Finite Group Databasebeta

← C2×#31917 #7506611

Subgroups — C2×#31917 #7506611

Order 512.

2020 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic, complemented
3#31917C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2067C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2012C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic, complemented
11C2×#1972C211yesmaximal, characteristic
12C2×#1940C211yesmaximal, characteristic, complemented
13C2×#1935C211yesmaximal, characteristic, complemented
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1879C211yesmaximal, characteristic
16C2×#1804C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2513—24no—
36#2513C2218yescomplemented
37#2447—24no—
38#2447C2218yescomplemented
39C2×#397—21no—
40C2×#397—21no—
41C2×#397—21no—
42C2×#397—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#385—21no—
54C2×#382C2211yescharacteristic
55C2×#381—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2097C2218yescomplemented
59#2067C2218yescomplemented
60#2067C2218yescomplemented
61#2055C2218yescomplemented
62#2055C2218yescomplemented
63#2012C2218yes—
64#1983C2218yescomplemented
65#1972C2218yes—
66#1940C2218yescomplemented
67#1935C2218yescomplemented
68#1920C2218yescomplemented
69#1879C2218yes—
70#1804C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—42no—
82C22×SD16—42no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C22×D8—24no—
86C22×D8—24no—
87C22×D8—42no—
88C22×D8C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#124—24no—
94C2×#124—22no—
95C2×#124—42no—
96C2×#124C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—42no—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
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×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#106—24no—
128C2×#106C2314yes—
129C2×C4⋊C8—42no—
130C2×C4⋊C8—42no—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103—24no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#97—42no—
152C2×#97C2314yes—
153#397—28no—
154#397—28no—
155#397—28no—
156#397—28no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#388—28no—
175#385—28no—
176#382—22no—
177#382C2314yes—
178#381—28no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D414yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊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×#95—21no—
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—22no—
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#4114yes—
224C22×D4C2412yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—44no—
246C2×SD16—416no—
247C2×D8—28no—
248C2×D8—28no—
249C2×D8—44no—
250C2×D8—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8#4114yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—28no—
268#124—44no—
269#124—48no—
270C4×D4—28no—
271C4×D4—44no—
272C4×D4—48no—
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—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D414yes—
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—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
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×C42—24no—
330C2×C42—82no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—44no—
338#106—28no—
339C4⋊C8—44no—
340C4⋊C8—44no—
341C4⋊C8—44no—
342#103—28no—
343#103—44no—
344#103—44no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350#97—44no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—24no—
363C24—22no—
364C24—22no—
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—28no—
393C2×D4—24no—
394C2×D4—28no—
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—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4C2×#4114yes—
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—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—48no—
443C22×C4—48no—
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—42no—
462C22×C4—84no—
463C22×C4—816no—
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—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—22no—
491C2×C8—22no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8—44no—
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—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—84no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23#53712yes—
565C23#54612yes—
566C23#55012yes—
567C23#57512yes—
568C23#57512yes—
569C23#57612yes—
570C23#57712yes—
571C23C23×D411yescharacteristic
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—84no—
577D4—24no—
578D4—24no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—48no—
584D4—42no—
585D4—42no—
586D4—84no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
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—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
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—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—816no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4#53712yes—
637C2×C4#53914yes—
638C2×C4#54214yes—
639C2×C4#54612yes—
640C2×C4#55012yes—
641C2×C4#55114yes—
642C2×C4#56014yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C8—24no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—816no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—168no—
669C22#270514yes—
670C22#325614yes—
671C22#325714yes—
672C22#330714yes—
673C22#331914yes—
674C22#340114yes—
675C22#340314yes—
676C22C2×#53711yescharacteristic
677C22C2×#54611yescharacteristic
678C22C2×#55011yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57611yescharacteristic
682C22C2×#57711yescharacteristic
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#270514yes—
696C4C2×#53911yescharacteristic
697C4C2×#54211yescharacteristic
698C4C2×#55111yescharacteristic
699C4C2×#56011yescharacteristic
700C2—84no—
701C2—162no—
702C2#3191718yesdirect product factor
703C2C2×#270511yescharacteristic
704C2C2×#325611yescharacteristic
705C2C2×#325711yescharacteristic
706C2C2×#330711yescharacteristic
707C2C2×#331911yescharacteristic
708C2C2×#340111yescharacteristic
709C2C2×#340311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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. C2×#2513C2semidirect
2. C2×#2447C2semidirect
3. #31917C2direct (N × H)
4. C2×#2097C2semidirect
5. C2×#2067C2semidirect
6. C2×#2067C2semidirect
7. C2×#2055C2semidirect
8. C2×#2055C2semidirect
10. C2×#1983C2semidirect
12. C2×#1940C2semidirect
13. C2×#1935C2semidirect
14. C2×#1920C2semidirect
16. C2×#1804C2semidirect
36. #2513C22semidirect
38. #2447C22semidirect
58. #2097C22semidirect
59. #2067C22semidirect
60. #2067C22semidirect
61. #2055C22semidirect
62. #2055C22semidirect
64. #1983C22semidirect
66. #1940C22semidirect
67. #1935C22semidirect
68. #1920C22semidirect
70. #1804C22semidirect
702. C2#31917direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#24472
#319172
C2×#20972
C2×#20672
C2×#20552
C2×#20122
C2×#19832
C2×#19722
C2×#19402
C2×#19352
C2×#19202
C2×#18792
C2×#18042