Finite Group Databasebeta

← #19469

Subgroups — #19469

Order 256.

305 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 109 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#3206C211yesmaximal, characteristic
2#3112C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic, complemented
4#2984C211yesmaximal, characteristic, complemented
5#1704C212yesmaximal, complemented
6#1686C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1666C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic, complemented
10#1556C211yesmaximal, characteristic, complemented
11#1551C211yesmaximal, characteristic
12#532C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#492C2212yes—
16#487C2212yes—
17#480—21no—
18#477C2211yescharacteristic
19#477C2211yescharacteristic
20#475—21no—
21#475—21no—
22#475C2212yes—
23#475C2212yes—
24#461—22no—
25#460C2211yescharacteristic
26#460C2211yescharacteristic
27#448—21no—
28#448—21no—
29#448—21no—
30#446—22no—
31#446C2212yes—
32C8×Q8C2212yes—
33#440—21no—
34#440—21no—
35#432—21no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431—21no—
39#431C2211yescharacteristic
40C2×#106—21no—
41C2×C4⋊C8C2211yescharacteristic
42C2×#103—21no—
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C22×Q8—21no—
51C22×Q8D411yescharacteristic
52C2×SD16—21no—
53C2×SD16—21no—
54C2×SD16—22no—
55C2×SD16—22no—
56C22×C8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2312yes—
61C4⋊Q8C2311yescharacteristic
62#127C2311yescharacteristic
63#124—21no—
64#121—21no—
65#121—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8D411yescharacteristic
69C4×Q8D411yescharacteristic
70C4×Q8C2311yescharacteristic
71C4×Q8C2311yescharacteristic
72C4×D4—21no—
73C4×D4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80#107—21no—
81#107—21no—
82#106—22no—
83#106—22no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8D412yes—
88C4⋊C8C2312yes—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92#103—22no—
93#103—22no—
94#103—21no—
95#103—21no—
96#103—22no—
97#103—22no—
98#102—21no—
99#102—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8D411yescharacteristic
103C22⋊C8D411yescharacteristic
104#97—22no—
105#97—21no—
106#97—21no—
107C4×C8—22no—
108C4×C8C2312yes—
109#41—82no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—82no—
117C2×Q8C2×D411yescharacteristic
118C2×Q8C2×D411yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4#4111yescharacteristic
130SD16—42no—
131SD16—42no—
132SD16—82no—
133SD16—82no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8C2×D412yes—
140C2×C8C2×D412yes—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4C2×D411yescharacteristic
153C4⋊C4C2×D411yescharacteristic
154C22⋊C4—41no—
155C22⋊C4—41no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42C2×D411yescharacteristic
161C42C2×D411yescharacteristic
162C42#4111yescharacteristic
163C42C2411yescharacteristic
164C23—41no—
165C23#12111yescharacteristic
166Q8—21no—
167Q8—21no—
168Q8—21no—
169Q8—42no—
170Q8—42no—
171Q8—42no—
172Q8—81no—
173Q8—82no—
174Q8#13712yes—
175D4—41no—
176D4—41no—
177D4—42no—
178D4—81no—
179D4—82no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#12111yescharacteristic
195C2×C4#12111yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4C22×D411yescharacteristic
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C8—22no—
201C8—22no—
202C8—41no—
203C8—42no—
204C8—41no—
205C8—42no—
206C22—21no—
207C22—21no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22#47112yes—
212C22C2×#12111yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—81no—
223C4C2×#13711yescharacteristic
224C4#57511yescharacteristic
225C4#57712yes—
226C2—22no—
227C2—161no—
228C2#304311yescharacteristic
229C2#305611yescharacteristic
230C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
3. C2×#475C2semidirect
4. #2984C2semidirect
5. #1704C2semidirect
6. #1686C2semidirect
7. #1684C2semidirect
8. #1666C2semidirect
9. #1573C2semidirect
10. #1556C2semidirect

Maximal subgroups

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

subgroupindex
#32062
#31122
C2×#4752
#29842
#17042
#16862
#16842
#16662
#15732
#15562
#15512