15372
#15062
#15052