Finite Group Databasebeta

← C2×#7830 #165755

Subgroups — C2×#7830 #165755

Order 512.

1321 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1321 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×#2095C211yesmaximal, characteristic, complemented
3C2×#2094C212yesmaximal, complemented
4#7830C218yesmaximal, direct product factor
5C2×#1272C211yesmaximal, characteristic, complemented
6C2×#1270C212yesmaximal, complemented
7C22×#124—21no—
8C22×#107—21no—
9C22×#106—22no—
10C22×#103—21no—
11C22×#102—22no—
12#2733—22no—
13#2733—24no—
14#2733C2214yes—
15C2×#394—21no—
16C2×#392—22no—
17C2×#391C2211yescharacteristic
18C2×#390C2212yes—
19C2×#388—22no—
20C2×#384C2211yescharacteristic
21#2095—22no—
22#2095C2214yescomplemented
23#2094—24no—
24#2094C2218yescomplemented
25C2×#336C2211yescharacteristic
26C2×#336C2212yes—
27#1272C2214yescomplemented
28#1270C2218yescomplemented
29C23×Q8D411yescharacteristic
30C23×D4—21no—
31C23×D4—21no—
32C23×D4D412yes—
33C23×C8—22no—
34C23×C8—21no—
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×#107—24no—
50C2×#107—44no—
51C2×#106—28no—
52C2×#106—48no—
53C2×#103—24no—
54C2×#103—44no—
55C2×#102—28no—
56C2×#102—48no—
57#394—24no—
58#394—24no—
59#392—28no—
60#392—28no—
61#391—22no—
62#391D412yes—
63#391C2312yes—
64#390—22no—
65#390—22no—
66#390D414yes—
67#390C2314yes—
68#388—28no—
69#388—28no—
70#384—22no—
71#384—21no—
72#384C2312yes—
73C2×#95—21no—
74C2×#95—22no—
75#336—22no—
76#336—21no—
77#336C2314yes—
78#336C2312yes—
79C25—22no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8SD1614yes—
85C22×Q8C2×D412yes—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—44no—
92C22×D4—44no—
93C22×D4—44no—
94C22×D4—82no—
95C22×D4—82no—
96C22×D4—84no—
97C22×D4—82no—
98C22×D4C2×D414yes—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—41no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C22×C8—28no—
106C22×C8—24no—
107C22×C8—24no—
108C22×C8—22no—
109C22×C8—44no—
110C22×C8—42no—
111C22×C8—44no—
112C22×C8—42no—
113#124—48no—
114#124—416no—
115#124—48no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D412yes—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—48no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—84no—
142C2×C42—24no—
143C2×C42—42no—
144C2×C42—44no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2411yescharacteristic
148#107—48no—
149#106—416no—
150#103—48no—
151#102—416no—
152#95—24no—
153#95—24no—
154#95—24no—
155C24—24no—
156C24—44no—
157C24—44no—
158C24—44no—
159C24—44no—
160C24—44no—
161C24—44no—
162C24—44no—
163C24#12011yescharacteristic
164C2×Q8—22no—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—48no—
168C2×Q8—42no—
169C2×Q8—84no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8C2×SD1614yes—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—416no—
178C2×D4—48no—
179C2×D4—44no—
180C2×D4—82no—
181C2×D4—82no—
182C2×D4—88no—
183C2×D4—84no—
184C2×D4—84no—
185C2×D4—88no—
186C2×D4—84no—
187C2×D4—816no—
188C2×D4—816no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—82no—
206C22×C4—84no—
207C22×C4#12011yescharacteristic
208C22×C4#12012yes—
209C22×C4C2×SD1612yes—
210C22×C4#13514yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13714yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D412yes—
215C2×C8—28no—
216C2×C8—24no—
217C2×C8—48no—
218C2×C8—416no—
219C2×C8—44no—
220C2×C8—42no—
221C2×C8—44no—
222C2×C8—42no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4—81no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C22⋊C4—416no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—48no—
239C22⋊C4—48no—
240C22⋊C4—88no—
241C42—42no—
242C42—41no—
243C42—41no—
244C42—48no—
245C23—44no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—88no—
253C23—88no—
254C23—88no—
255C23—88no—
256C23—88no—
257C23—88no—
258C23—88no—
259C23#45012yes—
260C23#45214yes—
261C23#45312yes—
262C23#45314yes—
263C23#45812yes—
264C23C2×#12011yescharacteristic
265Q8—44no—
266Q8—82no—
267Q8—84no—
268D4—48no—
269D4—84no—
270D4—88no—
271D4—816no—
272D4—816no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—88no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—82no—
298C2×C4—164no—
299C2×C4#45012yes—
300C2×C4#45214yes—
301C2×C4C22×SD1611yescharacteristic
302C2×C4C2×#13512yes—
303C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
304C2×C4C2×#13712yes—
305C8—44no—
306C8—48no—
307C22—84no—
308C22—84no—
309C22—84no—
310C22—84no—
311C22—84no—
312C22—84no—
313C22—84no—
314C22—1616no—
315C22#161314yes—
316C22#161818yes—
317C22#163118yes—
318C22#163414yes—
319C22#165514yes—
320C22C2×#45011yescharacteristic
321C22C2×#45212yes—
322C22C2×#45312yes—
323C22C2×#45311yescharacteristic
324C22C2×#45811yescharacteristic
325C4—21no—
326C4—21no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—24no—
331C4—44no—
332C4—82no—
333C4—84no—
334C4—162no—
335C2—164no—
336C2#783018yesdirect product factor
337C2C2×#161311yescharacteristic
338C2C2×#161812yes—
339C2C2×#163112yes—
340C2C2×#163411yescharacteristic
341C2C2×#165511yescharacteristic
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×#2095C2semidirect
3. C2×#2094C2semidirect
4. #7830C2direct (N × H)
5. C2×#1272C2semidirect
6. C2×#1270C2semidirect
22. #2095C22semidirect
24. #2094C22semidirect
27. #1272C22semidirect
28. #1270C22semidirect
336. C2#7830direct (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×#20952
C2×#20942
#78302
C2×#12722
C2×#12702