Finite Group Databasebeta

← #18877

Subgroups — #18877

Order 256.

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

normal subgroups only

This group has 461 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#3511C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3209C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1646C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1634C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9C2×#333C211yesmaximal, characteristic, complemented
10#1480C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12#556—21no—
13#555C2212yes—
14#554C2212yes—
15#553—22no—
16#548—22no—
17#547—21no—
18#542—22no—
19#537—21no—
20#533—21no—
21#533C2211yescharacteristic
22#532—21no—
23#532C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#494C2212yes—
28#487C2212yes—
29#479—21no—
30#478—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#476C2212yes—
34#475—21no—
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#465—22no—
38#462—22no—
39#461C2212yes—
40#460—21no—
41#451—22no—
42#431C2211yescharacteristic
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#333C2212yes—
48#333C2212yes—
49Q8⋊C8C2212yes—
50D4⋊C8C2212yes—
51D4⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41D411yescharacteristic
56C22×Q8D412yes—
57C2×Q16—24no—
58C2×SD16—22no—
59C22×C8—21no—
60C22×C8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8D412yes—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2312yes—
68#127D411yescharacteristic
69#126—24no—
70#125—22no—
71#125—21no—
72#124—22no—
73#124—24no—
74#124—21no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82#121—22no—
83#121—24no—
84#121—22no—
85#121—22no—
86#121—21no—
87#121D412yes—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8C2311yescharacteristic
94C4×D4—22no—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4D411yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2312yes—
103#117—22no—
104#117—22no—
105#117—21no—
106#117—21no—
107#117D411yescharacteristic
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113#107—21no—
114#107—21no—
115#106—21no—
116#106—21no—
117#106—22no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2312yes—
124#103—21no—
125#103—21no—
126#103—24no—
127#103—22no—
128#102—21no—
129#102—21no—
130#102—22no—
131C22⋊C8—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135#41—42no—
136#41—42no—
137#41—84no—
138C2×Q8—21no—
139C2×Q8—22no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—82no—
143C2×Q8C2×D411yescharacteristic
144C2×Q8C2×D414yes—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—24no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×D411yescharacteristic
151C2×D4C2×D412yes—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161Q16—84no—
162SD16—82no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—42no—
168C2×C8—81no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D411yescharacteristic
186C4⋊C4C2×D411yescharacteristic
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—22no—
193C22⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—41no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—41no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—21no—
206C23—22no—
207C23#12011yescharacteristic
208Q8—21no—
209Q8—21no—
210Q8—44no—
211Q8—44no—
212Q8—42no—
213Q8—42no—
214Q8—41no—
215Q8—41no—
216D4—21no—
217D4—21no—
218D4—22no—
219D4—22no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—42no—
224D4—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—84no—
243C2×C4#12011yescharacteristic
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4#13512yes—
247C2×C4#13712yes—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C8—42no—
252C8—41no—
253C8—41no—
254C8—42no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—42no—
263C22#45212yes—
264C22C2×#12011yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—41no—
272C4—44no—
273C4—81no—
274C4—81no—
275C4#45212yes—
276C4C2×#13511yescharacteristic
277C4C2×#13711yescharacteristic
278C4#57711yescharacteristic
279C4#57811yescharacteristic
280C2—22no—
281C2—81no—
282C2—82no—
283C2C2×#45211yescharacteristic
284C2#301311yescharacteristic
285C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3511C2
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
2. #3246C2semidirect
3. #3209C2semidirect
4. #3076C2semidirect
5. #1646C2semidirect
6. #1643C2semidirect
7. #1634C2semidirect
8. #1632C2semidirect
9. C2×#333C2semidirect
10. #1480C2semidirect
11. #1477C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#32462
#32092
#30762
#16462
#16432
#16342
#16322
C2×#3332
#14802
#14772