Finite Group Databasebeta

← #8574

Subgroups — #8574

Order 256.

307 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3C2×#328C212yesmaximal
4#1434C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8C411yescharacteristic, complemented
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8C2212yes—
11#391C411yescharacteristic, complemented
12#389C411yescharacteristic, complemented
13#387—21no—
14#385—21no—
15#380C2211yescharacteristic
16C4×C4⋊C4—21no—
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#328—24no—
24C22×Q8C2×C411yescharacteristic
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4C2311yescharacteristic
29C22×C8—22no—
30C22⋊Q8—21no—
31C22⋊Q8—21no—
32C22⋊Q8D412yes—
33C22⋊Q8Q812yes—
34C4×Q8—42no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C42—21no—
44C2×C42—21no—
45C2×C42—42no—
46C2×C42C2×C411yescharacteristic
47C2×C42C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C4⋊C8—42no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53#95—24no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—22no—
60#95—22no—
61#95—42no—
62C24C2×D411yescharacteristic
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8C4⋊C412yes—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×Q811yescharacteristic
94C22×C4#4111yescharacteristic
95C22×C4#4111yescharacteristic
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—42no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4C4⋊C412yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—42no—
110C42—22no—
111C42—22no—
112C42—41no—
113C42—41no—
114C42—42no—
115C42—42no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23C2×C22⋊C411yescharacteristic
121C23#12111yescharacteristic
122C23C22⋊Q811yescharacteristic
123C23#13512yes—
124C23#13712yes—
125Q8—42no—
126Q8—42no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4#9912yes—
167C2×C4#10412yes—
168C2×C4C2×C4⋊C411yescharacteristic
169C2×C4#11711yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4C22⋊Q811yescharacteristic
172C8—82no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22#38011yescharacteristic
182C22C2×#9911yescharacteristic
183C22#41011yescharacteristic
184C22#41611yescharacteristic
185C22#41911yescharacteristic
186C22C2×#10411yescharacteristic
187C22#42111yescharacteristic
188C22#46412yes—
189C22#47912yes—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—41no—
199C4—41no—
200C4—84no—
201C4—81no—
202C4—81no—
203C2—22no—
204C2—22no—
205C2#150011yescharacteristic
206C2#150811yescharacteristic
207C2#151511yescharacteristic
208C2#151611yescharacteristic
209C2#179911yescharacteristic
210C2#180511yescharacteristic
211C2#180611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8C4semidirect
11. #391C4semidirect
12. #389C4semidirect
17. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#17972
C2×#3282
#14342
#12942