Finite Group Databasebeta

← #18884

Subgroups — #18884

Order 256.

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

normal subgroups only

This group has 471 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#3227C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic
5#1653C212yesmaximal, complemented
6#1651C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1626C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×#135—21no—
13Q82—21no—
14#557C2212yes—
15#554—21no—
16#554—22no—
17D4×Q8—22no—
18D4×Q8C2212yes—
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#498C2212yes—
29#486C2212yes—
30#481—22no—
31#476—22no—
32#476—21no—
33#475C2212yes—
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—22no—
56C22×Q8—21no—
57#135—44no—
58C2×Q16—21no—
59C2×SD16—21no—
60C2×SD16—21no—
61C2×SD16—22no—
62C2×D8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8—24no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8D411yescharacteristic
73C4⋊Q8D412yes—
74C4⋊Q8D411yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2312yes—
77#127C2311yescharacteristic
78#125—22no—
79#125—24no—
80#125—22no—
81#125—21no—
82#124—21no—
83C22⋊Q8—22no—
84C22⋊Q8—24no—
85C22⋊Q8—24no—
86C22⋊Q8—21no—
87C22⋊Q8—22no—
88C22⋊Q8—24no—
89C22⋊Q8D412yes—
90#121—21no—
91#121—21no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—21no—
96C4×Q8D411yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×Q8C2312yes—
99C4×D4—24no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4D411yescharacteristic
106C4×D4C2311yescharacteristic
107#117—22no—
108#117—21no—
109#117—21no—
110#117D411yescharacteristic
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4D412yes—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117#106—22no—
118#106—24no—
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—42no—
136#41—42no—
137#41SD1614yescomplemented
138C2×Q8—24no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—21no—
143C2×Q8—81no—
144C2×Q8—81no—
145C2×Q8—82no—
146C2×Q8—82no—
147C2×Q8C2×D412yes—
148C2×Q8C2×D411yescharacteristic
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164Q16—42no—
165SD16—42no—
166SD16—42no—
167SD16—82no—
168D8—42no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8—81no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
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×D414yes—
194C4⋊C4C2×D412yes—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C42—24no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—22no—
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2×D411yescharacteristic
210C42C2411yescharacteristic
211C23—21no—
212C23—41no—
213C23#12011yescharacteristic
214Q8—21no—
215Q8—22no—
216Q8—22no—
217Q8—42no—
218Q8—42no—
219Q8—42no—
220Q8—41no—
221Q8—41no—
222Q8—82no—
223Q8—84no—
224Q8C2×SD1612yes—
225D4—24no—
226D4—21no—
227D4—42no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4C2×SD1612yes—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4#12011yescharacteristic
252C2×C4#12011yescharacteristic
253C2×C4#12011yescharacteristic
254C2×C4C2×SD1612yes—
255C2×C4#13712yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C8—42no—
260C8—41no—
261C8—41no—
262C8—42no—
263C22—21no—
264C22—21no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22#44812yes—
272C22C2×#12011yescharacteristic
273C4—22no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—41no—
278C4—42no—
279C4—44no—
280C4—41no—
281C4—42no—
282C4—81no—
283C4#44812yes—
284C4C22×SD1611yescharacteristic
285C4C2×#13711yescharacteristic
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—22no—
289C2—42no—
290C2—161no—
291C2C2×#44811yescharacteristic
292C2#301211yescharacteristic
293C2#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. #3503C2semidirect
2. #3246C2semidirect
3. #3227C2semidirect
5. #1653C2semidirect
6. #1651C2semidirect
7. #1627C2semidirect
8. #1626C2semidirect
9. C2×#332C2semidirect
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
#32272
#30352
#16532
#16512
#16272
#16262
C2×#3322
#14812
#14772