Finite Group Databasebeta

← C2×#9651 #204162

Subgroups — C2×#9651 #204162

Order 512.

2122 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C22×#324C212yesmaximal, complemented
5C22×#323C212yesmaximal
6#9651C218yesmaximal, direct product factor
7C23×C4⋊C4C2211yescharacteristic
8C23×C22⋊C4C411yescharacteristic
9C23×C22⋊C4C411yescharacteristic
10C23×C22⋊C4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8—21no—
16C2×#388—21no—
17C2×#388—21no—
18C2×#388—22no—
19C2×#388—21no—
20C2×#388—21no—
21C2×#388—22no—
22C2×#388C414yes—
23C2×#388C2214yes—
24C22×#95C411yescharacteristic
25C22×#95C411yescharacteristic
26#1797—22no—
27#1797C2214yescomplemented
28C4×C22⋊C8—22no—
29C4×C22⋊C8C2214yescomplemented
30C2×#336—21no—
31C2×#336—21no—
32C2×#324—28no—
33C2×#324C2218yescomplemented
34C2×#323—28no—
35C2×#323C2218yes—
36C26C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—44no—
67C22×C22⋊C4C818yescomplemented
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C414yes—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2312yes—
72C22×C42—44no—
73C22×C42C2×C412yes—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D412yes—
80C22×C42C2312yes—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—44no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8D418yes—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×C4×C8—24no—
92C2×C4×C8—44no—
93#388—24no—
94#388—24no—
95#388—24no—
96#388—24no—
97#388—24no—
98#388—24no—
99#388—24no—
100#388—24no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95C2×C414yes—
108C2×#95C2×C414yes—
109#336—24no—
110#336—24no—
111#324—216no—
112#323—216no—
113C25—21no—
114C25—21no—
115C25—22no—
116C25—24no—
117C25—24no—
118C25—42no—
119C25—42no—
120C25—42no—
121C25—42no—
122C25C2×C814yes—
123C25C22×C411yescharacteristic
124C25C22×C412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C2×C814yes—
154C23×C4C2×C814yes—
155C23×C4M1614yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D414yes—
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C22×C8—28no—
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—216no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4C2×C8116yescomplemented
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—44no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C4⋊C8—48no—
233C22⋊C8—28no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—44no—
238C22⋊C8C2×D418yes—
239C4×C8—48no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—216no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—48no—
260C24—48no—
261C24—42no—
262C24—48no—
263C24—48no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—48no—
276C24—88no—
277C24—88no—
278C24—88no—
279C24—88no—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C4×D414yes—
283C24C22×C812yes—
284C24C22×C814yes—
285C24C2×M1612yes—
286C24#13114yes—
287C24C23×C411yescharacteristic
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—216no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—416no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4C22⋊C814yes—
359C22×C4C22⋊C814yes—
360C22×C4C22⋊C818yes—
361C22×C4#9914yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C4×D414yes—
369C22×C4#12114yes—
370C22×C4#12114yes—
371C22×C4#12412yes—
372C22×C4#12412yes—
373C22×C4#12712yes—
374C22×C4#12712yes—
375C22×C4C22×C814yes—
376C22×C4C22×C814yes—
377C22×C4C2×M1614yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C2×#4112yes—
382C2×C8—28no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—416no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C2×C8—88no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C22⋊C4—216no—
396C22⋊C4—216no—
397C22⋊C4—48no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—48no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—28no—
421C23—28no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—28no—
427C23—28no—
428C23—24no—
429C23—28no—
430C23—44no—
431C23—48no—
432C23—44no—
433C23—48no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—48no—
443C23—48no—
444C23—48no—
445C23—48no—
446C23—42no—
447C23—832no—
448C23—84no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—832no—
453C23—84no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23#39012yes—
462C23#39012yes—
463C23C2×C22⋊C812yes—
464C23C2×C22⋊C812yes—
465C23#40812yes—
466C23#40812yes—
467C23C2×#9912yes—
468C23#41012yes—
469C23#41312yes—
470C23#41312yes—
471C23C8×D418yes—
472C23#43618yes—
473C23C22×C22⋊C411yescharacteristic
474C23C2×C4×D412yes—
475C23C23×C811yescharacteristic
476C23C22×M1611yescharacteristic
477C23C2×#13112yes—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4#39012yes—
533C2×C4#39012yes—
534C2×C4C2×C22⋊C814yes—
535C2×C4C2×C22⋊C814yes—
536C2×C4C2×#9914yes—
537C2×C4C2×#12112yes—
538C2×C4C2×#12411yescharacteristic
539C2×C4C2×#12711yescharacteristic
540C8—48no—
541C8—84no—
542C22—216no—
543C22—216no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—88no—
562C22—88no—
563C22—816no—
564C22—816no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22#145814yes—
570C22#145914yes—
571C22#146814yes—
572C22#147314yes—
573C22#195414yes—
574C22#195414yes—
575C22#196414yes—
576C22#196714yes—
577C22C2×#39011yescharacteristic
578C22C22×C22⋊C811yescharacteristic
579C22C2×#40811yescharacteristic
580C22C22×#9911yescharacteristic
581C22C2×#41011yescharacteristic
582C22C2×#41311yescharacteristic
583C22C2×#41311yescharacteristic
584C22C2×C8×D412yes—
585C22C2×#43612yes—
586C4—24no—
587C4—24no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—22no—
593C4—24no—
594C4—44no—
595C4—42no—
596C4—82no—
597C4—82no—
598C2—22no—
599C2—22no—
600C2—24no—
601C2—84no—
602C2#965118yesdirect product factor
603C2C2×#145811yescharacteristic
604C2C2×#145911yescharacteristic
605C2C2×#146811yescharacteristic
606C2C2×#147311yescharacteristic
607C2C2×#195411yescharacteristic
608C2C2×#196411yescharacteristic
609C2C2×#196711yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. C22×#324C2semidirect
6. #9651C2direct (N × H)
27. #1797C22semidirect
29. C4×C22⋊C8C22semidirect
33. C2×#324C22semidirect
67. C22×C22⋊C4C8semidirect
215. C2×C22⋊C4C2×C8semidirect
602. C2#9651direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3882
C2×#17972
C2×C4×C22⋊C82
C22×#3242
C22×#3232
#96512