Finite Group Databasebeta

← C2×#26479 #6444600

Subgroups — C2×#26479 #6444600

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×#2552C211yesmaximal, characteristic, complemented
4C2×#2367C211yesmaximal, characteristic
5#26479C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2026C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1959C211yesmaximal, characteristic
12C2×#1948C211yesmaximal, characteristic, complemented
13C2×#1940C211yesmaximal, characteristic, complemented
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#479—28no—
23C2×#479C2218yescomplemented
24C2×#478—28no—
25C2×#478C2218yes—
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#97C2211yescharacteristic
42#2552—24no—
43#2552C2218yescomplemented
44C2×#398—21no—
45C2×#398—21no—
46#2367—24no—
47#2367C2218yes—
48C2×#391C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#385—21no—
56C2×#384—21no—
57C2×#384C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4—21no—
62#2066C2218yescomplemented
63#2065C2218yes—
64#2026C2218yescomplemented
65#2025C2218yes—
66#1983C2218yes—
67#1959C2218yes—
68#1948C2218yescomplemented
69#1940C2218yescomplemented
70#1865C2218yescomplemented
71#1864C2218yes—
72#1776C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125—42no—
91C2×#125—42no—
92C2×#125C2314yes—
93C2×#125C2314yes—
94C2×#124—22no—
95C2×#124—24no—
96C2×#124Q818yes—
97C2×#124C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—24no—
100C2×C4×Q8—42no—
101C2×C4×Q8C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—42no—
105C2×C4×D4C2314yes—
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⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#479—232no—
135#478—232no—
136C2×#107D418yes—
137C2×#107C2314yes—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106—24no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—42no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—24no—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#97—24no—
167C2×#97C2314yes—
168#398—28no—
169#398—28no—
170#391—22no—
171#391C2314yes—
172#390—22no—
173#390C2314yes—
174#389—28no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#388—28no—
181#385—28no—
182#384—28no—
183#384—22no—
184#384C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D414yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C22⋊C4—28no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8C2×Q814yes—
214C22×Q8C2412yes—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Q814yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
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×C8C2×D414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#125—28no—
258#125—28no—
259#125—44no—
260#125—44no—
261#125—48no—
262#125—48no—
263#124—216no—
264#124—28no—
265#124C2×Q8116yes—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—48no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
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—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
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—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
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—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
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—82no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107C2×D4116yes—
347#106—28no—
348#106—28no—
349#106—28no—
350C4⋊C8—28no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—44no—
357#102—28no—
358#102—28no—
359#102—28no—
360#102—44no—
361#97—28no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—24no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8C22×Q814yes—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C22×Q814yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
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—84no—
452C22×C4—816no—
453C22×C4C22⋊Q814yes—
454C22×C4C22⋊Q814yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469C2×C8—28no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8C22×D414yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C22×D414yes—
512C4⋊C4C22×D414yes—
513C4⋊C4C2×#4114yes—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—48no—
526C42—48no—
527C42—84no—
528C42C22⋊Q818yes—
529C42C22×Q814yes—
530C42C2×#4114yes—
531C23—24no—
532C23—48no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23C2×C22⋊Q812yes—
546C23D4212yes—
547C23#54812yes—
548C23#57512yes—
549C23#57512yes—
550C23#57712yes—
551C23#57712yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—42no—
556Q8—42no—
557Q8—48no—
558D4—24no—
559D4—24no—
560D4—42no—
561D4—42no—
562D4—48no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
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—816no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4C2×C22⋊Q814yes—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4#53214yes—
618C2×C4#54214yes—
619C2×C4D4212yes—
620C2×C4#54812yes—
621C2×C4D4×Q814yes—
622C2×C4#55414yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C23×Q811yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C8—24no—
630C8—42no—
631C8—42no—
632C8—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—82no—
638C22—816no—
639C22—82no—
640C22—82no—
641C22#245214yes—
642C22#308014yes—
643C22#308414yes—
644C22#326714yes—
645C22#327114yes—
646C22#337114yes—
647C22#337614yes—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×D4211yescharacteristic
650C22C2×#54811yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57711yescharacteristic
654C22C2×#57711yescharacteristic
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#245214yes—
669C4C2×#53211yescharacteristic
670C4C2×#54211yescharacteristic
671C4C2×D4×Q811yescharacteristic
672C4C2×#55411yescharacteristic
673C2—84no—
674C2#2647918yesdirect product factor
675C2C2×#245211yescharacteristic
676C2C2×#308011yescharacteristic
677C2C2×#308411yescharacteristic
678C2C2×#326711yescharacteristic
679C2C2×#327111yescharacteristic
680C2C2×#337611yescharacteristic
681C2C2×#337111yescharacteristic
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. C22×#479C2semidirect
3. C2×#2552C2semidirect
5. #26479C2direct (N × H)
6. C2×#2066C2semidirect
8. C2×#2026C2semidirect
12. C2×#1948C2semidirect
13. C2×#1940C2semidirect
14. C2×#1865C2semidirect
16. C2×#1776C2semidirect
23. C2×#479C22semidirect
43. #2552C22semidirect
62. #2066C22semidirect
64. #2026C22semidirect
68. #1948C22semidirect
69. #1940C22semidirect
70. #1865C22semidirect
72. #1776C22semidirect
674. C2#26479direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4782
C2×#25522
C2×#23672
#264792
C2×#20662
C2×#20652
C2×#20262
C2×#20252
C2×#19832
C2×#19592
C2×#19482
C2×#19402
C2×#18652
C2×#18642
C2×#17762