Finite Group Databasebeta

← #24263

Subgroups — #24263

Order 256.

751 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×#458C212yesmaximal, complemented
3#2618C212yesmaximal, complemented
4#2322C211yesmaximal, characteristic, complemented
5#2044C212yesmaximal, complemented
6#2022C212yesmaximal, complemented
7#1964C212yesmaximal, complemented
8#1799C211yesmaximal, characteristic, complemented
9#1790C212yesmaximal, 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×#123C2211yescharacteristic
19C2×#123C2214yescomplemented
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic, complemented
24#458—28no—
25C2×#99C2212yes—
26C2×#99C2214yescomplemented
27C2×#99C2214yescomplemented
28#398—22no—
29#396—24no—
30#394—22no—
31#389—21no—
32#388—21no—
33#387—21no—
34#386C2211yescharacteristic
35#386C2214yescomplemented
36#385—21no—
37#385C2211yescharacteristic, complemented
38#381C2211yescharacteristic, complemented
39#380C2212yescomplemented
40C4×C22⋊C4C2211yescharacteristic, complemented
41#342C2212yescomplemented
42H(5,𝔽2)—24no—
43H(5,𝔽2)—24no—
44H(5,𝔽2)—44no—
45C2×#41—22no—
46C2×#41—24no—
47C22×D4—22no—
48C22×D4—24no—
49C22×D4—21no—
50C22×D4D412yes—
51C22×D4C2314yes—
52C22×D4C2311yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C4⋊Q8—21no—
58#127—21no—
59#124—22no—
60#123—24no—
61#123—22no—
62#123—24no—
63#123—28no—
64#123—44no—
65#123D414yescomplemented
66C22⋊Q8—24no—
67#121—22no—
68#121—22no—
69#121—24no—
70#121—22no—
71#121—22no—
72#121—22no—
73#121—22no—
74#121—22no—
75#121—22no—
76#120—24no—
77#120—24no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D412yes—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D414yes—
100C2×C22⋊C4C2314yes—
101C2×C22⋊C4C2311yescharacteristic
102C2×C22⋊C4C2311yescharacteristic
103C2×C22⋊C4C2311yescharacteristic
104C2×C42—21no—
105C2×C42D411yescharacteristic
106#99—24no—
107#99—28no—
108#99—44no—
109#95—21no—
110#95—22no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—24no—
116C24—22no—
117C24C2×D414yes—
118C24C2×D411yescharacteristic
119C24C2411yescharacteristic
120#41—48no—
121#41—48no—
122#41—44no—
123#41—44no—
124C2×Q8—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—24no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—48no—
140C2×D4—48no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4C2×D412yes—
146C2×D4#4112yes—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4C2×D414yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D412yescomplemented
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4—48no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—81no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—21no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—24no—
195C22⋊C4—48no—
196C22⋊C4—81no—
197C22⋊C4—84no—
198C22⋊C4C2×D412yescomplemented
199C42—21no—
200C42—21no—
201C42—81no—
202C23—21no—
203C23—21no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—82no—
215C23—82no—
216C23#12112yes—
217C23C22×D411yescharacteristic
218C23C22×D412yes—
219C23C22×D411yescharacteristic
220C23C2×#4111yescharacteristic
221C23H(5,𝔽2)12yes—
222Q8—44no—
223D4—24no—
224D4—24no—
225D4—44no—
226D4—44no—
227D4—48no—
228D4—44no—
229D4—44no—
230D4—44no—
231D4—44no—
232D4—82no—
233D4—82no—
234D4—82no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—48no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—44no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—81no—
270C2×C4#12011yescharacteristic
271C2×C4#12011yescharacteristic
272C2×C4#12012yes—
273C2×C4#12112yes—
274C22—21no—
275C22—22no—
276C22—21no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—81no—
289C22—84no—
290C22—84no—
291C22—164no—
292C22C2×#12011yescharacteristic
293C22C2×#12111yescharacteristic
294C22#53511yescharacteristic
295C22D4212yes—
296C22#54612yes—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—22no—
301C4—42no—
302C4—42no—
303C4—44no—
304C4—41no—
305C4—41no—
306C4—84no—
307C4—81no—
308C4—81no—
309C2—21no—
310C2—22no—
311C2—21no—
312C2—22no—
313C2—42no—
314C2—84no—
315C2#240811yescharacteristic
316C2#302611yescharacteristic
317C2#302811yescharacteristic
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. C2×#458C2semidirect
3. #2618C2semidirect
4. #2322C2semidirect
5. #2044C2semidirect
6. #2022C2semidirect
7. #1964C2semidirect
8. #1799C2semidirect
9. #1790C2semidirect
19. C2×#123C22semidirect
23. C22×C4⋊C4C22semidirect
26. C2×#99C22semidirect
27. C2×#99C22semidirect
35. #386C22semidirect
37. #385C22semidirect
38. #381C22semidirect
39. #380C22semidirect
40. C4×C22⋊C4C22semidirect
41. #342C22semidirect
65. #123D4semidirect
171. C22×C4C2×D4semidirect
198. C22⋊C4C2×D4semidirect

Maximal subgroups

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

subgroupindex
#34962
C2×#4582
#26182
#23222
#20442
#20222
#19642
#17992
#17902