Finite Group Databasebeta

← #25877

Subgroups — #25877

Order 256.

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

normal subgroups only

This group has 627 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#3500C211yesmaximal, characteristic
2#3142C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic, complemented
4#2861C211yesmaximal, characteristic, complemented
5#2038C212yesmaximal, complemented
6#2037C212yesmaximal, complemented
7#1967C211yesmaximal, characteristic, complemented
8#1962C211yesmaximal, characteristic, complemented
9#1793C212yesmaximal, complemented
10#1792C212yesmaximal, complemented
11C4×#104C211yesmaximal, characteristic, complemented
12C2×2−1+4C2211yescharacteristic
13C2×H(5,𝔽2)C2211yescharacteristic
14#548—22no—
15#546—22no—
16#546—24no—
17#536—21no—
18#536C2211yescharacteristic
19#535—21no—
20#535C2211yescharacteristic
21#532—21no—
22#532C2211yescharacteristic
23C2×#125C2211yescharacteristic, complemented
24C2×#124—21no—
25C2×#124C2211yescharacteristic
26C4×#41C2211yescharacteristic
27#491—21no—
28#490—21no—
29#488—21no—
30#488—21no—
31#479—22no—
32#478—22no—
33#428C2212yescomplemented
34#423C2211yescharacteristic, complemented
35C2×#104C2212yes—
36C2×#104C2212yes—
37C2×#104C2212yescomplemented
38#417C2212yes—
39#417C2212yes—
40#417C2212yescomplemented
41C2×#101C2212yescomplemented
42C2×#100C2212yes—
43C4×M16C2211yescharacteristic, complemented
44#391C2212yescomplemented
45#390C2212yes—
46#388—24no—
47C43C2211yescharacteristic, complemented
48#339C2212yescomplemented
492−1+4—24no—
502−1+4—42no—
51H(5,𝔽2)—24no—
52H(5,𝔽2)—42no—
53C2×#41—21no—
54C2×#41—21no—
55C2×#41—21no—
56C2×#41—21no—
57C2×#41—22no—
58C2×#41C2311yescharacteristic
59C2×#41C2311yescharacteristic
60C2×#41C2311yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C2×M16C2312yes—
69C2×M16C2312yes—
70C4⋊Q8—21no—
71C4⋊Q8D411yescharacteristic
72#127—21no—
73#127D411yescharacteristic
74#125—22no—
75#125—21no—
76#125—21no—
77#124—24no—
78#124—21no—
79#124—21no—
80#124—21no—
81#124—21no—
82#124—22no—
83#124D411yescharacteristic
84#124D411yescharacteristic
85#123—24no—
86#123—22no—
87C22⋊Q8—24no—
88C22⋊Q8—22no—
89#121—24no—
90#121—24no—
91#121—22no—
92#121—22no—
93#120—24no—
94#120—22no—
95C4×Q8D412yes—
96C4×D4—24no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4D412yes—
100#117—21no—
101#117C2311yescharacteristic
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2312yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4D414yes—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42C2312yes—
110C2×C42C2311yescharacteristic
111#107—22no—
112#106—22no—
113C4⋊C8—22no—
114#104—24no—
115#104—42no—
116#104—42no—
117#103—22no—
118#103—22no—
119#103—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#101—42no—
124#100—42no—
125#97—21no—
126C4×C8—21no—
127#95—24no—
128C24—22no—
129C24C2×D412yes—
130#41—22no—
131#41—22no—
132#41—22no—
133#41—22no—
134#41—22no—
135#41—44no—
136#41—48no—
137#41—44no—
138#41—42no—
139#41—42no—
140#41—42no—
141#41—42no—
142#41#4114yes—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—24no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—44no—
167C2×D4—48no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4C2×D411yescharacteristic
171C2×D4C2×D411yescharacteristic
172C2×D4C2×D411yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2411yescharacteristic
185M16—22no—
186M16—22no—
187M16—22no—
188M16—22no—
189C2×C8—22no—
190C2×C8—22no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4C2×D412yes—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—48no—
208C42—22no—
209C42—21no—
210C42—22no—
211C42—22no—
212C42—44no—
213C42C2×D411yescharacteristic
214C42C2×D411yescharacteristic
215C42C2×D412yes—
216C23—21no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—24no—
221C23—44no—
222C23—44no—
223C23—84no—
224C23C22×D411yescharacteristic
225Q8—21no—
226Q8—22no—
227Q8—22no—
228Q8—44no—
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8#12412yes—
233D4—24no—
234D4—21no—
235D4—22no—
236D4—22no—
237D4—44no—
238D4—44no—
239D4—44no—
240D4—42no—
241D4—42no—
242D4—41no—
243D4—41no—
244D4—42no—
245D4—82no—
246D4—84no—
247D4#12412yes—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4#12012yes—
270C2×C4#12012yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4112yes—
274C2×C4H(5,𝔽2)12yes—
275C8—42no—
276C8—42no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—44no—
285C22—42no—
286C22—84no—
287C22—164no—
288C22C2×#12011yescharacteristic
289C22#54612yes—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—82no—
299C4C2×#12411yescharacteristic
300C4#53511yescharacteristic
301C4#54612yes—
302C2—22no—
303C2—42no—
304C2—81no—
305C2—82no—
306C2—81no—
307C2#243511yescharacteristic
308C2#301812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3500C2
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. #3142C2semidirect
3. #3080C2semidirect
4. #2861C2semidirect
5. #2038C2semidirect
6. #2037C2semidirect
7. #1967C2semidirect
8. #1962C2semidirect
9. #1793C2semidirect
10. #1792C2semidirect
11. C4×#104C2semidirect
23. C2×#125C22semidirect
33. #428C22semidirect
34. #423C22semidirect
37. C2×#104C22semidirect
40. #417C22semidirect
41. C2×#101C22semidirect
43. C4×M16C22semidirect
44. #391C22semidirect
47. C43C22semidirect
48. #339C22semidirect

Maximal subgroups

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

subgroupindex
#35002
#31422
#30802
#28612
#20382
#20372
#19672
#19622
#17932
#17922
C4×#1042