Finite Group Databasebeta

← C2×#26428 #6441913

Subgroups — C2×#26428 #6441913

Order 512.

2158 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#26428C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#1988C211yesmaximal, characteristic, complemented
9C2×#1921C211yesmaximal, characteristic, complemented
10C2×#1894C211yesmaximal, characteristic
11C2×#1873C211yesmaximal, characteristic
12C2×#1863C211yesmaximal, characteristic, complemented
13C2×#1852C211yesmaximal, characteristic
14C2×#1851C211yesmaximal, characteristic
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1736C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2439—24no—
33#2439C2218yes—
34C2×#396—21no—
35C2×#393—21no—
36#2360—24no—
37#2360C2218yes—
38C2×#391—21no—
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2086C2218yescomplemented
54#2068C2218yescomplemented
55#2024C2218yescomplemented
56#1988C2218yescomplemented
57#1921C2218yescomplemented
58#1894C2218yes—
59#1873C2218yes—
60#1863C2218yescomplemented
61#1852C2218yes—
62#1851C2218yes—
63#1805C2218yescomplemented
64#1736C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×M16—24no—
82C22×M16—42no—
83C22×M16—42no—
84C22×M16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8Q818yescomplemented
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—24no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C2×#117—22no—
102C2×#117—24no—
103C2×#117—42no—
104C2×#117C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C2×#107—24no—
134C2×#107—42no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×C4⋊C8—42no—
138C2×C4⋊C8—42no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103—24no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2314yes—
151#396—28no—
152#393—28no—
153#391—22no—
154#391—28no—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—28no—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#387—28no—
164#387—28no—
165#385—28no—
166#382—22no—
167#382C2314yes—
168#380—22no—
169#380C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4—28no—
173C4×C22⋊C4D414yes—
174C4×C22⋊C4C2314yes—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25C2×D411yescharacteristic
199C22×Q8—21no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—82no—
209C22×Q8C2×Q814yes—
210C22×Q8C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×Q814yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C2×M16—28no—
238C2×M16—44no—
239C2×M16—44no—
240C2×M16—416no—
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—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C22⋊Q8—216no—
261C22⋊Q8—28no—
262C22⋊Q8C2×Q8116yescomplemented
263C4×Q8—28no—
264C4×Q8—44no—
265C4×Q8—48no—
266#117—28no—
267#117—44no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×Q814yes—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107—28no—
339#107—44no—
340C4⋊C8—44no—
341C4⋊C8—44no—
342#103—28no—
343#103—44no—
344#103—44no—
345#103—44no—
346C22⋊C8—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24C22⋊Q814yes—
365C24C22×D412yes—
366C24C22×D411yescharacteristic
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—816no—
383C2×Q8C22×Q814yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
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—22no—
418C22×C4—22no—
419C22×C4—48no—
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—48no—
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—42no—
440C22×C4—42no—
441C22×C4—84no—
442C22×C4—816no—
443C22×C4C22⋊Q814yes—
444C22×C4C22⋊Q818yes—
445C22×C4#13714yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4C2511yescharacteristic
461M16—416no—
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—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—44no—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
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—22no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4C22×Q814yes—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—216no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—48no—
519C42—84no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23C2×C22⋊Q814yes—
534C23C2×C22⋊Q812yes—
535C23#54612yes—
536C23#54612yes—
537C23C2×#13712yes—
538C23#57512yes—
539C23#57612yes—
540C23#57812yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—24no—
544Q8—42no—
545Q8—42no—
546Q8—48no—
547Q8—816no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—216no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
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—816no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q812yes—
606C2×C4#52914yes—
607C2×C4#53814yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#55414yes—
611C2×C4#55414yes—
612C2×C4C2×#13714yes—
613C2×C4C23×Q811yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C8—42no—
620C8—42no—
621C8—42no—
622C8—42no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22#245014yes—
632C22#307814yes—
633C22#308314yes—
634C22#331414yes—
635C22#331914yes—
636C22#332714yes—
637C22#332814yes—
638C22C22×C22⋊Q811yescharacteristic
639C22C2×#54611yescharacteristic
640C22C2×#54611yescharacteristic
641C22C22×#13711yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57611yescharacteristic
644C22C2×#57811yescharacteristic
645C4—24no—
646C4—24no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#245014yes—
659C4C2×#52911yescharacteristic
660C4C2×#53811yescharacteristic
661C4C2×#55411yescharacteristic
662C4C2×#55411yescharacteristic
663C2—44no—
664C2#2642818yesdirect product factor
665C2C2×#245011yescharacteristic
666C2C2×#307811yescharacteristic
667C2C2×#308311yescharacteristic
668C2C2×#331411yescharacteristic
669C2C2×#331911yescharacteristic
670C2C2×#332711yescharacteristic
671C2C2×#332811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26428C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2086C2semidirect
6. C2×#2068C2semidirect
7. C2×#2024C2semidirect
8. C2×#1988C2semidirect
9. C2×#1921C2semidirect
12. C2×#1863C2semidirect
15. C2×#1805C2semidirect
52. #2101C22semidirect
53. #2086C22semidirect
54. #2068C22semidirect
55. #2024C22semidirect
56. #1988C22semidirect
57. #1921C22semidirect
60. #1863C22semidirect
63. #1805C22semidirect
95. C2×C22⋊Q8Q8semidirect
262. C22⋊Q8C2×Q8semidirect
664. C2#26428direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23602
#264282
C2×#21012
C2×#20862
C2×#20682
C2×#20242
C2×#19882
C2×#19212
C2×#18942
C2×#18732
C2×#18632
C2×#18522
C2×#18512
C2×#18052
C2×#17362