Finite Group Databasebeta

← #9375

Subgroups — #9375

Order 256.

209 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 61 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#3238C211yesmaximal, characteristic
2#2992C211yesmaximal, characteristic
3#1988C211yesmaximal, characteristic, complemented
4#1558C211yesmaximal, characteristic, complemented
5#1509C211yesmaximal, characteristic
6#1290C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9#529C2211yescharacteristic
10#501C412yescomplemented
11#489C412yescomplemented
12#483—21no—
13#480—21no—
14#477—21no—
15#476—21no—
16#446—21no—
17#435—21no—
18#435—21no—
19C8×D4—21no—
20#433—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#424—21no—
25C2×C4⋊C8C2211yescharacteristic
26C4×C4⋊C4C2211yescharacteristic
27#340—21no—
28#340—21no—
29#333—21no—
30#333—21no—
31#328—21no—
32#327—21no—
33C22×Q8—21no—
34C2×M16—21no—
35C22×C8—21no—
36C22×C8D411yescharacteristic
37C22×C8D411yescharacteristic
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8C2×C412yes—
42#126—21no—
43#125C2×C411yescharacteristic
44#124C2×C411yescharacteristic
45#123—21no—
46C22⋊Q8D411yescharacteristic
47#121D411yescharacteristic
48C4×Q8D411yescharacteristic
49C4×D4—21no—
50C4×D4D411yescharacteristic
51#117—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#107—22no—
58#106—22no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D411yescharacteristic
66C4⋊C8D411yescharacteristic
67#103—22no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74#97—21no—
75#97—21no—
76#97—21no—
77#97C2×C412yes—
78C4×C8—21no—
79#95—21no—
80C2×Q8—22no—
81C2×Q8—41no—
82C2×Q8—81no—
83C2×Q8—81no—
84C2×Q8C22⋊C411yescharacteristic
85C2×D4—21no—
86C2×D4C22⋊C411yescharacteristic
87C22×C4—21no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4#4111yescharacteristic
93M16—81no—
94C2×C8—22no—
95C2×C8—21no—
96C2×C8—21no—
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—81no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4C22⋊C411yescharacteristic
114C4⋊C4C22⋊C411yescharacteristic
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—21no—
118C22⋊C4—21no—
119C22⋊C4—21no—
120C42—21no—
121C42—81no—
122C42C22×C411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42#4111yescharacteristic
126C23—21no—
127C23#12111yescharacteristic
128Q8—41no—
129Q8—82no—
130Q8—82no—
131D4—41no—
132D4—41no—
133D4—41no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4C2×C22⋊C411yescharacteristic
151C2×C4C4×D411yescharacteristic
152C2×C4C4×D411yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12311yescharacteristic
156C8—42no—
157C8—41no—
158C8—41no—
159C8—41no—
160C8—41no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22#38611yescharacteristic
168C22#47111yescharacteristic
169C22#47211yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—82no—
180C4#45511yescharacteristic
181C4#45611yescharacteristic
182C2—21no—
183C2—21no—
184C2—81no—
185C2#188211yescharacteristic
186C2#190711yescharacteristic
187C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#121D4
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
3. #1988C2semidirect
4. #1558C2semidirect
6. #1290C2semidirect
7. #1285C2semidirect
10. #501C4semidirect
11. #489C4semidirect

Maximal subgroups

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

subgroupindex
#32382
#29922
#19882
#15582
#15092
#12902
#12852