Finite Group Databasebeta

← C2×#27765 #6458556

Subgroups — C2×#27765 #6458556

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic
3#27765C2116yesmaximal, direct product factor
4C2×#2072C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic
6C2×#2043C211yesmaximal, characteristic, complemented
7C2×#2042C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic
9C2×#1959C211yesmaximal, characteristic
10C2×#1959C211yesmaximal, characteristic
11C2×#1956C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic
14C2×#1803C211yesmaximal, characteristic, complemented
15C2×#1771C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36C22×#97C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2552—24no—
39#2552C2218yescomplemented
40#2545—24no—
41#2545C2218yes—
42C2×#398—21no—
43C2×#398—21no—
44C2×#396—21no—
45C2×#396—21no—
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2072C2218yescomplemented
59#2071C2218yes—
60#2043C2218yescomplemented
61#2042C2218yes—
62#1983C2218yes—
63#1959C2218yes—
64#1959C2218yes—
65#1956C2218yescomplemented
66#1935C2218yescomplemented
67#1934C2218yes—
68#1803C2218yescomplemented
69#1771C2218yescomplemented
70#1763C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
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—28no—
89C2×#125—48no—
90C2×#124—24no—
91C2×#124—22no—
92C2×#124—22no—
93C2×#124—24no—
94C2×#124—24no—
95C2×#124—42no—
96C2×#124C2314yes—
97C2×#124C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
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⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—24no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102—24no—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102—42no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97—24no—
158C2×#97—42no—
159C2×#97C2314yes—
160C2×#97C2314yes—
161C2×C4×C8—42no—
162C2×C4×C8C2314yes—
163#398—28no—
164#398—28no—
165#396—28no—
166#396—28no—
167#391—22no—
168#391C2314yes—
169#390—22no—
170#390C2314yes—
171#389—28no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#388—28no—
177#388—28no—
178#384—22no—
179#384—22no—
180#384—22no—
181#384C2314yes—
182#384C2314yes—
183#384C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D414yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—21no—
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—22no—
214C22×Q8—22no—
215C22×Q8#4114yes—
216C22×Q8C2412yes—
217C22×Q8C2412yes—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8—22no—
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#4114yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—44no—
258C4⋊Q8—48no—
259#125—432no—
260#124—28no—
261#124—28no—
262#124—28no—
263#124—44no—
264#124—48no—
265#124—48no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×Q8—48no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
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—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
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×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42#4114yes—
339C2×C42#4114yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C4⋊C8—28no—
344#103—28no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—44no—
354#102—44no—
355#102—44no—
356#97—28no—
357#97—44no—
358C4×C8—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—22no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8C2×#4114yes—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
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—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
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—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4#12414yes—
457C22×C4#12414yes—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4#13714yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4C2511yescharacteristic
474C2×C8—28no—
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×C8C2×#4114yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—22no—
493C4⋊C4—22no—
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—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—48no—
531C42—48no—
532C42#12418yes—
533C42C2×#4114yes—
534C42C2×#4114yes—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23C2×#12412yes—
550C23#54612yes—
551C23#55012yes—
552C23C2×AGL(1,ℤ/8)12yes—
553C23C2×#13712yes—
554C23#57712yes—
555C23#57812yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—42no—
560Q8—42no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—84no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—84no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4C2×#12414yes—
621C2×C4C2×#12414yes—
622C2×C4C2×#12412yes—
623C2×C4#52914yes—
624C2×C4#54014yes—
625C2×C4#54612yes—
626C2×C4#55012yes—
627C2×C4#55314yes—
628C2×C4#55314yes—
629C2×C4C2×AGL(1,ℤ/8)14yes—
630C2×C4C2×#13714yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C8—24no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—168no—
649C22#249014yes—
650C22#313714yes—
651C22#314114yes—
652C22#330814yes—
653C22#331614yes—
654C22#339714yes—
655C22#339814yes—
656C22C22×#12411yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#55011yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57811yescharacteristic
663C4—24no—
664C4—24no—
665C4—24no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#249014yes—
677C4C2×#52911yescharacteristic
678C4C2×#54011yescharacteristic
679C4C2×#55311yescharacteristic
680C4C2×#55311yescharacteristic
681C2—162no—
682C2#2776518yesdirect product factor
683C2C2×#249011yescharacteristic
684C2C2×#313711yescharacteristic
685C2C2×#314111yescharacteristic
686C2C2×#330811yescharacteristic
687C2C2×#331611yescharacteristic
688C2C2×#339711yescharacteristic
689C2C2×#339811yescharacteristic
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
1. C2×#2552C2semidirect
3. #27765C2direct (N × H)
4. C2×#2072C2semidirect
6. C2×#2043C2semidirect
11. C2×#1956C2semidirect
12. C2×#1935C2semidirect
14. C2×#1803C2semidirect
15. C2×#1771C2semidirect
16. C2×#1763C2semidirect
39. #2552C22semidirect
58. #2072C22semidirect
60. #2043C22semidirect
65. #1956C22semidirect
66. #1935C22semidirect
68. #1803C22semidirect
69. #1771C22semidirect
70. #1763C22semidirect
682. C2#27765direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25452
#277652
C2×#20722
C2×#20712
C2×#20432
C2×#20422
C2×#19832
C2×#19592
C2×#19562
C2×#19352
C2×#19342
C2×#18032
C2×#17712
C2×#17632