Finite Group Databasebeta

← C2×#27011 #6445817

Subgroups — C2×#27011 #6445817

Order 512.

2060 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic, complemented
3#27011C2116yesmaximal, direct product factor
4C2×#2065C211yesmaximal, characteristic
5C2×#2059C211yesmaximal, characteristic, complemented
6C2×#2057C211yesmaximal, characteristic, complemented
7C2×#2025C211yesmaximal, characteristic
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1925C211yesmaximal, characteristic
11C2×#1925C211yesmaximal, characteristic
12C2×#1921C211yesmaximal, characteristic, complemented
13C2×#1919C211yesmaximal, characteristic, complemented
14C2×#1916C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2452—24no—
32#2452C2218yes—
33C2×#396—21no—
34C2×#393—21no—
35#2370—24no—
36#2370C2218yescomplemented
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2065C2218yes—
53#2059C2218yescomplemented
54#2057C2218yescomplemented
55#2025C2218yes—
56#1987C2218yescomplemented
57#1942C2218yescomplemented
58#1925C2218yes—
59#1925C2218yes—
60#1921C2218yescomplemented
61#1919C2218yescomplemented
62#1916C2218yescomplemented
63#1862C2218yes—
64C4×#102C2218yes—
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×Q8—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125—22no—
91C2×#125C2314yes—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—48no—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4C2314yes—
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⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—24no—
126C2×#107—24no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106—24no—
132C2×#106C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—42no—
138C2×#102Q818yescomplemented
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C4×C8—42no—
144C2×C4×C8C2314yes—
145#396—28no—
146#393—28no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#389—28no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#387—28no—
161#387—28no—
162#385—28no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4D414yes—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95D411yescharacteristic
180C2×#95D411yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×Q8—24no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—82no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×Q814yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—82no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8C2×Q814yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C4⋊Q8—28no—
241C4⋊Q8—44no—
242C4⋊Q8—48no—
243#125—216no—
244#125—28no—
245#125—28no—
246C22⋊Q8—432no—
247C4×D4—28no—
248C4×D4—44no—
249C4×D4—48no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
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—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—82no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—82no—
317C2×C42—82no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323C2×C42C2412yes—
324#107—28no—
325#107—28no—
326#107—44no—
327#106—28no—
328C4⋊C8—44no—
329#102—28no—
330#102—28no—
331#102—44no—
332#102C2×Q8116yescomplemented
333C4×C8—44no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95C2×D414yes—
342#95C2×D414yes—
343C24—24no—
344C24—22no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24C22×D411yescharacteristic
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—84no—
355C2×Q8—816no—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4C22×Q814yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
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—48no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
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—48no—
415C22×C4—84no—
416C22×C4—816no—
417C22×C4#13514yes—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q812yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C4C2511yescharacteristic
433C2×C8—28no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8C22×Q814yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4C22×Q814yes—
476C4⋊C4C2×#4114yes—
477C4⋊C4C2×#4114yes—
478C4⋊C4C2×#4114yes—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—48no—
494C42—84no—
495C42—84no—
496C23—24no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23#52512yes—
511C23#54612yes—
512C23D4×Q812yes—
513C23C2×#13512yes—
514C23C2×AGL(1,ℤ/8)12yes—
515C23#57612yes—
516C23#57812yes—
517C23C23×D411yescharacteristic
518Q8—816no—
519D4—24no—
520D4—24no—
521D4—42no—
522D4—42no—
523D4—48no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
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—816no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#52512yes—
576C2×C4#53314yes—
577C2×C4#54214yes—
578C2×C4#54612yes—
579C2×C4D4×Q812yes—
580C2×C4#55414yes—
581C2×C4#55614yes—
582C2×C4C2×#13514yes—
583C2×C4C2×AGL(1,ℤ/8)14yes—
584C2×C4C23×Q811yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C8—24no—
591C8—42no—
592C8—42no—
593C8—42no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—82no—
598C22—816no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22#246514yes—
603C22#310614yes—
604C22#312414yes—
605C22#330014yes—
606C22#331814yes—
607C22#338014yes—
608C22#338614yes—
609C22C2×#52511yescharacteristic
610C22C2×#54611yescharacteristic
611C22C2×D4×Q811yescharacteristic
612C22C22×#13511yescharacteristic
613C22C22×AGL(1,ℤ/8)11yescharacteristic
614C22C2×#57611yescharacteristic
615C22C2×#57811yescharacteristic
616C4—24no—
617C4—24no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4#246514yes—
630C4C2×#53311yescharacteristic
631C4C2×#54211yescharacteristic
632C4C2×#55411yescharacteristic
633C4C2×#55611yescharacteristic
634C2—84no—
635C2#2701118yesdirect product factor
636C2C2×#246511yescharacteristic
637C2C2×#310611yescharacteristic
638C2C2×#312411yescharacteristic
639C2C2×#330011yescharacteristic
640C2C2×#331811yescharacteristic
641C2C2×#338011yescharacteristic
642C2C2×#338611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2370C2semidirect
3. #27011C2direct (N × H)
5. C2×#2059C2semidirect
6. C2×#2057C2semidirect
8. C2×#1987C2semidirect
9. C2×#1942C2semidirect
12. C2×#1921C2semidirect
13. C2×#1919C2semidirect
14. C2×#1916C2semidirect
36. #2370C22semidirect
53. #2059C22semidirect
54. #2057C22semidirect
56. #1987C22semidirect
57. #1942C22semidirect
60. #1921C22semidirect
61. #1919C22semidirect
62. #1916C22semidirect
138. C2×#102Q8semidirect
332. #102C2×Q8semidirect
635. C2#27011direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#23702
#270112
C2×#20652
C2×#20592
C2×#20572
C2×#20252
C2×#19872
C2×#19422
C2×#19252
C2×#19212
C2×#19192
C2×#19162
C2×#18622
C2×C4×#1022