Finite Group Databasebeta

← #10008

Subgroups — #10008

Order 256.

327 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 51 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#3215C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic
4#1986C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1490C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×C22⋊Q8—21no—
11C2×#121—21no—
12C2×C4×D4C2211yescharacteristic
13#497—21no—
14#489—21no—
15#484—21no—
16#474—21no—
17#464—22no—
18#461—21no—
19#451—21no—
20#450—21no—
21#450—21no—
22#440—22no—
23#432C2211yescharacteristic
24#432C2211yescharacteristic
25C4×M16C2211yescharacteristic
26#399—21no—
27#392—21no—
28#390—22no—
29C4×C4⋊C4C2211yescharacteristic
30#339—22no—
31#329—22no—
32D4⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×D4—22no—
35C22×D4D411yescharacteristic
36C22×D4D411yescharacteristic
37C23×C4—21no—
38C2×Q16—22no—
39C2×SD16—22no—
40C2×SD16—22no—
41C2×M16—21no—
42C4⋊Q8D411yescharacteristic, complemented
43C4⋊Q8D411yescharacteristic, complemented
44#125—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48#124—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8—44no—
52#121—44no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59#117—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D411yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42D412yes—
69C2×C42C2311yescharacteristic
70#107—22no—
71C4⋊C8—21no—
72C4⋊C8D412yescomplemented
73#103—22no—
74#103—22no—
75#102—22no—
76#102—22no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79#97—21no—
80#97—21no—
81#97D412yescomplemented
82C4×C8—21no—
83#95—21no—
84C24—21no—
85C24—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—42no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—44no—
97C2×D4—42no—
98C2×D4—44no—
99C2×D4—82no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110Q16—82no—
111SD16—44no—
112SD16—82no—
113M16—22no—
114M16—41no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—41no—
118C4⋊C4—24no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—44no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C42—42no—
138C42—41no—
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C42C2×D411yescharacteristic
142C42#4111yescharacteristic
143C23—22no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23#12011yescharacteristic
156Q8—42no—
157Q8—42no—
158Q8—42no—
159Q8—81no—
160D4—22no—
161D4—22no—
162D4—41no—
163D4—41no—
164D4—42no—
165D4—81no—
166D4—88no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#12011yescharacteristic
184C2×C4#12011yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12111yescharacteristic
187C2×C4#12111yescharacteristic
188C2×C4#12711yescharacteristic
189C8—42no—
190C8—42no—
191C8—42no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—81no—
198C22—81no—
199C22—82no—
200C22—82no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22#39211yescharacteristic
207C22#45511yescharacteristic
208C22#45611yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4#45312yes—
220C2—21no—
221C2—21no—
222C2—82no—
223C2—161no—
224C2#200211yescharacteristic
225C2#200711yescharacteristic
226C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#2540C2
LCSG ▷ C2×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
1. #3215C2semidirect
2. #3110C2semidirect
4. #1986C2semidirect
5. #1527C2semidirect
6. #1490C2semidirect
7. #1285C2semidirect
42. C4⋊Q8D4semidirect
43. C4⋊Q8D4semidirect
72. C4⋊C8D4semidirect
81. #97D4semidirect

Maximal subgroups

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

subgroupindex
#32152
#31102
#25402
#19862
#15272
#14902
#12852