Finite Group Databasebeta

← C2×#27413 #6449359

Subgroups — C2×#27413 #6449359

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#27413C2116yesmaximal, direct product factor
4C2×#2086C211yesmaximal, characteristic, complemented
5C2×#2043C211yesmaximal, characteristic, complemented
6C2×#2036C211yesmaximal, characteristic
7C2×#2036C211yesmaximal, characteristic
8C2×#1981C211yesmaximal, characteristic, complemented
9C2×#1935C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1924C211yesmaximal, characteristic
12C2×#1918C211yesmaximal, characteristic, complemented
13C2×#1872C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117—21no—
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2429—24no—
38#2429C2218yes—
39C2×#398—21no—
40C2×#396—21no—
41#2344—24no—
42#2344C2218yes—
43C2×#391C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#387—21no—
50C2×#387—21no—
51C2×#381—21no—
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2086C2218yescomplemented
57#2043C2218yescomplemented
58#2036C2218yes—
59#2036C2218yes—
60#1981C2218yescomplemented
61#1935C2218yescomplemented
62#1928C2218yescomplemented
63#1924C2218yes—
64#1918C2218yescomplemented
65#1872C2218yes—
66#1872C2218yes—
67#1863C2218yescomplemented
68#1763C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8C2314yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—42no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—24no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C2×#117—28no—
102C2×#117—48no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
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⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107—42no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×C4⋊C8—42no—
137C2×C4⋊C8C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—24no—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8—42no—
155C2×C22⋊C8—42no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97—42no—
159C2×#97C2314yes—
160#398—28no—
161#396—28no—
162#391—22no—
163#391C2314yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—28no—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#388—28no—
172#387—28no—
173#387—28no—
174#381—28no—
175#380—22no—
176#380C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4D414yes—
182C4×C22⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
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—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
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—44no—
235C23×C4—44no—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
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×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—48no—
272#117—432no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4#4114yes—
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D414yes—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342#107—28no—
343#107—44no—
344C4⋊C8—44no—
345#103—28no—
346#103—28no—
347#103—28no—
348#103—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#103—44no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355#97—44no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24#13714yes—
373C24C22×D412yes—
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—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
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—24no—
426C22×C4—24no—
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—22no—
437C22×C4—48no—
438C22×C4—432no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
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×C4AGL(1,ℤ/8)14yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4C2511yescharacteristic
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—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4C2×#4114yes—
519C4⋊C4C2×#4114yes—
520C4⋊C4C2×#4114yes—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—48no—
533C42—48no—
534C42—48no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#52512yes—
549C23#54612yes—
550C23#54612yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23C2×#13714yes—
553C23C2×#13712yes—
554C23#57812yes—
555C23#57812yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—42no—
560Q8—42no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—84no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
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—432no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
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—48no—
608C2×C4—42no—
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—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#52512yes—
625C2×C4#52914yes—
626C2×C4#54014yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#55314yes—
630C2×C4#55514yes—
631C2×C4C2×AGL(1,ℤ/8)14yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22#247614yes—
651C22#311014yes—
652C22#312714yes—
653C22#330814yes—
654C22#331214yes—
655C22#331214yes—
656C22#333014yes—
657C22C2×#52511yescharacteristic
658C22C2×#54611yescharacteristic
659C22C2×#54611yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57811yescharacteristic
663C22C2×#57811yescharacteristic
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#247614yes—
678C4C2×#52911yescharacteristic
679C4C2×#54011yescharacteristic
680C4C2×#55311yescharacteristic
681C4C2×#55511yescharacteristic
682C2—44no—
683C2#2741318yesdirect product factor
684C2C2×#247611yescharacteristic
685C2C2×#311011yescharacteristic
686C2C2×#312711yescharacteristic
687C2C2×#330811yescharacteristic
688C2C2×#331211yescharacteristic
689C2C2×#331211yescharacteristic
690C2C2×#333011yescharacteristic
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. #27413C2direct (N × H)
4. C2×#2086C2semidirect
5. C2×#2043C2semidirect
8. C2×#1981C2semidirect
9. C2×#1935C2semidirect
10. C2×#1928C2semidirect
12. C2×#1918C2semidirect
15. C2×#1863C2semidirect
16. C2×#1763C2semidirect
56. #2086C22semidirect
57. #2043C22semidirect
60. #1981C22semidirect
61. #1935C22semidirect
62. #1928C22semidirect
64. #1918C22semidirect
67. #1863C22semidirect
68. #1763C22semidirect
683. C2#27413direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#23442
#274132
C2×#20862
C2×#20432
C2×#20362
C2×#19812
C2×#19352
C2×#19282
C2×#19242
C2×#19182
C2×#18722
C2×#18632
C2×#17632