Finite Group Databasebeta

← C2×#33101 #6629622

Subgroups — C2×#33101 #6629622

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#33101C2116yesmaximal, direct product factor
4C2×#1928C211yesmaximal, characteristic, complemented
5C2×#1928C211yesmaximal, characteristic, complemented
6C2×#1927C211yesmaximal, characteristic, complemented
7C2×#1927C211yesmaximal, characteristic, complemented
8C2×#1921C211yesmaximal, characteristic, complemented
9C2×#1919C211yesmaximal, characteristic, complemented
10C2×#1920C211yesmaximal, characteristic, complemented
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1873C211yesmaximal, characteristic
13C2×#1873C211yesmaximal, characteristic
14C2×#1854C211yesmaximal, characteristic
15C2×#1854C211yesmaximal, characteristic
16C2×#1817C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2855—24no—
32#2855C2218yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387—21no—
38C2×#387—21no—
39C2×#385—21no—
40C2×#381—21no—
41C2×#380C2211yescharacteristic
42C4×#117—24no—
43C4×#117C2218yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4—21no—
50C2×#376—21no—
51C2×C43—21no—
52#1928C2218yescomplemented
53#1928C2218yescomplemented
54#1927C2218yescomplemented
55#1927C2218yescomplemented
56#1921C2218yescomplemented
57#1920C2218yescomplemented
58#1919C2218yescomplemented
59#1918C2218yescomplemented
60#1873C2218yes—
61#1873C2218yes—
62#1854C2218yes—
63#1854C2218yes—
64#1817C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4D411yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2314yes—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117C2314yes—
96C2×#117C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42D414yes—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—24no—
126C2×#107—24no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106—24no—
130C2×#106—24no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—24no—
136C2×#103—24no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145#391—22no—
146#391—22no—
147#391C2314yes—
148#391C2314yes—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#387—28no—
154#387—28no—
155#385—28no—
156#381—28no—
157#380—22no—
158#380C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—28no—
169#376—28no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C43—28no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4#4114yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C22⋊Q8—48no—
252#117—216no—
253#117—216no—
254#117—28no—
255#117—28no—
256#117—28no—
257#117—28no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4#4114yes—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—216no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#107—28no—
328#107—28no—
329#106—28no—
330#106—28no—
331#103—28no—
332#103—28no—
333#103—28no—
334#103—28no—
335C22⋊C8—28no—
336C22⋊C8—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24#12314yes—
355C24C22×D412yes—
356C24C22×D411yescharacteristic
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4#12314yes—
433C22×C4#12318yes—
434C22×C4#13514yes—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—44no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4C2×#4114yes—
488C4⋊C4C2×#4114yes—
489C4⋊C4C2×#4114yes—
490C4⋊C4C2×#4114yes—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C42—216no—
496C42—216no—
497C42—28no—
498C42—24no—
499C42—24no—
500C42—28no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—48no—
506C42—48no—
507C42—48no—
508C42—48no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12314yes—
523C23C2×#12312yes—
524C23#52512yes—
525C23#52512yes—
526C23C2×#13512yes—
527C23C2×#13712yes—
528C23#57512yes—
529C23#57512yes—
530C23C23×D411yescharacteristic
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—84no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—216no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4C2×#12314yes—
593C2×C4C2×#12312yes—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4#52914yes—
597C2×C4#52914yes—
598C2×C4#52914yes—
599C2×C4#52914yes—
600C2×C4C2×#13514yes—
601C2×C4C2×#13714yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C8—42no—
609C8—42no—
610C8—42no—
611C8—42no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—416no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22#285314yes—
621C22#309114yes—
622C22#309514yes—
623C22#310214yes—
624C22#310714yes—
625C22#311114yes—
626C22#311614yes—
627C22C22×#12311yescharacteristic
628C22C2×#52511yescharacteristic
629C22C2×#52511yescharacteristic
630C22C22×#13511yescharacteristic
631C22C22×#13711yescharacteristic
632C22C2×#57511yescharacteristic
633C22C2×#57511yescharacteristic
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#285314yes—
648C4C2×#52911yescharacteristic
649C4C2×#52911yescharacteristic
650C4C2×#52911yescharacteristic
651C4C2×#52911yescharacteristic
652C2—44no—
653C2#3310118yesdirect product factor
654C2C2×#285311yescharacteristic
655C2C2×#309111yescharacteristic
656C2C2×#309511yescharacteristic
657C2C2×#310211yescharacteristic
658C2C2×#310711yescharacteristic
659C2C2×#311611yescharacteristic
660C2C2×#311111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33101C2direct (N × H)
4. C2×#1928C2semidirect
5. C2×#1928C2semidirect
6. C2×#1927C2semidirect
7. C2×#1927C2semidirect
8. C2×#1921C2semidirect
9. C2×#1919C2semidirect
10. C2×#1920C2semidirect
11. C2×#1918C2semidirect
52. #1928C22semidirect
53. #1928C22semidirect
54. #1927C22semidirect
55. #1927C22semidirect
56. #1921C22semidirect
57. #1920C22semidirect
58. #1919C22semidirect
59. #1918C22semidirect
653. C2#33101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×#1172
#331012
C2×#19282
C2×#19272
C2×#19212
C2×#19192
C2×#19202
C2×#19182
C2×#18732
C2×#18542
C2×#18172