Finite Group Databasebeta

← #8364

Subgroups — #8364

Order 256.

537 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#547C211yesmaximal, characteristic
2#1789C212yesmaximal, complemented
3C4×#99C211yesmaximal, characteristic, complemented
4#1508C212yesmaximal, complemented
5#1465C211yesmaximal, characteristic, complemented
6C22×#41C411yescharacteristic, complemented
7C22×#41C2211yescharacteristic
8#547—24no—
9#547—24no—
10C2×C4⋊Q8C411yescharacteristic, complemented
11C2×#123—21no—
12C2×#123—21no—
13C2×C22⋊Q8C411yescharacteristic, complemented
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121—21no—
16C2×C4×D4C411yescharacteristic, complemented
17C2×C4×D4C2211yescharacteristic
18C22×C4⋊C4—21no—
19C2×#104—24no—
20C2×#99—21no—
21#407C2212yes—
22C4×C22⋊C4C2212yes—
23#342—21no—
24#328—24no—
25#324—21no—
26#323—21no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—44no—
31C2×#41—44no—
32C2×#41D414yes—
33C22×Q8C2×C411yescharacteristic
34C22×Q8C2×C411yescharacteristic
35C22×D4—21no—
36C22×D4—21no—
37C22×D4C2×C411yescharacteristic
38C22×D4D411yescharacteristic
39C23×C4—21no—
40C23×C4—21no—
41C23×C4—21no—
42C23×C4C2×C411yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47#123—28no—
48#123—28no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8D414yes—
53#121—28no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C2×C4⋊C4—28no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2×C411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42—22no—
72C2×C42C2×C411yescharacteristic
73#104—48no—
74#99—24no—
75C22⋊C8—22no—
76C22⋊C8—22no—
77#95—22no—
78C24—21no—
79C24C22⋊C411yescharacteristic
80C24C2×D411yescharacteristic
81#41—28no—
82#41—44no—
83#41—48no—
84#41—48no—
85#41—44no—
86#41—44no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8C22⋊C412yes—
92C2×Q8C22⋊C412yes—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—44no—
101C2×D4—44no—
102C2×D4C22⋊C412yes—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C22⋊C411yescharacteristic
119C22×C4C22⋊C411yescharacteristic
120C22×C4C22⋊C411yescharacteristic
121C22×C4C22×C411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125M16—44no—
126C2×C8—42no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—24no—
133C4⋊C4C22⋊C412yes—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—24no—
140C22⋊C4—42no—
141C42—21no—
142C42—21no—
143C42—44no—
144C23—24no—
145C23—22no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23C2×C22⋊C411yescharacteristic
159C23#12012yes—
160Q8—24no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164D4—24no—
165D4—22no—
166D4—44no—
167D4—48no—
168D4—48no—
169D4#9914yescomplemented
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—48no—
182C2×C4—48no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—48no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#10412yescomplemented
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4#12012yes—
199C8—82no—
200C22—21no—
201C22—21no—
202C22—22no—
203C22—22no—
204C22—24no—
205C22—22no—
206C22—44no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22#37911yescharacteristic
218C22C2×#10411yescharacteristic
219C22#42111yescharacteristic
220C4—22no—
221C4—21no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—84no—
229C4C2×#9912yes—
230C2—21no—
231C2—21no—
232C2—42no—
233C2—44no—
234C2—81no—
235C2#178111yescharacteristic
236C2#178911yescharacteristic
237C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1789C2semidirect
3. C4×#99C2semidirect
4. #1508C2semidirect
5. #1465C2semidirect
6. C22×#41C4semidirect
10. C2×C4⋊Q8C4semidirect
13. C2×C22⋊Q8C4semidirect
16. C2×C4×D4C4semidirect
169. D4#99semidirect
195. C2×C4#104semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
#17892
C4×#992
#15082
#14652