Finite Group Databasebeta

← #18886

Subgroups — #18886

Order 256.

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

normal subgroups only

This group has 454 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#3509C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic
5#1650C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1628C212yesmaximal, complemented
8#1626C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×#135—21no—
13#557—21no—
14#557C2212yes—
15#556—21no—
16#556C2212yes—
17#552—22no—
18#552—22no—
19#541—22no—
20#536—21no—
21#533—21no—
22#533—21no—
23#533C2211yescharacteristic
24#532C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26#519—21no—
27C4×#41C2211yescharacteristic
28#500C2212yes—
29#486C2212yes—
30#480—22no—
31#476—21no—
32#475C2212yes—
33#474—22no—
34#465—22no—
35#464—21no—
36#463—21no—
37#462C2211yescharacteristic
38#461—21no—
39#461C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2212yes—
42#459C2211yescharacteristic
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—21no—
54C2×#41D411yescharacteristic
55C22×Q8—21no—
56#135—44no—
57C2×Q16—21no—
58C2×SD16—21no—
59C2×SD16—21no—
60C2×SD16—22no—
61C2×D8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C4⋊Q8—22no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D412yes—
69C4⋊Q8D411yescharacteristic
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2312yes—
72#127C2311yescharacteristic
73#126—24no—
74#126—24no—
75#125—22no—
76#125—24no—
77#125—22no—
78#125—24no—
79#125—21no—
80#124—22no—
81#124—21no—
82#123—24no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—21no—
88C22⋊Q8D412yes—
89#121—21no—
90#121—21no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—21no—
98C4×Q8D411yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×Q8C2312yes—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4D411yescharacteristic
106C4×D4C2311yescharacteristic
107#117—24no—
108#117—22no—
109#117—21no—
110#117—22no—
111#117—21no—
112#117D411yescharacteristic
113C2×C4⋊C4—21no—
114C2×C4⋊C4D412yes—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117#107—24no—
118#106—22no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8C2312yes—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125#103—22no—
126#103—21no—
127#103—21no—
128#102—21no—
129#102—21no—
130#102—22no—
131#102—24no—
132C22⋊C8—21no—
133C4×C8C2312yes—
134#41—22no—
135#41—22no—
136#41—42no—
137#41—42no—
138C2×Q8—21no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—21no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C2×D412yes—
145C2×Q8C2×D411yescharacteristic
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—41no—
151C2×D4C2×D411yescharacteristic
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160Q16—42no—
161SD16—42no—
162SD16—42no—
163SD16—82no—
164D8—42no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—42no—
170C2×C8—81no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4C2×D411yescharacteristic
188C4⋊C4C2×D411yescharacteristic
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D414yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—24no—
200C42—24no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—22no—
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2411yescharacteristic
209C23—21no—
210C23—41no—
211C23#12011yescharacteristic
212Q8—21no—
213Q8—21no—
214Q8—22no—
215Q8—22no—
216Q8—42no—
217Q8—42no—
218Q8—42no—
219Q8—44no—
220Q8—41no—
221Q8—41no—
222D4—21no—
223D4—21no—
224D4—42no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4#12011yescharacteristic
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4#13512yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C8—42no—
256C8—41no—
257C8—41no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#44912yes—
268C22C2×#12011yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—44no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—81no—
279C4#44912yes—
280C4C2×#13511yescharacteristic
281C4C2×AGL(1,ℤ/8)11yescharacteristic
282C4#57711yescharacteristic
283C4#57811yescharacteristic
284C2—22no—
285C2—81no—
286C2—161no—
287C2C2×#44911yescharacteristic
288C2#301311yescharacteristic
289C2#302011yescharacteristic
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. #3509C2semidirect
2. #3246C2semidirect
3. #3226C2semidirect
5. #1650C2semidirect
6. #1648C2semidirect
7. #1628C2semidirect
8. #1626C2semidirect
9. C2×#332C2semidirect
10. #1481C2semidirect
11. #1477C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#32462
#32262
#30352
#16502
#16482
#16282
#16262
C2×#3322
#14812
#14772