Finite Group Databasebeta

← #25900

Subgroups — #25900

Order 256.

703 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 129 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#3496C211yesmaximal, characteristic
2#2504C212yesmaximal, complemented
3#2487C211yesmaximal, characteristic, complemented
4#2044C214yesmaximal, complemented
5#1964C212yesmaximal, complemented
6#1790C214yesmaximal, complemented
7C4×#99C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)C2212yes—
9#547—22no—
10#546—24no—
11D42—22no—
12#537—21no—
13#534—21no—
14#534C2212yes—
15#532—21no—
16C2×#121—21no—
17C2×#121C2212yes—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic, complemented
20C2×#99C2212yes—
21C2×#99C2218yescomplemented
22#400—21no—
23#399—21no—
24#397—24no—
25#390—22no—
26#389—21no—
27#386C2212yes—
28#386C2218yescomplemented
29#385C2212yescomplemented
30#382—22no—
31#380—21no—
32#380C2211yescharacteristic, complemented
33C4×C22⋊C4C2212yescomplemented
34#342C2212yescomplemented
35H(5,𝔽2)—28no—
36H(5,𝔽2)—44no—
37C2×#41—22no—
38C2×#41—24no—
39C22×D4—22no—
40C22×D4—24no—
41C22×D4—21no—
42C22×D4D412yes—
43C22×D4C2314yes—
44C22×D4C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C4⋊Q8—21no—
49#127—21no—
50#124—22no—
51#123—24no—
52#123—24no—
53C22⋊Q8—24no—
54#121—22no—
55#121—22no—
56#121—24no—
57#121—22no—
58#121—22no—
59#121—22no—
60#121—22no—
61#121—22no—
62#121—22no—
63#120—24no—
64#120—24no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D412yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—28no—
81C2×C22⋊C4D411yescharacteristic
82C2×C22⋊C4D414yes—
83C2×C22⋊C4C2314yes—
84C2×C22⋊C4C2312yes—
85C2×C22⋊C4C2311yescharacteristic
86C2×C42—22no—
87C2×C42D411yescharacteristic
88#99—24no—
89#99—48no—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—28no—
94#95—22no—
95C24—22no—
96C24C2×D414yes—
97C24C2×D411yescharacteristic
98C24C2411yescharacteristic
99#41—416no—
100#41—44no—
101#41—44no—
102C2×Q8—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—22no—
107C2×D4—24no—
108C2×D4—24no—
109C2×D4—28no—
110C2×D4—28no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—44no—
116C2×D4—416no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—82no—
120C2×D4#4114yes—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—82no—
132C22×C4C2×D414yes—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—82no—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—22no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—88no—
160C22⋊C4—82no—
161C42—21no—
162C42—21no—
163C42—44no—
164C23—21no—
165C23—21no—
166C23—24no—
167C23—28no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—44no—
173C23—44no—
174C23—44no—
175C23—82no—
176C23—82no—
177C23#12412yes—
178C23C22×D411yescharacteristic
179C23C22×D412yes—
180C23C2×#4112yes—
181C23H(5,𝔽2)12yes—
182Q8—44no—
183D4—28no—
184D4—44no—
185D4—48no—
186D4—44no—
187D4—44no—
188D4—48no—
189D4—44no—
190D4—82no—
191D4—82no—
192D4—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—28no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—48no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—48no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—81no—
219C2×C4#12011yescharacteristic
220C2×C4#12012yes—
221C2×C4#12011yescharacteristic
222C2×C4#12412yes—
223C22—21no—
224C22—22no—
225C22—21no—
226C22—22no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—44no—
234C22—44no—
235C22—44no—
236C22—81no—
237C22—88no—
238C22—164no—
239C22C2×#12011yescharacteristic
240C22C2×#12411yescharacteristic
241C22#53511yescharacteristic
242C22#54614yes—
243C4—22no—
244C4—21no—
245C4—22no—
246C4—44no—
247C4—41no—
248C4—41no—
249C4—82no—
250C4—84no—
251C4—82no—
252C4#45812yes—
253C2—21no—
254C2—24no—
255C2—21no—
256C2—42no—
257C2—84no—
258C2#243511yescharacteristic
259C2C2×#45811yescharacteristic
260C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2504C2semidirect
3. #2487C2semidirect
4. #2044C2semidirect
5. #1964C2semidirect
6. #1790C2semidirect
7. C4×#99C2semidirect
19. C22×C4⋊C4C22semidirect
21. C2×#99C22semidirect
28. #386C22semidirect
29. #385C22semidirect
32. #380C22semidirect
33. C4×C22⋊C4C22semidirect
34. #342C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#25042
#24872
#20442
#19642
#17902
C4×#992