Finite Group Databasebeta

← #13317

Subgroups — #13317

Order 256.

276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#510C211yesmaximal, characteristic, complemented
2#3054C211yesmaximal, characteristic
3C2×#470C211yesmaximal, characteristic, complemented
4C2×#360C211yesmaximal, characteristic, complemented
5#2141C211yesmaximal, characteristic, complemented
6C2×#349C211yesmaximal, characteristic, complemented
7#1386C211yesmaximal, characteristic, complemented
8C2×#135C2211yescharacteristic
9C22×Q16C2211yescharacteristic
10C22×M16C2211yescharacteristic
11#525—21no—
12C2×C22⋊Q8—21no—
13#510—22no—
14#510—22no—
15C2×Q32—21no—
16C2×SD32—21no—
17C2×M32C2211yescharacteristic
18C2×M32C2211yescharacteristic
19#470—21no—
20#470—22no—
21#470—22no—
22#469—21no—
23#468—22no—
24#427C2211yescharacteristic
25C2×#106C2211yescharacteristic
26#416—21no—
27C2×#103—21no—
28#360—22no—
29#358—22no—
30#349—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C23×C4D411yescharacteristic
34#135—21no—
35#135—21no—
36#135—21no—
37#135—21no—
38C2×Q16—21no—
39C2×Q16—21no—
40C2×Q16—21no—
41C2×Q16—21no—
42C2×Q16—21no—
43C2×Q16D411yescharacteristic
44C2×Q16D412yescomplemented
45C2×SD16—21no—
46C2×D8D411yescharacteristic
47C2×M16—21no—
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—21no—
53C22×C8D411yescharacteristic
54C22×C8C2311yescharacteristic
55#123—41no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—42no—
59#121—22no—
60#120—41no—
61C4×D4—42no—
62#117—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—41no—
65C2×C22⋊C4—41no—
66Q32—24no—
67SD32—24no—
68M32—21no—
69M32—21no—
70M32—21no—
71M32D412yescomplemented
72C2×C16—21no—
73C2×C16—21no—
74#107D412yes—
75#106—21no—
76#106—21no—
77#106D412yescomplemented
78#103—24no—
79#103—22no—
80#102—22no—
81C24D811yescharacteristic
82#41—41no—
83#41—41no—
84#41—41no—
85#41—41no—
86C2×Q8—21no—
87C2×Q8—22no—
88C2×Q8—41no—
89C2×Q8—41no—
90C2×Q8—41no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×D4—21no—
94C2×D4—42no—
95C2×D4—41no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4D811yescharacteristic
106C22×C4C2×D411yescharacteristic
107Q16—22no—
108Q16—21no—
109Q16—22no—
110Q16—21no—
111Q16—41no—
112Q16—41no—
113SD16—41no—
114SD16—41no—
115D8—21no—
116D8—21no—
117M16—41no—
118M16—41no—
119M16—41no—
120M16—41no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C2×D411yescharacteristic
125C2×C8C2×D411yescharacteristic
126C2×C8C2×D411yescharacteristic
127C2×C8C2×D411yescharacteristic
128C2×C8C2×D411yescharacteristic
129C2×C8#4111yescharacteristic
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4—41no—
135C22⋊C4—42no—
136C22⋊C4—44no—
137C22⋊C4—42no—
138C22⋊C4—41no—
139C42—41no—
140C16—22no—
141C16—41no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—42no—
146C23—41no—
147C23C2×D811yescharacteristic
148Q8—41no—
149Q8—42no—
150Q8—42no—
151Q8—41no—
152Q8—81no—
153Q8—81no—
154D4—41no—
155D4—41no—
156D4—84no—
157D4—81no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4#12011yescharacteristic
176C2×C4#12011yescharacteristic
177C2×C4#12111yescharacteristic
178C2×C4#13511yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4AGL(1,ℤ/8)11yescharacteristic
181C8—41no—
182C8#12011yescharacteristic
183C8#12111yescharacteristic
184C8#12111yescharacteristic
185C8#12711yescharacteristic
186C22—22no—
187C22—42no—
188C22—42no—
189C22—41no—
190C22—81no—
191C22—82no—
192C22—81no—
193C22#44711yescharacteristic
194C22#44711yescharacteristic
195C22#46611yescharacteristic
196C4—41no—
197C4—82no—
198C4—82no—
199C4—81no—
200C4—82no—
201C4#39211yescharacteristic
202C4#44911yescharacteristic
203C4#44911yescharacteristic
204C4#46911yescharacteristic
205C2—21no—
206C2—21no—
207C2—81no—
208C2—161no—
209C2#199911yescharacteristic
210C2#219112yes—
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#510C2semidirect
3. C2×#470C2semidirect
4. C2×#360C2semidirect
5. #2141C2semidirect
6. C2×#349C2semidirect
7. #1386C2semidirect
44. C2×Q16D4semidirect
71. M32D4semidirect
77. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5102
#30542
C2×#4702
C2×#3602
#21412
C2×#3492
#13862