Finite Group Databasebeta

← C2×#1468 #17373

Subgroups — C2×#1468 #17373

Order 256.

745 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 199 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×#123C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3C22×C22⋊C8C211yesmaximal, characteristic, complemented
4#1468C218yesmaximal, direct product factor
5C2×#324C212yesmaximal, complemented
6C2×#323C212yesmaximal, complemented
7C23×D4C411yescharacteristic
8C24×C4C2211yescharacteristic
9C22×M16—21no—
10C23×C8—21no—
11C2×#123—21no—
12C2×#123—21no—
13C2×#123—22no—
14C2×#123—21no—
15C2×#123—21no—
16C2×#123—22no—
17C2×#123C414yes—
18C2×#123C2214yes—
19C22×C4⋊C4C411yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C411yescharacteristic
22C22×C22⋊C4C411yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24#407—22no—
25#407C2214yescomplemented
26C2×C22⋊C8—22no—
27C2×C22⋊C8—21no—
28C2×C22⋊C8—21no—
29C2×C22⋊C8C2212yes—
30C2×C22⋊C8C2212yes—
31C2×C22⋊C8C2214yescomplemented
32#324C2218yescomplemented
33#323C2218yescomplemented
34C25C2×C411yescharacteristic
35C25C2×C411yescharacteristic
36C22×D4—22no—
37C22×D4—21no—
38C22×D4—21no—
39C22×D4—44no—
40C22×D4C2×C414yes—
41C23×C4—44no—
42C23×C4C2×C411yescharacteristic
43C23×C4C2×C411yescharacteristic
44C23×C4C2×C411yescharacteristic
45C23×C4C2×C411yescharacteristic
46C23×C4C2×C412yes—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D412yes—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4D412yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C2×M16—24no—
56C2×M16—44no—
57C22×C8—22no—
58C22×C8—24no—
59C22×C8—22no—
60C22×C8—44no—
61#123—24no—
62#123—24no—
63#123—24no—
64#123—24no—
65#123—24no—
66#123—24no—
67#123—24no—
68#123—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2×C414yes—
77C2×C4⋊C4C2×C412yes—
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4C2×C414yes—
90C2×C22⋊C4C2×C414yes—
91C2×C22⋊C4C2×C412yes—
92C2×C22⋊C4C2312yes—
93C22⋊C8—22no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C22⋊C8—22no—
97C22⋊C8C2314yes—
98C22⋊C8C2314yes—
99C24—24no—
100C24—24no—
101C24—21no—
102C24—21no—
103C24—24no—
104C24—42no—
105C24—42no—
106C24—42no—
107C24C22⋊C411yescharacteristic
108C24C22⋊C412yes—
109C24C22⋊C411yescharacteristic
110C24C22×C412yes—
111C24C22×C411yescharacteristic
112C24C22×C412yes—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—48no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—48no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C22⋊C412yes—
139C22×C4C22⋊C412yes—
140C22×C4C22⋊C411yescharacteristic
141C22×C4C22⋊C411yescharacteristic
142C22×C4C22⋊C412yes—
143C22×C4C22⋊C412yes—
144C22×C4C22⋊C412yes—
145C22×C4C22×C412yes—
146C22×C4C22×C412yes—
147C22×C4C22×C412yes—
148C22×C4C22×C412yes—
149C22×C4C22×C411yescharacteristic
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159M16—48no—
160C2×C8—24no—
161C2×C8—24no—
162C2×C8—44no—
163C2×C8—48no—
164C2×C8—44no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—24no—
185C23—22no—
186C23—24no—
187C23—22no—
188C23—24no—
189C23—22no—
190C23—22no—
191C23—24no—
192C23—42no—
193C23—42no—
194C23—416no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—88no—
204C23#9914yes—
205C23C2×C22⋊C412yes—
206C23C2×C22⋊C411yescharacteristic
207C23C2×C22⋊C412yes—
208C23C2×C22⋊C411yescharacteristic
209C23#13114yes—
210C23C23×C411yescharacteristic
211D4—48no—
212D4—48no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4C2×C22⋊C411yescharacteristic
244C2×C4C2×C22⋊C411yescharacteristic
245C2×C4C2×C22⋊C412yes—
246C2×C4C2×C22⋊C412yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C8—44no—
250C8—44no—
251C22—22no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—28no—
258C22—28no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—48no—
263C22—48no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—84no—
269C22—84no—
270C22—84no—
271C22#40812yes—
272C22#40812yes—
273C22C2×#9914yes—
274C22C2×#9912yes—
275C22#41312yes—
276C22C22×C22⋊C411yescharacteristic
277C22C2×#13112yes—
278C4—21no—
279C4—21no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—42no—
287C2—22no—
288C2—22no—
289C2—24no—
290C2—42no—
291C2—82no—
292C2#146814yesdirect product factor
293C2C2×#40811yescharacteristic
294C2C22×#9911yescharacteristic
295C2C2×#41311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. C22×C22⋊C8C2semidirect
4. #1468C2direct (N × H)
5. C2×#324C2semidirect
6. C2×#323C2semidirect
25. #407C22semidirect
31. C2×C22⋊C8C22semidirect
32. #324C22semidirect
33. #323C22semidirect
292. C2#1468direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#4072
C22×C22⋊C82
#14682
C2×#3242
C2×#3232