Finite Group Databasebeta

← #18947

Subgroups — #18947

Order 256.

451 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 113 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#3548C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3235C211yesmaximal, characteristic, complemented
4#3231C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1647C212yesmaximal, complemented
7#1635C212yesmaximal, complemented
8#1635C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic, complemented
11#1480C211yesmaximal, characteristic, complemented
12#562—22no—
13#554—21no—
14#554—21no—
15#550—21no—
16#550—21no—
17#547C2212yes—
18#547C2212yes—
19#540—22no—
20#537—22no—
21#537—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#532C2211yescharacteristic
25#532C2211yescharacteristic
26#501—21no—
27#500C2211yescharacteristic
28#498C2211yescharacteristic
29#490—21no—
30#487C2211yescharacteristic
31#487C2211yescharacteristic
32#485—22no—
33#482—22no—
34#479—22no—
35#476C2212yes—
36#476C2212yes—
37#475—21no—
38#474C2212yes—
39#465—22no—
40#461C2212yes—
41#460—21no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45#332C2212yes—
46#332C2212yes—
47D4⋊C8C2212yes—
48D4⋊C8C2212yes—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—21no—
54C2×#41—21no—
55C2×SD16—22no—
56C22×C8—21no—
57C4⋊Q8D411yescharacteristic
58C4⋊Q8D411yescharacteristic
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#127—22no—
64#127D411yescharacteristic
65#127D411yescharacteristic
66#125—22no—
67#125—21no—
68#125D412yes—
69#124—21no—
70#124—21no—
71#123—24no—
72#123—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81#121—24no—
82#121—24no—
83#121—24no—
84#121—24no—
85#121—22no—
86#121—22no—
87#121—21no—
88#121—21no—
89#121D412yes—
90#121D412yes—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C2312yes—
100C4×D4C2312yes—
101#117—21no—
102#117D412yes—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#106—22no—
109#106—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—81no—
127#41—82no—
128#41—81no—
129#41—82no—
130#41—84no—
131#41—82no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×D412yes—
147C2×D4C2×D412yes—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155SD16—82no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—81no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C4⋊C4C2×D412yes—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—41no—
187C42—22no—
188C42—41no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2411yescharacteristic
193C23—22no—
194C23—22no—
195C23#12011yescharacteristic
196Q8—42no—
197Q8—41no—
198Q8—42no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—81no—
203Q8—81no—
204D4—22no—
205D4—24no—
206D4—22no—
207D4—42no—
208D4—42no—
209D4—81no—
210D4—81no—
211D4—82no—
212D4—82no—
213D4—84no—
214D4—84no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C8—41no—
237C8—41no—
238C8—42no—
239C8—42no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22C2×#12011yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4#57611yescharacteristic
261C4#57712yes—
262C4#57711yescharacteristic
263C2—41no—
264C2—82no—
265C2—82no—
266C2#300911yescharacteristic
267C2#301211yescharacteristic
268C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3548C2
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. #3235C2semidirect
4. #3231C2semidirect
5. #1654C2semidirect
6. #1647C2semidirect
7. #1635C2semidirect
8. #1635C2semidirect
9. #1552C2semidirect
10. #1496C2semidirect
11. #1480C2semidirect

Maximal subgroups

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

subgroupindex
#35482
#32462
#32352
#32312
#16542
#16472
#16352
#15522
#14962
#14802