Finite Group Databasebeta

← #18675

Subgroups — #18675

Order 256.

561 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 129 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#3497C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3167C211yesmaximal, characteristic, complemented
4#3078C211yesmaximal, characteristic, complemented
5#1651C212yesmaximal, complemented
6#1642C212yesmaximal, complemented
7#1637C212yesmaximal, complemented
8#1631C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1492C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×2−1+4—21no—
13#550—21no—
14#550C2212yes—
15D4×Q8—21no—
16D4×Q8C2212yes—
17#548—22no—
18#548—22no—
19#537—22no—
20#537—21no—
21#533C2211yescharacteristic
22#532—22no—
23#532C2211yescharacteristic
24#532C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26#498C2211yescharacteristic
27#492C2211yescharacteristic
28#487C2211yescharacteristic
29#486C2211yescharacteristic
30#479—22no—
31#477C2212yes—
32#476C2212yes—
33#475—21no—
34#475C2212yes—
35#465—22no—
36#465—22no—
37#461C2212yes—
38#460—21no—
39C4×SD16—22no—
40#428—21no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#419—21no—
44C2×C4×C8C2211yescharacteristic
45#332C2212yes—
46#331C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
512−1+4—44no—
522−1+4—44no—
53C2×#41—24no—
54C2×#41—22no—
55C2×#41—22no—
56C2×#41—21no—
57C2×#41—21no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8D411yescharacteristic
61C2×SD16—22no—
62C2×SD16—22no—
63C2×M16—21no—
64C22×C8—21no—
65C22×C8—21no—
66C4⋊Q8—22no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8C2311yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71#127—22no—
72#127D411yescharacteristic
73#127C2311yescharacteristic
74#125—21no—
75#124—24no—
76#124—21no—
77#124—21no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84#121—24no—
85#121—22no—
86#121—24no—
87#121—24no—
88#121—21no—
89#121—21no—
90#121—22no—
91#121D412yes—
92#121D412yes—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8D412yes—
98C4×Q8C2312yes—
99C4×D4—24no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4D412yes—
105C4×D4C2312yes—
106#117—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#107—22no—
112#106—22no—
113#106—22no—
114#106—22no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2312yes—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#102—22no—
124#102—22no—
125C4×C8—21no—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128#41—416no—
129#41—44no—
130#41—48no—
131#41—44no—
132#41—42no—
133#41—42no—
134#41—42no—
135#41—42no—
136C2×Q8—24no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—22no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—44no—
144C2×Q8—48no—
145C2×Q8—44no—
146C2×Q8SD1614yescomplemented
147C2×Q8C2×D412yes—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—24no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C2×D412yes—
157C2×D4C2×D412yes—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167SD16—44no—
168SD16—82no—
169M16—42no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—81no—
176C4⋊C4—24no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—41no—
198C42—22no—
199C42—22no—
200C42—41no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—22no—
206C23—22no—
207C23#12011yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—48no—
211Q8—42no—
212Q8—41no—
213Q8—41no—
214Q8—42no—
215Q8—41no—
216Q8—41no—
217Q8C2×SD1614yes—
218D4—22no—
219D4—22no—
220D4—42no—
221D4—48no—
222D4—44no—
223D4—44no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—42no—
229D4—81no—
230D4—84no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4#12011yescharacteristic
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4C2×SD1612yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C8—22no—
256C8—41no—
257C8—42no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22#45012yes—
268C22C2×#12011yescharacteristic
269C4—24no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4#45012yes—
279C4C22×SD1611yescharacteristic
280C4C2×AGL(1,ℤ/8)11yescharacteristic
281C4#57711yescharacteristic
282C4#57811yescharacteristic
283C2—22no—
284C2—82no—
285C2—82no—
286C2C2×#45011yescharacteristic
287C2#301311yescharacteristic
288C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3497C2
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. #3246C2semidirect
3. #3167C2semidirect
4. #3078C2semidirect
5. #1651C2semidirect
6. #1642C2semidirect
7. #1637C2semidirect
8. #1631C2semidirect
9. #1547C2semidirect
10. #1492C2semidirect
11. C2×Q8⋊C8C2semidirect
146. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#34972
#32462
#31672
#30782
#16512
#16422
#16372
#16312
#15472
#14922
C2×Q8⋊C82