Finite Group Databasebeta

← C2×#8750 #190100

Subgroups — C2×#8750 #190100

Order 512.

1518 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2312C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3#8750C218yesmaximal, direct product factor
4C2×#1304C214yesmaximal
5C23×C22⋊C4C2211yescharacteristic
6#2312—24no—
7#2312C414yescomplemented
8#2312C2214yes—
9C2×#388—24no—
10C2×#380C412yescomplemented
11C2×#379C2214yes—
12C2×#379C2212yes—
13C2×C4×C22⋊C4C412yescomplemented
14#1819—24no—
15#1819C2218yes—
16C2×#342—24no—
17#1304C22116yes—
18C26C2311yescharacteristic
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C22×C4⋊C4—22no—
22C22×C22⋊C4—24no—
23C22×C22⋊C4—24no—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—24no—
27C22×C22⋊C4—24no—
28C22×C22⋊C4—24no—
29C22×C22⋊C4C2×C412yes—
30C22×C22⋊C4C2×C414yes—
31C22×C22⋊C4D414yes—
32C22×C22⋊C4D414yes—
33C22×C22⋊C4C2312yes—
34C22×C42—22no—
35#388—216no—
36#388—216no—
37#380—24no—
38#380C2×C418yescomplemented
39#379—22no—
40#379—22no—
41#379—24no—
42#379—24no—
43#379—24no—
44#379C2×C414yes—
45#379C2318yes—
46#379C2314yes—
47C4×C22⋊C4—24no—
48C4×C22⋊C4C2×C418yescomplemented
49C2×#95—22no—
50C2×#95—22no—
51#342—216no—
52C25—44no—
53C25—82no—
54C25—82no—
55C25C22×C411yescharacteristic
56C25C22×C411yescharacteristic
57C25C2×D412yes—
58C25C2×D412yes—
59C25#4114yes—
60C25#4114yes—
61C25C2411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—44no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C412yes—
74C23×C4C22×C412yes—
75C23×C4C22×C412yes—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—84no—
78C2×C22⋊C4—28no—
79C2×C22⋊C4—28no—
80C2×C22⋊C4—28no—
81C2×C22⋊C4—28no—
82C2×C22⋊C4—28no—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—28no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—48no—
93C2×C22⋊C4—416no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—84no—
98C2×C22⋊C4—84no—
99C2×C22⋊C4—88no—
100C2×C22⋊C4—88no—
101C2×C22⋊C4C22⋊C418yes—
102C2×C22⋊C4C22×C418yes—
103C2×C22⋊C4C2×D418yes—
104C2×C42—28no—
105C2×C42—84no—
106#95—28no—
107#95—28no—
108C24—24no—
109C24—28no—
110C24—42no—
111C24—42no—
112C24—44no—
113C24—44no—
114C24—48no—
115C24—48no—
116C24—42no—
117C24—44no—
118C24—48no—
119C24—48no—
120C24—42no—
121C24—81no—
122C24—81no—
123C24—82no—
124C24—82no—
125C24—84no—
126C24—84no—
127C24—81no—
128C24—81no—
129C24—832no—
130C24C2×C22⋊C411yescharacteristic
131C24C2×C22⋊C412yes—
132C24C2×C22⋊C411yescharacteristic
133C24#11712yes—
134C24#11712yes—
135C24#11714yes—
136C24#12314yes—
137C24#12314yes—
138C24#12314yes—
139C24#12314yes—
140C24C23×C411yescharacteristic
141C24C22×D412yes—
142C24C2×#4114yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—88no—
167C22×C4—88no—
168C22×C4—88no—
169C22×C4C2×C22⋊C414yes—
170C22×C4C2×C22⋊C414yes—
171C4⋊C4—88no—
172C22⋊C4—216no—
173C22⋊C4—216no—
174C22⋊C4—48no—
175C22⋊C4—48no—
176C22⋊C4—416no—
177C22⋊C4—84no—
178C22⋊C4—84no—
179C22⋊C4—84no—
180C22⋊C4—84no—
181C22⋊C4—88no—
182C22⋊C4—88no—
183C22⋊C4—88no—
184C22⋊C4—88no—
185C22⋊C4—88no—
186C42—88no—
187C23—22no—
188C23—22no—
189C23—28no—
190C23—24no—
191C23—24no—
192C23—24no—
193C23—24no—
194C23—28no—
195C23—28no—
196C23—22no—
197C23—22no—
198C23—28no—
199C23—24no—
200C23—24no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—44no—
210C23—44no—
211C23—42no—
212C23—42no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—44no—
219C23—44no—
220C23—44no—
221C23—44no—
222C23—44no—
223C23—44no—
224C23—44no—
225C23—416no—
226C23—816no—
227C23—832no—
228C23—88no—
229C23—816no—
230C23—832no—
231C23—88no—
232C23#38114yes—
233C23#38112yes—
234C23#38111yescharacteristic
235C23#38111yescharacteristic
236C23#41014yes—
237C23C22×C22⋊C411yescharacteristic
238C23C2×#11712yes—
239C23C2×#12314yes—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—48no—
249C2×C4—48no—
250C2×C4—48no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—816no—
256C2×C4—88no—
257C2×C4—88no—
258C2×C4—88no—
259C2×C4—1616no—
260C22—28no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—22no—
274C22—22no—
275C22—24no—
276C22—24no—
277C22—44no—
278C22—44no—
279C22—48no—
280C22—416no—
281C22—48no—
282C22—84no—
283C22—84no—
284C22—88no—
285C22—88no—
286C22—88no—
287C22—88no—
288C22—88no—
289C22#182012yes—
290C22#182012yes—
291C22#182812yes—
292C22#182812yes—
293C22C2×#38111yescharacteristic
294C22C2×#41012yes—
295C4—44no—
296C4—44no—
297C4—88no—
298C4—168no—
299C2—22no—
300C2—22no—
301C2—44no—
302C2—42no—
303C2—42no—
304C2—44no—
305C2#875014yesdirect product factor
306C2C2×#182011yescharacteristic
307C2C2×#182811yescharacteristic
308C2C2×#182811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8750C2direct (N × H)
7. #2312C4semidirect
10. C2×#380C4semidirect
13. C2×C4×C22⋊C4C4semidirect
38. #380C2×C4semidirect
48. C4×C22⋊C4C2×C4semidirect
305. C2#8750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23122
C2×#18192
#87502
C2×#13042