Finite Group Databasebeta

← #9079

Subgroups — #9079

Order 256.

568 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×C22⋊C8C211yesmaximal, characteristic, complemented
2C2×#386C211yesmaximal, characteristic
3#1779C211yesmaximal, characteristic, complemented
4C2×#323C212yesmaximal, complemented
5#1280C212yesmaximal, complemented
6C23×D4C411yescharacteristic
7C24×C4C2211yescharacteristic
8C23×C8—21no—
9C22×C22⋊C4C411yescharacteristic
10C22×C22⋊C4C411yescharacteristic
11C22×C22⋊C4C2211yescharacteristic
12C2×C22⋊C8—21no—
13C2×C22⋊C8—24no—
14C2×C22⋊C8—21no—
15C2×C22⋊C8—21no—
16C2×C22⋊C8C2212yes—
17C2×C22⋊C8C2212yes—
18#386—21no—
19#386—21no—
20#386—21no—
21#386—21no—
22C2×#95C411yescharacteristic
23C2×#95C2211yescharacteristic
24#323—24no—
25C25C2×C411yescharacteristic
26C25C2×C411yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4—22no—
30C22×D4—42no—
31C22×D4—42no—
32C22×D4C814yescomplemented
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—42no—
36C23×C4—42no—
37C23×C4C2×C411yescharacteristic
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C411yescharacteristic
40C23×C4C2×C411yescharacteristic
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C22×C8—24no—
47C22×C8—22no—
48C22×C8—22no—
49C22×C8—42no—
50C22×C8—42no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C22⋊C8—24no—
62C22⋊C8—22no—
63C22⋊C8—22no—
64C22⋊C8—22no—
65C22⋊C8—42no—
66C22⋊C8—42no—
67C22⋊C8D414yescomplemented
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72C24—22no—
73C24—22no—
74C24—22no—
75C24—22no—
76C24—21no—
77C24—21no—
78C24—22no—
79C24—22no—
80C24—41no—
81C24—41no—
82C24—41no—
83C24—41no—
84C24—41no—
85C24—41no—
86C24C22⋊C411yescharacteristic
87C24C22⋊C411yescharacteristic
88C24C2×C812yes—
89C24C2×C812yescomplemented
90C24M1612yes—
91C24C22×C411yescharacteristic
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—48no—
95C2×D4—42no—
96C2×D4—42no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—48no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—48no—
126C22×C4—48no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4C22⋊C411yescharacteristic
134C22×C4C22⋊C411yescharacteristic
135C22×C4C2×C812yescomplemented
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4#4112yes—
140C2×C8—24no—
141C2×C8—48no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—41no—
152C22⋊C4—42no—
153C22⋊C4—41no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—24no—
163C23—24no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—24no—
171C23—24no—
172C23—22no—
173C23—24no—
174C23—42no—
175C23—42no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23—82no—
192C23—82no—
193C23—82no—
194C23—82no—
195C23C22⋊C812yes—
196C23C22⋊C812yes—
197C23#9912yes—
198C23#10012yes—
199C23C2×C22⋊C411yescharacteristic
200C23C4×D412yes—
201C23C22×C811yescharacteristic
202C23C2×M1611yescharacteristic
203C23#13112yes—
204D4—48no—
205D4—48no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4#9912yes—
238C2×C4#12011yescharacteristic
239C2×C4#12112yes—
240C2×C4#12311yescharacteristic
241C8—44no—
242C8—82no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—48no—
261C22—48no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—84no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22#32314yescomplemented
276C22#38611yescharacteristic
277C22C2×C22⋊C811yescharacteristic
278C22#40811yescharacteristic
279C22C2×#9911yescharacteristic
280C22C2×#9911yescharacteristic
281C22C2×#10011yescharacteristic
282C22#41311yescharacteristic
283C22C8×D412yes—
284C22#43512yes—
285C4—21no—
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—42no—
291C4—81no—
292C4—81no—
293C2—22no—
294C2—22no—
295C2—24no—
296C2—42no—
297C2—82no—
298C2C2×#32311yescharacteristic
299C2#145911yescharacteristic
300C2#146811yescharacteristic
301C2#147111yescharacteristic
302C2#186911yescharacteristic
303C2#189811yescharacteristic
304C2#190011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC23×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8C2semidirect
3. #1779C2semidirect
4. C2×#323C2semidirect
5. #1280C2semidirect
32. C22×D4C8semidirect
67. C22⋊C8D4semidirect
89. C24C2×C8semidirect
135. C22×C4C2×C8semidirect
275. C22#323semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C82
C2×#3862
#17792
C2×#3232
#12802