Finite Group Databasebeta

← C2×#30827 #7501963

Subgroups — C2×#30827 #7501963

Order 512.

2324 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic, complemented
3C2×#2552C211yesmaximal, characteristic, complemented
4C2×#2445C211yesmaximal, characteristic
5#30827C2116yesmaximal, direct product factor
6C2×#2093C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2041C211yesmaximal, characteristic
11C2×#2025C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic, complemented
15C2×#1917C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#123—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#490—28no—
23C2×#490C2218yes—
24C2×#478—28no—
25C2×#478C2218yescomplemented
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2552—24no—
41#2552C2218yescomplemented
42#2445—24no—
43#2445C2218yes—
44C2×#398—21no—
45C2×#398—21no—
46C2×#394—21no—
47C2×#394—21no—
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#386—21no—
56C2×#384—21no—
57C2×#384C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2093C2218yescomplemented
61#2071C2218yescomplemented
62#2065C2218yescomplemented
63#2059C2218yescomplemented
64#2041C2218yes—
65#2025C2218yescomplemented
66#1944C2218yescomplemented
67#1926C2218yescomplemented
68#1926C2218yescomplemented
69#1917C2218yescomplemented
70#1862C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×#127—24no—
89C2×#127—22no—
90C2×#127—42no—
91C2×#127C2314yes—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—22no—
95C2×#125—22no—
96C2×#125—42no—
97C2×#125—42no—
98C2×#125C2314yes—
99C2×#125C2314yes—
100C2×#123—28no—
101C2×#123—48no—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—42no—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#490—232no—
135#478—232no—
136C2×#107—24no—
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2314yes—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102—42no—
153C2×#102D418yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#97—24no—
163C2×#97C2314yes—
164#398—28no—
165#398—28no—
166#394—28no—
167#394—28no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389—28no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#386—28no—
182#384—28no—
183#384—22no—
184#384C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4D414yes—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2314yes—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C25—21no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
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—44no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—82no—
225C22×D4—82no—
226C22×D4C2×D414yes—
227C22×D4C2412yes—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—82no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
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×C8C2×D414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262#127—28no—
263#127—44no—
264#127—48no—
265#125—28no—
266#125—28no—
267#125—44no—
268#125—44no—
269#125—48no—
270#125—48no—
271#123—432no—
272C4×D4—28no—
273C4×D4—44no—
274C4×D4—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
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—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4#4114yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—28no—
350#107—44no—
351#106—28no—
352#106—28no—
353C4⋊C8—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—28no—
360#102—44no—
361#102C2×D4116yes—
362#97—28no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95C2×D414yes—
368#95C2×D414yes—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D411yescharacteristic
386C2×D4—28no—
387C2×D4—24no—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
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—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—816no—
418C2×D4—816no—
419C2×D4C22×D414yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
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—48no—
446C22×C4—48no—
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—42no—
463C22×C4—84no—
464C22×C4—816no—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
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×C8C22×D414yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—22no—
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—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C22×D414yes—
524C4⋊C4C2×#4114yes—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—48no—
542C23—24no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23#53612yes—
571C23D4212yes—
572C23#54812yes—
573C23C2×AGL(1,ℤ/8)12yes—
574C23#57512yes—
575C23#57612yes—
576C23#57712yes—
577C23C23×D411yescharacteristic
578D4—24no—
579D4—24no—
580D4—48no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—816no—
588D4—816no—
589D4—84no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
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—48no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—44no—
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—42no—
633C2×C4—816no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4#53612yes—
639C2×C4#53614yes—
640C2×C4#54214yes—
641C2×C4D4212yes—
642C2×C4#54714yes—
643C2×C4#54812yes—
644C2×C4#56314yes—
645C2×C4C2×AGL(1,ℤ/8)14yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C8—24no—
653C8—42no—
654C8—42no—
655C8—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—816no—
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#267414yes—
673C22#323414yes—
674C22#324114yes—
675C22#326714yes—
676C22#327414yes—
677C22#336814yes—
678C22#337014yes—
679C22C2×#53611yescharacteristic
680C22C2×D4211yescharacteristic
681C22C2×#54811yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57611yescharacteristic
685C22C2×#57711yescharacteristic
686C4—24no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#267414yes—
699C4C2×#53611yescharacteristic
700C4C2×#54211yescharacteristic
701C4C2×#54711yescharacteristic
702C4C2×#56311yescharacteristic
703C2—84no—
704C2—162no—
705C2#3082718yesdirect product factor
706C2C2×#267411yescharacteristic
707C2C2×#323411yescharacteristic
708C2C2×#324111yescharacteristic
709C2C2×#326711yescharacteristic
710C2C2×#327411yescharacteristic
711C2C2×#336811yescharacteristic
712C2C2×#337011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. C22×#478C2semidirect
3. C2×#2552C2semidirect
5. #30827C2direct (N × H)
6. C2×#2093C2semidirect
7. C2×#2071C2semidirect
8. C2×#2065C2semidirect
9. C2×#2059C2semidirect
11. C2×#2025C2semidirect
12. C2×#1944C2semidirect
13. C2×#1926C2semidirect
14. C2×#1926C2semidirect
15. C2×#1917C2semidirect
16. C2×#1862C2semidirect
25. C2×#478C22semidirect
41. #2552C22semidirect
60. #2093C22semidirect
61. #2071C22semidirect
62. #2065C22semidirect
63. #2059C22semidirect
65. #2025C22semidirect
66. #1944C22semidirect
67. #1926C22semidirect
68. #1926C22semidirect
69. #1917C22semidirect
70. #1862C22semidirect
705. C2#30827direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4902
C22×#4782
C2×#25522
C2×#24452
#308272
C2×#20932
C2×#20712
C2×#20652
C2×#20592
C2×#20412
C2×#20252
C2×#19442
C2×#19262
C2×#19172
C2×#18622