Finite Group Databasebeta

← C2×#9461 #201280

Subgroups — C2×#9461 #201280

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4C2×#1511C212yesmaximal
5#9461C218yesmaximal, direct product factor
6C2×#1284C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2502—22no—
11#2502—22no—
12#2502C414yescomplemented
13#2502C2214yes—
14C2×#398—22no—
15#2313—24no—
16#2313—22no—
17#2313C2214yes—
18C2×#391C412yescomplemented
19C2×#387—24no—
20C2×#385—21no—
21C2×#385—21no—
22C2×#384C412yescomplemented
23C2×#381C2212yes—
24C2×#380—21no—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1821—22no—
28#1821C2214yes—
29C2×#343—22no—
30#1511—24no—
31#1511C2218yes—
32C2×#328—24no—
33#1284C2218yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42C2×C412yes—
67#407—22no—
68#407—22no—
69#407—22no—
70#407C2314yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73#398—28no—
74#398—28no—
75#391—24no—
76#391C2×C418yescomplemented
77#387—216no—
78#387—216no—
79#385—24no—
80#385—24no—
81#385—24no—
82#385—24no—
83#384—24no—
84#384C2×C418yescomplemented
85#381—22no—
86#381—22no—
87#381D414yes—
88#381C2314yes—
89#380—24no—
90#380—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4C2312yes—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95D414yes—
104#343—28no—
105#328—216no—
106C25C2×D411yescharacteristic
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8C22×C412yes—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C2×M16—24no—
144C2×M16—42no—
145C2×M16—48no—
146C22×C8—42no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—48no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—44no—
200C2×C42—84no—
201C2×C42C22×C414yes—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#95—28no—
207#95—28no—
208#95—22no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—24no—
215#95C2×D418yes—
216C24—24no—
217C24—24no—
218C24—42no—
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C411yescharacteristic
221C24#12011yescharacteristic
222C24#12011yescharacteristic
223C24#12311yescharacteristic
224C24#12311yescharacteristic
225C24C22×D411yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—42no—
230C2×Q8—84no—
231C2×Q8—84no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C414yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12012yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12312yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C2×#4112yes—
302M16—48no—
303M16—84no—
304C2×C8—44no—
305C2×C8—84no—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C4⋊C4—88no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C42—44no—
326C42—44no—
327C42—88no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—416no—
348C23#38611yescharacteristic
349C23#38611yescharacteristic
350C23#38611yescharacteristic
351C23#38611yescharacteristic
352C23#41014yes—
353C23#45614yes—
354C23C22×C22⋊C411yescharacteristic
355C23C2×#12011yescharacteristic
356C23C2×#12311yescharacteristic
357Q8—84no—
358Q8—84no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—416no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—816no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4C2×C4×D412yes—
410C2×C4C2×#12112yes—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—28no—
419C22—28no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22#188612yes—
434C22#188612yes—
435C22#189912yes—
436C22#189912yes—
437C22#191112yes—
438C22C2×#38611yescharacteristic
439C22C2×#41012yes—
440C22C2×#45612yes—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—48no—
446C4—44no—
447C4—44no—
448C4—88no—
449C4—82no—
450C4—82no—
451C2—22no—
452C2—22no—
453C2—22no—
454C2—22no—
455C2—41no—
456C2—41no—
457C2#946114yesdirect product factor
458C2C2×#188611yescharacteristic
459C2C2×#189911yescharacteristic
460C2C2×#191111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9461C2direct (N × H)
12. #2502C4semidirect
18. C2×#391C4semidirect
22. C2×#384C4semidirect
76. #391C2×C4semidirect
84. #384C2×C4semidirect
457. C2#9461direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#23132
C2×#18212
C2×#15112
#94612
C2×#12842