Finite Group Databasebeta

← #28184

Subgroups — #28184

Order 256.

382 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 113 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×#473C211yesmaximal, characteristic, complemented
2#3059C211yesmaximal, characteristic, complemented
3#2955C211yesmaximal, characteristic
4#2950C211yesmaximal, characteristic, complemented
5#2082C212yesmaximal, complemented
6#2009C212yesmaximal
7#1912C212yesmaximal, complemented
8#1905C212yesmaximal, complemented
9#1833C211yesmaximal, characteristic, complemented
10#1814C211yesmaximal, characteristic, complemented
11#1812C211yesmaximal, characteristic, complemented
12#575C2211yescharacteristic
13C2×#137C2212yes—
14C22×Q16C2211yescharacteristic
15C2×#131C2211yescharacteristic
16C2×#131C2211yescharacteristic
17#536C2212yes—
18#519C2212yes—
19#473—22no—
20#473—22no—
21#470—22no—
22#467—22no—
23#444—22no—
24#443—22no—
25#441—22no—
26C4×Q16—22no—
27#430C2211yescharacteristic
28C2×#108C2211yescharacteristic
29#427C2211yescharacteristic
30#427C2211yescharacteristic
31#421C2211yescharacteristic
32#421C2211yescharacteristic
33#421C2212yes—
34#419C2212yes—
35#419C2212yes—
36C2×#101C2212yes—
37#410C2212yes—
38#405C2211yescharacteristic
39#344C2212yes—
40#344C2212yes—
41H(3,ℤ/4)C2212yes—
42H(3,ℤ/4)C2212yes—
43C2×#41C2311yescharacteristic
44C2×#41C2311yescharacteristic
45C22×Q8C2312yes—
46#137—21no—
47#137—21no—
48#137—24no—
49#137—42no—
50AGL(1,ℤ/8)—21no—
51AGL(1,ℤ/8)—21no—
52#135—22no—
53C2×Q16—22no—
54C2×Q16—21no—
55C2×Q16—21no—
56C2×Q16—21no—
57C2×Q16—21no—
58C2×Q16D412yes—
59C2×Q16D412yes—
60C2×SD16—22no—
61#131—21no—
62#131—21no—
63#131—21no—
64#131—21no—
65#131—21no—
66#131—21no—
67#131—21no—
68#131—21no—
69C2×M16—21no—
70C2×M16—21no—
71C2×M16C2311yescharacteristic
72C2×M16C2311yescharacteristic
73C2×M16C2312yes—
74C2×M16C2311yescharacteristic
75C22×C8—21no—
76C22×C8—21no—
77C22×C8C2311yescharacteristic
78C4⋊Q8—22no—
79#124—22no—
80#123—42no—
81C22⋊Q8—24no—
82C4×Q8—24no—
83C4×Q8—42no—
84#117—42no—
85#117C2312yes—
86#117C2312yes—
87#117C2311yescharacteristic
88#108—21no—
89#108—21no—
90#108—22no—
91#107—21no—
92#107—21no—
93#106—21no—
94#106—21no—
95#104—22no—
96#104—22no—
97#104—42no—
98#103—24no—
99#103—24no—
100#101—24no—
101#99—42no—
102#97D412yes—
103C4×C8D412yes—
104#41—21no—
105#41—21no—
106#41—21no—
107#41—21no—
108#41—21no—
109#41—21no—
110#41—21no—
111#41—21no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D414yes—
118C2×Q8#4111yescharacteristic
119C2×Q8#4111yescharacteristic
120C2×D4—21no—
121C2×D4—21no—
122C2×D4#4111yescharacteristic
123C2×D4#4111yescharacteristic
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C2411yescharacteristic
128Q16—22no—
129Q16—22no—
130Q16—22no—
131Q16—22no—
132Q16—22no—
133Q16—44no—
134SD16—22no—
135SD16—22no—
136SD16—44no—
137D8—22no—
138M16—21no—
139M16—21no—
140M16—21no—
141M16—22no—
142M16—21no—
143M16—21no—
144M16—21no—
145M16—22no—
146M16—41no—
147M16—41no—
148M16—41no—
149M16—41no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—22no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8C2×D412yes—
158C2×C8C2×D411yescharacteristic
159C2×C8C2×D411yescharacteristic
160C2×C8C2×D411yescharacteristic
161C2×C8C2×D411yescharacteristic
162C2×C8#4111yescharacteristic
163C2×C8#4111yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C22⋊C4—21no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—44no—
173C42—22no—
174C42—22no—
175C42—44no—
176C42C2×D412yes—
177C23—21no—
178C23—21no—
179C23H(5,𝔽2)11yescharacteristic
180Q8—24no—
181Q8—21no—
182Q8—21no—
183Q8—24no—
184Q8—21no—
185Q8—21no—
186Q8—42no—
187Q8—42no—
188D4—21no—
189D4—21no—
190D4—21no—
191D4—21no—
192D4—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)11yescharacteristic
214C8—22no—
215C8—41no—
216C8—41no—
217C8—42no—
218C8#12111yescharacteristic
219C8#12111yescharacteristic
220C8#12111yescharacteristic
221C8#12111yescharacteristic
222C22—41no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22#53511yescharacteristic
228C22#54611yescharacteristic
229C22#55311yescharacteristic
230C4—42no—
231C4—41no—
232C4—41no—
233C4—44no—
234C4—82no—
235C4—82no—
236C4C2×#12111yescharacteristic
237C4#52511yescharacteristic
238C4#54611yescharacteristic
239C4#55311yescharacteristic
240C2—21no—
241C2—21no—
242C2—21no—
243C2—81no—
244C2—81no—
245C2#253511yescharacteristic
subgroupG/H
center, socleC2#2535
commutator, Frattini, cosocleC22×C4C24
prime radical#575C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#473C2semidirect
2. #3059C2semidirect
4. #2950C2semidirect
5. #2082C2semidirect
7. #1912C2semidirect
8. #1905C2semidirect
9. #1833C2semidirect
10. #1814C2semidirect
11. #1812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30592
#29552
#29502
#20822
#20092
#19122
#19052
#18332
#18142
#18122