Finite Group Databasebeta

← C2×#30353 #6434261

Subgroups — C2×#30353 #6434261

Order 512.

1984 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2C2×#2645C211yesmaximal, characteristic, complemented
3#30353C2116yesmaximal, direct product factor
4C2×#2067C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2059C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2041C211yesmaximal, characteristic
10C2×#2026C211yesmaximal, characteristic, complemented
11C2×#2014C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic, complemented
13C2×#1968C211yesmaximal, characteristic, complemented
14C2×#1936C211yesmaximal, characteristic, complemented
15C2×#1934C211yesmaximal, characteristic, complemented
16C2×#1768C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2645—24no—
36#2645C2218yescomplemented
37#2641—24no—
38#2641C2218yes—
39C2×#401—21no—
40C2×#400—21no—
41C2×#400—21no—
42C2×#400—21no—
43C2×#394—21no—
44C2×#394—21no—
45C2×#394—21no—
46C2×#394—21no—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#2067C2218yescomplemented
59#2065C2218yescomplemented
60#2059C2218yescomplemented
61#2055C2218yescomplemented
62#2055C2218yescomplemented
63#2041C2218yes—
64#2026C2218yescomplemented
65#2014C2218yes—
66#1973C2218yescomplemented
67#1968C2218yescomplemented
68#1936C2218yescomplemented
69#1934C2218yescomplemented
70#1768C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#127—22no—
95C2×#127—42no—
96C2×#127—42no—
97C2×#127C2314yes—
98C2×#125—22no—
99C2×#125—42no—
100C2×#125—42no—
101C2×#125C2314yes—
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×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×C4⋊C8—24no—
128C2×C4⋊C8—24no—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103—42no—
132C2×#103—42no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#97—42no—
148C2×#97C2314yes—
149C2×C4×C8—42no—
150C2×C4×C8C2314yes—
151#401—28no—
152#400—28no—
153#400—28no—
154#400—28no—
155#394—28no—
156#394—28no—
157#394—28no—
158#394—28no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#376D414yes—
180#376C2314yes—
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—
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8C2412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—24no—
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×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—82no—
229C22×D4—82no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—416no—
254C2×SD16—416no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8#4114yes—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#127—44no—
272#127—44no—
273#127—48no—
274#125—44no—
275#125—44no—
276#125—48no—
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—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
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—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
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—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—44no—
336#106—44no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#103—44no—
340#103—44no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345#102—44no—
346#102—44no—
347#97—44no—
348C4×C8—44no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
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—48no—
405C2×D4—48no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—816no—
425C2×D4—816no—
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—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
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—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482SD16—416no—
483SD16—416no—
484C2×C8—28no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8C2×#4114yes—
499C2×C8C2×#4114yes—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23#53212yes—
566C23#53512yes—
567C23#53712yes—
568C23#57612yes—
569C23#57712yes—
570C23#57712yes—
571C23#57712yes—
572C23C23×D411yescharacteristic
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—84no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—816no—
587D4—816no—
588D4—84no—
589D4—84no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
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—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
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—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4#53212yes—
639C2×C4#53512yes—
640C2×C4#53712yes—
641C2×C4#53814yes—
642C2×C4#53814yes—
643C2×C4#54214yes—
644C2×C4#54214yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—24no—
653C8—42no—
654C8—42no—
655C22—82no—
656C22—82no—
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—168no—
670C22—168no—
671C22#266214yes—
672C22#317414yes—
673C22#317914yes—
674C22#322214yes—
675C22#322414yes—
676C22#325714yes—
677C22#326014yes—
678C22C2×#53211yescharacteristic
679C22C2×#53511yescharacteristic
680C22C2×#53711yescharacteristic
681C22C2×#57611yescharacteristic
682C22C2×#57711yescharacteristic
683C22C2×#57711yescharacteristic
684C22C2×#57711yescharacteristic
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#266214yes—
698C4C2×#53811yescharacteristic
699C4C2×#53811yescharacteristic
700C4C2×#54211yescharacteristic
701C4C2×#54211yescharacteristic
702C2—162no—
703C2—162no—
704C2#3035318yesdirect product factor
705C2C2×#266211yescharacteristic
706C2C2×#317411yescharacteristic
707C2C2×#317911yescharacteristic
708C2C2×#322211yescharacteristic
709C2C2×#322411yescharacteristic
710C2C2×#325711yescharacteristic
711C2C2×#326011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2645C2semidirect
3. #30353C2direct (N × H)
4. C2×#2067C2semidirect
5. C2×#2065C2semidirect
6. C2×#2059C2semidirect
7. C2×#2055C2semidirect
8. C2×#2055C2semidirect
10. C2×#2026C2semidirect
12. C2×#1973C2semidirect
13. C2×#1968C2semidirect
14. C2×#1936C2semidirect
15. C2×#1934C2semidirect
16. C2×#1768C2semidirect
36. #2645C22semidirect
58. #2067C22semidirect
59. #2065C22semidirect
60. #2059C22semidirect
61. #2055C22semidirect
62. #2055C22semidirect
64. #2026C22semidirect
66. #1973C22semidirect
67. #1968C22semidirect
68. #1936C22semidirect
69. #1934C22semidirect
70. #1768C22semidirect
704. C2#30353direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#26452
#303532
C2×#20672
C2×#20652
C2×#20592
C2×#20552
C2×#20412
C2×#20262
C2×#20142
C2×#19732
C2×#19682
C2×#19362
C2×#19342
C2×#17682