Finite Group Databasebeta

← C2×#30804 #7501886

Subgroups — C2×#30804 #7501886

Order 512.

2140 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#482C211yesmaximal, characteristic
3C2×#2490C211yesmaximal, characteristic
4C2×#2439C211yesmaximal, characteristic, complemented
5#30804C2116yesmaximal, direct product factor
6C2×#2092C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2034C211yesmaximal, characteristic
11C2×#2023C211yesmaximal, characteristic, complemented
12C2×#1949C211yesmaximal, characteristic, complemented
13C2×#1936C211yesmaximal, characteristic, complemented
14C2×#1924C211yesmaximal, characteristic, complemented
15C2×#1920C211yesmaximal, characteristic, complemented
16C2×#1854C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#501—28no—
22C2×#501C2218yescomplemented
23C2×#482—28no—
24C2×#482C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×#97C2211yescharacteristic
39#2490—24no—
40#2490C2218yes—
41#2439—24no—
42#2439C2218yescomplemented
43C2×#397—21no—
44C2×#396—21no—
45C2×#394—21no—
46C2×#393—21no—
47C2×#391—21no—
48C2×#390—21no—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#387—21no—
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2092C2218yescomplemented
61#2075C2218yescomplemented
62#2067C2218yescomplemented
63#2059C2218yescomplemented
64#2034C2218yes—
65#2023C2218yescomplemented
66#1949C2218yescomplemented
67#1936C2218yescomplemented
68#1924C2218yescomplemented
69#1920C2218yescomplemented
70#1854C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2314yes—
91C2×#125—24no—
92C2×#125—22no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×#121—22no—
96C2×#121—24no—
97C2×#121—42no—
98C2×#121C2314yes—
99C2×#117—22no—
100C2×#117—24no—
101C2×#117—42no—
102C2×#117C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#501—232no—
130#482—232no—
131C2×#107—24no—
132C2×#107—24no—
133C2×#107—24no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106—42no—
139C2×#106D418yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—42no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×#97—24no—
158C2×#97C2314yes—
159#397—28no—
160#396—28no—
161#394—28no—
162#393—28no—
163#391—28no—
164#390—28no—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#387—28no—
175#382—22no—
176#382C2314yes—
177#380—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4D414yes—
184C4×C22⋊C4C2314yes—
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#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
203C25C2×D411yescharacteristic
204C22×Q8—24no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—82no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—82no—
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2412yes—
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×C8C2×D414yes—
252C22×C8#4114yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260#125—28no—
261#125—44no—
262#125—48no—
263#121—28no—
264#121—44no—
265#121—48no—
266#117—28no—
267#117—44no—
268#117—48no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
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—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—82no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#107—28no—
341#107—28no—
342#106—28no—
343#106—44no—
344#106C2×D4116yes—
345C4⋊C8—28no—
346#102—28no—
347#102—28no—
348#102—28no—
349#102—44no—
350C22⋊C8—28no—
351C22⋊C8—28no—
352#97—28no—
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—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—816no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—84no—
399C2×D4—816no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
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—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
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—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469C2×C8—28no—
470C2×C8—28no—
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×C8C22×D414yes—
484C2×C8C2×#4114yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
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—22no—
498C4⋊C4—22no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—84no—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D414yes—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—48no—
528C42—84no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23#53612yes—
558C23D4212yes—
559C23#54712yes—
560C23#57512yes—
561C23#57512yes—
562C23#57612yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—816no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—816no—
571D4—84no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—48no—
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—42no—
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—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#53612yes—
626C2×C4#53614yes—
627C2×C4#54214yes—
628C2×C4D4212yes—
629C2×C4#54712yes—
630C2×C4#54814yes—
631C2×C4#56314yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—24no—
639C8—24no—
640C8—42no—
641C8—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—88no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—168no—
658C22#267414yes—
659C22#324014yes—
660C22#324114yes—
661C22#326814yes—
662C22#327514yes—
663C22#334714yes—
664C22#335014yes—
665C22C2×#53611yescharacteristic
666C22C2×D4211yescharacteristic
667C22C2×#54711yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57611yescharacteristic
671C22C2×#57711yescharacteristic
672C4—42no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#267414yes—
685C4C2×#53611yescharacteristic
686C4C2×#54211yescharacteristic
687C4C2×#54811yescharacteristic
688C4C2×#56311yescharacteristic
689C2—82no—
690C2—162no—
691C2#3080418yesdirect product factor
692C2C2×#267411yescharacteristic
693C2C2×#324011yescharacteristic
694C2C2×#324111yescharacteristic
695C2C2×#326811yescharacteristic
696C2C2×#327511yescharacteristic
697C2C2×#334711yescharacteristic
698C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
4. C2×#2439C2semidirect
5. #30804C2direct (N × H)
6. C2×#2092C2semidirect
7. C2×#2075C2semidirect
8. C2×#2067C2semidirect
9. C2×#2059C2semidirect
11. C2×#2023C2semidirect
12. C2×#1949C2semidirect
13. C2×#1936C2semidirect
14. C2×#1924C2semidirect
15. C2×#1920C2semidirect
16. C2×#1854C2semidirect
22. C2×#501C22semidirect
42. #2439C22semidirect
60. #2092C22semidirect
61. #2075C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
65. #2023C22semidirect
66. #1949C22semidirect
67. #1936C22semidirect
68. #1924C22semidirect
69. #1920C22semidirect
70. #1854C22semidirect
691. C2#30804direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4822
C2×#24902
C2×#24392
#308042
C2×#20922
C2×#20752
C2×#20672
C2×#20592
C2×#20342
C2×#20232
C2×#19492
C2×#19362
C2×#19242
C2×#19202
C2×#18542