Finite Group Databasebeta

← #9260

Subgroups — #9260

Order 256.

297 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 71 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×#480C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1823C211yesmaximal, characteristic, complemented
5#1519C211yesmaximal, characteristic
6#1289C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#117—21no—
10C22×C4⋊C4C2211yescharacteristic
11#480—22no—
12#480—21no—
13#480—21no—
14C2×#107C412yescomplemented
15C2×#102C412yescomplemented
16C2×#99—21no—
17C2×#99—21no—
18C2×C22⋊C8C2211yescharacteristic
19C2×C22⋊C8C2211yescharacteristic
20#389—21no—
21#387—22no—
22#381—21no—
23#380C2211yescharacteristic
24#380C2211yescharacteristic
25C4×C4⋊C4—21no—
26C4×C22⋊C4C2211yescharacteristic
27#340—22no—
28#327—22no—
29C22×D4—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C23×C4C2311yescharacteristic
33C22×C8—21no—
34C22×C8C2×C412yes—
35#121—21no—
36#121—21no—
37#121—21no—
38#121—21no—
39#117—42no—
40#117—42no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4C2×C411yescharacteristic
48C2×C4⋊C4C2×C412yes—
49C2×C4⋊C4D412yes—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4D411yescharacteristic
56C2×C22⋊C4D411yescharacteristic
57C2×C42—22no—
58C2×C42—21no—
59#107—22no—
60#107—22no—
61#102—22no—
62#102—22no—
63#99—42no—
64#99—42no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69#95—22no—
70#95—22no—
71#95—21no—
72#95—21no—
73#95D412yes—
74C24—21no—
75C24C2×D411yescharacteristic
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—42no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C22×C411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4111yescharacteristic
104C22×C4#4111yescharacteristic
105C2×C8—22no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—41no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4C22⋊C412yes—
119C4⋊C4C22⋊C412yes—
120C22⋊C4—21no—
121C22⋊C4—21no—
122C22⋊C4—21no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4D812yes—
130C42—42no—
131C42—44no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—81no—
138C23—81no—
139C23C4×D411yescharacteristic
140C23#12011yescharacteristic
141C23#12111yescharacteristic
142C23C2×D811yescharacteristic
143C23#13511yescharacteristic
144C23AGL(1,ℤ/8)11yescharacteristic
145C23AGL(1,ℤ/8)11yescharacteristic
146D4—42no—
147D4—81no—
148D4—81no—
149D4—81no—
150D4—81no—
151D4—81no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4C2×C22⋊C411yescharacteristic
184C2×C4C4×D411yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12311yescharacteristic
187C8—42no—
188C8—81no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22—161no—
204C22#38611yescharacteristic
205C22C4×D811yescharacteristic
206C22#44211yescharacteristic
207C22#44711yescharacteristic
208C22#44911yescharacteristic
209C22#45911yescharacteristic
210C22#46311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—81no—
220C4—161no—
221C4#41012yes—
222C2—21no—
223C2—21no—
224C2—21no—
225C2—21no—
226C2—21no—
227C2—21no—
228C2—161no—
229C2#187911yescharacteristic
230C2#189911yescharacteristic
231C2#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. #2344C2semidirect
4. #1823C2semidirect
6. #1289C2semidirect
7. #1282C2semidirect
14. C2×#107C4semidirect
15. C2×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
#23442
#19642
#18232
#15192
#12892
#12822