Finite Group Databasebeta

← #9919

Subgroups — #9919

Order 256.

358 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 67 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#2618C211yesmaximal, characteristic
3C2×#398C211yesmaximal, characteristic, complemented
4#1823C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic
6C2×#324C211yesmaximal, characteristic, complemented
7#1294C211yesmaximal, characteristic, complemented
8C2×#123—21no—
9C2×#121C2211yescharacteristic
10C22×C4⋊C4—21no—
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13#480—22no—
14#480—22no—
15C2×#107—21no—
16C2×#102—21no—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#398—21no—
20#398—22no—
21#398—22no—
22#396—22no—
23#394—21no—
24#386—21no—
25#380C2211yescharacteristic
26C2×#95—21no—
27C2×#95C2211yescharacteristic
28#340—22no—
29#327—22no—
30#324—22no—
31C22×D4—21no—
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—21no—
39C22×C8—21no—
40#123—42no—
41#123—42no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D412yes—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68#107—24no—
69#102—24no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8D412yes—
74#95—24no—
75#95—22no—
76#95—21no—
77#95—21no—
78#95D412yes—
79C24—21no—
80C24C2×D411yescharacteristic
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C22×C4—24no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4D812yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4111yescharacteristic
122C2×C8—22no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—41no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—42no—
131C4⋊C4—48no—
132C4⋊C4—44no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C22⋊C4—44no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—82no—
145C23—22no—
146C23—22no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23#12011yescharacteristic
153C23#12011yescharacteristic
154C23#12111yescharacteristic
155C23C2×D811yescharacteristic
156C23#13511yescharacteristic
157C23AGL(1,ℤ/8)11yescharacteristic
158C23AGL(1,ℤ/8)11yescharacteristic
159D4—42no—
160D4—81no—
161D4—81no—
162D4—81no—
163D4—81no—
164D4—81no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4#12011yescharacteristic
201C2×C4#12111yescharacteristic
202C2×C4#12111yescharacteristic
203C2×C4#12711yescharacteristic
204C8—42no—
205C8—81no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—82no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#39211yescharacteristic
220C22#44711yescharacteristic
221C22#44711yescharacteristic
222C22#44911yescharacteristic
223C22#44911yescharacteristic
224C22#45612yes—
225C22#45812yes—
226C22#46611yescharacteristic
227C22#46911yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—82no—
235C4—82no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2—21no—
241C2—21no—
242C2—161no—
243C2#160912yes—
244C2#161012yes—
245C2#199911yescharacteristic
246C2#200511yescharacteristic
247C2#202211yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#398C2semidirect
4. #1823C2semidirect
6. C2×#324C2semidirect
7. #1294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
#26182
C2×#3982
#18232
C2×#3272
C2×#3242
#12942