Finite Group Databasebeta

← C2×#11535 #480340

Subgroups — C2×#11535 #480340

Order 512.

1202 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1202 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×#1465C212yesmaximal
3#11535C218yesmaximal, direct product factor
4C2×#1325C212yesmaximal, complemented
5C2×#1319C212yesmaximal, complemented
6C22×#124C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#407—22no—
10#2326—24no—
11#2326C414yes—
12#2326C2214yes—
13C2×#391C2211yescharacteristic
14C2×#390C411yescharacteristic
15C2×#388—22no—
16C2×#385—22no—
17C2×#384C2211yescharacteristic
18C2×#376C411yescharacteristic
19#1465—24no—
20#1465C2218yes—
21C2×#324C2214yes—
22C2×#323—22no—
23#1325C2218yescomplemented
24#1319C2218yescomplemented
25C23×Q8C2×C411yescharacteristic
26C23×D4C2×C411yescharacteristic
27C24×C4D412yes—
28C22×M16—42no—
29C2×#124—24no—
30C2×#124—22no—
31C2×#124—24no—
32C2×#124C2×C414yes—
33C2×C4×Q8—22no—
34C2×C4×Q8—42no—
35C2×C4×Q8—42no—
36C2×C4×Q8C2×C414yes—
37C2×C4×D4—42no—
38C2×C4×D4—42no—
39C2×C4×D4C2×C412yes—
40C2×C4×D4D414yes—
41C2×C4×D4C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4D412yes—
49C22×C42—21no—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53#407—28no—
54#407—44no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—42no—
57#391—22no—
58#391C2×C412yes—
59#391C2312yes—
60#390—22no—
61#390C2×C414yes—
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—21no—
73C2×#95—22no—
74#324—24no—
75#324C2318yes—
76#323—28no—
77C25C22⋊C412yes—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—42no—
83C22×Q8M1614yescomplemented
84C22×Q8C22×C412yes—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4C22⋊C412yes—
91C22×D4C22×C412yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—84no—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C414yes—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D414yes—
106C2×M16—48no—
107C2×M16—88no—
108C22×C8—44no—
109#124—216no—
110#124—28no—
111#124—28no—
112C4×Q8—48no—
113C4×Q8—48no—
114C4×D4—44no—
115C4×D4—44no—
116C4×D4—44no—
117C4×D4—44no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C22×C412yes—
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2411yescharacteristic
133C2×C22⋊C4—216no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—84no—
139C2×C22⋊C4C22⋊C414yes—
140C2×C22⋊C4C2×D414yes—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42—21no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42C22⋊C412yes—
151C2×C42M1614yescomplemented
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C22⋊C8—28no—
156C22⋊C8—44no—
157C22⋊C8—48no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162C24—24no—
163C24—22no—
164C24—22no—
165C24—24no—
166C24—44no—
167C24—44no—
168C24C2×C22⋊C411yescharacteristic
169C24C2×C22⋊C414yes—
170C2×Q8—22no—
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—42no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8C2×M1614yescomplemented
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—88no—
228C22×C4#9912yes—
229C22×C4#9912yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×M1612yes—
235C22×C4C2×M1612yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238M16—816no—
239C2×C8—48no—
240C2×C8—88no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C22⋊C4—232no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—88no—
253C42—22no—
254C42—24no—
255C42—24no—
256C42—24no—
257C42—42no—
258C42—42no—
259C42—48no—
260C42C2×M1614yescomplemented
261C23—22no—
262C23—22no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—44no—
267C23—44no—
268C23—416no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—44no—
273C23—82no—
274C23—82no—
275C23—88no—
276C23#40712yes—
277C23C2×#9911yescharacteristic
278C23C2×#9911yescharacteristic
279C23#41312yes—
280C23C22×C22⋊C411yescharacteristic
281Q8—44no—
282Q8—44no—
283Q8—48no—
284D4—44no—
285D4—44no—
286D4—48no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#40712yes—
330C2×C4C2×#9914yes—
331C2×C4C22×M1611yescharacteristic
332C2×C4C22×M1611yescharacteristic
333C8—88no—
334C22—24no—
335C22—24no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—48no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—84no—
345C22—88no—
346C22—88no—
347C22#146512yes—
348C22#146512yes—
349C22#212712yes—
350C22#213512yes—
351C22C2×#40711yescharacteristic
352C22C22×#9911yescharacteristic
353C22C2×#41311yescharacteristic
354C4—24no—
355C4—21no—
356C4—21no—
357C4—22no—
358C4—44no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—44no—
363C4—82no—
364C4—82no—
365C2—21no—
366C2—21no—
367C2—22no—
368C2—84no—
369C2#1153514yesdirect product factor
370C2C2×#146511yescharacteristic
371C2C2×#212711yescharacteristic
372C2C2×#213511yescharacteristic
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×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11535C2direct (N × H)
4. C2×#1325C2semidirect
5. C2×#1319C2semidirect
23. #1325C22semidirect
24. #1319C22semidirect
83. C22×Q8M16semidirect
151. C2×C42M16semidirect
180. C2×Q8C2×M16semidirect
260. C42C2×M16semidirect
369. C2#11535direct (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×#14652
#115352
C2×#13252
C2×#13192