Finite Group Databasebeta

← C2×#12860 #493870

Subgroups — C2×#12860 #493870

Order 512.

1958 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1958 conjugacy classes of subgroups — too many to draw. Only its 274 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×#1784C212yesmaximal
3#12860C218yesmaximal, direct product factor
4C2×#1394C214yesmaximal, complemented
5C24×Q8C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C42C411yescharacteristic
8C22×#101—22no—
9C22×#101C2214yes—
10C2×#407—22no—
11C2×#391—22no—
12C2×#391—24no—
13C2×#391C414yes—
14C2×#391C412yes—
15C2×#391C2212yes—
16C2×#391C2214yes—
17C22×#95—22no—
18#1784—24no—
19#1784C2218yes—
20#1394C22116yescomplemented
21C23×Q8—28no—
22C23×Q8—41no—
23C23×Q8—41no—
24C23×Q8—42no—
25C23×Q8—44no—
26C23×Q8C2×C411yescharacteristic
27C23×Q8C2×C412yes—
28C23×Q8D412yes—
29C23×Q8D414yes—
30C23×Q8C2312yes—
31C23×Q8C2311yescharacteristic
32C24×C4—21no—
33C24×C4—21no—
34C24×C4C2×C411yescharacteristic
35C24×C4D412yes—
36C22×M16—24no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4—41no—
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4C2×C412yes—
44C22×C42—22no—
45C22×C42—21no—
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42C2×C414yes—
49C2×#101—216no—
50C2×#101—28no—
51C2×#101—24no—
52C2×#101—48no—
53C2×#101C2318yes—
54#407—28no—
55#407—44no—
56C2×C22⋊C8—42no—
57C2×C22⋊C8—42no—
58#391—28no—
59#391—24no—
60#391—28no—
61#391—24no—
62#391C2×C418yes—
63#391C2×C414yes—
64#391C2314yes—
65C2×#95—24no—
66C2×#95—28no—
67C2×#95—28no—
68C2×#95—24no—
69C25C22⋊C411yescharacteristic
70C22×Q8—28no—
71C22×Q8—216no—
72C22×Q8—28no—
73C22×Q8—28no—
74C22×Q8—24no—
75C22×Q8—28no—
76C22×Q8—42no—
77C22×Q8—42no—
78C22×Q8—42no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—42no—
82C22×Q8—44no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—41no—
90C22×Q8—44no—
91C22×Q8—48no—
92C22×Q8—44no—
93C22×Q8—416no—
94C22×Q8—416no—
95C22×Q8—432no—
96C22×Q8C22⋊C414yes—
97C22×Q8C22⋊C412yes—
98C22×Q8C22×C412yes—
99C22×Q8C22×C411yescharacteristic
100C22×Q8C2×D414yes—
101C22×Q8C2×D412yes—
102C22×Q8#4118yes—
103C22×Q8C2411yescharacteristic
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—28no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—28no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—48no—
122C23×C4—42no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C414yes—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D414yes—
132C23×C4C2×D412yes—
133C2×M16—28no—
134C2×M16—44no—
135C2×M16—48no—
136C2×M16—48no—
137C22×C8—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C22×C414yes—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—44no—
152C2×C42—48no—
153C2×C42—44no—
154C2×C42—48no—
155C2×C42C22×C414yes—
156#101—216no—
157#101—416no—
158#101—48no—
159C22⋊C8—48no—
160C22⋊C8—48no—
161#95—216no—
162#95—28no—
163#95—28no—
164C24—24no—
165C24—24no—
166C24—22no—
167C24—22no—
168C24#9912yes—
169C24#9911yescharacteristic
170C24#9911yescharacteristic
171C24C2×C22⋊C412yes—
172C24C2×C22⋊C411yescharacteristic
173C2×Q8—24no—
174C2×Q8—28no—
175C2×Q8—28no—
176C2×Q8—24no—
177C2×Q8—28no—
178C2×Q8—28no—
179C2×Q8—42no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—48no—
190C2×Q8—48no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—416no—
194C2×Q8—416no—
195C2×Q8—416no—
196C2×Q8—416no—
197C2×Q8—432no—
198C2×Q8—432no—
199C2×Q8—416no—
200C2×Q8—416no—
201C2×Q8—48no—
202C2×Q8—48no—
203C2×Q8#11714yes—
204C2×Q8#12318yes—
205C2×Q8C2×#4114yes—
206C22×C4—216no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—216no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—416no—
245C22×C4—48no—
246C22×C4—432no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—416no—
251C22×C4—48no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—88no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4#9914yes—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4#11714yes—
265C22×C4#12318yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268M16—48no—
269M16—48no—
270M16—88no—
271C2×C8—48no—
272C2×C8—88no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C42—44no—
277C42—44no—
278C42—48no—
279C23—216no—
280C23—216no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23#35514yes—
295C23#35614yes—
296C23#38112yes—
297C23C2×#9911yescharacteristic
298C23C2×#9912yes—
299C23C2×#9912yes—
300C23C2×#9912yes—
301C23C2×#9911yescharacteristic
302C23C22×C22⋊C411yescharacteristic
303Q8—44no—
304Q8—48no—
305Q8—44no—
306Q8—416no—
307Q8—416no—
308Q8—48no—
309Q8—48no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—28no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—432no—
339C2×C4—416no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—88no—
348C2×C4—88no—
349C2×C4—84no—
350C2×C4#38112yes—
351C2×C4#38114yes—
352C2×C4C2×#9914yes—
353C2×C4C2×#11712yes—
354C2×C4C2×#12314yes—
355C8—88no—
356C22—28no—
357C22—216no—
358C22—28no—
359C22—24no—
360C22—28no—
361C22—28no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22#181912yes—
372C22#181912yes—
373C22C2×#35514yes—
374C22C2×#35512yes—
375C22C2×#35614yes—
376C22C2×#35612yes—
377C22C2×#38111yescharacteristic
378C22C22×#9911yescharacteristic
379C22C22×#9911yescharacteristic
380C4—24no—
381C4—22no—
382C4—22no—
383C4—24no—
384C4—24no—
385C4—48no—
386C4—44no—
387C4—84no—
388C2—21no—
389C2—22no—
390C2—24no—
391C2—24no—
392C2—21no—
393C2#1286014yesdirect product factor
394C2C2×#181911yescharacteristic
395C2C22×#35511yescharacteristic
396C2C22×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12860C2direct (N × H)
4. C2×#1394C2semidirect
20. #1394C22semidirect
393. C2#12860direct (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×#17842
#128602
C2×#13942