Finite Group Databasebeta

← C2×#9410 #200620

Subgroups — C2×#9410 #200620

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3C2×#1507C212yesmaximal
4C2×#1463C211yesmaximal, characteristic, complemented
5#9410C218yesmaximal, direct product factor
6C2×#1297C212yesmaximal, complemented
7C22×#121C2211yescharacteristic
8C22×#117—21no—
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2591—22no—
12#2591—22no—
13#2591C414yescomplemented
14#2591C2214yes—
15C2×#400C412yescomplemented
16C2×#394C412yescomplemented
17C2×#388—22no—
18C2×#386—21no—
19C2×#386—21no—
20#2274—24no—
21#2274—22no—
22#2274C2214yescomplemented
23C2×C4×C4⋊C4—24no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27#1507—24no—
28#1507C2218yes—
29C2×#327—24no—
30#1463—22no—
31#1463C2214yescomplemented
32C2×#324—22no—
33#1297C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C2×#121—21no—
41C2×#121—21no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121C2×C412yes—
47C2×#121C2312yes—
48C2×#117—216no—
49C2×#117—24no—
50C2×#117—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—21no—
72#407—22no—
73#407—22no—
74#407—22no—
75#407C2314yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#400—24no—
79#400C2×C418yescomplemented
80#394—24no—
81#394C2×C418yescomplemented
82#388—28no—
83#388—28no—
84#386—24no—
85#386—24no—
86#386—24no—
87#386—24no—
88C4×C4⋊C4—216no—
89C4×C4⋊C4—216no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D414yes—
98C4×C22⋊C4C2312yes—
99C4×C22⋊C4C2314yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95C2×C412yes—
104C2×#95D414yes—
105#327—216no—
106#324—28no—
107C25—21no—
108C25C2×D411yescharacteristic
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—48no—
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—82no—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—28no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C2×M16—24no—
152C2×M16—42no—
153C2×M16—84no—
154C22×C8—42no—
155#121—24no—
156#121—24no—
157#121—24no—
158#121—24no—
159#121—42no—
160#121—42no—
161#121—42no—
162#121—42no—
163#117—232no—
164#117—232no—
165C2×C4⋊C4—216no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—84no—
201C2×C42—28no—
202C2×C42—216no—
203C2×C42—28no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—84no—
209C2×C42—82no—
210C2×C42—84no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—22no—
219#95—24no—
220#95C22×C414yes—
221#95C2×D418yes—
222C24—24no—
223C24—24no—
224C24—22no—
225C24—42no—
226C24—44no—
227C24—44no—
228C24—41no—
229C24—41no—
230C24—44no—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24#12011yescharacteristic
234C24#12011yescharacteristic
235C24#12311yescharacteristic
236C24#12311yescharacteristic
237C24C22×D411yescharacteristic
238C2×D4—24no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—84no—
247C2×D4—82no—
248C2×D4—84no—
249C2×D4—82no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—84no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—216no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—44no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—82no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4#10418yes—
304C22×C4C2×C22⋊C414yes—
305C22×C4C2×C22⋊C414yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12012yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12312yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4112yes—
322M16—84no—
323M16—84no—
324C2×C8—44no—
325C2×C8—84no—
326C4⋊C4—232no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C22⋊C4—216no—
334C22⋊C4—216no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—416no—
340C22⋊C4—88no—
341C42—232no—
342C42—82no—
343C42—84no—
344C42—84no—
345C42—82no—
346C42—88no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—416no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—82no—
372C23—82no—
373C23—816no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23#38611yescharacteristic
379C23#38611yescharacteristic
380C23#38611yescharacteristic
381C23#38611yescharacteristic
382C23C2×#10414yes—
383C23#45414yes—
384C23C22×C22⋊C411yescharacteristic
385C23C2×#12011yescharacteristic
386C23C2×#12311yescharacteristic
387D4—84no—
388D4—84no—
389D4—84no—
390D4—84no—
391D4—84no—
392D4—84no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—216no—
405C2×C4—24no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—432no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—88no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#38612yes—
439C2×C4#38612yes—
440C2×C4C2×#10418yes—
441C2×C4C2×C4×D412yes—
442C2×C4C2×#12112yes—
443C8—84no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—28no—
449C22—28no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—88no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—164no—
470C22—164no—
471C22#188812yes—
472C22#188812yes—
473C22#189612yes—
474C22#189612yes—
475C22#191312yes—
476C22C2×#38611yescharacteristic
477C22C22×#10412yes—
478C22C2×#45412yes—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—28no—
484C4—416no—
485C4—44no—
486C4—84no—
487C4—82no—
488C2—22no—
489C2—22no—
490C2—22no—
491C2—22no—
492C2—41no—
493C2—41no—
494C2—162no—
495C2#941014yesdirect product factor
496C2C2×#188811yescharacteristic
497C2C2×#189611yescharacteristic
498C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2274C2semidirect
4. C2×#1463C2semidirect
5. #9410C2direct (N × H)
6. C2×#1297C2semidirect
13. #2591C4semidirect
15. C2×#400C4semidirect
16. C2×#394C4semidirect
22. #2274C22semidirect
31. #1463C22semidirect
33. #1297C22semidirect
79. #400C2×C4semidirect
81. #394C2×C4semidirect
495. C2#9410direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#22742
C2×#15072
C2×#14632
#94102
C2×#12972