Finite Group Databasebeta

← C2×#11534 #480341

Subgroups — C2×#11534 #480341

Order 512.

1158 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2326C211yesmaximal, characteristic
2C2×#1467C212yesmaximal, complemented
3#11534C218yesmaximal, direct product factor
4C2×#1325C212yesmaximal, complemented
5C2×#1318C212yesmaximal
6C22×#124C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×C4×D4C411yescharacteristic
9C2×#432—22no—
10#2326—24no—
11#2326C414yes—
12#2326C2214yes—
13C2×#391C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#388—22no—
16C2×#385—22no—
17C2×#384C2211yescharacteristic
18C2×#376C411yescharacteristic
19#1467—24no—
20#1467C2218yescomplemented
21C2×#324—22no—
22C2×#324C2214yes—
23#1325C2218yescomplemented
24#1318C2218yes—
25C23×Q8C2×C411yescharacteristic
26C23×D4C2×C411yescharacteristic
27C24×C4—21no—
28C2×#124—22no—
29C2×#124—24no—
30C2×#124—24no—
31C2×#124C2×C414yes—
32C2×C4×Q8—42no—
33C2×C4×Q8—42no—
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8D414yes—
36C2×C4×Q8C2312yes—
37C2×C4×D4—22no—
38C2×C4×D4—42no—
39C2×C4×D4—42no—
40C2×C4×D4C2×C414yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42D412yes—
52#432—28no—
53#432—44no—
54C2×C4⋊C8—42no—
55C2×C22⋊C8—24no—
56C2×#97—42no—
57#391—22no—
58#391C2×C414yes—
59#390—22no—
60#390C2×C412yes—
61#390C2312yes—
62#388—28no—
63#388—28no—
64#385—28no—
65#385—28no—
66#384—22no—
67#384C2×C412yes—
68#384C2312yes—
69#376—22no—
70#376C2×C414yes—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—21no—
74#324—28no—
75#324—24no—
76#324C2318yes—
77C25—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—42no—
83C22×Q8C22⋊C412yes—
84C22×Q8C22×C412yes—
85C22×D4—21no—
86C22×D4—22no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4M1614yes—
91C22×D4C22×C412yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—84no—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C22×C8—44no—
105#124—216no—
106#124—28no—
107#124—28no—
108C4×Q8—44no—
109C4×Q8—44no—
110C4×Q8—44no—
111C4×Q8—44no—
112C4×D4—48no—
113C4×D4—48no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4C22⋊C414yes—
125C2×C4⋊C4C22×C412yes—
126C2×C4⋊C4C22×C411yescharacteristic
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2411yescharacteristic
130C2×C22⋊C4—216no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—84no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—24no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C414yes—
149C2×C42M1614yes—
150C2×C42C22×C412yes—
151C2×C42C22×C412yes—
152C2×C42C22×C412yes—
153C2×C42C2×D414yes—
154C4⋊C8—48no—
155C22⋊C8—28no—
156C22⋊C8—44no—
157#97—48no—
158#95—24no—
159#95—24no—
160#95—28no—
161C24—22no—
162C24—24no—
163C24—44no—
164C24—44no—
165C24—41no—
166C24—41no—
167C24—42no—
168C24C2×C22⋊C411yescharacteristic
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×D4—22no—
181C2×D4—24no—
182C2×D4—42no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4C2×M1614yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—48no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—88no—
219C22×C4—88no—
220C22×C4—88no—
221C22×C4#9912yes—
222C22×C4#9912yes—
223C22×C4C2×C22⋊C414yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×M1612yes—
229C22×C4C2×M1612yes—
230C22×C4C23×C411yescharacteristic
231C22×C4C22×D412yes—
232C2×C8—48no—
233C2×C8—88no—
234C4⋊C4—44no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C22⋊C4—232no—
244C22⋊C4—48no—
245C22⋊C4—84no—
246C22⋊C4—84no—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—24no—
251C42—24no—
252C42—24no—
253C42—24no—
254C42—24no—
255C42—42no—
256C42—42no—
257C42C2×M1614yes—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—44no—
262C23—416no—
263C23—44no—
264C23—42no—
265C23—42no—
266C23—44no—
267C23—44no—
268C23—82no—
269C23—88no—
270C23—82no—
271C23—82no—
272C23—82no—
273C23#40712yes—
274C23C2×#9911yescharacteristic
275C23C2×#9911yescharacteristic
276C23#41312yes—
277C23C22×C22⋊C411yescharacteristic
278Q8—44no—
279Q8—44no—
280Q8—48no—
281D4—44no—
282D4—44no—
283D4—48no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—816no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#40712yes—
324C2×C4C2×#9914yes—
325C2×C4C22×M1611yescharacteristic
326C2×C4C22×M1611yescharacteristic
327C8—88no—
328C22—24no—
329C22—24no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—48no—
335C22—44no—
336C22—84no—
337C22—88no—
338C22—88no—
339C22—82no—
340C22—82no—
341C22#146512yes—
342C22#146512yes—
343C22#212812yes—
344C22#213312yes—
345C22C2×#40711yescharacteristic
346C22C22×#9911yescharacteristic
347C22C2×#41311yescharacteristic
348C4—24no—
349C4—21no—
350C4—21no—
351C4—22no—
352C4—44no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—44no—
357C4—82no—
358C4—82no—
359C2—21no—
360C2—21no—
361C2—22no—
362C2—84no—
363C2#1153414yesdirect product factor
364C2C2×#146511yescharacteristic
365C2C2×#212811yescharacteristic
366C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1467C2semidirect
3. #11534C2direct (N × H)
4. C2×#1325C2semidirect
20. #1467C22semidirect
23. #1325C22semidirect
363. C2#11534direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#14672
#115342
C2×#13252
C2×#13182