Finite Group Databasebeta

← #25875

Subgroups — #25875

Order 256.

715 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 131 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#3498C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic, complemented
4#2858C211yesmaximal, characteristic, complemented
5#2037C214yesmaximal, complemented
6#1965C211yesmaximal, characteristic, complemented
7#1962C211yesmaximal, characteristic, complemented
8#1792C214yesmaximal, complemented
9C4×#104C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)C2212yes—
11#550—21no—
12#550—21no—
13D42—22no—
14D42—22no—
15D42—21no—
16D42—21no—
17#535—22no—
18#535C2212yes—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic, complemented
22C2×#127—21no—
23C2×#127C2211yescharacteristic
24C2×#124—22no—
25C4×#41C2211yescharacteristic
26#489—22no—
27#487—21no—
28#486—21no—
29#477—21no—
30#476—21no—
31#475—21no—
32#474—21no—
33#428C2212yescomplemented
34#423C2211yescharacteristic, complemented
35C2×#104C2212yes—
36C2×#104C2214yescomplemented
37#417C2212yes—
38#417C2214yescomplemented
39C2×#100C2214yescomplemented
40C4×M16C2211yescharacteristic, complemented
41#390—22no—
42#390C2214yescomplemented
43C43C2211yescharacteristic, complemented
44#339C2212yescomplemented
45H(5,𝔽2)—28no—
46H(5,𝔽2)—44no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41C2312yes—
50C2×#41C2311yescharacteristic
51C22×Q8—21no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4D412yes—
58C22×D4D412yes—
59C22×D4C2312yes—
60C2×M16C2312yes—
61C2×M16C2312yes—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D411yescharacteristic
67#127—22no—
68#127—22no—
69#127—22no—
70#127—21no—
71#127—21no—
72#127—21no—
73#127—21no—
74#127D411yescharacteristic
75#124—22no—
76#124—48no—
77#124D412yes—
78#121—22no—
79#121—22no—
80#121—24no—
81#121—24no—
82#121—21no—
83#121—21no—
84#121—22no—
85#121—22no—
86#120—24no—
87#120—24no—
88#120—24no—
89C4×Q8D411yescharacteristic
90C4×Q8D411yescharacteristic
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4D411yescharacteristic
98C4×D4D411yescharacteristic
99#117—21no—
100#117C2311yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2312yes—
103C2×C22⋊C4—28no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2312yes—
108C2×C42C2311yescharacteristic
109#107—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113#104—24no—
114#104—44no—
115#103—22no—
116#103—24no—
117#102—22no—
118#102—24no—
119#100—44no—
120#97—21no—
121C4×C8—21no—
122C24—24no—
123C24C2×D414yes—
124#41—22no—
125#41—24no—
126#41—24no—
127#41—48no—
128#41—44no—
129#41—44no—
130#41#4114yes—
131C2×Q8—44no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8C2×D412yes—
135C2×Q8C2×D411yescharacteristic
136C2×D4—21no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—48no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—416no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—84no—
159C2×D4—84no—
160C2×D4C2×D412yes—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172M16—22no—
173M16—22no—
174M16—24no—
175C2×C8—22no—
176C2×C8—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4C2×D411yescharacteristic
186C4⋊C4C2×D411yescharacteristic
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—416no—
195C42—24no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—22no—
200C42—22no—
201C42—42no—
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C23—21no—
207C23—24no—
208C23—22no—
209C23—28no—
210C23—48no—
211C23—44no—
212C23—44no—
213C23—84no—
214C23—84no—
215C23C22×D411yescharacteristic
216Q8—21no—
217Q8—24no—
218Q8—82no—
219Q8—82no—
220Q8#12412yescomplemented
221D4—24no—
222D4—21no—
223D4—24no—
224D4—48no—
225D4—48no—
226D4—44no—
227D4—42no—
228D4—42no—
229D4—44no—
230D4—44no—
231D4—42no—
232D4—44no—
233D4—42no—
234D4—82no—
235D4—82no—
236D4—84no—
237D4—84no—
238D4#12412yescomplemented
239C2×C4—21no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—48no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4#12011yescharacteristic
257C2×C4#12011yescharacteristic
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4C2×#4112yes—
263C2×C4H(5,𝔽2)12yes—
264C8—42no—
265C8—42no—
266C22—21no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—44no—
273C22—44no—
274C22—42no—
275C22—44no—
276C22—42no—
277C22—88no—
278C22—168no—
279C22C2×#12011yescharacteristic
280C22#54612yes—
281C4—22no—
282C4—21no—
283C4—21no—
284C4—44no—
285C4—42no—
286C4—41no—
287C4—41no—
288C4—82no—
289C4#45312yes—
290C4#45512yes—
291C4C2×#12411yescharacteristic
292C4#53511yescharacteristic
293C4#54612yes—
294C2—22no—
295C2—42no—
296C2—84no—
297C2—82no—
298C2#243511yescharacteristic
299C2C2×#45311yescharacteristic
300C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3498C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3141C2semidirect
3. #3079C2semidirect
4. #2858C2semidirect
5. #2037C2semidirect
6. #1965C2semidirect
7. #1962C2semidirect
8. #1792C2semidirect
9. C4×#104C2semidirect
21. C2×C4⋊Q8C22semidirect
33. #428C22semidirect
34. #423C22semidirect
36. C2×#104C22semidirect
38. #417C22semidirect
39. C2×#100C22semidirect
40. C4×M16C22semidirect
42. #390C22semidirect
43. C43C22semidirect
44. #339C22semidirect
220. Q8#124semidirect
238. D4#124semidirect

Maximal subgroups

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

subgroupindex
#34982
#31412
#30792
#28582
#20372
#19652
#19622
#17922
C4×#1042