Finite Group Databasebeta

← #13316

Subgroups — #13316

Order 256.

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

normal subgroups only

This group has 302 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×#509C211yesmaximal, characteristic, complemented
2#3054C211yesmaximal, characteristic, complemented
3C2×#469C211yesmaximal, characteristic, complemented
4C2×#360C211yesmaximal, characteristic, complemented
5#2140C211yesmaximal, characteristic, complemented
6C2×#349C211yesmaximal, characteristic, complemented
7#1386C211yesmaximal, characteristic, complemented
8C2×#135C2211yescharacteristic
9C22×D8C2211yescharacteristic
10C22×M16C2211yescharacteristic
11#525—21no—
12C2×#121—21no—
13#509—22no—
14#509—22no—
15C2×SD32—21no—
16C2×D16—21no—
17C2×M32C2211yescharacteristic
18C2×M32C2211yescharacteristic
19#470—21no—
20#469—22no—
21#469—21no—
22#469—22no—
23#468—22no—
24#427C2211yescharacteristic
25C2×#106C2211yescharacteristic
26#416—21no—
27C2×#102—21no—
28#360—22no—
29#357—22no—
30#349—22no—
31C2×#41—21no—
32C22×D4—21no—
33C22×D4—41no—
34C23×C4D411yescharacteristic
35#135—21no—
36#135—21no—
37#135—21no—
38#135—21no—
39C2×Q16D411yescharacteristic, complemented
40C2×SD16—21no—
41C2×D8—21no—
42C2×D8—21no—
43C2×D8—21no—
44C2×D8—21no—
45C2×D8—21no—
46C2×D8D412yescomplemented
47C2×D8D411yescharacteristic, complemented
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—21no—
53C2×M16—21no—
54C22×C8D411yescharacteristic
55C22×C8C2311yescharacteristic
56#123—41no—
57C22⋊Q8—22no—
58#121—24no—
59#121—22no—
60#121—42no—
61#120—41no—
62C4×D4—42no—
63#117—21no—
64C2×C4⋊C4—21no—
65C2×C22⋊C4—41no—
66SD32—24no—
67D16—24no—
68M32—21no—
69M32—21no—
70M32—21no—
71M32D412yescomplemented
72C2×C16—21no—
73C2×C16—21no—
74#107D412yescomplemented
75#106—21no—
76#106—21no—
77#106D412yescomplemented
78#103—22no—
79#102—24no—
80#102—22no—
81C24—41no—
82C24D811yescharacteristic
83#41—41no—
84#41—41no—
85#41—41no—
86#41—41no—
87C2×Q8—21no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—42no—
91C2×D4—44no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—84no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4D811yescharacteristic
108C22×C4C2×D411yescharacteristic
109Q16—21no—
110Q16—21no—
111SD16—41no—
112SD16—41no—
113D8—21no—
114D8—22no—
115D8—22no—
116D8—21no—
117D8—41no—
118D8—41no—
119M16—41no—
120M16—41no—
121M16—41no—
122M16—41no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8C2×D411yescharacteristic
127C2×C8C2×D411yescharacteristic
128C2×C8C2×D411yescharacteristic
129C2×C8C2×D411yescharacteristic
130C2×C8C2×D411yescharacteristic
131C2×C8#4111yescharacteristic
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—41no—
136C22⋊C4—44no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—41no—
140C42—41no—
141C16—22no—
142C16—41no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—42no—
147C23—42no—
148C23—41no—
149C23—82no—
150C23—82no—
151C23—81no—
152C23—81no—
153C23C2×D811yescharacteristic
154Q8—41no—
155Q8—41no—
156D4—42no—
157D4—41no—
158D4—42no—
159D4—41no—
160D4—84no—
161D4—88no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4#12011yescharacteristic
180C2×C4#12011yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#13511yescharacteristic
183C2×C4AGL(1,ℤ/8)11yescharacteristic
184C2×C4AGL(1,ℤ/8)11yescharacteristic
185C8—41no—
186C8#12011yescharacteristic
187C8#12111yescharacteristic
188C8#12111yescharacteristic
189C8#12711yescharacteristic
190C22—22no—
191C22—42no—
192C22—42no—
193C22—41no—
194C22—81no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—81no—
199C22—82no—
200C22—162no—
201C22—162no—
202C22#44711yescharacteristic
203C22#44711yescharacteristic
204C22#46611yescharacteristic
205C4—41no—
206C4—82no—
207C4—82no—
208C4—81no—
209C4#39211yescharacteristic
210C4#44911yescharacteristic
211C4#44911yescharacteristic
212C4#46911yescharacteristic
213C2—21no—
214C2—21no—
215C2—81no—
216C2—161no—
217C2—162no—
218C2#199911yescharacteristic
219C2#219012yes—
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
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×#509C2semidirect
2. #3054C2semidirect
3. C2×#469C2semidirect
4. C2×#360C2semidirect
5. #2140C2semidirect
6. C2×#349C2semidirect
7. #1386C2semidirect
39. C2×Q16D4semidirect
46. C2×D8D4semidirect
47. C2×D8D4semidirect
71. M32D4semidirect
74. #107D4semidirect
77. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5092
#30542
C2×#4692
C2×#3602
#21402
C2×#3492
#13862