Finite Group Databasebeta

← #18816

Subgroups — #18816

Order 256.

459 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 117 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#3503C211yesmaximal, characteristic, complemented
2#3216C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4#3115C211yesmaximal, characteristic
5#1645C212yesmaximal, complemented
6#1644C212yesmaximal, complemented
7#1623C212yesmaximal, complemented
8#1622C212yesmaximal, complemented
9#1558C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic, complemented
12Q82C2212yes—
13#557—21no—
14#554—22no—
15#554C2212yes—
16D4×Q8—22no—
17D4×Q8—21no—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#497—21no—
28#492C2212yes—
29#489—21no—
30#487C2212yes—
31#482—21no—
32#475C2211yescharacteristic
33#474C2211yescharacteristic
34#462C2212yes—
35#461—22no—
36#460C2211yescharacteristic
37#460C2212yes—
38#459C2211yescharacteristic
39#452—22no—
40#449—21no—
41#448—22no—
42#442—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
47#333C2212yes—
48#333C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—21no—
54C2×#41D411yescharacteristic
55C22×Q8—22no—
56C22×Q8D412yes—
57C2×Q16—22no—
58C2×SD16—21no—
59C2×SD16—24no—
60C2×SD16—22no—
61C2×D8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—24no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8D412yes—
70C4⋊Q8D411yescharacteristic
71C4⋊Q8D411yescharacteristic
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2312yes—
74#127C2311yescharacteristic
75#125—22no—
76#125—24no—
77#125—22no—
78#125—21no—
79#124—21no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83C22⋊Q8—22no—
84C22⋊Q8—24no—
85C22⋊Q8—21no—
86C22⋊Q8D412yes—
87#121—21no—
88#121—21no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—21no—
93C4×Q8D411yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×Q8C2312yes—
96C4×D4—24no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4D411yescharacteristic
103C4×D4C2311yescharacteristic
104#117—22no—
105#117—21no—
106#117—21no—
107#117D411yescharacteristic
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#106—21no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—24no—
122#103—22no—
123#102—21no—
124#102—21no—
125#102—21no—
126#102—22no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135#41—82no—
136C2×Q8—24no—
137C2×Q8—22no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—84no—
141C2×Q8—82no—
142C2×Q8C2×D412yes—
143C2×Q8C2×D411yescharacteristic
144C2×Q8C2×D414yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—41no—
150C2×D4C2×D411yescharacteristic
151C22×C4—21no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160Q16—82no—
161SD16—42no—
162SD16—82no—
163SD16—84no—
164D8—42no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D411yescharacteristic
188C4⋊C4C2×D411yescharacteristic
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—22no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—21no—
206C23—41no—
207C23#12011yescharacteristic
208Q8—21no—
209Q8—22no—
210Q8—22no—
211Q8—44no—
212Q8—44no—
213Q8—42no—
214Q8—42no—
215Q8—84no—
216Q8#13712yes—
217D4—24no—
218D4—21no—
219D4—42no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—82no—
224D4#13712yes—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4#12011yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C8—42no—
249C8—41no—
250C8—41no—
251C8—42no—
252C22—21no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22C2×#12011yescharacteristic
262C4—21no—
263C4—22no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—44no—
271C4—42no—
272C4—81no—
273C4C2×#13711yescharacteristic
274C4C2×#13711yescharacteristic
275C4#57712yes—
276C2—41no—
277C2—42no—
278C2—161no—
279C2#300411yescharacteristic
280C2#301011yescharacteristic
281C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3115C2
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
1. #3503C2semidirect
2. #3216C2semidirect
3. #3206C2semidirect
5. #1645C2semidirect
6. #1644C2semidirect
7. #1623C2semidirect
8. #1622C2semidirect
9. #1558C2semidirect
10. #1497C2semidirect
11. #1488C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#32162
#32062
#31152
#16452
#16442
#16232
#16222
#15582
#14972
#14882