Finite Group Databasebeta

← #9262

Subgroups — #9262

Order 256.

375 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 75 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×#448C211yesmaximal, characteristic
2Q8×C22⋊C4C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1787C211yesmaximal, characteristic, complemented
5#1519C211yesmaximal, characteristic
6#1289C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C23×Q8C2211yescharacteristic
9C22×SD16C412yescomplemented
10C2×#121C2211yescharacteristic
11C2×C4×Q8—21no—
12#448—21no—
13#448—22no—
14#448—21no—
15C2×#103—22no—
16C2×#103C412yescomplemented
17C2×#99—21no—
18C2×#99—21no—
19C2×C22⋊C8C2211yescharacteristic
20C2×C22⋊C8C2211yescharacteristic
21#391—21no—
22#391—22no—
23#380—21no—
24#380C2211yescharacteristic
25#380C2211yescharacteristic
26C4×C22⋊C4—21no—
27C4×C22⋊C4C2211yescharacteristic
28#327—22no—
29C22×Q8—21no—
30C22×Q8—21no—
31C22×Q8—21no—
32C22×Q8—44no—
33C22×Q8C2×C412yes—
34C22×Q8D411yescharacteristic
35C22×Q8D412yes—
36C22×Q8D411yescharacteristic
37C22×D4—21no—
38C22×D4C2×C411yescharacteristic
39C23×C4—21no—
40C23×C4C2311yescharacteristic
41C2×SD16—22no—
42C2×SD16—22no—
43C2×SD16—44no—
44C22×C8—21no—
45C22×C8C2×C412yes—
46#121—21no—
47#121—21no—
48#121—21no—
49#121—21no—
50C4×Q8—44no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2×C411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4D411yescharacteristic
59C2×C22⋊C4D411yescharacteristic
60C2×C42—22no—
61C2×C42—21no—
62#103—22no—
63#103—22no—
64#103—44no—
65#99—42no—
66#99—42no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71#95—22no—
72#95—22no—
73#95—21no—
74#95—21no—
75#95D412yes—
76C24—21no—
77C24C2×D411yescharacteristic
78C2×Q8—24no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—24no—
84C2×Q8—24no—
85C2×Q8—42no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—44no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8C22⋊C412yes—
92C2×Q8C22⋊C412yes—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—41no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—41no—
113C22×C4C22×C411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4#4111yescharacteristic
118C22×C4#4111yescharacteristic
119SD16—44no—
120SD16—44no—
121C2×C8—22no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—41no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4SD1612yescomplemented
140C42—44no—
141C42—42no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—81no—
148C23—81no—
149C23C4×D411yescharacteristic
150C23#12011yescharacteristic
151C23#12111yescharacteristic
152C23C2×SD1611yescharacteristic
153C23#13511yescharacteristic
154C23AGL(1,ℤ/8)11yescharacteristic
155C23#13711yescharacteristic
156Q8—24no—
157Q8—22no—
158Q8—22no—
159Q8—22no—
160Q8—44no—
161Q8—44no—
162Q8—42no—
163Q8#9914yescomplemented
164D4—42no—
165D4—81no—
166D4—81no—
167D4—81no—
168D4—81no—
169D4—81no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C4×D411yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12311yescharacteristic
201C8—42no—
202C8—81no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—161no—
217C22—161no—
218C22#38611yescharacteristic
219C22C4×SD1611yescharacteristic
220C22#44011yescharacteristic
221C22#44811yescharacteristic
222C22#44911yescharacteristic
223C22#46011yescharacteristic
224C22#46411yescharacteristic
225C4—21no—
226C4—24no—
227C4—21no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4—161no—
235C4C2×#9912yes—
236C2—21no—
237C2—21no—
238C2—21no—
239C2—21no—
240C2—21no—
241C2—21no—
242C2—161no—
243C2#188011yescharacteristic
244C2#189811yescharacteristic
245C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
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. Q8×C22⋊C4C2semidirect
4. #1787C2semidirect
6. #1289C2semidirect
7. #1282C2semidirect
9. C22×SD16C4semidirect
16. C2×#103C4semidirect
139. C22⋊C4SD16semidirect
163. Q8#99semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
Q8×C22⋊C42
#19642
#17872
#15192
#12892
#12822