Finite Group Databasebeta

← C2×#29246 #6461189

Subgroups — C2×#29246 #6461189

Order 512.

2264 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#29246C2116yesmaximal, direct product factor
4C2×#2087C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic, complemented
8C2×#1921C211yesmaximal, characteristic, complemented
9C2×#1918C211yesmaximal, characteristic, complemented
10C2×#1875C211yesmaximal, characteristic
11C2×#1874C211yesmaximal, characteristic
12C2×#1864C211yesmaximal, characteristic, complemented
13C2×#1862C211yesmaximal, characteristic, complemented
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1852C211yesmaximal, characteristic, complemented
16C2×#1798C211yesmaximal, characteristic, complemented
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2324—24no—
35#2324C2218yes—
36#2316—24no—
37#2316C2218yescomplemented
38C2×#390—21no—
39C2×#390C2211yescharacteristic
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#388—21no—
48C2×#386—21no—
49C2×#385—21no—
50C2×#380—21no—
51C2×#380—21no—
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2088C2218yescomplemented
55#2087C2218yescomplemented
56#2067C2218yescomplemented
57#2067C2218yescomplemented
58#1921C2218yescomplemented
59#1918C2218yescomplemented
60#1875C2218yes—
61#1874C2218yes—
62#1864C2218yescomplemented
63#1862C2218yescomplemented
64#1854C2218yescomplemented
65#1852C2218yescomplemented
66#1798C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#121—22no—
87C2×#121—24no—
88C2×#121Q818yes—
89C2×#121C2314yes—
90C2×C4×D4—22no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4C2314yes—
94C2×#117—22no—
95C2×#117—42no—
96C2×#117—42no—
97C2×#117—42no—
98C2×#117—42no—
99C2×#117D414yes—
100C2×#117C2314yes—
101C2×#117C2314yes—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106—42no—
140C2×#106C2314yes—
141C2×C4⋊C8—42no—
142C2×C4⋊C8—42no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157#390—28no—
158#390—22no—
159#390C2314yes—
160#389—28no—
161#389—28no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#388—28no—
173#386—28no—
174#385—28no—
175#380—22no—
176#380—28no—
177#380—28no—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C25C2×D411yescharacteristic
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—82no—
209C22×D4—82no—
210C22×D4C2×Q814yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4—82no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×Q814yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#121—216no—
258#121—28no—
259#121C2×Q8116yes—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—48no—
263#117—48no—
264#117—44no—
265#117—44no—
266#117—44no—
267#117—44no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×Q814yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—82no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—28no—
345#107—28no—
346#107—44no—
347#106—44no—
348C4⋊C8—44no—
349C4⋊C8—44no—
350#102—28no—
351#102—28no—
352#102—44no—
353#102—44no—
354C22⋊C8—28no—
355C22⋊C8—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22⋊Q814yes—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380C2×D4—216no—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—816no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4C22×Q814yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—816no—
456C22×C4—816no—
457C22×C4C22⋊Q814yes—
458C22×C4C22⋊Q818yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8—44no—
489C2×C8—44no—
490C4⋊C4—216no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C22×Q814yes—
517C4⋊C4C2×#4114yes—
518C4⋊C4C2×#4114yes—
519C22⋊C4—216no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—48no—
534C42—84no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23C2×C22⋊Q814yes—
563C23C2×C22⋊Q812yes—
564C23#52512yes—
565C23#53412yes—
566C23#57612yes—
567C23#57612yes—
568C23#57712yes—
569C23#57812yes—
570C23C23×D411yescharacteristic
571D4—24no—
572D4—24no—
573D4—48no—
574D4—42no—
575D4—42no—
576D4—816no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—216no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—816no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×C22⋊Q814yes—
632C2×C4C2×C22⋊Q812yes—
633C2×C4#52512yes—
634C2×C4#53412yes—
635C2×C4#55314yes—
636C2×C4#55314yes—
637C2×C4#55414yes—
638C2×C4#55414yes—
639C2×C4C23×Q811yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C8—42no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—416no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#257914yes—
666C22#308114yes—
667C22#308314yes—
668C22#312114yes—
669C22#312214yes—
670C22#319814yes—
671C22#320314yes—
672C22C22×C22⋊Q811yescharacteristic
673C22C2×#52511yescharacteristic
674C22C2×#53411yescharacteristic
675C22C2×#57611yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57811yescharacteristic
679C4—24no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#257914yes—
692C4C2×#55311yescharacteristic
693C4C2×#55311yescharacteristic
694C4C2×#55411yescharacteristic
695C4C2×#55411yescharacteristic
696C2—44no—
697C2—84no—
698C2#2924618yesdirect product factor
699C2C2×#257911yescharacteristic
700C2C2×#308111yescharacteristic
701C2C2×#308311yescharacteristic
702C2C2×#312111yescharacteristic
703C2C2×#312211yescharacteristic
704C2C2×#319811yescharacteristic
705C2C2×#320311yescharacteristic
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×#2316C2semidirect
3. #29246C2direct (N × H)
4. C2×#2087C2semidirect
5. C2×#2088C2semidirect
6. C2×#2067C2semidirect
7. C2×#2067C2semidirect
8. C2×#1921C2semidirect
9. C2×#1918C2semidirect
12. C2×#1864C2semidirect
13. C2×#1862C2semidirect
14. C2×#1854C2semidirect
15. C2×#1852C2semidirect
16. C2×#1798C2semidirect
37. #2316C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
56. #2067C22semidirect
57. #2067C22semidirect
58. #1921C22semidirect
59. #1918C22semidirect
62. #1864C22semidirect
63. #1862C22semidirect
64. #1854C22semidirect
65. #1852C22semidirect
66. #1798C22semidirect
698. C2#29246direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23242
C2×#23162
#292462
C2×#20872
C2×#20882
C2×#20672
C2×#19212
C2×#19182
C2×#18752
C2×#18742
C2×#18642
C2×#18622
C2×#18542
C2×#18522
C2×#17982