Finite Group Databasebeta

← #13308

Subgroups — #13308

Order 256.

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

normal subgroups only

This group has 266 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
2C2×#473C211yesmaximal, characteristic, complemented
3#3061C211yesmaximal, characteristic
4C2×#359C211yesmaximal, characteristic, complemented
5C2×#358C211yesmaximal, characteristic, complemented
6C2×#350C211yesmaximal, characteristic, complemented
7#1387C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9C2×#135C2211yescharacteristic
10C22×Q16C2211yescharacteristic
11C2×#131C2211yescharacteristic
12#537—21no—
13#510—22no—
14#510—22no—
15C2×Q32—21no—
16C2×SD32—21no—
17C2×M32C2211yescharacteristic
18C22×C16C2211yescharacteristic
19#473—22no—
20#473—22no—
21#468—22no—
22#467—21no—
23#466—21no—
24C2×#108C2211yescharacteristic
25C2×#107C2211yescharacteristic
26#417—21no—
27C2×#101—21no—
28#359—22no—
29#358—22no—
30#350—22no—
31C2×#41—21no—
32C2×#41D411yescharacteristic
33C22×Q8—21no—
34#137—24no—
35#137—42no—
36#135—21no—
37#135—21no—
38#135—21no—
39#135—21no—
40C2×Q16—21no—
41C2×Q16—21no—
42C2×Q16—21no—
43C2×Q16—21no—
44C2×Q16—21no—
45C2×Q16—41no—
46C2×Q16D411yescharacteristic
47C2×Q16D412yescomplemented
48C2×SD16—21no—
49C2×SD16—41no—
50C2×D8D411yescharacteristic
51#131—21no—
52#131—21no—
53#131—21no—
54#131—21no—
55C2×M16—21no—
56C2×M16—21no—
57C2×M16D411yescharacteristic
58C22×C8—21no—
59C22×C8C2311yescharacteristic
60C22⋊Q8—22no—
61C22⋊Q8—41no—
62#121—22no—
63#121—42no—
64#121—41no—
65C2×C4⋊C4—21no—
66Q32—24no—
67SD32—24no—
68M32—21no—
69M32D412yescomplemented
70C2×C16—21no—
71C2×C16—21no—
72C2×C16—21no—
73C2×C16—21no—
74#108—21no—
75#108D412yescomplemented
76#107—21no—
77#107D412yes—
78#103—22no—
79#102—22no—
80#101—24no—
81#41—21no—
82#41—21no—
83#41—21no—
84#41—21no—
85#41—41no—
86#41—41no—
87#41—41no—
88#41—41no—
89C2×Q8—22no—
90C2×Q8—21no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8D811yescharacteristic
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—42no—
101C2×D4D811yescharacteristic
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C2×D411yescharacteristic
107Q16—22no—
108Q16—21no—
109Q16—22no—
110Q16—21no—
111Q16—44no—
112Q16—41no—
113Q16—41no—
114SD16—44no—
115SD16—41no—
116SD16—41no—
117D8—21no—
118D8—21no—
119M16—21no—
120M16—22no—
121M16—21no—
122M16—41no—
123M16—41no—
124M16—41no—
125C2×C8—21no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8C2×D411yescharacteristic
130C2×C8C2×D411yescharacteristic
131C2×C8C2×D411yescharacteristic
132C2×C8C2×D411yescharacteristic
133C2×C8C2×D411yescharacteristic
134C2×C8#4111yescharacteristic
135C4⋊C4—22no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C16—22no—
141C16—41no—
142C23—21no—
143C23—41no—
144C23AGL(1,ℤ/8)11yescharacteristic
145Q8—21no—
146Q8—21no—
147Q8—41no—
148Q8—42no—
149Q8—42no—
150Q8—41no—
151Q8—81no—
152Q8—81no—
153D4—21no—
154D4—21no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—81no—
160D4—81no—
161D4—162no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—161no—
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4C2×D811yescharacteristic
177C2×C4#13511yescharacteristic
178C2×C4AGL(1,ℤ/8)11yescharacteristic
179C8—41no—
180C8—42no—
181C8#12011yescharacteristic
182C8#12111yescharacteristic
183C8#12111yescharacteristic
184C8#12711yescharacteristic
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#44711yescharacteristic
194C22#44911yescharacteristic
195C22#46911yescharacteristic
196C4—41no—
197C4—82no—
198C4—81no—
199C4—82no—
200C4#39211yescharacteristic
201C4#44711yescharacteristic
202C4#44911yescharacteristic
203C4#46611yescharacteristic
204C2—21no—
205C2—21no—
206C2—81no—
207C2—161no—
208C2#199911yescharacteristic
209C2#219411yescharacteristic
210C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3061C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#510C2semidirect
2. C2×#473C2semidirect
4. C2×#359C2semidirect
5. C2×#358C2semidirect
6. C2×#350C2semidirect
7. #1387C2semidirect
47. C2×Q16D4semidirect
69. M32D4semidirect
75. #108D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5102
C2×#4732
#30612
C2×#3592
C2×#3582
C2×#3502
#13872