Finite Group Databasebeta

← C2×#7810 #165754

Subgroups — C2×#7810 #165754

Order 512.

1277 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1277 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×#2733C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2071C212yesmaximal, complemented
4#7810C218yesmaximal, direct product factor
5C2×#1272C211yesmaximal, characteristic, complemented
6C2×#1270C212yesmaximal, complemented
7C22×#124—21no—
8C22×#106—21no—
9C22×#103—21no—
10C22×#102—22no—
11#2733—22no—
12#2733—24no—
13#2733C2214yes—
14C2×#394—21no—
15C2×#392—22no—
16C2×#391C2211yescharacteristic
17C2×#390C2212yes—
18C2×#388—22no—
19C2×#384C2211yescharacteristic
20#2096—22no—
21#2096C2214yescomplemented
22#2071—24no—
23#2071C2218yescomplemented
24C2×#340—22no—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27#1272C2214yescomplemented
28#1270C2218yescomplemented
29C23×Q8D411yescharacteristic
30C23×D4—21no—
31C23×D4—21no—
32C23×D4D412yes—
33C23×C8—21no—
34C23×C8—22no—
35C2×#124—24no—
36C2×#124—24no—
37C2×#124—44no—
38C2×#124—44no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D412yes—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—22no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49C2×#106—24no—
50C2×#106—44no—
51C2×#103—24no—
52C2×#103—44no—
53C2×#102—28no—
54C2×#102—48no—
55#394—24no—
56#394—24no—
57#392—28no—
58#392—28no—
59#391—22no—
60#391D412yes—
61#391C2312yes—
62#390—22no—
63#390—22no—
64#390D414yes—
65#390C2314yes—
66#388—28no—
67#388—28no—
68#384—22no—
69#384—21no—
70#384C2312yes—
71C2×#95—21no—
72C2×#95—22no—
73#340—28no—
74#336—22no—
75#336—21no—
76#336C2314yes—
77#336C2312yes—
78C25—22no—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8SD1614yes—
84C22×Q8C2×D412yes—
85C22×D4—24no—
86C22×D4—24no—
87C22×D4—22no—
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—44no—
91C22×D4—44no—
92C22×D4—44no—
93C22×D4—82no—
94C22×D4—82no—
95C22×D4—82no—
96C22×D4—84no—
97C22×D4C2×D414yes—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C22×C8—24no—
105C22×C8—22no—
106C22×C8—24no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—44no—
110C22×C8—44no—
111C22×C8—44no—
112#124—48no—
113#124—416no—
114#124—48no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—84no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—44no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2411yescharacteristic
147#106—48no—
148#103—48no—
149#102—416no—
150#95—24no—
151#95—24no—
152#95—24no—
153C24—24no—
154C24—44no—
155C24—44no—
156C24—44no—
157C24—44no—
158C24—44no—
159C24—44no—
160C24—44no—
161C24#12011yescharacteristic
162C2×Q8—22no—
163C2×Q8—24no—
164C2×Q8—42no—
165C2×Q8—48no—
166C2×Q8—42no—
167C2×Q8—84no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8C2×SD1614yes—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—416no—
176C2×D4—48no—
177C2×D4—44no—
178C2×D4—82no—
179C2×D4—82no—
180C2×D4—88no—
181C2×D4—84no—
182C2×D4—84no—
183C2×D4—84no—
184C2×D4—88no—
185C2×D4—816no—
186C2×D4—816no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—84no—
204C22×C4—82no—
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4C2×SD1612yes—
208C22×C4#13514yes—
209C22×C4AGL(1,ℤ/8)14yes—
210C22×C4AGL(1,ℤ/8)12yes—
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D412yes—
213C2×C8—24no—
214C2×C8—48no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—44no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—88no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C22⋊C4—416no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C22⋊C4—88no—
239C42—42no—
240C42—41no—
241C42—41no—
242C42—48no—
243C23—44no—
244C23—44no—
245C23—44no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—88no—
251C23—88no—
252C23—88no—
253C23—88no—
254C23—88no—
255C23—88no—
256C23—88no—
257C23#44914yes—
258C23#45012yes—
259C23#45312yes—
260C23#45514yes—
261C23#45812yes—
262C23C2×#12011yescharacteristic
263Q8—44no—
264Q8—82no—
265Q8—84no—
266D4—48no—
267D4—84no—
268D4—88no—
269D4—816no—
270D4—816no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—88no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—164no—
297C2×C4#44914yes—
298C2×C4#45012yes—
299C2×C4C22×SD1611yescharacteristic
300C2×C4C2×#13512yes—
301C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C4C2×AGL(1,ℤ/8)12yes—
303C8—44no—
304C8—84no—
305C22—84no—
306C22—84no—
307C22—84no—
308C22—84no—
309C22—84no—
310C22—84no—
311C22—84no—
312C22—1616no—
313C22#161218yes—
314C22#161314yes—
315C22#162414yes—
316C22#164218yes—
317C22#165914yes—
318C22C2×#45011yescharacteristic
319C22C2×#44912yes—
320C22C2×#45311yescharacteristic
321C22C2×#45512yes—
322C22C2×#45811yescharacteristic
323C4—21no—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—24no—
329C4—44no—
330C4—82no—
331C4—84no—
332C4—162no—
333C2—164no—
334C2#781018yesdirect product factor
335C2C2×#161311yescharacteristic
336C2C2×#161212yes—
337C2C2×#162411yescharacteristic
338C2C2×#164212yes—
339C2C2×#165911yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2733C2
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×#2096C2semidirect
3. C2×#2071C2semidirect
4. #7810C2direct (N × H)
5. C2×#1272C2semidirect
6. C2×#1270C2semidirect
21. #2096C22semidirect
23. #2071C22semidirect
27. #1272C22semidirect
28. #1270C22semidirect
334. C2#7810direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#20962
C2×#20712
#78102
C2×#12722
C2×#12702