Finite Group Databasebeta

← #18679

Subgroups — #18679

Order 256.

473 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 121 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#3506C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic, complemented
3#3167C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1652C212yesmaximal, complemented
6#1641C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1631C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, characteristic, complemented
12#559—22no—
13#556C2212yes—
14#555—21no—
15#553—22no—
16#550C2212yes—
17#547—21no—
18#538—22no—
19#535—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#525—22no—
24#520—21no—
25C2×C4×D4C2211yescharacteristic
26#498C2211yescharacteristic, complemented
27#492C2211yescharacteristic, complemented
28#487C2211yescharacteristic, complemented
29#486C2211yescharacteristic, complemented
30#478—22no—
31#477—21no—
32#477C2212yescomplemented
33#476C2212yes—
34#474C2212yes—
35#467—22no—
36#465—22no—
37#462—21no—
38#462C2212yescomplemented
39C4×SD16—22no—
40#428—21no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#418—21no—
44C2×C4×C8C2211yescharacteristic
45#332C2212yescomplemented
46#331C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—22no—
52C2×#41—21no—
53C22×D4D411yescharacteristic
54C23×C4—21no—
55C2×Q16—22no—
56C2×SD16—22no—
57C2×M16—21no—
58C22×C8—21no—
59C22×C8—21no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65#127—22no—
66#127C2311yescharacteristic
67#126—24no—
68#125—21no—
69#125—21no—
70#124—24no—
71#124—22no—
72#124—21no—
73#123—24no—
74#123—24no—
75C22⋊Q8—22no—
76C22⋊Q8—21no—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79C22⋊Q8D412yes—
80#121—22no—
81#121—22no—
82#121—24no—
83#121—21no—
84#121—22no—
85#121D412yes—
86#120—24no—
87C4×Q8—22no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8C2312yes—
92C4×D4—24no—
93C4×D4—24no—
94C4×D4—22no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D412yes—
99C4×D4D412yes—
100C4×D4C2312yes—
101#117—22no—
102#117—21no—
103#117—22no—
104#117—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4D411yescharacteristic
107C2×C22⋊C4—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#106—22no—
112#106—22no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2312yes—
116C4⋊C8C2312yes—
117#103—22no—
118#103—22no—
119#103—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—21no—
127#41—42no—
128#41—42no—
129#41—82no—
130#41—82no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8C2×D412yes—
136C2×D4—22no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×D412yes—
145C2×D4C2×D412yes—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156Q16—82no—
157SD16—44no—
158M16—42no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—81no—
165C4⋊C4—24no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—41no—
191C42—24no—
192C42—22no—
193C42—41no—
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2×D411yescharacteristic
197C42C2411yescharacteristic
198C23—22no—
199C23—22no—
200C23—42no—
201C23—81no—
202C23—81no—
203C23#12011yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—81no—
212D4—22no—
213D4—22no—
214D4—42no—
215D4—44no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220D4—42no—
221D4—84no—
222D4—84no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—84no—
238C2×C4#12011yescharacteristic
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#13512yes—
242C2×C4#13712yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C8—22no—
247C8—41no—
248C8—42no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—84no—
259C22#45212yes—
260C22C2×#12011yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—81no—
270C4#45212yes—
271C4C2×#13511yescharacteristic
272C4C2×#13711yescharacteristic
273C4#57611yescharacteristic
274C4#57711yescharacteristic
275C2—22no—
276C2—82no—
277C2—82no—
278C2C2×#45211yescharacteristic
279C2#301211yescharacteristic
280C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3506C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3249C2semidirect
3. #3167C2semidirect
4. #3077C2semidirect
5. #1652C2semidirect
6. #1641C2semidirect
7. #1638C2semidirect
8. #1631C2semidirect
9. #1547C2semidirect
10. #1493C2semidirect
11. C2×D4⋊C8C2semidirect
26. #498C22semidirect
27. #492C22semidirect
28. #487C22semidirect
29. #486C22semidirect
32. #477C22semidirect
38. #462C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35062
#32492
#31672
#30772
#16522
#16412
#16382
#16312
#15472
#14932
C2×D4⋊C82