Finite Group Databasebeta

← C2×#30340 #6434255

Subgroups — C2×#30340 #6434255

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic, complemented
2C2×#2860C211yesmaximal, characteristic, complemented
3#30340C2116yesmaximal, direct product factor
4C2×#2068C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2059C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic, complemented
8C2×#2054C211yesmaximal, characteristic
9C2×#2042C211yesmaximal, characteristic, complemented
10C2×#2025C211yesmaximal, characteristic, complemented
11C2×#2012C211yesmaximal, characteristic
12C2×#1970C211yesmaximal, characteristic
13C2×#1968C211yesmaximal, characteristic, complemented
14C2×#1934C211yesmaximal, characteristic, complemented
15C2×#1933C211yesmaximal, characteristic, complemented
16C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#126—21no—
20C22×#125C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2861—24no—
37#2861C2218yescomplemented
38#2860—24no—
39#2860C2218yescomplemented
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#388—21no—
53C2×#388—21no—
54C2×#388—21no—
55C2×#388—21no—
56C2×#388—21no—
57C2×C43C2211yescharacteristic
58#2068C2218yescomplemented
59#2065C2218yescomplemented
60#2059C2218yescomplemented
61#2055C2218yescomplemented
62#2054C2218yes—
63#2042C2218yescomplemented
64#2025C2218yescomplemented
65#2012C2218yes—
66#1970C2218yes—
67#1968C2218yescomplemented
68#1934C2218yescomplemented
69#1933C2218yescomplemented
70C4×C4⋊C8C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C22×SD16—24no—
81C22×SD16—42no—
82C22×SD16—42no—
83C22×SD16C2314yes—
84C22×D8—24no—
85C22×D8—42no—
86C22×D8—42no—
87C22×D8C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×#126—28no—
93C2×#126—48no—
94C2×#125—22no—
95C2×#125—42no—
96C2×#125—42no—
97C2×#125C2314yes—
98C2×#124—22no—
99C2×#124—42no—
100C2×#124—42no—
101C2×#124C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107—42no—
132C2×#107C2314yes—
133C2×#106—42no—
134C2×#106C2314yes—
135C2×C4⋊C8—24no—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#103—24no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C4×C8—42no—
156C2×C4×C8—42no—
157C2×C4×C8C2314yes—
158C2×C4×C8C2314yes—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#388—28no—
181#388—28no—
182#388—28no—
183#388—28no—
184#388—28no—
185#388—28no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C43D414yes—
194C43C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2412yes—
226C22×D4C2412yes—
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
240C2×SD16—28no—
241C2×SD16—44no—
242C2×SD16—44no—
243C2×SD16—416no—
244C2×D8—28no—
245C2×D8—44no—
246C2×D8—44no—
247C2×D8—416no—
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×C8—22no—
257C22×C8—22no—
258C22×C8#4114yes—
259C22×C8#4114yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264#126—432no—
265#125—44no—
266#125—44no—
267#125—48no—
268#124—44no—
269#124—44no—
270#124—48no—
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—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42C2×D412yes—
336C2×C42C2×D414yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#106—44no—
347C4⋊C8—28no—
348C4⋊C8—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357C4×C8—44no—
358C4×C8—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366C24—22no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D411yescharacteristic
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—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
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—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
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—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4#13514yes—
462C22×C4#13514yes—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D412yes—
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×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478SD16—416no—
479D8—416no—
480C2×C8—28no—
481C2×C8—28no—
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—22no—
491C2×C8—22no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8C2×#4114yes—
495C2×C8C2×#4114yes—
496C4⋊C4—22no—
497C4⋊C4—22no—
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—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—28no—
532C42—28no—
533C42—28no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#53212yes—
570C23#53512yes—
571C23#53712yes—
572C23C2×#13512yes—
573C23C2×#13512yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23#57512yes—
576C23C23×D411yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—84no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—84no—
591D4—84no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4#53212yes—
638C2×C4#53512yes—
639C2×C4#53712yes—
640C2×C4#53814yes—
641C2×C4#53814yes—
642C2×C4#54214yes—
643C2×C4#54214yes—
644C2×C4C2×#13514yes—
645C2×C4C2×#13514yes—
646C2×C4C2×AGL(1,ℤ/8)14yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—24no—
654C8—24no—
655C8—42no—
656C8—42no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—168no—
672C22—168no—
673C22#266214yes—
674C22#316814yes—
675C22#317114yes—
676C22#320714yes—
677C22#321214yes—
678C22#324714yes—
679C22#325014yes—
680C22C2×#53211yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#53711yescharacteristic
683C22C22×#13511yescharacteristic
684C22C22×#13511yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C2×#57511yescharacteristic
687C4—24no—
688C4—24no—
689C4—24no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#266214yes—
700C4C2×#53811yescharacteristic
701C4C2×#53811yescharacteristic
702C4C2×#54211yescharacteristic
703C4C2×#54211yescharacteristic
704C2—162no—
705C2—162no—
706C2#3034018yesdirect product factor
707C2C2×#266211yescharacteristic
708C2C2×#316811yescharacteristic
709C2C2×#317111yescharacteristic
710C2C2×#320711yescharacteristic
711C2C2×#321211yescharacteristic
712C2C2×#324711yescharacteristic
713C2C2×#325011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#2861C2semidirect
2. C2×#2860C2semidirect
3. #30340C2direct (N × H)
4. C2×#2068C2semidirect
5. C2×#2065C2semidirect
6. C2×#2059C2semidirect
7. C2×#2055C2semidirect
9. C2×#2042C2semidirect
10. C2×#2025C2semidirect
13. C2×#1968C2semidirect
14. C2×#1934C2semidirect
15. C2×#1933C2semidirect
16. C2×C4×C4⋊C8C2semidirect
37. #2861C22semidirect
39. #2860C22semidirect
58. #2068C22semidirect
59. #2065C22semidirect
60. #2059C22semidirect
61. #2055C22semidirect
63. #2042C22semidirect
64. #2025C22semidirect
67. #1968C22semidirect
68. #1934C22semidirect
69. #1933C22semidirect
70. C4×C4⋊C8C22semidirect
706. C2#30340direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28612
C2×#28602
#303402
C2×#20682
C2×#20652
C2×#20592
C2×#20552
C2×#20542
C2×#20422
C2×#20252
C2×#20122
C2×#19702
C2×#19682
C2×#19342
C2×#19332
C2×C4×C4⋊C82