Finite Group Databasebeta

← C2×#11506 #480256

Subgroups — C2×#11506 #480256

Order 512.

1854 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1854 conjugacy classes of subgroups — too many to draw. Only its 250 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×#391C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11506C218yesmaximal, direct product factor
4C2×#1319C214yesmaximal, complemented
5C24×Q8C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C23×C42C411yescharacteristic
8C2×#407—22no—
9C2×#391—22no—
10C2×#391—24no—
11C2×#391C414yes—
12C2×#391C412yes—
13C2×#391C2212yes—
14C2×#391C2214yes—
15C22×#95—22no—
16#1463—24no—
17#1463C2218yes—
18C2×#324—22no—
19C2×#324C2214yes—
20#1319C22116yescomplemented
21C23×Q8—28no—
22C23×Q8—22no—
23C23×Q8—21no—
24C23×Q8—44no—
25C23×Q8—42no—
26C23×Q8—41no—
27C23×Q8—41no—
28C23×Q8C2×C414yes—
29C23×Q8C2×C412yes—
30C24×C4—21no—
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D412yes—
34C22×M16—42no—
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D414yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C4⋊C4C2312yes—
44C22×C42—22no—
45C22×C42—21no—
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42C2×C414yes—
49#407—28no—
50#407—44no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—42no—
53#391—24no—
54#391—28no—
55#391—24no—
56#391—28no—
57#391C2×C414yes—
58#391C2×C418yes—
59#391C2314yes—
60C2×#95—28no—
61C2×#95—24no—
62C2×#95—24no—
63C2×#95—28no—
64#324—28no—
65#324—24no—
66#324C2318yes—
67C25C22⋊C411yescharacteristic
68C22×Q8—28no—
69C22×Q8—216no—
70C22×Q8—216no—
71C22×Q8—22no—
72C22×Q8—24no—
73C22×Q8—24no—
74C22×Q8—48no—
75C22×Q8—432no—
76C22×Q8—416no—
77C22×Q8—416no—
78C22×Q8—44no—
79C22×Q8—44no—
80C22×Q8—42no—
81C22×Q8—42no—
82C22×Q8—48no—
83C22×Q8—44no—
84C22×Q8—44no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—44no—
88C22×Q8—44no—
89C22×Q8M1618yescomplemented
90C22×Q8C22×C414yes—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—28no—
97C23×C4—28no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—44no—
106C23×C4—44no—
107C23×C4—44no—
108C23×C4—42no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—48no—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C414yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C2×M16—48no—
121C2×M16—88no—
122C22×C8—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C414yes—
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2411yescharacteristic
142C2×C42—22no—
143C2×C42—24no—
144C2×C42—44no—
145C2×C42—48no—
146C2×C42—44no—
147C2×C42—48no—
148C2×C42C22×C414yes—
149C22⋊C8—28no—
150C22⋊C8—44no—
151C22⋊C8—48no—
152#95—28no—
153#95—216no—
154#95—28no—
155C24—24no—
156C24—24no—
157C24—22no—
158C24—22no—
159C24#9912yes—
160C24#9911yescharacteristic
161C24#9911yescharacteristic
162C24C2×C22⋊C412yes—
163C24C2×C22⋊C411yescharacteristic
164C2×Q8—216no—
165C2×Q8—24no—
166C2×Q8—28no—
167C2×Q8—432no—
168C2×Q8—432no—
169C2×Q8—432no—
170C2×Q8—432no—
171C2×Q8—44no—
172C2×Q8—48no—
173C2×Q8—44no—
174C2×Q8—416no—
175C2×Q8—416no—
176C2×Q8—48no—
177C2×Q8—48no—
178C2×Q8—48no—
179C2×Q8—48no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—48no—
183C2×Q8C2×M1618yescomplemented
184C22×C4—216no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—216no—
202C22×C4—28no—
203C22×C4—216no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—48no—
216C22×C4—432no—
217C22×C4—48no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—416no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—84no—
235C22×C4—88no—
236C22×C4#10014yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×M1614yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244M16—816no—
245C2×C8—48no—
246C2×C8—88no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—41no—
254C42—44no—
255C42—44no—
256C42—48no—
257C23—216no—
258C23—216no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—24no—
264C23—22no—
265C23—24no—
266C23—22no—
267C23—24no—
268C23—22no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23#35414yes—
273C23#35614yes—
274C23#40712yes—
275C23C2×#9912yes—
276C23C2×#9912yes—
277C23C2×#9911yescharacteristic
278C23C2×#9911yescharacteristic
279C23C2×#10012yes—
280C23C22×C22⋊C411yescharacteristic
281Q8—48no—
282Q8—48no—
283Q8—416no—
284Q8—432no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—432no—
305C2×C4—44no—
306C2×C4—416no—
307C2×C4—416no—
308C2×C4—416no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4#40712yes—
326C2×C4C2×#10014yes—
327C2×C4C22×M1612yes—
328C8—88no—
329C22—28no—
330C22—28no—
331C22—216no—
332C22—21no—
333C22—22no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—28no—
342C22—28no—
343C22—24no—
344C22#146212yes—
345C22#146212yes—
346C22C2×#35414yes—
347C22C2×#35412yes—
348C22C2×#35614yes—
349C22C2×#35612yes—
350C22C2×#40711yescharacteristic
351C22C22×#9911yescharacteristic
352C22C22×#10011yescharacteristic
353C4—22no—
354C4—22no—
355C4—24no—
356C4—48no—
357C4—42no—
358C4—42no—
359C4—44no—
360C4—48no—
361C2—21no—
362C2—21no—
363C2—24no—
364C2—24no—
365C2—22no—
366C2#1150614yesdirect product factor
367C2C2×#146211yescharacteristic
368C2C22×#35411yescharacteristic
369C2C22×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11506C2direct (N × H)
4. C2×#1319C2semidirect
20. #1319C22semidirect
89. C22×Q8M16semidirect
183. C2×Q8C2×M16semidirect
366. C2#11506direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3912
C2×#14632
#115062
C2×#13192