Finite Group Databasebeta

← C2×#30020 #6433338

Subgroups — C2×#30020 #6433338

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#30020C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2056C211yesmaximal, characteristic, complemented
9C2×#2036C211yesmaximal, characteristic
10C2×#2035C211yesmaximal, characteristic
11C2×#2001C211yesmaximal, characteristic
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1797C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2855—24no—
36#2855C2218yes—
37C4×C22⋊Q8—24no—
38C4×C22⋊Q8C2218yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#387—21no—
47C2×#387—21no—
48C2×#385—21no—
49C2×#385—21no—
50C2×#381—21no—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56#2088C2218yescomplemented
57#2077C2218yes—
58#2068C2218yescomplemented
59#2066C2218yescomplemented
60#2056C2218yescomplemented
61#2036C2218yes—
62#2035C2218yes—
63#2001C2218yes—
64#1927C2218yescomplemented
65#1918C2218yescomplemented
66#1878C2218yescomplemented
67#1865C2218yescomplemented
68#1797C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16—24no—
83C22×Q16—42no—
84C22×Q16—42no—
85C22×Q16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8—42no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—42no—
100C2×C4×Q8—42no—
101C2×C4×Q8C2314yes—
102C22×C4⋊C4—21no—
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×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—42no—
121C22×C42—42no—
122C22×C42D414yes—
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C2×#107—24no—
129C2×#107C2314yes—
130C2×#106—42no—
131C2×#106—42no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—42no—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8—42no—
151C2×C22⋊C8—42no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#391C2314yes—
159#391C2314yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#387—28no—
167#387—28no—
168#387—28no—
169#385—28no—
170#385—28no—
171#381—28no—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176C4×C4⋊C4—28no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—28no—
180C2×#95—21no—
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×#95—21no—
191C2×#95—21no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C23×C4—24no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—24no—
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×C4—21no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D414yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—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—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22⋊Q8—216no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
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—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—48no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#106—44no—
341#106—44no—
342C4⋊C8—44no—
343C4⋊C8—44no—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350C22⋊C8—44no—
351C22⋊C8—44no—
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#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24#13714yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8C2×#4114yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
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—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
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—42no—
464C22×C4—42no—
465C22×C4#13514yes—
466C22×C4#13714yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480Q16—416no—
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—22no—
493C2×C8—44no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8—44no—
497C4⋊C4—216no—
498C4⋊C4—24no—
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—22no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4C2×#4114yes—
521C22⋊C4—216no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—44no—
533C42—48no—
534C42—48no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#52512yes—
549C23#53412yes—
550C23#53712yes—
551C23C2×#13512yes—
552C23C2×#13714yes—
553C23C2×#13712yes—
554C23C2×#13712yes—
555C23#57512yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—48no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—84no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
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—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#52512yes—
625C2×C4#53412yes—
626C2×C4#53712yes—
627C2×C4#53814yes—
628C2×C4#53914yes—
629C2×C4#54014yes—
630C2×C4#54214yes—
631C2×C4C2×#13514yes—
632C2×C4C2×#13714yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—416no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22#263014yes—
652C22#310214yes—
653C22#311314yes—
654C22#319014yes—
655C22#319614yes—
656C22#324814yes—
657C22#325214yes—
658C22C2×#52511yescharacteristic
659C22C2×#53411yescharacteristic
660C22C2×#53711yescharacteristic
661C22C22×#13511yescharacteristic
662C22C22×#13711yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57511yescharacteristic
665C4—24no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#263014yes—
679C4C2×#53811yescharacteristic
680C4C2×#53911yescharacteristic
681C4C2×#54011yescharacteristic
682C4C2×#54211yescharacteristic
683C2—44no—
684C2#3002018yesdirect product factor
685C2C2×#263011yescharacteristic
686C2C2×#310211yescharacteristic
687C2C2×#311311yescharacteristic
688C2C2×#319011yescharacteristic
689C2C2×#319611yescharacteristic
690C2C2×#324811yescharacteristic
691C2C2×#325211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30020C2direct (N × H)
4. C2×#2088C2semidirect
6. C2×#2068C2semidirect
7. C2×#2066C2semidirect
8. C2×#2056C2semidirect
12. C2×#1927C2semidirect
13. C2×#1918C2semidirect
14. C2×#1878C2semidirect
15. C2×#1865C2semidirect
56. #2088C22semidirect
58. #2068C22semidirect
59. #2066C22semidirect
60. #2056C22semidirect
64. #1927C22semidirect
65. #1918C22semidirect
66. #1878C22semidirect
67. #1865C22semidirect
684. C2#30020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×C22⋊Q82
#300202
C2×#20882
C2×#20772
C2×#20682
C2×#20662
C2×#20562
C2×#20362
C2×#20352
C2×#20012
C2×#19272
C2×#19182
C2×#18782
C2×#18652
C2×#17972