Finite Group Databasebeta

← C2×#29916 #6434091

Subgroups — C2×#29916 #6434091

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#29916C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2094C211yesmaximal, characteristic, complemented
7C2×#2092C211yesmaximal, characteristic, complemented
8C2×#2084C211yesmaximal, characteristic, complemented
9C2×#2085C211yesmaximal, characteristic, complemented
10C2×#2073C211yesmaximal, characteristic, complemented
11C2×#2071C211yesmaximal, characteristic, complemented
12C2×#1875C211yesmaximal, characteristic
13C2×#1874C211yesmaximal, characteristic
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1853C211yesmaximal, characteristic, complemented
16C2×#1818C211yesmaximal, characteristic, complemented
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2624—24no—
32#2624C2218yes—
33C2×#399—21no—
34C2×#397—21no—
35C2×#394—21no—
36C2×#394—21no—
37#2317—24no—
38#2317C2218yescomplemented
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389—21no—
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380C2211yescharacteristic
52#2101C2218yescomplemented
53#2099C2218yescomplemented
54#2094C2218yescomplemented
55#2092C2218yescomplemented
56#2085C2218yescomplemented
57#2084C2218yescomplemented
58#2073C2218yescomplemented
59#2071C2218yescomplemented
60#1875C2218yes—
61#1874C2218yes—
62#1854C2218yescomplemented
63#1853C2218yescomplemented
64#1818C2218yescomplemented
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×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#121—22no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121C2314yes—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—22no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D414yes—
95C2×#117C2314yes—
96C2×#117C2314yes—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—42no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#107—42no—
130C2×#107C2314yes—
131C2×#106—42no—
132C2×#106—42no—
133C2×#106—42no—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8—24no—
146C2×C22⋊C8—24no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149#399—28no—
150#397—28no—
151#394—28no—
152#394—28no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2314yes—
157#389—28no—
158#389—28no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#385—28no—
164#385—28no—
165#384—22no—
166#384—22no—
167#384C2314yes—
168#384C2314yes—
169#382—22no—
170#382—22no—
171#382C2314yes—
172#382C2314yes—
173#380—22no—
174#380C2314yes—
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—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—82no—
206C22×D4—82no—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—82no—
222C23×C4—82no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
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—
249#121—28no—
250#121—28no—
251#121—48no—
252#117—216no—
253#117—28no—
254#117—28no—
255#117—44no—
256#117—44no—
257#117—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
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—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—44no—
336#106—44no—
337#106—44no—
338#106—44no—
339#102—44no—
340#102—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—28no—
344C22⋊C8—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24#12314yes—
367C24C22×D412yes—
368C24C22×D411yescharacteristic
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—48no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—816no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—48no—
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—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—816no—
442C22×C4#12314yes—
443C22×C4#12318yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—48no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23C2×#12314yes—
549C23C2×#12312yes—
550C23#53512yes—
551C23#53612yes—
552C23#57612yes—
553C23#57712yes—
554C23#57712yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—816no—
562D4—84no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—216no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4C2×#12314yes—
617C2×C4C2×#12312yes—
618C2×C4#53512yes—
619C2×C4#53612yes—
620C2×C4#53914yes—
621C2×C4#53914yes—
622C2×C4#53914yes—
623C2×C4#53914yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—42no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—416no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—168no—
650C22#262314yes—
651C22#309614yes—
652C22#309914yes—
653C22#322114yes—
654C22#322414yes—
655C22#324114yes—
656C22#324414yes—
657C22C22×#12311yescharacteristic
658C22C2×#53511yescharacteristic
659C22C2×#53611yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57711yescharacteristic
663C22C2×#57711yescharacteristic
664C4—24no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#262314yes—
677C4C2×#53911yescharacteristic
678C4C2×#53911yescharacteristic
679C4C2×#53911yescharacteristic
680C4C2×#53911yescharacteristic
681C2—44no—
682C2—162no—
683C2#2991618yesdirect product factor
684C2C2×#262311yescharacteristic
685C2C2×#309611yescharacteristic
686C2C2×#309911yescharacteristic
687C2C2×#322411yescharacteristic
688C2C2×#322111yescharacteristic
689C2C2×#324111yescharacteristic
690C2C2×#324411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C2×#2317C2semidirect
3. #29916C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2099C2semidirect
6. C2×#2094C2semidirect
7. C2×#2092C2semidirect
8. C2×#2084C2semidirect
9. C2×#2085C2semidirect
10. C2×#2073C2semidirect
11. C2×#2071C2semidirect
14. C2×#1854C2semidirect
15. C2×#1853C2semidirect
16. C2×#1818C2semidirect
38. #2317C22semidirect
52. #2101C22semidirect
53. #2099C22semidirect
54. #2094C22semidirect
55. #2092C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
62. #1854C22semidirect
63. #1853C22semidirect
64. #1818C22semidirect
683. C2#29916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26242
C2×#23172
#299162
C2×#21012
C2×#20992
C2×#20942
C2×#20922
C2×#20842
C2×#20852
C2×#20732
C2×#20712
C2×#18752
C2×#18742
C2×#18542
C2×#18532
C2×#18182