Finite Group Databasebeta

← C2×#33105 #6629629

Subgroups — C2×#33105 #6629629

Order 512.

2076 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#33105C2116yesmaximal, direct product factor
4C2×#1928C211yesmaximal, characteristic, complemented
5C2×#1927C211yesmaximal, characteristic, complemented
6C2×#1927C211yesmaximal, characteristic, complemented
7C2×#1927C211yesmaximal, characteristic, complemented
8C2×#1921C211yesmaximal, characteristic, complemented
9C2×#1919C211yesmaximal, characteristic, complemented
10C2×#1919C211yesmaximal, characteristic, complemented
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1873C211yesmaximal, characteristic
13C2×#1872C211yesmaximal, characteristic
14C2×#1854C211yesmaximal, characteristic
15C2×#1853C211yesmaximal, characteristic
16C2×#1818C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2625—24no—
32#2625C2218yes—
33C2×#400—21no—
34C2×#398—21no—
35C2×#397—21no—
36C2×#397—21no—
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#380C2211yescharacteristic
42#2274—24no—
43#2274C2218yes—
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4—21no—
52#1928C2218yescomplemented
53#1927C2218yescomplemented
54#1927C2218yescomplemented
55#1927C2218yescomplemented
56#1921C2218yescomplemented
57#1920C2218yescomplemented
58#1919C2218yescomplemented
59#1919C2218yescomplemented
60#1873C2218yes—
61#1872C2218yes—
62#1854C2218yes—
63#1853C2218yes—
64#1818C2218yes—
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×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2314yes—
88C2×#117—22no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117D414yes—
94C2×#117C2314yes—
95C2×#117C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—24no—
126C2×#107C2314yes—
127C2×#106—24no—
128C2×#106—24no—
129C2×#106—24no—
130C2×#106C2314yes—
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#400—28no—
146#398—28no—
147#397—28no—
148#397—28no—
149#391—22no—
150#391—22no—
151#391C2314yes—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#380—22no—
158#380C2314yes—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—28no—
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—
181C2×#95—21no—
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—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
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×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—48no—
248#117—216no—
249#117—216no—
250#117—28no—
251#117—28no—
252#117—28no—
253#117—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—82no—
318C2×C42—82no—
319C2×C42—82no—
320C2×C42—82no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325#107—28no—
326#106—28no—
327#106—28no—
328#106—28no—
329#103—28no—
330#103—28no—
331#103—28no—
332#103—28no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335#95—24no—
336#95—24no—
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—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24#12314yes—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
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—84no—
426C22×C4—84no—
427C22×C4—816no—
428C22×C4#12314yes—
429C22×C4#12318yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4114yes—
486C4⋊C4C2×#4114yes—
487C4⋊C4C2×#4114yes—
488C22⋊C4—216no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C42—216no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—84no—
501C42—84no—
502C42—84no—
503C42—84no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C2×#12314yes—
518C23C2×#12312yes—
519C23#52512yes—
520C23#52512yes—
521C23#57712yes—
522C23#57712yes—
523C23#57712yes—
524C23#57812yes—
525C23C23×D411yescharacteristic
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—84no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—216no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
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—816no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C2×#12314yes—
589C2×C4C2×#12312yes—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#52914yes—
593C2×C4#52914yes—
594C2×C4#52914yes—
595C2×C4#52914yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C8—42no—
603C8—42no—
604C8—42no—
605C8—42no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—416no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22#285314yes—
615C22#309714yes—
616C22#309914yes—
617C22#312014yes—
618C22#312114yes—
619C22#312314yes—
620C22#312514yes—
621C22C22×#12311yescharacteristic
622C22C2×#52511yescharacteristic
623C22C2×#52511yescharacteristic
624C22C2×#57711yescharacteristic
625C22C2×#57711yescharacteristic
626C22C2×#57711yescharacteristic
627C22C2×#57811yescharacteristic
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#285314yes—
642C4C2×#52911yescharacteristic
643C4C2×#52911yescharacteristic
644C4C2×#52911yescharacteristic
645C4C2×#52911yescharacteristic
646C2—44no—
647C2#3310518yesdirect product factor
648C2C2×#285311yescharacteristic
649C2C2×#309711yescharacteristic
650C2C2×#309911yescharacteristic
651C2C2×#312011yescharacteristic
652C2C2×#312111yescharacteristic
653C2C2×#312311yescharacteristic
654C2C2×#312511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33105C2direct (N × H)
4. C2×#1928C2semidirect
5. C2×#1927C2semidirect
6. C2×#1927C2semidirect
7. C2×#1927C2semidirect
8. C2×#1921C2semidirect
9. C2×#1919C2semidirect
10. C2×#1919C2semidirect
11. C2×#1920C2semidirect
52. #1928C22semidirect
53. #1927C22semidirect
54. #1927C22semidirect
55. #1927C22semidirect
56. #1921C22semidirect
57. #1920C22semidirect
58. #1919C22semidirect
59. #1919C22semidirect
647. C2#33105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26252
C2×#22742
#331052
C2×#19282
C2×#19272
C2×#19212
C2×#19192
C2×#19202
C2×#18732
C2×#18722
C2×#18542
C2×#18532
C2×#18182