Finite Group Databasebeta

← #8567

Subgroups — #8567

Order 256.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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#2293C211yesmaximal, characteristic
2#1842C211yesmaximal, characteristic, complemented
3C4×C4⋊C8C211yesmaximal, characteristic, complemented
4C2×D4⋊C8C212yesmaximal
5#1279C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#117—21no—
8C2×C4⋊C8—21no—
9C2×C4⋊C8—21no—
10C2×C4⋊C8—21no—
11C2×C4⋊C8C2212yes—
12C2×C4×C8C2212yes—
13#390C411yescharacteristic
14#390C411yescharacteristic
15#389C411yescharacteristic
16#389C411yescharacteristic
17#388—21no—
18#388—21no—
19C4×C4⋊C4C2211yescharacteristic
20C4×C22⋊C4—21no—
21C43C2211yescharacteristic
22#336—21no—
23D4⋊C8—24no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C22×C8—22no—
27C22×C8—22no—
28C4×D4—21no—
29C4×D4—21no—
30C4×D4D412yes—
31C4×D4Q812yes—
32#117—42no—
33#117—42no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42—21no—
45C2×C42C2×C411yescharacteristic
46C2×C42C2×C411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42C2311yescharacteristic
52C4⋊C8—24no—
53C4⋊C8—22no—
54C4⋊C8—22no—
55C4⋊C8—42no—
56C4⋊C8—42no—
57C4×C8—22no—
58C4×C8—22no—
59#95—21no—
60#95—21no—
61#95—21no—
62C24—21no—
63C2×D4—21no—
64C2×D4—22no—
65C2×D4—22no—
66C2×D4C4⋊C412yes—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C2×C8—24no—
84C2×C8—42no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—42no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4C4⋊C412yes—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C42—22no—
102C42—22no—
103C42—22no—
104C42—21no—
105C42—21no—
106C42—41no—
107C42—41no—
108C42—41no—
109C42D812yes—
110C42SD1612yes—
111C42C2×D411yescharacteristic
112C42C2×Q811yescharacteristic
113C42#4111yescharacteristic
114C42#4111yescharacteristic
115C23—22no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23C2×C22⋊C411yescharacteristic
123D4—24no—
124D4—42no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4#10212yes—
150C2×C4#10212yes—
151C2×C4#10412yes—
152C2×C4C2×C4⋊C411yescharacteristic
153C2×C4#11711yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4C22⋊Q811yescharacteristic
157C2×C4C22⋊Q811yescharacteristic
158C2×C4#13112yes—
159C2×C4#13112yes—
160C2×C4C2×D811yescharacteristic
161C2×C4C2×SD1611yescharacteristic
162C2×C4AGL(1,ℤ/8)11yescharacteristic
163C2×C4#13711yescharacteristic
164C8—44no—
165C8—82no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22#38011yescharacteristic
174C22#40811yescharacteristic
175C22#40811yescharacteristic
176C22C2×#10211yescharacteristic
177C22#41711yescharacteristic
178C22C2×#10411yescharacteristic
179C22#42111yescharacteristic
180C4—21no—
181C4—21no—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—42no—
186C4—41no—
187C4—81no—
188C4—81no—
189C4#42412yes—
190C4#43312yes—
191C4#45911yescharacteristic
192C4#46111yescharacteristic
193C4#47411yescharacteristic
194C4#47611yescharacteristic
195C2—82no—
196C2#148011yescharacteristic
197C2#148311yescharacteristic
198C2#149211yescharacteristic
199C2#149511yescharacteristic
200C2#179811yescharacteristic
201C2#180211yescharacteristic
202C2#180611yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1842C2semidirect
3. C4×C4⋊C8C2semidirect
5. #1279C2semidirect

Maximal subgroups

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

subgroupindex
#22932
#18422
C4×C4⋊C82
C2×D4⋊C82
#12792