Finite Group Databasebeta

← #18717

Subgroups — #18717

Order 256.

492 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 115 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#3230C211yesmaximal, characteristic, complemented
3#3227C211yesmaximal, characteristic, complemented
4#3152C211yesmaximal, characteristic, complemented
5#1632C214yesmaximal, complemented
6#1631C214yesmaximal, complemented
7#1543C211yesmaximal, characteristic, complemented
8#1485C211yesmaximal, characteristic, complemented
9#1484C211yesmaximal, characteristic, complemented
10C2×#137—22no—
11#562—22no—
12#554—21no—
13#554C2212yes—
14#550—21no—
15#550C2212yes—
16#547—21no—
17#547—21no—
18#540—22no—
19#537—22no—
20#537—22no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25#500—21no—
26#498—21no—
27#497—22no—
28#494C2212yes—
29#492C2212yes—
30#487—21no—
31#486—21no—
32#484—22no—
33#481—22no—
34#477C2214yes—
35#476C2214yes—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38C4×M16C2211yescharacteristic
39#333C2212yes—
40#331C2212yes—
41Q8⋊C8C2214yes—
42D4⋊C8C2214yes—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—21no—
46C2×#41D412yes—
47C22×Q8—21no—
48#137—48no—
49C2×Q16—24no—
50C2×SD16—24no—
51C2×M16—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D411yescharacteristic
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58C4⋊Q8C2312yes—
59C4⋊Q8C2311yescharacteristic
60#127—22no—
61#127D411yescharacteristic
62#127C2311yescharacteristic
63#125—22no—
64#125—21no—
65#124—21no—
66#124D412yes—
67#123—24no—
68#123—24no—
69C22⋊Q8—22no—
70C22⋊Q8—24no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73C22⋊Q8—21no—
74C22⋊Q8D412yes—
75#121—24no—
76#121—24no—
77#121—22no—
78#121—24no—
79#121—24no—
80#121—21no—
81#121—21no—
82#121—21no—
83#121—22no—
84#121D412yes—
85C4×Q8—21no—
86C4×Q8C2312yes—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4C2312yes—
93#117—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C42C2311yescharacteristic
99#107—24no—
100#106—24no—
101C4⋊C8C2312yes—
102C4⋊C8C2312yes—
103#103—24no—
104#102—24no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107#97—21no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111#41—44no—
112#41—82no—
113#41—84no—
114#41—84no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—44no—
119C2×Q8—41no—
120C2×Q8C2×D412yes—
121C2×Q8C2×D412yes—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4C2×D412yes—
131C2×D4C2×D412yes—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140Q16—48no—
141SD16—48no—
142M16—24no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—22no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—22no—
175C23—22no—
176C23#12011yescharacteristic
177Q8—24no—
178Q8—42no—
179Q8—44no—
180Q8—42no—
181Q8—81no—
182Q8—82no—
183D4—24no—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—81no—
189D4—82no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4—84no—
205C2×C4—82no—
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4#12011yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C8—24no—
213C8—42no—
214C8—42no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×#12011yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—82no—
233C4#45312yes—
234C4#57712yes—
235C4#57812yes—
236C2—41no—
237C2—82no—
238C2—82no—
239C2#300911yescharacteristic
240C2#301111yescharacteristic
241C2C2×#45311yescharacteristic
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. #3230C2semidirect
3. #3227C2semidirect
4. #3152C2semidirect
5. #1632C2semidirect
6. #1631C2semidirect
7. #1543C2semidirect
8. #1485C2semidirect
9. #1484C2semidirect

Maximal subgroups

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

subgroupindex
#35482
#32302
#32272
#31522
#16322
#16312
#15432
#14852
#14842