Finite Group Databasebeta

← C2×#29417 #6464166

Subgroups — C2×#29417 #6464166

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#29417C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic
5C2×#2036C211yesmaximal, characteristic
6C2×#2035C211yesmaximal, characteristic
7C2×#2001C211yesmaximal, characteristic
8C2×#1928C211yesmaximal, characteristic, complemented
9C2×#1927C211yesmaximal, characteristic, complemented
10C2×#1919C211yesmaximal, characteristic, complemented
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1878C211yesmaximal, characteristic, complemented
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1817C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C4×C22⋊Q8—24no—
35C4×C22⋊Q8—24no—
36C4×C22⋊Q8C2218yes—
37C4×C22⋊Q8C2218yes—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#387—21no—
44C2×#385—21no—
45C2×#385—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4—21no—
54#2077C2218yes—
55#2036C2218yes—
56#2035C2218yes—
57#2001C2218yes—
58#1928C2218yescomplemented
59#1927C2218yescomplemented
60#1919C2218yescomplemented
61#1918C2218yescomplemented
62#1878C2218yescomplemented
63#1878C2218yescomplemented
64#1865C2218yescomplemented
65#1865C2218yescomplemented
66#1817C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16—24no—
83C22×Q16—24no—
84C22×Q16—42no—
85C22×Q16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42—42no—
127C22×C42—42no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D414yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C2×#107—24no—
135C2×#107C2314yes—
136C2×#106—24no—
137C2×#106—42no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#103—24no—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103—24no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8—42no—
153C2×C22⋊C8—42no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#387—28no—
165#387—28no—
166#385—28no—
167#385—28no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—28no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4—28no—
179C4×C22⋊C4—28no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8#4114yes—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C23×C4—24no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D414yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—28no—
249C2×Q16—44no—
250C2×Q16—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22⋊Q8—216no—
268C22⋊Q8—216no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
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×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—48no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—28no—
347#106—28no—
348#106—44no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24#13714yes—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8C2×#4114yes—
401C2×Q8C2×#4114yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4#13514yes—
469C22×C4#13714yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4C2511yescharacteristic
483Q16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8—44no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4C2×#4114yes—
523C4⋊C4C2×#4114yes—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—44no—
536C42—48no—
537C42—48no—
538C42—48no—
539C42—48no—
540C23—28no—
541C23—28no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—28no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23#52512yes—
554C23#52512yes—
555C23#53412yes—
556C23C2×#13512yes—
557C23C2×#13714yes—
558C23C2×#13712yes—
559C23C2×#13712yes—
560C23#57512yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—24no—
565Q8—24no—
566Q8—48no—
567Q8—48no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
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—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4#53412yes—
632C2×C4#55114yes—
633C2×C4#55314yes—
634C2×C4#55314yes—
635C2×C4#55914yes—
636C2×C4C2×#13514yes—
637C2×C4C2×#13714yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—416no—
656C22#260014yes—
657C22#310214yes—
658C22#310714yes—
659C22#311314yes—
660C22#311714yes—
661C22#319014yes—
662C22#319614yes—
663C22C2×#52511yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#53411yescharacteristic
666C22C22×#13511yescharacteristic
667C22C22×#13711yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C4—24no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#260014yes—
684C4C2×#55111yescharacteristic
685C4C2×#55311yescharacteristic
686C4C2×#55311yescharacteristic
687C4C2×#55911yescharacteristic
688C2—44no—
689C2#2941718yesdirect product factor
690C2C2×#260011yescharacteristic
691C2C2×#310211yescharacteristic
692C2C2×#310711yescharacteristic
693C2C2×#311711yescharacteristic
694C2C2×#311311yescharacteristic
695C2C2×#319011yescharacteristic
696C2C2×#319611yescharacteristic
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. #29417C2direct (N × H)
8. C2×#1928C2semidirect
9. C2×#1927C2semidirect
10. C2×#1919C2semidirect
11. C2×#1918C2semidirect
12. C2×#1878C2semidirect
13. C2×#1878C2semidirect
14. C2×#1865C2semidirect
15. C2×#1865C2semidirect
58. #1928C22semidirect
59. #1927C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
62. #1878C22semidirect
63. #1878C22semidirect
64. #1865C22semidirect
65. #1865C22semidirect
689. C2#29417direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#294172
C2×#20772
C2×#20362
C2×#20352
C2×#20012
C2×#19282
C2×#19272
C2×#19192
C2×#19182
C2×#18782
C2×#18652
C2×#18172