Finite Group Databasebeta

← #25901

Subgroups — #25901

Order 256.

703 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 127 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#2591C212yesmaximal, complemented
3#2579C211yesmaximal, characteristic, complemented
4#2044C212yesmaximal, complemented
5#2044C212yesmaximal, complemented
6#1964C212yesmaximal, complemented
7#1790C212yesmaximal, complemented
8#1790C212yesmaximal, complemented
9C4×#99C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)C2212yes—
11#547—22no—
12#546—24no—
13D42—22no—
14#537—21no—
15#534—21no—
16#534C2212yes—
17#532—21no—
18C2×#121—21no—
19C2×#121C2212yes—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic, complemented
22C2×#99C2212yes—
23C2×#99C2214yescomplemented
24C2×#99C2214yescomplemented
25#400—22no—
26#398—21no—
27#396—21no—
28#394—22no—
29#388—24no—
30#387—21no—
31#386C2211yescharacteristic
32#386C2211yescharacteristic
33#386C2214yescomplemented
34#386C2214yescomplemented
35#385—21no—
36#385C2211yescharacteristic, complemented
37#385C2211yescharacteristic, complemented
38#380C2211yescharacteristic, complemented
39C4×C22⋊C4C2212yescomplemented
40#342C2212yescomplemented
41H(5,𝔽2)—24no—
42H(5,𝔽2)—24no—
43H(5,𝔽2)—44no—
44C2×#41—22no—
45C2×#41—24no—
46C22×D4—22no—
47C22×D4—24no—
48C22×D4—21no—
49C22×D4D412yes—
50C22×D4C2314yes—
51C22×D4C2311yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C4⋊Q8—21no—
57#127—21no—
58#124—22no—
59#123—24no—
60#123—24no—
61C22⋊Q8—24no—
62#121—22no—
63#121—22no—
64#121—24no—
65#121—22no—
66#121—22no—
67#121—22no—
68#121—22no—
69#121—22no—
70#121—22no—
71#120—24no—
72#120—24no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D412yes—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4D414yes—
94C2×C22⋊C4C2314yes—
95C2×C22⋊C4C2311yescharacteristic
96C2×C22⋊C4C2311yescharacteristic
97C2×C22⋊C4C2311yescharacteristic
98C2×C42—22no—
99C2×C42D411yescharacteristic
100#99—24no—
101#99—44no—
102#99—44no—
103#95—22no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—24no—
110#95—24no—
111C24—22no—
112C24C2×D414yes—
113C24C2×D411yescharacteristic
114C24C2411yescharacteristic
115#41—48no—
116#41—48no—
117#41—44no—
118#41—44no—
119C2×Q8—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—48no—
135C2×D4—48no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—82no—
139C2×D4—82no—
140C2×D4#4114yes—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4C2×D414yes—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D411yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—24no—
185C22⋊C4—48no—
186C22⋊C4—42no—
187C22⋊C4—84no—
188C22⋊C4—81no—
189C42—21no—
190C42—21no—
191C42—82no—
192C23—21no—
193C23—21no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—82no—
205C23—82no—
206C23#12412yes—
207C23C22×D411yescharacteristic
208C23C22×D412yes—
209C23C2×#4112yes—
210C23H(5,𝔽2)12yes—
211Q8—44no—
212D4—28no—
213D4—44no—
214D4—44no—
215D4—48no—
216D4—44no—
217D4—44no—
218D4—44no—
219D4—44no—
220D4—82no—
221D4—82no—
222D4—82no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—81no—
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4#12012yes—
261C2×C4#12412yes—
262C22—21no—
263C22—22no—
264C22—21no—
265C22—22no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—44no—
273C22—44no—
274C22—44no—
275C22—81no—
276C22—84no—
277C22—84no—
278C22—164no—
279C22C2×#12011yescharacteristic
280C22C2×#12411yescharacteristic
281C22#53511yescharacteristic
282C22#54614yes—
283C4—22no—
284C4—21no—
285C4—21no—
286C4—22no—
287C4—44no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—84no—
294C4—81no—
295C2—21no—
296C2—24no—
297C2—21no—
298C2—42no—
299C2—84no—
300C2#243511yescharacteristic
301C2#302511yescharacteristic
302C2#302611yescharacteristic
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. #2591C2semidirect
3. #2579C2semidirect
4. #2044C2semidirect
5. #2044C2semidirect
6. #1964C2semidirect
7. #1790C2semidirect
8. #1790C2semidirect
9. C4×#99C2semidirect
21. C22×C4⋊C4C22semidirect
23. C2×#99C22semidirect
24. C2×#99C22semidirect
33. #386C22semidirect
34. #386C22semidirect
36. #385C22semidirect
37. #385C22semidirect
38. #380C22semidirect
39. C4×C22⋊C4C22semidirect
40. #342C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#25912
#25792
#20442
#19642
#17902
C4×#992