Finite Group Databasebeta

← C2×#7824 #165733

Subgroups — C2×#7824 #165733

Order 512.

1093 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1093 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2724C211yesmaximal, characteristic
2C2×#2100C212yesmaximal, complemented
3C2×#2094C211yesmaximal, characteristic
4#7824C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1270C211yesmaximal, characteristic
7C22×#123—21no—
8C22×#106—21no—
9C22×#102—21no—
10#2724—22no—
11#2724—24no—
12#2724C2214yes—
13C2×#398—21no—
14C2×#397—22no—
15C2×#390C2211yescharacteristic
16C2×#389C2212yes—
17C2×#385—22no—
18C2×#384C2211yescharacteristic
19#2100—24no—
20#2100C2218yescomplemented
21#2094—22no—
22#2094C2214yes—
23C2×#340—22no—
24C2×#340—22no—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27#1271C2218yescomplemented
28#1270C2214yes—
29C23×D4D411yescharacteristic
30C24×C4—21no—
31C23×C8—21no—
32C23×C8—22no—
33C2×#123—24no—
34C2×#123—24no—
35C2×#123—44no—
36C2×#123—44no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C42C2311yescharacteristic
48C2×#106—24no—
49C2×#106—44no—
50C2×#102—24no—
51C2×#102—44no—
52#398—24no—
53#398—24no—
54#397—28no—
55#397—28no—
56#390—22no—
57#390D412yes—
58#390C2312yes—
59#389—22no—
60#389—22no—
61#389D414yes—
62#389C2314yes—
63#385—28no—
64#385—28no—
65#384—21no—
66#384—22no—
67#384C2312yes—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71#340—28no—
72#340—28no—
73#336—21no—
74#336—22no—
75#336C2312yes—
76#336C2314yes—
77C25—21no—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—42no—
81C22×D4—42no—
82C22×D4SD1614yes—
83C22×D4C2×D412yes—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—44no—
89C23×C4—41no—
90C23×C4—82no—
91C23×C4—82no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—44no—
100C22×C8—44no—
101C22×C8—44no—
102#123—48no—
103#123—416no—
104#123—48no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—48no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C42—42no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42C2411yescharacteristic
140#106—48no—
141#102—48no—
142#95—28no—
143#95—24no—
144#95—24no—
145#95—24no—
146C24—22no—
147C24—24no—
148C24—44no—
149C24—42no—
150C24—44no—
151C24—42no—
152C24#12011yescharacteristic
153C2×D4—22no—
154C2×D4—24no—
155C2×D4—42no—
156C2×D4—48no—
157C2×D4—42no—
158C2×D4—84no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4C2×SD1614yes—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4—816no—
182C22×C4—82no—
183C22×C4—84no—
184C22×C4#12011yescharacteristic
185C22×C4#12012yes—
186C22×C4C2×SD1612yes—
187C22×C4#13514yes—
188C22×C4AGL(1,ℤ/8)12yes—
189C22×C4#13714yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192C2×C8—24no—
193C2×C8—48no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—88no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—81no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C22⋊C4—48no—
217C22⋊C4—416no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—48no—
221C42—42no—
222C42—41no—
223C42—41no—
224C23—24no—
225C23—48no—
226C23—416no—
227C23—44no—
228C23—42no—
229C23—44no—
230C23—42no—
231C23—88no—
232C23—84no—
233C23—84no—
234C23#45012yes—
235C23#45214yes—
236C23#45312yes—
237C23#45414yes—
238C23#45812yes—
239C23C2×#12011yescharacteristic
240D4—44no—
241D4—82no—
242D4—84no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—816no—
264C2×C4—88no—
265C2×C4—84no—
266C2×C4—88no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—164no—
270C2×C4#45012yes—
271C2×C4#45214yes—
272C2×C4C22×SD1611yescharacteristic
273C2×C4C2×#13512yes—
274C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
275C2×C4C2×#13712yes—
276C8—44no—
277C8—84no—
278C22—48no—
279C22—44no—
280C22—82no—
281C22—82no—
282C22—816no—
283C22—84no—
284C22#161314yes—
285C22#161518yes—
286C22#163014yes—
287C22#163518yes—
288C22#165714yes—
289C22C2×#45011yescharacteristic
290C22C2×#45212yes—
291C22C2×#45311yescharacteristic
292C22C2×#45412yes—
293C22C2×#45811yescharacteristic
294C4—21no—
295C4—21no—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—24no—
300C4—82no—
301C4—84no—
302C4—84no—
303C4—162no—
304C2—84no—
305C2#782418yesdirect product factor
306C2C2×#161311yescharacteristic
307C2C2×#161512yes—
308C2C2×#163011yescharacteristic
309C2C2×#163512yes—
310C2C2×#165711yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
4. #7824C2direct (N × H)
5. C2×#1271C2semidirect
20. #2100C22semidirect
27. #1271C22semidirect
305. C2#7824direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#21002
C2×#20942
#78242
C2×#12712
C2×#12702