Finite Group Databasebeta

← #18849

Subgroups — #18849

Order 256.

429 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 113 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#3507C211yesmaximal, characteristic
2#3254C211yesmaximal, characteristic
3#3231C211yesmaximal, characteristic, complemented
4#3113C211yesmaximal, characteristic, complemented
5#1652C212yesmaximal, complemented
6#1650C212yesmaximal, complemented
7#1644C212yesmaximal, complemented
8#1643C212yesmaximal, complemented
9#1559C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic
11#1487C211yesmaximal, characteristic, complemented
12#559—21no—
13#557—21no—
14#556—22no—
15#556C2212yes—
16#554C2212yes—
17#552—22no—
18#539—22no—
19#536—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#529—22no—
24#520—21no—
25C2×C4×Q8C2211yescharacteristic
26#500C2211yescharacteristic
27#498C2211yescharacteristic
28#487C2211yescharacteristic
29#487C2211yescharacteristic
30#485—22no—
31#483—21no—
32#479—22no—
33#477—21no—
34#475C2212yes—
35#474C2212yes—
36#468—22no—
37#462C2212yes—
38#461C2212yes—
39#460—21no—
40#451—21no—
41#441—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44#423C2211yescharacteristic
45#333C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8D411yescharacteristic
53C2×Q16—22no—
54C2×SD16—22no—
55C2×M16—21no—
56C4⋊Q8—22no—
57C4⋊Q8D411yescharacteristic
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62#127D411yescharacteristic
63#126—24no—
64#126—24no—
65#125—24no—
66#125—22no—
67#125—21no—
68#125—21no—
69#124—22no—
70#124—22no—
71#124—21no—
72#123—24no—
73#123—24no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—24no—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8D412yes—
81#121—21no—
82#121D412yes—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8D412yes—
91C4×Q8D412yes—
92C4×Q8C2312yes—
93C4×D4—24no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4C2312yes—
98#117—24no—
99#117—21no—
100#117—22no—
101#117—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—44no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—42no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C2×D412yes—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146Q16—44no—
147SD16—82no—
148M16—81no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C42—24no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—41no—
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2411yescharacteristic
190C23—22no—
191C23#12011yescharacteristic
192Q8—22no—
193Q8—22no—
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—41no—
198Q8—41no—
199Q8—42no—
200Q8—41no—
201Q8—41no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—44no—
206D4—81no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—81no—
221C2×C4—84no—
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#12011yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C8—41no—
229C8—41no—
230C8—42no—
231C8—42no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22C2×#12011yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4#57512yes—
251C4#57711yescharacteristic
252C4#57811yescharacteristic
253C2—41no—
254C2—82no—
255C2#300511yescharacteristic
256C2#301311yescharacteristic
257C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
3. #3231C2semidirect
4. #3113C2semidirect
5. #1652C2semidirect
6. #1650C2semidirect
7. #1644C2semidirect
8. #1643C2semidirect
9. #1559C2semidirect
11. #1487C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32542
#32312
#31132
#16522
#16502
#16442
#16432
#15592
#14962
#14872