Finite Group Databasebeta

← #18872

Subgroups — #18872

Order 256.

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

normal subgroups only

This group has 477 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#3503C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic
5#1645C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1623C212yesmaximal, complemented
8#1622C212yesmaximal, complemented
9C2×#333C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×#135—21no—
13Q82C2212yes—
14#557—21no—
15#554—22no—
16#554C2212yes—
17D4×Q8—22no—
18D4×Q8—21no—
19#544—22no—
20#544—21no—
21#533—21no—
22#533C2211yescharacteristic
23#532C2211yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8C2211yescharacteristic
26#520—21no—
27C4×#41C2211yescharacteristic
28#492C2212yes—
29#487C2212yes—
30#476—21no—
31#475C2212yes—
32#465—22no—
33#464—21no—
34#463—21no—
35#462C2211yescharacteristic
36#461—22no—
37#461—21no—
38#461C2211yescharacteristic
39#460C2212yes—
40#460C2211yescharacteristic
41#459C2211yescharacteristic
42#448—22no—
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#333C2212yes—
48#333C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—21no—
54C2×#41D411yescharacteristic
55C22×Q8—22no—
56C22×Q8D412yes—
57#135—44no—
58C2×Q16—21no—
59C2×SD16—21no—
60C2×SD16—21no—
61C2×SD16—24no—
62C2×SD16—22no—
63C2×D8—21no—
64C22×C8—21no—
65C22×C8—21no—
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68C4⋊Q8—24no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8D412yes—
74C4⋊Q8D411yescharacteristic
75C4⋊Q8D411yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2312yes—
78#127C2311yescharacteristic
79#125—22no—
80#125—24no—
81#125—22no—
82#125—21no—
83#124—21no—
84C22⋊Q8—22no—
85C22⋊Q8—24no—
86C22⋊Q8—24no—
87C22⋊Q8—22no—
88C22⋊Q8—24no—
89C22⋊Q8—21no—
90C22⋊Q8D412yes—
91#121—21no—
92#121—21no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—21no—
97C4×Q8D411yescharacteristic
98C4×Q8C2311yescharacteristic
99C4×Q8C2312yes—
100C4×D4—24no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4D411yescharacteristic
107C4×D4C2311yescharacteristic
108#117—22no—
109#117—21no—
110#117—21no—
111#117D411yescharacteristic
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118#106—22no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8C2312yes—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125#103—24no—
126#103—22no—
127#103—21no—
128#103—21no—
129#102—21no—
130#102—21no—
131#102—22no—
132C22⋊C8—21no—
133C4×C8C2312yes—
134#41—22no—
135#41—42no—
136#41—42no—
137#41SD1614yescomplemented
138C2×Q8—24no—
139C2×Q8—22no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—84no—
143C2×Q8—82no—
144C2×Q8C2×D412yes—
145C2×Q8C2×D411yescharacteristic
146C2×Q8C2×D414yes—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4C2×D411yescharacteristic
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162Q16—42no—
163SD16—42no—
164SD16—42no—
165SD16—82no—
166SD16—84no—
167D8—42no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—42no—
173C2×C8—81no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D412yes—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C42—24no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—22no—
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2411yescharacteristic
210C23—21no—
211C23—41no—
212C23#12011yescharacteristic
213Q8—21no—
214Q8—22no—
215Q8—22no—
216Q8—44no—
217Q8—44no—
218Q8—42no—
219Q8—41no—
220Q8—41no—
221Q8—84no—
222Q8C2×SD1612yes—
223D4—24no—
224D4—21no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—82no—
231D4C2×SD1612yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4#12011yescharacteristic
252C2×C4C2×SD1612yes—
253C2×C4#13712yes—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C8—42no—
258C8—41no—
259C8—41no—
260C8—42no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22#44812yes—
270C22C2×#12011yescharacteristic
271C4—22no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—44no—
279C4—42no—
280C4—81no—
281C4#44812yes—
282C4C22×SD1611yescharacteristic
283C4C2×#13711yescharacteristic
284C4#57711yescharacteristic
285C4#57811yescharacteristic
286C2—22no—
287C2—42no—
288C2—161no—
289C2C2×#44811yescharacteristic
290C2#301311yescharacteristic
291C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3035C2
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. #3246C2semidirect
3. #3206C2semidirect
5. #1645C2semidirect
6. #1643C2semidirect
7. #1623C2semidirect
8. #1622C2semidirect
9. C2×#333C2semidirect
10. #1481C2semidirect
11. #1477C2semidirect
137. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#35032
#32462
#32062
#30352
#16452
#16432
#16232
#16222
C2×#3332
#14812
#14772