Finite Group Databasebeta

← C2×#27744 #6458432

Subgroups — C2×#27744 #6458432

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic, complemented
3#27744C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic
7C2×#2071C211yesmaximal, characteristic
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1987C211yesmaximal, characteristic, complemented
10C2×#1936C211yesmaximal, characteristic
11C2×#1934C211yesmaximal, characteristic
12C2×#1836C211yesmaximal, characteristic, complemented
13C2×#1834C211yesmaximal, characteristic, complemented
14C2×#1804C211yesmaximal, characteristic
15C2×#1771C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2570—24no—
30#2570C2218yescomplemented
31#2542—24no—
32#2542C2218yes—
33C2×#400—21no—
34C2×#399—21no—
35C2×#399—21no—
36C2×#398—21no—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#387—21no—
42C2×#387—21no—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2218yescomplemented
51#2099C2218yescomplemented
52#2073C2218yes—
53#2071C2218yes—
54#1993C2218yescomplemented
55#1987C2218yescomplemented
56#1936C2218yes—
57#1934C2218yes—
58#1836C2218yescomplemented
59#1834C2218yescomplemented
60#1804C2218yes—
61#1771C2218yescomplemented
62#1762C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
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×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2314yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—22no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
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×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×C4⋊C8—24no—
124C2×C4⋊C8C2314yes—
125C2×#102—24no—
126C2×#102—24no—
127C2×#102—24no—
128C2×#102—24no—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#97—24no—
134C2×#97—24no—
135C2×#97C2314yes—
136C2×#97C2314yes—
137C2×C4×C8—24no—
138C2×C4×C8C2314yes—
139#400—28no—
140#399—28no—
141#399—28no—
142#398—28no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#387—28no—
152#387—28no—
153#384—22no—
154#384—22no—
155#384—22no—
156#384C2314yes—
157#384C2314yes—
158#384C2314yes—
159#382—22no—
160#382—22no—
161#382C2314yes—
162#382C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4D414yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95D411yescharacteristic
175C2×#95D411yescharacteristic
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25—21no—
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4#4114yes—
202C22×D4C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—24no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
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#4114yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—48no—
240C22⋊Q8—432no—
241C22⋊Q8—432no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—48no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—82no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42#4114yes—
315C2×C42#4114yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319C4⋊C8—28no—
320#102—28no—
321#102—28no—
322#102—28no—
323#102—28no—
324#97—28no—
325#97—28no—
326C4×C8—28no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—24no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24C22×D411yescharacteristic
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—816no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4C2×#4114yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
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—48no—
407C22×C4—816no—
408C22×C4#12414yes—
409C22×C4#12414yes—
410C22×C4AGL(1,ℤ/8)14yes—
411C22×C4AGL(1,ℤ/8)14yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C42−1+412yes—
424C22×C42−1+412yes—
425C22×C4C2511yescharacteristic
426C2×C8—28no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8C2×#4114yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C42—24no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—84no—
486C42#12418yes—
487C42C2×#4114yes—
488C42C2×#4114yes—
489C23—24no—
490C23—48no—
491C23—48no—
492C23—42no—
493C23—42no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23C2×#12412yes—
504C23#54712yes—
505C23#54812yes—
506C23C2×AGL(1,ℤ/8)12yes—
507C23C2×AGL(1,ℤ/8)12yes—
508C23#57712yes—
509C23#57812yes—
510C23C23×D411yescharacteristic
511Q8—816no—
512D4—24no—
513D4—24no—
514D4—42no—
515D4—42no—
516D4—48no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—816no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4C2×#12414yes—
567C2×C4C2×#12414yes—
568C2×C4C2×#12412yes—
569C2×C4#52914yes—
570C2×C4#54114yes—
571C2×C4#54712yes—
572C2×C4#54812yes—
573C2×C4#55214yes—
574C2×C4#55214yes—
575C2×C4C2×AGL(1,ℤ/8)14yes—
576C2×C4C2×AGL(1,ℤ/8)14yes—
577C2×C4C22×#4111yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×2−1+411yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C8—24no—
584C8—42no—
585C8—42no—
586C8—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22#248914yes—
596C22#313714yes—
597C22#313814yes—
598C22#334314yes—
599C22#334514yes—
600C22#336714yes—
601C22#337314yes—
602C22C22×#12411yescharacteristic
603C22C2×#54711yescharacteristic
604C22C2×#54811yescharacteristic
605C22C22×AGL(1,ℤ/8)11yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C2×#57711yescharacteristic
608C22C2×#57811yescharacteristic
609C4—24no—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#248914yes—
623C4C2×#52911yescharacteristic
624C4C2×#54111yescharacteristic
625C4C2×#55211yescharacteristic
626C4C2×#55211yescharacteristic
627C2—84no—
628C2#2774418yesdirect product factor
629C2C2×#248911yescharacteristic
630C2C2×#313711yescharacteristic
631C2C2×#313811yescharacteristic
632C2C2×#334311yescharacteristic
633C2C2×#334511yescharacteristic
634C2C2×#336711yescharacteristic
635C2C2×#337311yescharacteristic
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×#2570C2semidirect
3. #27744C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2099C2semidirect
8. C2×#1993C2semidirect
9. C2×#1987C2semidirect
12. C2×#1836C2semidirect
13. C2×#1834C2semidirect
15. C2×#1771C2semidirect
30. #2570C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
54. #1993C22semidirect
55. #1987C22semidirect
58. #1836C22semidirect
59. #1834C22semidirect
61. #1771C22semidirect
628. C2#27744direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#25702
#277442
C2×#21002
C2×#20992
C2×#20732
C2×#20712
C2×#19932
C2×#19872
C2×#19362
C2×#19342
C2×#18362
C2×#18342
C2×#18042
C2×#17712
C2×#17622