Finite Group Databasebeta

← C2×#29194 #6463638

Subgroups — C2×#29194 #6463638

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#29194C2116yesmaximal, direct product factor
4C2×#2068C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic
7C2×#2065C211yesmaximal, characteristic
8C2×#1958C211yesmaximal, characteristic
9C2×#1958C211yesmaximal, characteristic
10C2×#1865C211yesmaximal, characteristic, complemented
11C2×#1863C211yesmaximal, characteristic, complemented
12C2×#1864C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic
14C2×#1846C211yesmaximal, characteristic, complemented
15C2×#1835C211yesmaximal, characteristic, complemented
16C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2864—24no—
37#2864C2218yescomplemented
38C4×#124—24no—
39C4×#124C2218yes—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48C2×#387—21no—
49C2×#387—21no—
50C2×#387—21no—
51C2×#387—21no—
52C2×#382C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56C2×C4×C22⋊C4—21no—
57C2×C43C2211yescharacteristic
58#2068C2218yescomplemented
59#2067C2218yes—
60#2066C2218yescomplemented
61#2065C2218yes—
62#1958C2218yes—
63#1958C2218yes—
64#1865C2218yescomplemented
65#1864C2218yes—
66#1863C2218yescomplemented
67#1862C2218yes—
68#1846C2218yescomplemented
69#1835C2218yescomplemented
70C4×C4⋊C8C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#124—22no—
85C2×#124—24no—
86C2×#124Q818yes—
87C2×#124C2314yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—42no—
90C2×C4×Q8—42no—
91C2×C4×Q8C2314yes—
92C2×C4×D4—22no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
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⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C42—21no—
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C2×#107—42no—
121C2×#107—42no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106—42no—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8—24no—
129C2×C4⋊C8—24no—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C4×C8—24no—
149C2×C4×C8—24no—
150C2×C4×C8C2314yes—
151C2×C4×C8C2314yes—
152#391—22no—
153#391C2314yes—
154#390—22no—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#388—28no—
165#388—28no—
166#387—28no—
167#387—28no—
168#387—28no—
169#387—28no—
170#382—22no—
171#382—22no—
172#382—22no—
173#382C2314yes—
174#382C2314yes—
175#382C2314yes—
176C4×C22⋊C4—28no—
177C4×C22⋊C4—28no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C43D414yes—
191C43C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C22×Q8—21no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8C2×Q814yes—
206C22×Q8C2412yes—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4C2×Q814yes—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8#4114yes—
244C22×C8#4114yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249#124—216no—
250#124—28no—
251#124C2×Q8116yes—
252C4×Q8—44no—
253C4×Q8—44no—
254C4×Q8—48no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—48no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
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—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D412yes—
319C2×C42C2×Q814yes—
320C2×C42#4114yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323C2×C42C2412yes—
324#107—44no—
325#107—44no—
326#106—44no—
327#106—44no—
328C4⋊C8—28no—
329C4⋊C8—28no—
330#103—28no—
331#103—28no—
332#103—44no—
333#103—44no—
334#102—28no—
335#102—28no—
336#102—44no—
337#102—44no—
338C4×C8—28no—
339C4×C8—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24—24no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24C22×D411yescharacteristic
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8C22×Q814yes—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4C22×Q814yes—
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—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—816no—
433C22×C4C22⋊Q814yes—
434C22×C4C22⋊Q814yes—
435C22×C4#13514yes—
436C22×C4#13514yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q812yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4C2511yescharacteristic
451C2×C8—28no—
452C2×C8—28no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—44no—
465C2×C8C2×#4114yes—
466C2×C8C2×#4114yes—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C42—24no—
496C42—24no—
497C42—28no—
498C42—28no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—44no—
504C42—44no—
505C42—48no—
506C42C22⋊Q818yes—
507C42C22×Q814yes—
508C42C2×#4114yes—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23C2×C22⋊Q812yes—
524C23#53212yes—
525C23#53512yes—
526C23C2×#13512yes—
527C23C2×#13512yes—
528C23#57512yes—
529C23#57512yes—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—48no—
534Q8—42no—
535Q8—42no—
536D4—24no—
537D4—24no—
538D4—48no—
539D4—42no—
540D4—42no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—816no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C2×C22⋊Q814yes—
591C2×C4C2×C22⋊Q814yes—
592C2×C4C2×C22⋊Q812yes—
593C2×C4#53212yes—
594C2×C4#53512yes—
595C2×C4#55314yes—
596C2×C4#55314yes—
597C2×C4#55414yes—
598C2×C4#55414yes—
599C2×C4C2×#13514yes—
600C2×C4C2×#13514yes—
601C2×C4C23×Q811yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C8—24no—
608C8—24no—
609C8—42no—
610C8—42no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22#256614yes—
620C22#307614yes—
621C22#307614yes—
622C22#316814yes—
623C22#318314yes—
624C22#321114yes—
625C22#321214yes—
626C22C22×C22⋊Q811yescharacteristic
627C22C2×#53211yescharacteristic
628C22C2×#53511yescharacteristic
629C22C22×#13511yescharacteristic
630C22C22×#13511yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57511yescharacteristic
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4#256614yes—
647C4C2×#55311yescharacteristic
648C4C2×#55311yescharacteristic
649C4C2×#55411yescharacteristic
650C4C2×#55411yescharacteristic
651C2—84no—
652C2#2919418yesdirect product factor
653C2C2×#256611yescharacteristic
654C2C2×#307611yescharacteristic
655C2C2×#307611yescharacteristic
656C2C2×#316811yescharacteristic
657C2C2×#318311yescharacteristic
658C2C2×#321111yescharacteristic
659C2C2×#321211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2864C2semidirect
3. #29194C2direct (N × H)
4. C2×#2068C2semidirect
5. C2×#2066C2semidirect
10. C2×#1865C2semidirect
11. C2×#1863C2semidirect
14. C2×#1846C2semidirect
15. C2×#1835C2semidirect
16. C2×C4×C4⋊C8C2semidirect
37. #2864C22semidirect
58. #2068C22semidirect
60. #2066C22semidirect
64. #1865C22semidirect
66. #1863C22semidirect
68. #1846C22semidirect
69. #1835C22semidirect
70. C4×C4⋊C8C22semidirect
652. C2#29194direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28642
C2×C4×#1242
#291942
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#19582
C2×#18652
C2×#18632
C2×#18642
C2×#18622
C2×#18462
C2×#18352
C2×C4×C4⋊C82